ARTeMIS Extractor
Output-only Modal Identification "Output-only modal identification" is when the modal properties are identified from mea
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Output-only Modal Identification "Output-only modal identification" is when the modal properties are identified from measured responses only. In mechanical engineering it is normal to use the term "operational modal identification" or "operational modal" for the same situation. In civil engineering the terms "ambient identification" or "ambient response analysis" are often used.
What is modal identification? Modal identification means: to determine the modal parameters from experimental data. The modal parameters are: mode shapes (the way the structure moves at a certain resonance frequency), natural frequencies (the resonance frequencies) and damping ratios (the degree to which the structure itself is able of damping out vibrations). In input-output modal identification the modal parameters are found by fitting a model to the so-called Frequency Response Function, a function relating excitation and response. The traditional techniques in input-output modal identification is described in many textbooks, see for instance the references [1] and [2] below. When modal identification is based on the measured response (output) only, things become more complicated for several reasons: The excitation (input) is unknown. The measured response (output) is often noisy For these reasons, in the past, doing output-only modal identification was a job only for the few specialists. However, ARTeMIS Extractor offers you all the known techniques available in a user-friendly software design - the software is doing most of the job for you. Now every engineer can do it himself !
What is output-only modal Identification? Output-only modal identification is used for large civil engineering structures, operating machinery or other structures that are not easily excited artificially. If a structure is easily excited it is always - or nearly always - an advantage to excite the structure artificially and measure the excitation and the response at the same time. As mentioned above this technology is called input-output modal identification, and has been used extensively in mechanical engineering for the last 20 years.
Large civil engineering structures are often excited by natural loads that cannot easily be controlled, for instance wave loads (offshore structures), wind loads (Buildings) or traffic loads (bridges). For operating machinery the problems are the same. They are also excited by natural sources like noise from bearings or vibrations from the environment around the structure. The same accounts for on-the-road vehicle testing where the road and the engine excites the vehicle. In these cases, it is an advantage to use output-only modal identification. Instead of exciting the structure artificially and dealing with the natural excitation as an unwanted noise source, the natural excitation is used as the excitation source. The idea of output-only modal identification is illustrated in the following figure:
The unknown loads are assumed to be produced by a virtual system loaded by white noise. The white noise is not assumed to drive the structural system but the total system consisting of the real structural system and the virtual loading system. Thus, in the identification process, the user identifies not only the structure itself, but might also identify some 'modes' that belongs to the virtual loading system. Also, the user might identify computational modes that appear because the signals are contaminated with noise. This means, that the art of output-only modal identification is the art of identifying all modes, and then to be able to separate the structural modes from the noise modes and excitation modes. The main advantages of this kind of testing are: Testing is cheap and fast, since the equipment for excitation is unnecessary. Testing does not interfere with the operation of the structure. The measured response is representative for the real operating conditions of the structure. However, using this kind of testing, the response of the structure is small and often partly covered in noise. Further, the excitation is unknown, and thus, the identification becomes more difficult than in input-output modal identification. The main difficulties are: Sensitive equipment is needed.
Careful data analysis is needed. These drawbacks are the main reasons that output-only modal identification has not been used in a large scale in the past. However, the problems are vanishing: The last few years prices on high quality equipment has dropped significantly, analysis techniques are developing, and the necessary large scale computer analysis can now be performed on a PC. The conclusions are: Equipment of the necessary quality is now avalilable from many vendors. The analysis tools that you need is now available in this piece of software.
How do you perform an output-only modal identification To perform a good output-only modal identification you need good equipment, experience and good planning. Some guide lines will be given here. Once you have bought your accelerometers, supplied them with high quality cables or a wireless transmission system, all hooked up to a high quality data acquisition system, you need to know how to plan your measurements: Measured Degrees-Of-Freedom's (Measurement Points): You need to decide in how many point you want to know the mode shapes, i.e. the spatial resolution of the mode. For small structures it might be just 5-10 points, for large structures it might be 100-200 points. However, there are no clear guidelines for this choice. It is mainly up to you what you want. If you have no close modes and if you are not really interested in the mode shapes, but only in the natural frequencies and the damping, you can do with only one measurement point. Data Sets (Setups): You need to decide how to distribute the many measurement points in groups, the so-called data sets or setups. When you want to know the mode shape in say 100 points, you could use 100 transducers - typically accelerometers. However, this is too expensive and also not practical. Typically, you have a smaller set of accelerometers, like 8, 12 or 16. You keep 2 or 3 at the same location (the reference points) moving the other ones around until you have time series from all the measurement points. Only one issue is important here: If you expect close modes, you must have the close modes well represented in all data sets - and for every data set the corresponding pieces of mode shape for the close modes must be easily
distinguished. Reference Points: The reference points are the measurement points that are common in all data sets. The main rule is, that you place the reference points in such a way, that all modes contribute well to the response signal at the reference points. If close modes are expected, you pay some extra attention to these modes having them well represented in the reference signals. Concerning this matter, you need to have some ideas of the modes shapes. This you can get from a finite element analysis. Data Acquisition: You need to decide how to filter and how to sample the signals, and you need to decide how many data points you want in your time series. The good rule is to sample a little higher than you really need and then decimate the signal afterwards. How many points do you need ? - there is no final answer. However, a good rule of thumb is to say, that you need at least points corresponding to 500 cycles of the lowest natural frequency that you expect. This is when your data is totally noise-free and when you have no close modes. In most cases it is necessary to take time series corresponding to say 1000 cycles of the lowest natural frequency that you expect or more. Analysis and validation: After you have taken your measurements, and on the spot checked your data to see if you do not have clipping, outliers and other measurement errors, when you have checked that you have a suitable amount of information, i.e. a good signal to noise ratio and a suitable length on the data records, then you can start the analysis. If you are testing a structure that is difficult to access, it is a good idea to perform a preliminar analysis on the test site with the ARTeMIS Extractor software to check if the desired information can be obtained from the data. If you can conclude that the data are good enough, then you take them back home and you start analyzing them using the ARTeMIS Extractor software starting with the simplest techniques going trough a series of identifications that ends with a validation of your results.
References 1. D.J. Ewins: Modal Testing: Theory and Practise, John Wiley & Sons Inc. 1995 2. Editors Maia, Nuno M.M. and Júlio M.M. Silva: Theoretical and Experimental Modal Analysis, John Wiley & Sons Inc., 1997.
Validation of Results Validation is the key to all good modal identification. The value of any estimation is determined solely on basis of the validation. Validation is an evaluation of the quality of the results and is performed on two levels: Validation of individual models. Validation of the estimated modal results.
Validation of individual models. Now, when we say 'models' we do not necessarily mean parametric models. Models in this sense is a more generalized concept, a way of thinking, a way of processing data. At this step we try to justify the model, the way of thinking or what ever we are doing. If it is a parametric model, we might look at how this individual model fits the data. For instance we might look at how the model fits the spectral density functions, the correlations functions, or even the raw time series. If it is a non-parametric model, we might look at how well the results of the different signal processing comply with what we expect. For instance, in the Enhanced Frequency Domain Decomposition (EFDD) technique, we try to identify the single-degree-of freedom contribution to the singular value decomposition of the spectral density matrix. In this case, we know that this contribution is expected to be a nice single-bell spectral density function. Thus, in this case, the model validation is an evaluation of how the different bell-functions comply with our expectations.
Validation of the estimated modal results. An estimation should never be performed using only one model or one technique only. If only one technique is used, the user should perform several estimations using the same technique but spreading over a suitable variation of user choices. If the user wants to use a parametric model, he should perform identifications using several model orders. If the user wants to apply non-parametric estimation, he can vary some user choices to obtain different modal estimates. The best is to use several different techniques, for instance to use a non-parametric technique and one or more parametric techniques, and to vary the user choices in order to
obtain several estimates for each technique. At the end the user validates the modal estimates against each other. This is the final and most important validation. Without this validation the user cannot know if the identification is reliable. However, the modal estimates that appear in all estimations with small deviations are the most reliable. The larger deviations, the smaller the reliability, and modes that only appear in some identifications have the lowest reliability. The user should validate all the results of the modal identification, the natural frequencies, the damping ratios and the mode shapes.
About ARTeMIS Extractor ARTeMIS Extractor is the tool for effective modal identification in the cases where only the output is known. The software allows you to perform accurate modal identification under operational conditions and in situations where the structure is impossible or difficult to excite by externally applied forces. Uses Modal identification from responses only. Modal identification of structures under real operating or ambient conditions. Modal analysis without use of shaker or hammer excitation. Mode shape animation and comparison of different mode shape estimates. Features Direct geometry and data transfer from ARTeMIS Testor and Brüel & Kjær PULSE using OLE Automation. Guided measurement procedures using ARTeMIS Testor and Brüel & Kjær Modal Test Consultant. Processed number of channels and amount of data only limited by PC. Fast identification in frequency domain of mode shapes, natural frequencies and damping ratios. Time domain data-driven algorithms for unbiased identification of mode shapes, natural frequencies and damping ratios. Handle multiple data-sets and multiple reference points, including automatic mode shape merging. Effective and easy-to-use signal processing configuration wizard. Two Frequency Domain Decomposition algorithms, producing immediate mode shape animation. Three time domain Stochastic Subspace Identification algorithms, Principal Component,
Unweighted Principal Component and Canonical Variate Analysis. Stabilization diagrams to discriminate between physical and computational modes. Synthesis of response spectra and correlation functions for validation. Modal Assurance Criterion, calculates MAC matrix between mode shapes from different projects. OLE support for graphics and tables. OLE automation support for e.g. Excel and MATLAB. Allows also the user to call the software from his own application. Open Based on Windows NT Technology, ARTeMIS Extractor easily integrates into your vibration lab and its computing environment. Data Sharing: ARTeMIS Extractor runs under Windows 2000/XP, so all your data is easily available to everyone else on the network. Microsoft Office: Once you’ve generated data and results, ARTeMIS Extractor makes it easy to export them to the popular Microsoft® Office applications like Word. This lets you create professional reports. File Transfers: With ARTeMIS Extractor, your data will never get caught in a dead-end. That’s because ARTeMIS Extractor supports both Universal File Format (UFF) as well the SVS format for exporting and importing your data to and from other popular analysis applications. Programmability: ARTeMIS Extractor is programmable via ActiveX™ (OLE Automation) technology using off-the-shelf programming tools such as Visual Basic® and Visual C++®. This is an invaluable feature for automating routine tasks and creating customised and systems integration solutions.
About Pro, Handy and Light Versions ARTeMIS Extractor is produced in three different versions. The difference between the versions is only what is included of modal identification techniques. The graphical interface and the overall functionality is the same. For this reason all three versions are using this help system. In the table below it is shown what the different version includes of modal identification techniques
Version Modal Identification Techniques
FDD Frequency Domain Decomposition Peak Picking EFDD Enhanced Frequency Domain Decomposition Peak Picking Pro
UPC Stochastic Subspace Identification PC Stochastic Subspace Identification CVA Stochastic Subspace Identification
FDD Frequency Domain Decomposition Peak Picking Handy EFDD Enhanced Frequency Domain Decomposition Peak Picking
FDD Frequency Domain Decomposition
Modal Results
Natural frequencies, damping ratios and mode shapes using both frequency and time domain techniques. This version is for those who want all the best techniques available, needs the highest accuracy and the best validation of the results. With this version you can perform accurate identifications in time domain as well as in the frequency domain using the strongest identification tools available today. The safest validation one can perform is to validate frequency and time domain results against one another.
Natural frequencies, damping ratios and mode shapes using both frequency domain techniques. This version is for those who wants efficiency but can compromise a bit on the possibility to get optimum results in any case.
Natural frequencies and mode shapes using a frequency domain
Light
Peak Picking
technique. For those who only need a fast and easy identification of mode shapes and natural frequencies.
If a help topic is specific to one or more of the version it will be marked just below the title of the topic. An example could be:
Stochastic Subspace Identification (SSI) (Pro version only)
About Help - Contents Main Help This is accessed from the Help menu. There are a number of "books" in the Help Contents. Selecting one of these provides a listing of the main topics for that section. The Help also contains an alphabetical index and full search facilities with wild cards. Where relevant, help topics give access to information available on the web sites of Structural Vibration Solutions, for example www.svibs.com. Context-sensitive Help Context-sensitive Help is built into ARTeMIS Extractor. This accesses the topic in the main Help that is related to menu items, dialog boxes, etc., for which you request help.
About Help - Conventions Throughout the help system, a number of conventions are used: Menu items and buttons in dialog boxes are indicated in bold, e.g., Select File, Save, Press OK Text in dialog boxes is indicated in italics, e.g., In the Save As dialog box, enter a File name. Select the Color tab Actions taken using the keyboard are indicated in bold with "" symbols, e.g., Press , Press
About Help - Accessing Help To obtain help on a subject, you can obtain context-sensitive help, or display the Help Contents tab pages using the Help menu and then select the Contents, Index or Search tab page to search for the help you require. Navigate through the Help system by clicking on words that are highlighted and are related to your interest, or by clicking the Related Topics button to obtain a pop-up menu of related topics. If a topic title contains words in parentheses, these words generally relate to the menu from which the command is available or to the main subject under which the topic is located.
Context-sensitive Help To obtain help related to a selected menu item, window or dialog box: Select the item or control of interest, then press . Point on the item or control of interest, then press the right-hand side mouse button. On simple controls such as radio buttons etc. the context sensitive help appear in a small text-box. For windows the context sensitive help is displayed by selecting the specify a record field: Setups
[ ]
In the first record you can enter a string with a short description of the data set. In the second record you must specify the name, and optionally include the path, of the file that contains the measurements of this data set. Record three is the DOF definition record. The fields of this record are: 1. The node number where the transducer is mounted as integer. 2. The X-component of the directional vector that describes the direction of the transducer. This value can be any floating-point. 3. The Y-component of the directional vector that describes the direction of the transducer. This value can be any floating-point. 4. The Z-component of the directional vector that describes the direction of the transducer. This value can be any floating-point. 5. Reference value used to normalize e.g. spectral densities with when presented in dB displays. This value can be any positive floating-point. 6. String describing the unit of the measurements, e.g. m, m/s or m/s². 7. String describing the type of measurement, e.g. Displacement, Velocity, Acceleration. 8. A label string that serves as an ID of this particular transducer, e.g. Transducer #1. Field 1 is the location of the specified DOF, and fields 2 to 4 is the vector that define the direction in which the transducer is pointing. Note: The directional vector defined by fields 2 to 4 should usually be defined as a unit
vector. The estimated mode shapes will be divided into the X, Y and Z directions by multiplication of these directional components. Thus, if the directional vector has a length different from unity this will affect the mode shape value in the point accordingly. The fields 5 to 8 can be omitted. However, omitting one field imply that all remaining field also must be omitted. In other words, if you leave out field 5 you must also leave out 6 to 8. The DOF definition records are repeated for all degrees of freedom in the data set. These records must be defined in the same order as the measurements of the DOF's are stored in the data file. The default data file format is a standard ASCII format where the measurements of the DOF's are stored column-wise. So the first DOF definition record corresponds to the first column, the second DOF definition record corresponds to the second column and so on. If you want to use the SVS binary data file format you must specify /binary after the file name in the record. All the above records define one data set only. If you have multiple data sets all the above records are repeated with values describing the next data set. Note: In the case of multiple data sets there must be at least one reference transducer that is located in the same node and in the same direction in all data sets. That means in all the data set definitions there must be at least one record in which the at least first four fields are the same. The building model example consist of two data sets with two reference transducer, which are located in nodes 5 and 6. The definition of the data sets is shown below. To the right of the definition there are comments describing the record and which data set it belong to: Setups
Setup 1
Data set label. (Data Set 1)
mes31set.asc
Name of the file containing the measured data. (Data Set 1)
2 0 1 0 0.000001 m/s² Acceleration Transducer 1
1st. DOF definition record. (Data Set 1)
5 0 1 0 0.000001 m/s² Acceleration Transducer 2
2nd. DOF definition record. (Data Set 1)
11 -1 0 0 0.000001 m/s² Acceleration Transducer 3
3rd. DOF definition record. (Data Set 1)
6 0 1 0 0.000001 m/s² Acceleration Transducer 4
4th. DOF definition record. (Data Set 1)
Setup 2
Data set label. (Data Set 2)
mes32set.asc
Name of the file containing the measured data. (Data Set 2)
5 0 1 0 0.000001 m/s² Acceleration Transducer 1
1st. DOF definition record. (Data Set 2)
3 0 1 0 0.000001 m/s² Acceleration Transducer 2
2nd. DOF definition record. (Data Set 2)
6 0 -1 0 0.000001 m/s² Acceleration Transducer 3
3rd. DOF definition record. (Data Set 2)
In the following the geometry as well as the location of the transducers of the two data sets are shown. The green arrows indicate the transducers relating only to the specific data set, whereas the blue arrows indicate the reference transducers. The direction of the arrows indicate the orientation of the directional vectors. If the length of one of these arrows is different from the others, it is an indication that the length of this directional vector is different from the others.
Setup 1 (Data Set 1)
Setup 2 (Data Set 2)
The Data File Format The ASCII data file format used to import measured data and export processed data is a general text format that can be imported e.g. into Microsoft Excel or MATLABTM (MathWorks Inc). It is the similar way MATLABTM stores a data matrix in an ASCII file. The measurements of a specific transducer are stored as a single column in the ASCII file. So the data in the ASCII file can be interpreted as a matrix whose columns are the different transducers and the rows the measurements of these. An example (Examples Folder: \Building Model\Mes31set.asc) of a data file is show below:
In this case there are four columns corresponding to four transducers and 18 rows which corresponds to 18 samples per transducer. The measurements of the first transducer is column number 1 from the left. The following should noted: All transducers must have the same number of samples. There must not be any other information stored in the file, such as comments in the beginning. Formatting such as scientific or fixed formats are allowed.
The delimiters between the samples must be either blank spaces or tabs. No other delimiters are allowed.
The SVS Binary Data File Format The SVS binary data file format used to import measured data in a fast and compressed format. If you want to make use of this format you must store the following information (here explained using the C-language variable types) in the specified order. 1. Save the number of samples (measurements) per channels (transducers) as a long integer. 2. Save the number of channels as a long integer. 3. Save all samples of channel number one as either float or double values. 4. Save all samples of channel number two as either float or double values. 5. ... and so on until all channels have been saved for a specific data set. The following should noted: All transducers must have the same number of samples. There must not be any other information stored in the file, such as comments in the beginning. You must create a new file for each of the data sets.
The Equations Keyword (Optionally) The group that starts with the Equations keyword is used to define linear combinations of the measured motions. The slave node equations can be used for definitions of rigid body motions and slave nodes. This is very helpfull when the mode shapes are animated. Even though the measurements only are available in few nodes of the geometry the equations can be used to make the complete geometry move. Each record in the group contains only one field which is a string of any length. This string is one equation, which imply that equations must be defined in one line only. All equations start in the same way as shown below: node( , ) =
This equation defines the the motion of the node specified by in the direction specified by . The field is a number from 1 to 3. 1 correponds to X direction, 2 correponds to Y direction and 3 correponds to Z direction. Note: To define that the motion is in the negative direction must be performed on the righthand side of the equation sign. In other words, the motion in one direction of a node must always be done in the positive direction.
Making a Nodes Direction Static If you want to set the nodes motion in a specific direction to a static values you can simply write an equation like node( , ) = 3.1415926535
This equation sets the motion of the node specified by in the direction specified by to pi. Incidently, pi is actually the only predefined constant available which means that in the present case you also write: node( , ) = pi
Making a Nodes Direction a Linear Combination of other Nodes You can also make a nodes motion in a specific direction a linear combination of other nodes by writing an equation like 1.5*node( , ) + 2*pi node( , ) =
cos( node( , ) )
This equation sets the motion of the node specified by in the direction specified by equal to a motion that is a linear combination of the motion of in and the cosine of in . Please note that the splitting of the equation into two lines as shown above is not allowed in ARTeMIS Extractor. The following functions are available and can be used with numerical or functional arguments:
node( , )
Returns the motion node
in direction
/
Division
pi
The number Pi = 3.141592653589793108624468950438
()
Brackets with possibility to have up to 20 bracked terms inside each other.
Building Model Example For the building model example the group of equations is like this: Equations node(8,1)
= node(11,1)
node(8,2)
= node(2,2)
node(11,2) = node(5,2) node(9,1)
= node(12,1)
node(9,2)
= node(3,2)
node(12,2) = node(6,2) node(5,1)
= node(11,1) - node(2,2) + node(5,2)
node(2,1)
= node(5,1)
node(6,1)
= node(12,1) - node(3,2) + node(6,2)
node(3,1)
= node(6,1)
Here the motion of node 8 in direction 1 (x-direction) is defined to be equal to node 11 direction 1. A more sofisticated example is that the motion of node 5 in direction 1 is specified as: node(5,1)
= node(11,1) - node(2,2) + node(5,2)
In order to understand the equations in the example, consider the following figure illustrating the movements of one of the deck plate in the building model:
It is assumed that the deck plate moves as a rigid body, i.e. the movement can be described by two displacements and one angle. The deformation angle is given by:
where a is the depth of the plate. Now, assuming the plate is moving like a rigid body, the deformation in the x-direction of the lower right corner of the plate is given by
or if we introduce the formula for the angle
and this is what the 7th equation says. Exchange u(5,x) with node(5,1), u(5,y) with node(5,2) etc. and you arrive at the equation in the 7th line. The idea is simply to specify the motions of the nodes that are not defined directly by the measurements by the rigid body motions. If you want to postpone the equation definition or if you have found an error you can always reload the equations from the File menu. You will then be asked to specify a SVS Configuration File containing the equations to be loaded. In this case you can still use you existing configuration file now with equations added or modified. However, in the file you load only the Equations group needs to be present. You can also add the equations directly in the Project Control window in the Slave Node Equations Editor at any time after the project has been created and compile them immediately.
Trouble Shooting Load Errors If you get an error while loading the SVS Configuration File you will get an error meassage indicating the kind of error and where in the file you have the error. The line number given in the error message is either where you have the error or the last line of the group in which the error has been located. If you get load errors that relate to problems with reading the file, try to load at the file in e.g. the Microsoft Notepad and verify that everything is good. These error can be caused by the use of third party editors that inserts unsupported escape characters.
Creating a Project from Universal File Format It is possible to import a multiple data set ARTeMIS Extractor project from a set of files with data stored using the so-called ASCII Universal File Format (ASCII UFF), see e.g WEB-site of the Structural Dynamics Research Laboratory at University of Cincinnati, Ohio. In the Universal File Format, data is stored in various so-called UFF data sets that has different numbers. The UFF data sets that are necessary in order to create a full ARTeMIS Extractor project are: 1. A UFF Data Set Number 15. This data set contains the node definitions and is equal to the group in the SVS Configuration File format that starts with the keyword Nodes. 2. UFF Data Sets Number 82. These data sets contain the trace line definitions and are equal to the group in the SVS Configuration File format that starts with the keyword Lines. These data sets are optionally. 3. A UFF Data Set Number 2412. This data set contains the triangular surface definitions and is equal to the group in the SVS Configuration File format that starts with the keyword Surfaces. This data set is optionally. 4. UFF Data Sets Number 58. These data sets contains the data at a specific node in a specific direction and is closely related to the group in the SVS Configuration File format that starts with the keyword Setups. These data sets are optionally. In the following the example located in Examples\BK_Plate will be used. This example contains geometry and measurements of a simple plate. The plate is modelled using 9 nodes, which is the places where the accelerometers were mounted. The measurements were organized in two data sets using three reference accelerometers. The nodes, trace lines and surfaces of the geometry is located in the file BK_Plate_Geometry.uff. The UFF data sets number 58 containing the measured data of the first ARTeMIS Extractor data set is located in the file BK_Plate_Meas1.uff, whereas the UFF data sets number 58 containing the measured data of the second ARTeMIS Extractor data set is located in the file BK_Plate_Meas2.uff. To create a project from UFF ASCII format select the Universal File Format option in the New Project dialog shown below:
After pressing OK the dialog shown below will then appear:
Loading Nodes, Trace Lines and Surfaces The first step in the creation process is to load information about the nodes, trace lines and surfaces. All this information must be stored in the same file and concist of a UFF data set
#15 (Nodes), and optionally one or more UFF data sets #82 (Trace Lines) and one UFF data set #2412 (Surface). The file containing all this information is specified in the edit field of the upper group called UFF Data Sets for Geometry Definition. In the example the geometrical information is stored in the file BK_Plate_Geometry.uff.
Optionally, you can use the browse button to the right. When succesfully loaded it will be shown how many nodes, trace lines and surfaces that has been loaded. Note: If there are multiple UFF data set # 82 in the file they are all loaded and used when drawing the geometry. However, the color specifications of the data sets are disregarded and all lines are drawn with the same color. It is possible to stop here and create an ARTeMIS Extractor project that only contains a geometry. This can sometimes be a good idea if you are not sure about the quality of the geometry. To stop just press the OK button.
Loading Measured Data The last step is to open the files containing the measured data. Measured time series are in Universal File Format stored in ASCII files containing the UFF data set # 58. The ARTeMIS Extractor data sets must be stored in separate files, one for each data set. The current version only support loading of the ASCII version of UFF data sets 58 storing real time response of either single or double precision. In the example all UFF data sets 58 belonging to the first ARTeMIS Extractor data set is stored in the file BK_Plate_Meas1.uff, and all belonging to the second ARTeMIS Extractor data set is stored in BK_Plate_Meas2.uff. The data sets can be loaded sequentially but also by multiple selection by pressing the browse button to the right of the edit field of the group called UFF Data Sets with Time Response Data, see below:
Pressing the button starts the open file dialog shown below:
and the two files with the time response data for the two ARTeMIS Extractor data sets can be selected. When the Open button is pressed the data is uploaded. When the data is uploaded the corresponding ARTeMIS Extractor data set are presented as shown below:
As seen it might happen that the first ARTeMIS Extractor data set does not correspond to the first data file. This is matter of the order the files have been selected. If you want to be absolutely sure to select the files in a specific order you should do the upload sequentially. In other words, browse and open one file before continuing with the next file. When all files are uploaded you simply press the OK button. General Notes: Within each ARTeMIS Extractor data set the order of the transducers will be the same as the order the UFF data set 58 are stored in the file. Please note that since the data is loaded from an ASCII file it may take some time to load large data sets. If you have very long paths and/or file names and you are using multiple selection of the files you might experience that nothing is loaded. In this case please upload the files sequentially. The error is caused by a limitation in the Open Dialog used when browsing for the files. The Universal File Format does not support the definition of slave node equations. However, these can be loaded into a project afterwards from the File, Import, Slave Node Equations. You can also type in the equations directly in the Project Control window in the Slave Node Equations Editor.
Trouble Shooting Load Errors If you have load errors when uploading the measured data please check the following field in your UFF data sets #58: Record 6 - Field 1: Function Type. Must be 1 (Time Response). Record 6 - Field 6: Response Node. The node must be present in the uploaded UFF data set #15. Record 7 - Field 1: Data Type. Must be either 2 or 4 (Real single or double precision). Record 7 - Field 3: Abscissa Spacing. Must 1 (Even spacing). Record 8 - Field 1: Specific Abscissa Data Type. Must be 17 (Time). There might of course be other reasons for errors but typically the data set have been exported with some of the above options set differently.
UFF Data Set Number 15 This is the Universal File Format data set for Nodes. For more information see e.g WEBsite of the Structural Dynamics Research Laboratory at University of Cincinnati, Ohio.
Record 1: FORMAT(4I10,1P3E13.5) Field 1 Node label. Field 2 Definition coordinate system number. (Not used) Field 3 Displacement coordinate system number. (Not used) Field 4 Color. (Not used) Field 5-7
3 - Dimensional coordinates of node in the definition system.
Note: Repeat record for each node. Only one data set number 15 with global node definitions is supported.
UFF Data Set Number 55 This is the Universal File Format data set for Data at Nodes. For more information see e.g WEB-site of the Structural Dynamics Research Laboratory at University of Cincinnati, Ohio. In ARTeMIS Extractor it is used for export of modal results.
RECORD 1: Format (40A2) Field 1 ID Line 1.
RECORD 2: Format (40A2) Field 1 ID Line 2.
RECORD 3: Format (40A2) Field 1 ID Line 3.
RECORD 4: Format (40A2) Field 1 ID Line 4.
RECORD 5: Format (40A2) Field 1 ID Line 5.
RECORD 6: Format (6I10) Data Definition Parameters Model Type 0: Unknown Field 1 1: Structural 2: Heat Transfer 3: Fluid Flow Analysis Type 0: Unknown 1: Static 2: Normal Mode Field 2 3: Complex eigenvalue first order 4: Transient 5: Frequency Response 6: Buckling 7: Complex eigenvalue second order Data Characteristic 0: Unknown 1: Scalar Field 3 2: 3 DOF Global Translation Vector 3: 6 DOF Global Translation & Rotation Vector
4: Symmetric Global Tensor 5: General Global Tensor Specific Data Type 0: Unknown 1: General 2: Stress 3: Strain 4: Element Force 5: Temperature 6: Heat Flux 7: Strain Energy 8: Displacement Field 4 9: Reaction Force 10: Kinetic Energy 11: Velocity 12: Acceleration 13: Strain Energy Density 14: Kinetic Energy Density 15: Hydro-Static Pressure 16: Heat Gradient 17: Code Checking Value 18: Coefficient Of Pressure Data Type
Field 5 2: Real 5: Complex
Field 6 Number Of Data Values Per Node (NDV)
Records 7 And 8 Are Analysis Type Specific General Form RECORD 7: Format (8I10) Number Of Integer Data Values Field 1 1 < Or = Nint < Or = 10 Number Of Real Data Values Field 2 1 < Or = Nrval < Or = 12 Field 3-N
Type Specific Integer Parameters
RECORD 8: Format (6E13.5) Field 1-N
Type Specific Real Parameters
For Analysis Type = 0, Unknown RECORD 7: Field 1 1
Field 2 1 Field 3 ID Number RECORD 8: Field 1 0.0
For Analysis Type = 1, Static RECORD 7: Field 1 1 Field 2 1 Field 3 Load Case Number RECORD 8: Field 1 0.0
For Analysis Type = 2, Normal Mode RECORD 7: Field 1 2 Field 2 4
Field 3 Load Case Number Field 4 Mode Number RECORD 8: Field 1 Frequency (Hertz) Field 2 Modal Mass Field 3 Modal Viscous Damping Ratio Field 4 Modal Hysteretic Damping Ratio
For Analysis Type = 3, Complex Eigenvalue RECORD 7: Field 1 2 Field 2 6 Field 3 Load Case Number Field 4 Mode Number RECORD 8: Field 1 Real Part Eigenvalue
Field 2 Imaginary Part Eigenvalue Field 3 Real Part Of Modal A Field 4 Imaginary Part Of Modal A Field 5 Real Part Of Modal B Field 6 Imaginary Part Of Modal B
For Analysis Type = 4, Transient RECORD 7: Field 1 2 Field 2 1 Field 3 Load Case Number Field 4 Time Step Number RECORD 8: Field 1 Time (Seconds)
For Analysis Type = 5, Frequency Response RECORD 7:
Field 1 2 Field 2 1 Field 3 Load Case Number Field 4 Frequency Step Number RECORD 8: Field 1 Frequency (Hertz)
For Analysis Type = 6, Buckling RECORD 7: Field 1 1 Field 2 1 Field 3 Load Case Number RECORD 8: Field 1 Eigenvalue
RECORD 9: Format (I10)
Field 1 Node Number
RECORD 10: Format (6E13.5) Field 1-N
Data At This Node (NDV Real Or Complex Values)
Records 9 And 10 Are Repeated For Each Node.
UFF Data Set Number 58 This is the Universal File Format data set used for importing/exporting the data of a transducer sitting in a specific node and pointing in a specific direction. In the case of ARTeMIS Extractor it is the raw time series of a transducer. For more information see e.g WEB-site of the Structural Dynamics Research Laboratory at University of Cincinnati, Ohio.
Record 1: Format(80A1) Field 1 ID Line 1. Note: ID Line 1 is generally used for the function description.
Record 2: Format(80A1) Field 1 ID Line 2.
Record 3: Format(80A1) Field 1 ID Line 3. Note: ID Line 3 is generally used to identify when the function was created. The date is in the form DD-MMM-YY, and the time is in the form HH:MM:SS, with a general Format(9A1,1X,8A1).
Record 4: Format(80A1) Field 1 ID Line 4.
Record 5: Format(80A1) Field 1 ID Line 5.
Record 6: Format(2(I5,I10),2(1X,10A1,I10,I4)) DOF Identification Function Type 0: General or Unknown 1: Time Response (Must be used) 2: Auto Spectrum 3: Cross Spectrum 4: Frequency Response Function 5: Transmissibility 6: Coherence 7: Auto Correlation 8: Cross Correlation 9: Power Spectral Density (PSD) 10: Energy Spectral Density (ESD) 11: Probability Density Function 12: Spectrum Field 1 13: Cumulative Frequency Distribution 14: Peaks Valley
15: Stress/Cycles 16: Strain/Cycles 17: Orbit 18: Mode Indicator Function 19: Force Pattern 20: Partial Power 21: Partial Coherence 22: Eigenvalue 23: Eigenvector 24: Shock Response Spectrum 25: Finite Impulse Response Filter 26: Multiple Coherence 27: Order Function Field 2 Function Identification Number Field 3 Version Number, or sequence number Load Case Identification Number Field 4 0: Single Point Excitation Field 5 Response Entity Name ("NONE" if unused) Field 6 Response Node (Must be present in the associate data set #15)
Response Direction
0: Scalar 1: +X Translation 4: +X Rotation (Not allowed) -1: -X Translation -4: -X Rotation (Not allowed) Field 7 2: +Y Translation 5: +Y Rotation (Not allowed) -2: -Y Translation -5: -Y Rotation (Not allowed) 3: +Z Translation 6: +Z Rotation (Not allowed) -3: -Z Translation -6: -Z Rotation (Not allowed)
Field 8 Reference Entity Name ("NONE" if unused) Field 9 Reference Node Field 10
Reference Direction (same as field 7)
Note: Fields 8, 9, and 10 are only relevant if field 4 is zero.
Record 7: Format(3I10,3E13.5) Data Form
Ordinate Data Type 2: Real, single precision Field 1 4: Real, double precision 5: Complex, single precision (Not allowed) 6: Complex, double precision (Not allowed)
Field 2
Number of data pairs for uneven abscissa spacing, or number of data values for even abscissa spacing Abscissa Spacing
Field 3 0: uneven (Not allowed) 1: even (no abscissa values stored) Field 4 Abscissa minimum (0.0 if spacing uneven) Field 5 Abscissa increment (0.0 if spacing uneven) Field 6 Z-axis value (0.0 if unused)
Record 8: Format(I10,3I5,2(1X,20A1)) Abscissa Data Characteristics Specific Data Type 0: unknown 1: general 2: stress
3: strain 5: temperature 6: heat flux 8: displacement Field 1
9: reaction force 11: velocity 12: acceleration 13: excitation force 15: pressure 16: mass 17: time (Must be used) 18: frequency 19: rpm 20: order
Field 2 Length units exponent Field 3 Force units exponent Temperature units exponent Field 4 Note: Fields 2, 3 and 4 are relevant only if the Specific Data Type is General, or in the case of ordinates, the response/reference direction is a scalar, or the functions are being used for nonlinear connectors in System Dynamics Analysis. Field 5 Axis label ("NONE" if not used) Field 6 Axis units label ("NONE" if not used)
Note: If fields 5 and 6 are supplied, they take precendence over program generated labels and units.
Record 9: Format(I10,3I5,2(1X,20A1)) Ordinate (or ordinate numerator) Data Characteristics
Record 10: Format(I10,3I5,2(1X,20A1)) Ordinate Denominator Data Characteristics
Record 11: Format(I10,3I5,2(1X,20A1)) Z-axis Data Characteristics
Note: Records 9, 10, and 11 are always included and have fields the same as record 8. If records 10 and 11 are not used, set field 1 to zero.
Record 12: Data Values Ordinate
Abscissa
Case
Type
Precision
Spacing
Format
1
real
single
even
6E13.5
2
real
single
uneven
6E13.5
3
complex
single
even
6E13.5
4
complex
single
uneven
6E13.5
5
real
double
even
4E20.12
6
real
double
uneven
2(E13.5,E20.12)
7
complex
double
even
4E20.12
8
complex
double
uneven
E13.5,2E20.12
General Note: ID lines may not be blank. If no information is required, the word "NONE" must appear in columns 1 through 4.
UFF Data Set Number 82 This is the Universal File Format data set for Trace Lines. For more information see e.g WEB-site of the Structural Dynamics Research Laboratory at University of Cincinnati, Ohio.
Record 1: FORMAT(3I10) Field 1 Trace line number. (Not used) Field 2 Number of nodes defining the trace line. (maximum of 250) Field 3 Color. (Not used)
Record 2: FORMAT(80A1) Field 1 Identification line. (Not used)
Record 3: FORMAT(8I10) Nodes defining trace line: Field 1 > 0 draw line to node. 0 move to node (a move to the first node is implied) Notes: 1. MODAL-PLUS node numbers must not exceed 8000. 2. Identification line may not be blank. 3. Systan only uses the first 60 characters of the identification text.
4. MODAL-PLUS does not support trace lines longer than 125 nodes. 5. Supertab only uses the first 40 characters of the identification line for a name.
UFF Data Set Number 2412 This is the Universal File Format data set for Elements. For more information see e.g WEBsite of the Structural Dynamics Research Laboratory at University of Cincinnati, Ohio. The surfaces used is modelled as Thin Shell Linear Triangles. The definition below reflects this restriction.
Record 1: FORMAT(6I10) Field 1 Element label (Not used) Field 2 FE descriptor ID (Must be 91 for Thin Shell Linear Triangle) Field 3 Physical property table number (Not used) Field 4 Material property table number (Not used) Field 5 Color (Not used) Field 6 Number of nodes on element (Must be 3) Record 2: FORMAT(8I10) Field 1 Node label 1 Field 2 Node label 2 Field 3 Node label 3 Field 4 (Not used)
Field 5 (Not used) Field 6 (Not used) Field 7 (Not used) Field 8 (Not used) Note: Records 1 and 2 are repeated for every triangular surface.
Creating a Project using the OLE Automation Interface Object Linking and Embedding (OLE) 2.0 automation is supported in ARTeMIS Extractor for project creation. This allows you to use visual programming tools, such as Visual BASIC or Visual C++, to develop custom programs for initializing ARTeMIS Extractor. Such programs allow a predefined sequence of events to be performed automatically. For example, clicking a start button in a custom Visual BASIC program could start the execution of ARTeMIS Extractor, create a new project and flush data directly from the custom made program into the newly created project. The custom program acts as the user interface. This interface can be very simple, for example, it may only have a start and a cancel button allowing unskilled workers to use it. In such a custom program, ARTeMIS Extractor becomes the server for the external custom program and performs defined tasks as a slave under the control of it. It is through this OLE automation interface that ARTeMIS Extractor is controlled by e.g the Brüel & Kjær Modal Test Consultant Type 7753. A type library and an example showing how the automation works with MATLABTM is included with ARTeMIS Extractor. The type library is the artx.tlb file located in the Bin folder. The example file to run is artx.m located in the Examples folder: MATLAB.
Working with Slave Node Equations The slave node equations can be used for definitions of rigid body motions and slave nodes. This is very helpful when the mode shapes are animated. Even though the measurements only are available in few nodes of the geometry the equations can be used to make the complete geometry move. The slave node equations can be entered directly into the project using the Slave Node Equations editor in the Project Control window, or they can be loaded together with the rest of the project when the project is created using the SVS Configuration File, see here to have a detailed explanation of the possibilities. Finally, you can load a new set of equation that will replace all other previously entered equations from the File menu.
Project Control Window The first window to appear when a new project is created or an existing project is opened is the Project Control window. In case of the example mes32set.axp the windows as shown below:
As seen the window contain several tabs in the bottom and a tree to the left. The four lowest tabs called Project, Data set, Geometry and Signal Processing are all referred to as the main group tabs of the Project Control window. As seen above some of the groups also include a set of sub-tabs. Below the different part of the Project Control windowis explained. The Data Sets Tree To the left in the Project Control window there is a Data Sets Tree that list all the available data sets in the project. The open data set is the one that is currently active. When you have to change the current data set this is a place to do it. The Project Main Tab What you see in the above example is the default main group called Project. This group
presents and accepts information general to the whole project. The Data Set Main Tab The next main group is located under the Data Set main tab. This window presents information and accepts information related to the currently selected data set. The Geometry Tab When pressing the tab called Geometry the project geometry is presented. The Signal Processing Settings Main Tab This tab present the Trial Signal Processing Configuration and the Signal Processing Configuration Selected for Analysis. The Signal Processing Log Main Tab This tab present all the Signal Processing Configurations Selected for Analysis that has been applied to the data from the creation of the project.
The Data Sets Tree The Data Sets Tree is the tree control located in the left-hand side of the Project Control window as shown below:
The Data Sets Tree has several functions. These are Selection of current data set. Renaming of data set labels. Enabling / disabling data sets.
Select Current Data Set The Data Set Tree is one of the controls that can be used to select the current data set. The current data set is what will be presented in e.g. the View Processed Data window and all other windows that present information of a specific data set. Below is shown how the first of two data sets is selected:
The tree item having the blue background is the currently selected. You can also use the following controls located in the Project Toolbar and in the View menu: Select current data set from the drop down list.
Select previous data set as current. Select next data set as current.
Rename Data Set Labels If you want to rename the label of a data set the Data Set Tree is where you do it. In the Data Set Tree shown below you simply double-click on the label you would like to rename:
When you have double clicked a edit box appear:
where you can enter the new label. To accept the new label press or click somewhere outside the edit box. To abort the change press .
Enabling / Disabling Data Sets The most advanced feature of the Data Set Tree is the ability to enable and disable data sets. To the left of the data set label there is a check box as shown below:
In the above both data sets are enabled which is indicated by the check marks and . You can disable a data set by unchecking the corresponding check box. The data set will then become inaccessible which is indicated by . In the project geometry tab window the transducers of the inaccesseible data sets will be drawn as red arrows to indicate that they are disabled. The suggested usage of this feature is primarely for closeup inspection of mode shape parts and to disable data sets containing bad measurements. The enabling / disabling of data sets has different effects depending on whether the disabling / enabling is performed before or after an operation. Disabling a data set before an operation Below it is describe what happens if a data set is disabled before a specific action: Action
Consequence
None. All data sets are processed. Processing Measured Data
Only enabled data sets will be included in the modal estimation. In other words, mode shapes will be zero for transducers belonging to disabled data sets, and natural frequencies will only be calculated solely on the basis of picked frequencies of enabled data sets.
FDD Mode Estimation
The disabled data sets will be prepared to use the frequency picked in the averaged SVD display and the corresponding mode shape. If the data sets later on are enabled the Spectral Density Matrices of these data sets will be updated automatically (if necessary) and the natural frequencies / mode shapes determined.
Only enabled data sets will be included in the modal estimation. In other words, mode shapes will be zero for transducers belonging to disabled data sets, and natural frequencies and damping ratios will only be calculated solely on the basis of picked frequencies of enabled data sets.
EFDD Mode Estimation
The disabled data sets will be prepared to use the frequency picked in the averaged SVD display and the corresponding mode shape in the enhanced estimation of the modal parameters. In addition, the MAC rejection level and the correlation limits used in the enhanced estimation will be initialized to the values setup up by the Preference dialog. If the data sets later on are enabled the Spectral Density Matrices of these data sets will be updated automatically (if necessary) and the natural frequencies / damping ratios / mode shapes determined.
The estimation of parametric models can only be performed on enabled data sets. In the Select and Link Modes editor only the selected parametric models of the enabled data sets are presented and included in the link procedure. In other words, mode shapes will be zero for transducers belonging to
SSI Mode Estimation
disabled data sets, and natural frequencies and damping ratios will only be calculated solely on the basis of picked frequencies of enabled data sets. If the data set are enabled later on the modes of the selected models of these data sets must be selected and linked afterwards. This part is not done automatically as with the non-parametric estimators (FDD and EFDD).
Disabling a data set after an operation Action
Consequence
None. All data sets are processed. Processing Measured Data
FDD Mode Estimation
The modes are re-estimated. Only enabled data sets will be included in this modal reestimation. In other words, mode shapes will be zero for transducers belonging to disabled data sets, and natural frequencies will only be calculated solely on the basis of picked frequencies of enabled data sets. The information needed to include the disabled data sets if they are enabled later on is preserved.
The modes are re-estimated. Only enabled data sets will be included in this modal re-
EFDD Mode Estimation
estimation. In other words, mode shapes will be zero for transducers belonging to disabled data sets, and natural frequencies and damping ratios will only be calculated solely on the basis of picked frequencies of enabled data sets. The information needed to include the disabled data sets if they are enabled later on is preserved.
SSI Mode Estimation
Any parametric models of the disabled data sets already estimated are preserved. However, in the Select and Link Modes editor only the selected parametric models of the enabled data sets are presented and included in the link procedure. In other words, mode shapes will be zero for transducers belonging to disabled data sets, and natural frequencies and damping ratios will only be calculated solely on the basis of picked frequencies of enabled data sets.
The Project Main Tab What you see in the below example is the main group called Project.
This group presents and accepts information general to the whole project. This group has a list of sub-tabs which are listed below: General. Nodes. Lines. Surfaces. Comments. Slave Node Equations.
The General Tab (Project Main Tab) The General sub-tab window located under the Project main tab presents the current status of the project. In the case of the Building Model example the General tab looks as shown below right after the project has been created using the SVS Configuration File:
There are four different items presented. These are: General. In this item the project title is shown in a editable field. Double-click with the mouse on the field and you will be able to edit the text. In the next field the operator or author can enter his name. Also, the SVS Configuration File used to create this project is shown below. The project title as well as the author is show as summary information in the Windows Explore, see below. Geometry. List how many nodes and trace lines that are presented in the geometry displays. Data. Presents how many data sets there are in the project. Also, the number of reference transducers are presented. This number will be zero if only one data set is available. Finally, the sampling interval, sampling frequency and Nyquist frequency are presented.
No. Estimated Modes by. Present the number of modes currently estimated using the different estimators. This table supports the general table options for copy and print. Example showing the summery information tooltip of the Windows Explore for the above project.
The Nodes Tab (Project Main Tab) The Nodes sub-tab window located under the Project main tab presents a list of the nodes and their X, Y and Z coordinates. In the case of the Building Model example the Nodes tab looks as shown below when the project has been created using the SVS Configuration File:
The coordinates are presented in the same units as they have been entered into the project. This table supports the general table options for copy and print.
The Lines Tab (Project Main Tab) The Lines sub-tab window located under the Project main tab presents a list of the lines defined by the connecting nodes. In the case of the Building Model example the Lines tab looks as shown below when the project has been created using the SVS Configuration File:
The lines are presented in the same order as they have been entered into the project. This table supports the general table options for copy and print. If you want to hide some of the lines it can be accomplished by double-clicking at the line in the Visible column.
The Surfaces Tab (Project Main Tab) The Surfaces sub-tab window located under the Project main tab presents a list of the surfaces defined by the connecting nodes. In the case of the Building Model example the Surfaces tab looks as shown below when the project has been created using the SVS Configuration File:
The surfaces are presented in the same order as they have been entered into the project. This table supports the general table options for copy and print. If you want to hide some of the surfaces it can be accomplished by double-clicking at the line in the Visible column.
The Comments Tab (Project Main Tab) The Comments sub-tab window located under the Project main tab is an editor where you can write or paste information relating to the project in general. Comments specific to the individual data set should not be written here but rather under the Comments sub-tab located under the Data Set main tab. This editor supports the general editor options for copy, cut, paste and print. Since this is a Rich Text Format window all text formatting is preserved when copying text from e.g. Microsoft Word.
The Slave Node Equations Tab (Project Main Tab) The Slave Node Equations sub-tab window located under the Project main tab is an editor where you can write or paste new Slave Node Equations. In the case of the Building Model example the Slave Node Equations tab looks as shown below right after the project has been created using the SVS Configuration File:
You can modify existing or add new equations while you are animating the mode shapes, and immediately see the change. In the case of the above example, we have estimated a mode using the Frequency Domain Decomposition Peak Picking estimator a put the mode shape animation of this mode beside the Slave Node Equations editor. We have also put the Project Toolbar next to it.
Now, try and add a new constraint value of -10 cm in the Z-direction of node 1:
When you start edit the Update Modes button on the Project Toolbar becomes active. Pressing the button immediately updates the displayed mode:
The update of course depends on whether the new equation has been correctly entered. If an error occurs a message will appear explaining the error, and the line where the error occurred will be highlighted. You can also update the equations/modes using the contextsensitive menu by right-clicking the mouse inside the editor. If you have created the project using the SVS Configuration File and you want to return to the original equations, use the File, Import, Slave Node Equations. This import will overwrite all current equations. This editor supports the general editor options for copy, cut, paste and print.
The Data Set Main Tab What you see in the below example is the main group called Data Set.
This group presents and accepts information general to the currently selected data set. This group has a list of sub-tabs which are listed below: General. Transducers. Measured DOFs. Comments.
The General Tab (Data Set Main Tab) The General sub-tab window located under the Data Set main tab presents information as well as the status of the currently active data set. In the case of the Building Model example the General tab looks as shown below right after all the signal processing has been completed:
There are four different items presented. These are: General. In this item the label and number of the data set is presented as well as the status of the data set. Also the state of the data set is shown. If the data set is checked in the Data Sets Tree the status will say Enabled and Disabled otherwise. Data. Presents how many measured records or DOF's that are in the data set. This number includes the records of the reference transducers. Also, the length of record both in samples and time is presented. Data Source. Display information about the origin of the data that forms the data set. In this case the data comes from the file mes31set.asc. In addition to the file name the size, type and creation date and time are presented. Signal Processing Status. Display information about the signal processing status of
this particular data set. If all parts of the signal processing has been applied to the data the status is marked Complete. If some parts are missing the status will be Not Complete, and the missing parts are shown below with a label saying Not Applied. This table supports the general table options for copy and print.
The Transducers Tab (Data Set Main Tab) The Transducers sub-tab window located under the Data Set main tab list information about the transducers that forms the currently selected data set. In the case of the Building Model example the Transducers tab looks as shown below.
There are seven columns presented. These are: Transducer. This column presents the label of the transducer. The label of a transducer can be edited by double-clicking on the label. By doing so an edit box appear. Type in the new label and press to close the edit box. If you press the edit box disappear and the original label is restored. Node. Display the node where the transducer is located. Status. Display the status of a particular transducer. The transducer status can either be Enabled or Disabled. The status can be changed by double-clicking in the column on the particular transducer. If you change the status of one or more transducers all previous estimated modes will be delete. Also, any parametric models of the specific data set is deleted and you will have to re-estimate the Common SSI Input Matrix for this specific data set. If the transducer is a reference the status of the same transducer of all the other data sets will be changed accordingly.
Reference. Identified whether the transducer is a reference or not. If the transducer is a reference the label will say Yes and No otherwise. dB Reference. Lists the reference values to be used in dB displays of e.g. spectral densities. Record Unit. Shows the unit of the measured record. Record Type. Presents what type of measurement the transducer has returned. This table supports the general table options for copy and print.
The Measured DOFs Tab (Data Set Main Tab) The Measured DOFs sub-tab window located under the Data Set main tab list information about the which degrees-of-freedom (DOF) that has been measured using the transducers of the currently selected data set. In the case of the Building Model example the Measured DOFs tab looks as shown below right after the project has been created using the SVS Configuration File:
There are six columns presented. These are: Transducer. This column presents the label of the transducer. The label of a transducer can be edited by double-clicking on the label. By doing so an edit box appear. Type in the new label and press to close the edit box. If you press the edit box disappear and the original label is restored. Node. Display the node where the transducer is located. X-Component. List the X-component of the directional vector of the transducer. Y-Component. List the Y-component of the directional vector of the transducer. Z-Component. List the Z-component of the directional vector of the transducer.
Vector Length. Display the resulting length of the directional vectors. Usually, all numbers in this column should be equal to 1. This table supports the general table options for copy and print.
The Comments Tab (Data Set Main Tab) The Comments sub-tab window located under the Data Set main tab is an editor where you can write or paste information relating to the currently selected data set. Comments that relate to the hole project should not be written here but rather under the Comments sub-tab located under the Project main tab. This editor supports the general editor options for copy, cut, paste and print. Since this is a Rich Text Format window all text formatting is preserved when copying text from e.g. Microsoft Word.
The Geometry Tab The Geometry main tab presents geometry as well as the transducer location and orientation of the currently active data set. The 3D - View Tab In the case of the Building Model example the geometry looks as shown below right after the project has been created using the SVS Configuration File:
In the corner where the X-, Y- and Z-coordinate is smallest the arrows of the standard coordinate system are shown in cyan colors. Together with the geometry the transducers of the currently selected data set are presented as arrows pointing in the positive direction of the transducer. The blue arrows indicates the transducers that are common and represented in all data sets. The green arrows represent the rest of the transducers of the current data set that are enabled. Red arrows are the disabled transducers of the current data set. In the above example all transducers are seen to be enabled. Besides the general 3D-display options the following features are available:
The node numbers can be hidden by releasing the button in the modal toolbar. If you have selected some nodes in the Nodes tab of the Project Control Window these nodes remain on the display. If you point at an object (node, line or surface) it will be highlighted and the X, Y and Z coordinates pointed at, on the object, are presented just below the title. The color of the background, lines, coordinate arrows, transducer arrows can be changed in the properties dialog.
The Quad View Tab The second tab is the Quad View which presents the same infomation from three fixed angles and one general angle.
The three fixed angles are: 1. Side (+Y). From the side looking in the positive Y direction 2. Side (+X). From the side looking in the positive X direction
3. Top (+Z). Looking from the bottom towards the top in the positive Z direction The genral view can be manipulated just like the 3D-View described above.
The Properties Dialog When the Geometry tab of the Project Control Window is active you can display the properties dialog associated with it.
By pressing the color-well buttons you can change the colors in the active 3D display of: Background. Coordinate system arrows. The geometry (lines and surfaces). The inactive transducers. The active transducers. The reference transducers.
The Signal Processing Settings Tab The Signal Processing Settings main tab present the Trial Signal Processing Configuration and the Signal Processing Configuration Selected for Analysis. Both of these are initialized and edited by the Signal Processing Configuration wizard. Using the above example you will get the following shown when both the Trial Signal Processing Configuration and the Signal Processing Configuration Selected for Analysis are initialized using the Signal Processing Configuration wizard. Before the initialization both displays of this tab will be empty.
To the left the trial signal processing configuration is shown. The items presented (Decimation, Filtering, Spectral Density Matrices Estimation, Correlation Function Estimation and Common SSI Input Matrix Estimation) all display the corresponding information selected using the Signal Processing Configuration wizard and saved as the project trial configuration. On the right-hand side is the equivalent presentation of the signal processing configuration selected for analysis. To emphasize the irreversible part of the signal processing the headline of these items explicitly says Irreversible Action.
In the File, Export menu items or from the context sensitive menu you can export the signal processing configuration of the active one of the two views. So, if your point at the Trial Signal Processing Configuration window this will be the exported configuration. In this way you can re-use the configuration in other projects. The configuration is saved in the SVS Signal Processing Configuration format.
The Signal Processing Log Tab The last main tab is called Signal Processing Log. This tab present the Trial Signal Processing Configuration and the Signal Processing Configuration Selected for Analysis. Both of these are initialized and edited by the Signal Processing Configuration wizard. Using the above example you will get the following shown when both the Trial Signal Processing Configuration and the Signal Processing Configuration Selected for Analysis are initialized using the Signal Processing Configuration wizard. Before the initialization both displays of this tab will be empty.
This tab present all applied signal processing configuration selected for analysis. In addition, it is marked which of the signal processing configuration items that has been applied to the data of all data sets. In case it has not been applied the right-aligned text says Not Applied and otherwise it say Applied. So in the above example everything has currently been applied. If multiple configurations has been applied they will appear in separate sub tabs. The label of the tab will be a number starting from 1 as well as the date and time of the creation of
the configuration.
Processing Measured Data In the ARTeMIS Extractor software there is a so-called Signal Processing Configuration wizard available which can be shown by clicking the button Define Sign. Proc. Conf. in the Data task bar pane or from the Project menu item. This wizard is available for several reasons: You might want to filter or decimate the data before you start using it in the analysis. The modal identification estimators need to have the measured data processed in different ways before it can be used. The signal processing configuration wizard is used to specify basic signal processing like filtering and decimation of the signals. Once this kind of processing has been performed, the raw data has been changed, and the only way to get back to the original data is to read in the raw data files once again. For this reason this kind of signal processing is called irreversible signal processing. The wizard is also used to specify estimation of spectral density functions, correlation functions and the estimation of the common input matrix that is used by the Stochastic Subspace Identification techniques. Since these results are derived from the measured data, they can be changed at a later stage in the process. For this reason this kind of signal processing is called reversible signal processing. However, even though you can change the settings of the reversible signal processing later on without having to reload the measured data, the modes that depends on the parts of the signal processing that are changed will be deleted. There are two kinds of signal processing configurations available in a project. These are Trial Signal Processing Configuration. Signal Processing Configuration Selected for Analysis. When a new project is created both of these configurations are empty. The idea with these two configuration is to have one to play with in a safe mode that will not destroy the data, and one that is selected for use in the analysis once the user has convinced himself about the right signal processing choices. Note: You cannot do any signal processing or modal estimation before you have initialized the Signal Processing Configuration Selected for Analysis using the Signal Processing Configuration wizard. Also note that you cannot use the Trial Signal Processing Configuration before it has been initialized using the wizard. The signal processing configuration contain information about how to process the data. This
information can be put into the following groups:
Processing Action Order
1.
Decimation: The decimation is an irreversible signal processing used to narrow the frequency range to work with. The configuration stores the information needed to perform this action. This information is the number of time to decimate. If the configuration is selected for analysis the decimation is reset when the last data set has been decimated. To be reset means not to be applied later on.
Irreversible/Reversible
Irreversible
Note: Changing this setting will delete all modes already estimated.
2.
Filtering: After decimation but before any of the reversible signal processing it is possible to filter the data using a filter that you define yourself. The configuration stores the information needed to perform this action. This information is the filter type, order and cut-off frequencies. If the configuration is selected for analysis the filter is reset when the last data set has been filtered. To be reset means not to be applied later on.
Irreversible
Note: Changing these settings will delete all modes already estimated.
Spectral Density Matrices Estimation: After all the irreversible signal processing has been completed the spectral density matrices can be estimated. The configuration stores the information needed to perform this action. This information is the frequency resolution. 3.
Reversible
Note: Changing this setting will delete all Frequency Domain Decomposition modes (FDD and EFDD) already estimated.
4.
Correlation Functions Estimation: After all the irreversible signal processing has been completed the correlation function matrices can be estimated. The configuration stores the information needed to perform this action. This information is the number samples in the estimated correlation function.
Common SSI Input Matrix Estimation: After all the irreversible signal processing has been completed the Common SSI input matrix for use with all the stochastic subspace identification techniques is estimated. The 5. configuration stores the information needed to perform this action. This information is the expected number of (Pro version structural modes, harmonics, and noise modes. only) Note: Changing these settings will delete all Stochastic Subspace Identification modes already estimated.
Reversible
Reversible
The irreversible signal processing is processed first for all data sets and cannot be stopped. This is necessary in order preserve consistency of the raw data. When the irreversible signal processing is started it always starts with the part of the configuration that has been modified that has the lowest Processing Order. The reversible processing of the spectral density matrices, the correlation functions and the common SSI input matrix are running in parallel and can be stopped whenever you like. Everything that has been processed at the time of stopping is saved and can be used fior analysis. The signal procesing is started in the Process Data dialog, and the status of the signal processing can be monitered in the Processing Status tab of the Control Panel. When modifying a signal processing configuration using the Signal Processing Configuration wizard the order in which the configuration is modified reflects the above mentioned order.
The Trial Signal Processing Configuration When the signal processing applied in "safe mode" the signal processing is performed using the trial signal processing configuration. This signal processing configuration is applied to a copy of the measured data of the currently selected data set. So the irreversible as well as the reversible signal processing is only applied on this copy of the data. The result of using the trial signal processing configuration is presented in the Test Trial Signal Processing Configuration window. You should play around with the settings in the trial signal processing configuration using the Signal Processing Configuration wizard and inspect how the settings affects the data in the Test Trial Signal Processing Configuration window until you are sure which settings to use. You should inspect how the settings work in all data sets for two reasons. First of all, some of the settings might be good in some data sets and might not be good in other data sets. This could e.g. be the frequency resolution that need to be adjusted due to more noise in one data set compared to another. Secondly, by forcing yourself to inspect the configuration using all data sets, you also force yourself to check the quality of the data. In this your will be able to discover unexpected errors in the data, such as dead channels etc. Note: Be sure to actually define your own trial signal processing configuration. In earlier releases a default configuration would have been used. However, now you cannot start the Test Trial Signal Processing Configuration window before this configuration is initialized using the Signal Processing Configuration wizard.
The Signal Processing Configuration Selected for Analysis When you are sure that you have a good trial signal processing configuration you can save it as the configuration selected for analysis using the Signal Processing Configuration wizard. Any existing configuration that were selected for analysis before is then overwritten and all the modes that are related to the part of the configuration that has changed will be deleted. In the Signal Processing window in Project Control it will be indicated which parts of the configuration that has been applied, and already is up to date, and which have not been applied. If you know how to set up the signal processing to use in the analysis in advance, you could e.g. be doing repeated testing of the same structure, then you do not need to create a trial signal processing configuration first. In this case you can use the Signal Processing Configuration wizard to set up the configuration selected for analysis directly using the wizard dialogs or by importing a signal processing configuration from a file. Note: Be sure to actually define your own configuration selected for analysis. In earlier releases a default configuration would have been used. However, now you cannot start any signal processing before this configuration is initialized using the Signal Processing Configuration wizard.
The Signal Processing Configuration Wizard The Signal Processing Configuration wizard is a versatile tool. The idea is that you initialize the wizard with a copy of an already defined configuration. You then modify this configuration and export the modified configuration to a desired location. The input and output of the Signal Processing Configuration wizard is shown below: Wizard Input
Wizard
Wizard Output
Trial Signal Processing Configuration (If already initialized) Trial Signal Processing Configuration Signal Processing Configuration Selected for Analysis (If already initialized)
Preset Signal Processing Configuration
-> Configure -> Signal Processing Configuration Selected for Analysis
Export a Signal Processing Configuration to a File
Import a Signal Processing Configuration from a File
The left-hand side of the above figure list the possible input to the wizard, and the righthand side list the possible output of the wizard. You can only select one input to the wizard but as output you can save the configuration to all the listed outputs. So, let's say you want to modify the trial signal processing configuration of the project then you will select this as your input to the wizard. You can then modify the copy of the trial configuration in the wizard, and then export this modified copy as the new trial configuration of the project. If you just want to save the trial configuration as the configuration selected for analysis you will still import the trial configuration into the wizard and step through all the steps of the wizard. When you reach the end you can then save the copy of the trial configuration as the one selected for analysis.
Note: When a new project is created there will not be a trial signal processing configuration or a signal processing configuration selected for analysis available. At this point there will only be a preset signal processing configuration available as well as the ability to import a signal processing configuration from a file. The wizard consists of seven pages. These are: 1. Define Signal Processing Configuration. 2. Decimation. 3. Filtering. 4. Spectral Density Matrices Estimation. 5. Correlation Function Estimation. 6. Common SSI Input Matrix Estimation. 7. Finish Signal Processing Configuration.
Define Signal Processing Configuration (Step 1) When you have launched the Signal Processing Configuration wizard the first page to appear is the page where you select which signal processing configuration to use a copy of inside the wizard. In the case of the Building Model example, the page looks as below:
Here you have four options for choosing the signal processing configuration. These are: Trial configuration. Configuration selected for analysis. Preset configuration. Import configuration from a file. You can only select one of these options and at this state you do not have to be concerned about whether it should be saved as one signal processing configuration or another. If you select the import from file option the browse button on the page will be enabled to allow you to browse for the file.
The Preset Signal Processing Configuration (Step 1) If you need to restore one of the signal processing configuration in the project you can initialize the Signal Processing Configuration wizard with a preset signal processing configuration. This configuration is the typically starting point when you want to establish a Trial Signal Processing Configuration as well as the Signal Processing Configuration Selected for Analysis.
Importing a Signal Processing Configuration (Step 1) If you have created a signal processing configuration in one project and exported it from e.g. the Project Control window in an SVS Signal Processing Configuration format, you can import this configuration into another project using the Import configuration from a file option on the first page of the Signal Processing Configuration wizard. If you select this option a button becomes active so that you can browse for the desired file. The extension of this type of file is .spc. When you have uploaded a configuration you are allowed to go to the last step of the wizard. You can do this pressing the button .
The SVS Signal Processing Configuration Format To show the SVS Signal Processing Configuration Format we can try and see how the Trial Signal Processing Configuration presented in the Project Control window below appear in the file format:
When save to a file using e.g. the context-menu the result looks as below:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % This file contains the ASCII file representation of a Signal Processing Configuration. % % This configuration was created by the project: % C:\Development\Installation Files\Examples\Version 3.1\Building Model\mes32set.axp
% % % EDIT THIS FILE WITH EXTREME CAUTION % ONLY EDIT INFORMATION TO THE RIGHT OF THE ":" SYMBOL. %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
DECIMATION Number of Decimations : 1
FILTERING Filter (Not Editable) : Butterworth Order : 0 Type : 0 Nyquist Frequency : 7.497500e+001 Lower Cut-Off Frequency : 0.000000e+000 Upper Cut-Off Frequency : 7.497500e+001
SPECTRAL_DENSITY Segment Length : 2048 Segment Overlap : 1365 Window Function : 0
CORRELATION Correlation Length : 256
SSI Number of Structural Modes: 10 Number of Harmonics : 0 Number of Noise Modes : 30
Note: This file is only displayed so that you can see how the different settings are saved. You should not try to edit this file manually but only use it to transfer a signal processing configuration from one project to another. This type of file has the extension .spc.
Decimation (Step 2) The first signal processing to perform on the measured data is the decimation. The purpose of decimation is to reduce the frequency range to the frequency range of interest. For instance, if a signal has been sampled at 200 Hz, i.e. 200 samples per second, then the Nyquist frequency is 100 Hz and the spectral density estimates will defined from 0-100 Hz. Now, if only you are interested in the modal properties under 10 Hz, then you should decimate the signal 10 times. If you want to work in the frequency domain it is desirable to decimate to zoom to the range of interest. If you are working with parametric models in the time domain, you should decimate in order to introduce poles only in the frequency range of interest. The decimation is basically a process where some samples are discarded and some samples are kept. However, in order to prevent aliasing the signal is first low-pass filtered and then the excessive samples are discarded. If the signal is decimated 2 times then every 2nd sample is kept and the rest is discarded. If the signal is decimated 3 times every 3rd sample is kept the rest is discarded, etc. The following figures show how a signal looks before and after it is decimated by 2:
Now, since the anti-aliasing filter is active into the frequency range of the decimated signal, you should only use the modal results in the frequency range from 0 to 80 % of the new Nyquist frequency. In ARTeMIS Extractor the applied anti-aliasing filter is an 8th order Chebyshev Type 1 low-pass filter. Note: Decimation is an irreversible process which will change the data of the project permanently when applied in the signal processing configuration selected for analysis. After the decimation has been applied to the data the decimation of this configuration is reset and if the project contains any estimated modes they will all be deleted.
Setting up the Decimation The setting up of the decimation is performed as the second step in the Signal Processing Configuration wizard. In the case of the Building Model example, the page looks as below:
If you do not want to perform any decimation but just continue press the button. The order of decimation is selected from the combo box located on the left-hand side of the page. The default order of decimation is 1 indicating that no decimation should be performed at all. To the right the result of the selected order of decimation is presented in terms of the new sampling interval and the new sampling frequency. The possible orders of decimation are listed in the table below together with the influence the chosen decimation has on the original sampling interval Ts and the original sampling frequency Fs. Order of Decimation
New Sampling Interval
New Sampling Frequency
1
Ts
Fs
2
2 Ts
Fs / 2
3
3 Ts
Fs / 3
4
4 Ts
Fs / 4
5
5 Ts
Fs / 5
10
10 Ts
Fs / 10
20
20 Ts
Fs / 20
30
30 Ts
Fs / 30
40
40 Ts
Fs / 40
50
50 Ts
Fs / 50
100
100 Ts
Fs / 100
200
200 Ts
Fs / 200
300
300 Ts
Fs / 300
400
400 Ts
Fs / 400
500
500 Ts
Fs / 500
1000
1000 Ts
Fs / 1000
When you have selected the desired order of selection press the button to continue. Note: When you change the order of decimation you also change all maximum values that relates to the maximum number of samples and the sampling interval in the following steps of the wizard.
Filtering (Step 3) Filtering is used to shape the signal in the frequency domain. Typical reason for filtering could e.g. be to high-pass filter the signal to get rid of a slowly varying zero-offset of the signal, or to band-pass filter the signal to reduce the necessary parametric model order for cases with high modal density. In ARTeMIS Extractor you have the posibility to apply one of the following types of filters: low-pass, high-pass, band-pass and band-stop. In each case the filter is of the Butterworth type and you can select filter orders between 1 and 50. The Butterworth filter is a good allround filter that is simple to use. At the specified cut-off frequencies it attenuates 3 dB. In the case of the Building Model example, the Filtering page looks by default as below:
As seen it is by default disabled. If you do not want to specify any filter but just continue press the button. To enable filtering check the box called Enable. The controls will the be active:
The default filter type to appear is an 8th order low-pass filter which is a reasonable order to start from. You can then select different types of filters from the Type combo box and different order from the Order edit field. The order depended slope of the filter is presented both in Slope per octave and Slope per decade. In case you are specifying a low-pass filter you will have to specify an Upper cut-off frequency. In case it is a high-pass filter you will to specify a Lower cut-off frequency. For the band-stop and stop-band filter types you must specify both. Note: If the filter is unrealizable you will get an error when pressing either the or the button. The most commom reason for a filter to be unrealizable is that the filter is un-stable. To reduce the risk of a filter becomming un-stable, choose a lower model order or move the cut-off frequency further away from the frequency boundaries, i.e. towards the midle of the frequency band from zero to the Nyquist frequency. Also note that filtering is an irreversible process which will change the data of the project permanently when applied in the signal processing configuration selected for analysis. After the filtering has been applied to the data the filter settings of this configuration is reset and if the project contains any estimated modes they will all be deleted. You can view the selected filter characteristic while working with the trial signal processing configuration using the Test Trial Signal Processing Configuration window.
Projection Channels (Step 4) To avoid too much redundant cross information when estimating Correlation Functions, Spectral Density Matrices and the Common SSI Input Matrix you can manually select the number of projection channels to use. Technically, what is selected is actually the column space to make use of in the estimation of Correlation Functions and Spectral Density Matrices. In case of the Common SSI Input Matrix estimation it corresponds to the selection of the row space to project the measurements on. The selection of the projection channels are done automatically. The first columns chosen are the reference channels if they are available. The rest of the columns are the ones that on average correlates the least with the reference channels. This is done to maximize the amount of independent information. In the case of the Building Model example, the page looks as below when 3 projection channels has been chosen for both data sets:
You activate the option by clicking the check box called Enable. By doing so the drop-down box is activated and the available number of channels are presented. If multiple data sets are available it is not allowed to select less than the number of reference channels available. It is never recommended to use less than 3 projection channels. The philosophy behind this is that in case of two close modes at least two projection channels are needed to separate the modes plus one additional channel to account for the measurement noise. NOTE: Since selecting a limited amount of projection channels will remove redundant information, it is likely that the number of noise modes in the configuration of the Common
SSI Input Matrix Estimation can be decreased significantly.
Spectral Density Matrices Estimation (Step 5) The fifth step of the Signal Processing Configuration wizard is to configure the spectral density matrices estimation. The spectral density matrices are the estimate of the spectral density function at discrete equally spaced frequency lines in a range between zero (DC) frequency and the Nyquist frequency. In order to set this estimation up it is necessary to specify how many discrete frequency lines (DC excluded from this number) to estimate. In the case of the Building Model example, the Spectral Density Matrices Estimation page looks by default as below:
The only action you need to make is to select the number of Frequency Lines in the combo box. This is a radix-2 number due to the use of the Fast Fourier Transform (FFT). Below the combo box you can get some information about the Sampling Interval, the Sampling Frequency and the Nyquist Frequency. Please observe that these values as well as the maximum number of frequency lines depends on the chosen order of decimation. To the right, the Spectral Analysis Configuration is listed. Here you can see the Frequency Line Spacing, i.e. the distance between two estimates on the frequency axis. You can also see which Window Function that is used. The window function has been choosen permanently to be a Hanning window. Finally, you can also see that the Overlap between two data segments is 66.67%. The window function is introduced to minimize the wrap-around bias introduced by the periodicity assumption introduced by using the FFT algorithm. The bias introduced this way can be reduced but not removed, and it will appear as a leakage bias in the frequency domain, i.e. spectral peaks will be blunted. The overlap is introduced in order to
compensate for the loss of information due to tapering of the data segments when the data segments are multiplied by the window function. The estimated spectral density matrices are primarely used to estimate the modes in the Frequency Domain Decomposition technique but also as validation tool for the Stochastic Subspace Identification techniques. Note: Changing the number of Frequency Lines is an reversible action that does not change the measured data when applied in the signal processing configuration selected for analysis. In this case, only the modes estimated using the Frequency Domain Decomposition technique will be deleted. The less number of frequency lines you use the more averages you get, and the more smooth your spectral densities will look. The smooth appearance is caused by averaging out the noise. However, not only does it look nicer, your mode shape estimates obtained from the Frequency Domain Decomposition will also be more accurate. So the advice is not to use too many frequency lines, but allow the estimation algorithm to make some averages. Use only the resolution that is necessary to clearly distinguish the individual peaks of the modes. The default number of frequency lines, 256, is a number that is reasonable if only a few modes are to be identified. If you are dealing with a problem of high modal density, for instance having something like 20 modes to identify, then you should use a much higher number of frequency lines, and thus in order to have enough averages to minimize the noise on the spectral densities, you might need longer data segments. Note: The number of frequency lines choosen corresponds to the number of samples in the correlation function estimation. By default the estimation of the correlation functions is simply done by an inverse Fourier transformation of the spectral densities. This approach is the fastest way to obtain the estimates but the result is biased because of the leakage bias in the spectral estimates. If the check box Use unbiased estimation is checked the correlation functions are estimated using an unbiased estimation technique based on doubling the length of each data segment by zero padding before using the FFT algorithm. This changes the unpredictable wrap-around bias into the well-defined basic lag window bias. This bias can be removed in the time domain by dividing with the basic lag window, which is also called a Bow-Tie correction. The correlation functions are only used as a validation tool for the Stochastic Subspace Identification techniques. If an estimated parametric model is optimal in a statistical sense the covariance functions of the data and the model will coincide. The model is said to be covariance equivalent. It is therefore a good validation tool to use for the Stochastic Subspace Identification techniques. You can also view the correlation functions in the View Processed Data window, the Compare Processed Reference Data window and in the Test Trial Signal Processing Configuration window.
Common SSI Input Matrix Estimation (Step 6) (Pro version only) The initialization of the Common SSI Input matrix Estimation is something that you cannot really do before you have looked at the data using e.g. the Test Trial Signal Processing Configuration window, or processed what is needed to inspect the spectral densities in the View Processed Data window. The reason is that you need to have an idea of how many modes that are present in the data. The modes that are estimated using the Stochastic Subspace Identification techniques can be grouped into the following types: Structural Modes. Harmonics. Noise Modes. Usually, it is the structural modes that we are interested in since these describe the dynamics of the structure we are analyzing. Besides, these modes there will also be harmonics in the data as soon as there rotating parts in the structure or in the excitation. Finally, there will always be noise modes or computational modes present. They are not always easy to see in a spectral density plot because of their tendency of being highly damped. However, you will always have computational modes arising e.g. from the antialiasing filter. Computational modes also appear if your system is not completely linear, or if the excitation is far from being a realization of a shaped Gaussian white noise, or if in other ways the assumption for using the estimation techniques are not met. The computational modes are then added by the estimation algorithm to take account of the discrepancies. It is usually a very good idea to inspect the Singular Value Decomposition of the Spectral Density Matrices because this very elegantly reveal the structural modes, the harmonics and any closely spaced modes. By counting the structural modes and the harmonics you will be well prepared for making the initialization of the Common SSI Input matrix Estimation. In the case of the Building Model example, the Common SSI Input matrix Estimation page looks by default as below:
Even though the estimation of the common input matrix only requires one number which is the sum of all modes, we have divided it in to the above mentioned categories. So, in the edit field called No. of Structural Modes you enter how many structural modes you believe there are in the data. If you have detected any harmonics they are entered in the edit field No. of Harmonics. Finally, you will have to guess on how many noise modes you are suspecting. In this case you should always over-specify instead of under-specifying the number - the quality of the estimate improves as this number is increased. A good rule of thumb is to multiply the sum of structural modes and harmonics by 3. As an absolute minimum you should never specify less computational modes than the sum of the structural modes and harmonics. The total number of modes is then presented as the Maximum No. of Modes. Since the Stochastic Subspace Identification estimators returns parametric model in the so-called stochastic state space format, the Maximum State Space Dimension is also presented. The state space dimension is the same as the number of eigenvalues or poles in the dynamic model. Note: Remember if your Stochastic Subspace Identification estimates turns out to be inaccurate, or if the state space dimension of the best-fit model comes to close to the maximum state space dimension, you should go back to the Signal Processing Wizard and increase the number of modes. Also, note that changing the number of modes in any of the above edit field forces ARTeMIS Extractor to reestimate the common SSI input matrix and consequently delete all modes estimated by any of the stochastic subspace identification techniques.
Finish Signal Processing Configuration (Step 7) This is the final step in the Signal Processing Configuration wizard. This is where you specify which signal processing configuration that should be overwritten with the modified copy of the wizard.
So, now you simply check the boxes of the configurations you want to overwrite as shown below:
Check the first box to save the configuration As the trial configuration of the project. Check the second box to save it As the configuration selected for analysis. Finally, check the last box to save the configuration To a signal processing configuration file. Note: If you have imported a configuration from a file in the first step of the wizard, and if you are absolutely sure you want to use this configuration. This imported configuration can then be selected for analysis by checking the second box.
This last page of the wizard is the only place where you can modify the signal processing configurations of the project. It is on purpose that you are forced to go through the hole wizard every time, since this will decrease the probability that you overwrite the configuration selected for analysis by accident. Because once you have overwritten the existing configuration selected for analysis the already estimated modes might be deleted.
The Test Trial Signal Processing Configuration window The Test Trial Signal Processing Configuration window is used to view how the Trial Signal Processing configuration affects the data of the currently selected data set. The testing is performed on a copy of the data so that the original data remain unchanged. You can view the results of all the parameter choices made in the trial signal processing configuration except the parameters relating to the estimation of the Common SSI Input Matrix. When started from either the Data Pane of the Task Bar or from the Project menu the window first becomes visible when the copy of the data has been processed with the Trial Signal Processing Configuration. The signal processing can be monitored in the Status Bar. When the signal processing is finished the window appear. Note: You have to initialize the Trial Signal Processing Configuration using the Signal Processing Configuration Wizard. In the case of the building model example the window look as below:
In this example we have specified a 10th order band-pass filter with cut-off frequencies at 10 Hz and 70 hz. As seen it is a splitted window that enable you to view two different representations of the data at the same time. You switch among the different view using the tabs located below the views. All the views are 2D data displays and they all support the general 2D display
options. You can view the following representations of the data: Magnitude of Spectral Densities. Phase Angle of Spectral Densities. Coherence. Singular Value Decomposition of Spectral Density Matrices. Average of Diagonal Elements of Spectral Density Matrices. Average of All Elements of Spectral Density Matrices. Normalized Correlation Functions. Magnitude of Filter Transfer Function. Unwrapped Phase Angle of Filter Transfer Function. All representations of the data except the two relating to the filter are also used for presentation of the processed data in the View Processed Data window.
The Magnitude Tab This display presents the magnitude of the estimated spectral density function between two measured degrees of freedom (DOF's).
The two DOF's being displayed are presented in the title in terms of the labels of the transducers and the currently selected data set, in this case Free Transducer 1, Ref. Transducer 1 and Measurement 1, respectively. This entry in the spectral density matrices has been selected using the 2D-Displays tab of the Control Panel. In the present case the displayed is a cross-spectral density function. If it were the same transducer, e.g. a display of Free Transducer 1 and Free Transducer 1, then it would be the auto-spectral density function which is also the power spectral density function. Note: The above plot uses a dB scale but can also use a linear scale. You can use the general 2D data display options with this display.
The Phase Tab This display presents the wrapped phase angle in degrees of the estimated spectral density function between two measured degrees of freedom (DOF's).
The two DOF's being displayed are presented in the title in terms of the labels of the transducers and the currently selected data set, in this case Free Transducer 1, Ref. Transducer 1 and Measurement 1, respectively. This entry in the spectral density matrices has been selected using the 2D-Displays tab of the Control Panel. In the present case the displayed is a cross-spectral density function. If it were the same transducer, e.g. a display of Free Transducer 1 and Free Transducer 1, then it would be the auto-spectral density function having a zero phase angle. Note: You can use the general 2D data display options with this display.
The Coherence Tab This display presents the coherence of the estimated spectral density function between two measured degrees of freedom (DOF's).
The two DOF's being displayed are presented in the title in terms of the labels of the transducers and the currently selected data set, in this case Free Transducer 1, Ref. Transducer 1 and Measurement 1, respectively. This entry in the spectral density matrices has been selected using the 2D-Displays tab of the Control Panel. In the present case the displayed is coherence between two different transducers. If it were the same transducer, e.g. a display of Free Transducer 1 and Free Transducer 1, then the coherence would exactly 1. Note: You can use the general 2D data display options with this display.
The coherence function
The coherence function between two signals, a(t) and b(t) relates how much of the
measured output signal, a(t) is linearly related to the measured output signal, b(t) at any given frequency. A coherence of 1 indicates a perfect linear relationship, and 0, no relationship. The coherence is always bounded between 0 and 1. Coherence function is very similar to the Correlation Coefficient Function between two stochastic variables, and is defined as the covariance of the two variables divided by the product of the standard variation of the two variables. At each given frequency the Coherence function corresponds to the Correlation Coefficient Function squared. Coherence less than one can be due to one or more of the following situations. 1. Uncorrelated noise in the measurement of a(t) and/or b(t). 2. Non-linearity of the system under investigation. 3. Leakage in the analysis (resolution bias error). 4. Delays in the system not compensation for. Coherence function is widely used for input/output measurement for validating Frequency Response Function measurements. For output only measurements Coherence Function is also a very useful function since it is expected that the coherence will take high values at resonance frequencies (except at node points / lines), where a strong vibration pattern exist and a high signal to noise ratio is found. Note: The MAC (Modal Assurance Criteria) is defined in a similar way as the Coherence, and expresses the degree of linear relationship (similarity) between mode shapes.
The SVD Tab This display presents the singular values of the matrices of the estimated spectral density function. At each frequency there are as many singular values as there are measured degrees of freedom (DOF's) in the currently selected data set.
This display is a very usefull tool because you can detect close modes, but also because the frequency content is presented in a single display. The singular value display the rank of each of the spectral density matrices. If only one mode is dominating at the frequency only one singular value will be dominating at this frequency. This is why only one singular value pop up at about 34 Hz in the example below. There is only one dominating mode at this frequency. If you have close or repeated modes you will see as many dominating singular values as there are close or repeated modes. Above you can see two singular values pop up at around 20 Hz and again around 55 Hz. At both frequencies there are two close modes. Note: You can use the general 2D data display options with this display.
The Average Diagonal Tab This display presents the average of the diagonal elements of each of the matrices of the estimated spectral density function of the currently selected data set. In other words, it presents the average of all auto-spectral density functions.
This average line is an alternative to the Singular Value Decomposition of the estimated matrices of the spectral density function. It provides a fast indication of where the most dominating modes are located. However, you cannot see the presence of close modes. Note: You can use the general 2D data display options with this display.
The Average All Tab This display presents the average of all the elements of each of the matrices of the estimated spectral density function of the currently selected data set. In other words, it presents the average of all auto-spectral density functions and all cross-spectral density functions.
This average line is an alternative to the Singular Value Decomposition of the estimated matrices of the spectral density function. It provides a fast indication of where the most dominating modes are located. However, you cannot see the presence of close modes. Note: You can use the general 2D data display options with this display.
The Correlation Tab This display presents the correlation function between two measured degrees of freedom (DOF's). The presented correlation function have been normalized so that the autocorrelation function starts in 1 or -1.
The correlation function is only used as a validation tool in the Stochastic Subspace Identification editor and in the Compare Processed Reference Data window. In order to take full advantage of these validation features you should make sure that the time lag is large enough. Note: You can use the general 2D data display options with this display.
The Filter Magnitude Tab This display presents the magnitude of the transfer function of the filter that can be used as a part of the signal processing. In the example below the filter is an 10th order band-pass filter with cut-off frequencies at 10 Hz and 70 Hz. The dB scale is always presented with a reference value equal to 1. The filter is a Butterworth type filter and at the cut-off frequencies it attenuates 3 dB.
Note: You can use the general 2D data display options with this display.
The Filter Phase Angle Tab This display presents the unwrapped phase angle of the transfer function of the filter that can be used as a part of the signal processing. In the example below the filter is an 10th order band-pass filter with cut-off frequencies at 10 Hz and 70 Hz.
Note: You can use the general 2D data display options with this display.
The Process Data Dialog Once you have set up the Signal Processing Configuration Selected for Analysis using the Signal Processing Configuration Wizard you can start processing the data. This is done using the Process Data dialog. The first time this dialog is started it will have the appearance shown below: (Handy and Light versions)
(Pro version)
In this dialog you can specify what to process. By default everything is processed for all data sets when you press the button. However, in certain situations it is beneficial to be able only to do a part of the processing. The first check box from the top allows you to select the Spectral Density Matrices
Estimation as a part of the processing. As seen this will delete all modes estimated by the Frequency Domain Decomposition techniques (FDD and EFDD peak picking). In the table below it is shown which editor / windows that require the spectral density matrices estimated. The table is divided into operations that require all enabled data sets and which that only require the currently selected data set processed. Required means that this operation cannot work without the processed data. Optionally means that the operation can work without the processed data. Selected Data Set
All Enabled Data Sets
View Processed Data window
Optionally
-
Compare Processed Reference window
Optionally
Optionally
FDD Peak Picking editor
-
Required
EFDD Peak Picking editor
-
Required
Required
-
-
Required
Operation
Stochastic Subspace Identification editor Select and Link Modes editor
The second check box allows you to select the Correlation Functions Estimation as a part of the processing. In the table below it is shown which editor / windows that require the correlation functions estimated.
Operation
View Processed Data window
Selected Data Set
All Enabled Data Sets
Optionally
-
Compare Processed Reference window
Optionally
Optionally
FDD Peak Picking editor
-
-
EFDD Peak Picking editor
-
-
Required
-
-
-
Stochastic Subspace Identification editor Select and Link Modes editor
In the Pro version the third check box allows you to select the Common SSI Input Matrix Estimation as a part of the processing. As seen this will delete all modes estimated by the Stochastic Subspace Identification (SSI) techniques. It is only the Stochastic Subspace Identification editor that require this operation to be completed for the selected data set. If the box is checked you will be allowed to activate the Automatic Model Estimation, which will then enable an automatic model estimation as a part of the processing. In the Estimators group you select the SSI estimators to apply (UPC, PC and CVA). In the group called State Space Dimensions, you specify the smallest and largest model order (state space dimension) as well as the increment in state space dimension between two consequtive models. This action is the most time consuming part of the processing. So, a good advice is to wait with this part of the processing until you have inspected the data in e.g. the View Processed Data window and perhaps also completed the modal identification using the FDD Peak Picking Editor and the EFDD Peak Picking Editor. When you have done this and found that the signal processing configuration is working good you can go back and do this processing. If you have many data sets and if you only need to process the current data set you can check the box labeled Only process currently selected data set. The requested processing actions will then only be performed for the selected data set. Note: If you have a window open that require a signal processing action performed for the selected or all data sets, then if you change the data set from the Project Control window the signal processing will be performed automatically. Therefore, in order to concentrate the waiting for completion of the signal processing it is a good practice to complete as much as possible in the first step. So the advice is: Process the Spectral Density Matrices
Estimation action and the Correlation Functions Estimation action immediately the first time for all data sets. If you are trying to open a window that require a specific unfinished processing action you will get an error message informing you about what to process. If you try to start the Stochastic Subspace Identification editor without having completed any processing actions you will get the following error message: Unable to open the requested window. The signal processing is incomplete. Please perform the following processing: Processing on currently selected data set: - Spectral Density Matrices Estimation. - Correlation Functions Estimation. - Common SSI Input Matrix Estimation. You will then have to go back to the Process Data dialog and perform the requested processing actions.
The View Processed Data Window The View Processed Data window presents the results of the signal processing of the data of the currently selected data set. You can view the results of all the parameter choices made in the signal processing configuration selected for analysis except the parameters relating to the estimation of the Common SSI Input Matrix. The window can be started either from the Data Pane of the Task Bar or from the Project menu. In the case of the building model example the window look as below:
As seen it is a splitted window that enable you to view two different representations of the data at the same time. You switch among the different view using the tabs located below the views. All the views are 2D data displays and they all support the general 2D display options. You can view the following representations of the data: Magnitude of Spectral Densities. Phase Angle of Spectral Densities. Coherence. Singular Value Decomposition of Spectral Density Matrices.
Average of Diagonal Elements of Spectral Density Matrices. Average of All Elements of Spectral Density Matrices. Normalized Correlation Functions. All representations of the data are also used for presentation of the processed data in the Test Trial Signal Processing Configuration window.
The Compare Processed Reference Data Window The Compare Processed Reference Data window presents the processed data results of the reference transducers of all enabled and signal processed data sets. You can view the results of all the parameter choices made in the signal processing configuration selected for analysis except the parameters relating to the estimation of the Common SSI Input Matrix. The window can be started either from the Data Pane of the Task Bar or from the Project menu. In the case of the HCT building in Vancouver, Canada example (Examples Folder: HCT Building\hct.axp) the window look as below:
In this window you can compare the quality of the measured data across the data sets. There are essentially two things that you can analyze using this window. You can verify that all the modes are represented properly in all the data sets. If a mode is badly represented in one or more data sets, you must expect that at least some of the modal parameters of the mode will be poorly estimated in these data sets. You can also verify if the energy level of the measurements changes significantly from data set to data set. In such cases you should also expect modal estimates of bad quality in the low energy level data sets. As seen it is a splitted window that enable you to view two different representations of the
data at the same time. You switch among the different view using the tabs located below the views. All the views are 2D data displays and they all support the general 2D display options. You can view the following representations of the data: Magnitude of Spectral Densities. Phase Angle of Spectral Densities. Coherence. Average of Diagonal Elements of Spectral Density Matrices. Average of All Elements of Spectral Density Matrices. Normalized Correlation Functions. The different tabs of this window are highly related to the tab windows of the Test Trial Signal Processing Configuration window and the View Processed Data window.
The Magnitude Tab (Compare Processed Reference Data Window) This display presents the magnitude of the estimated spectral density function between two measured degrees of freedom (DOF's) of all signal processed and enabled data sets. In the case of the HCT building in Vancouver, Canada example (Examples Folder: HCT Building\hct.axp) the window look as below:
The two DOF's being displayed are presented in the title in terms of the labels of the reference transducers, in this case Ref. Transducer 2 and Ref. Transducer 1. This entry in the spectral density matrices has been selected using the 2D-Displays tab of the Control Panel. The color coding of the four enabled and signal processed data sets are documented in the legend to the right. In the present case the displayed is a cross-spectral density function. If it were the same transducer, e.g. a display of Ref. Transducer 1 and Ref. Transducer 1, then it would be the auto-spectral density function which is also the power spectral density function. If the label of a specific reference node is the same in all data sets, then this label is used. If not, a default label will be assigned to it. Note: The above plot uses a dB scale but can also use a linear scale. You can use the general 2D data display options with this display.
The Phase Angle Tab (Compare Processed Reference Data Window) This display presents the wrapped phase angle in degrees of the estimated spectral density function between two measured degrees of freedom (DOF's) of all signal processed and enabled data sets. In the case of the HCT building in Vancouver, Canada example (Examples Folder: HCT Building\hct.axp) the window look as below:
The two DOF's being displayed are presented in the title in terms of the labels of the reference transducers, in this case Ref. Transducer 2 and Ref. Transducer 1. This entry in the spectral density matrices has been selected using the 2D-Displays tab of the Control Panel. The color coding of the four enabled and signal processed data sets are documented in the legend to the right. In the present case the displayed is a cross-spectral density function. If it were the same transducer, e.g. a display of Ref. Transducer 1 and Ref. Transducer 1, then it would be the auto-spectral density function which is also the power spectral density function having zero phase. If the label of a specific reference node is the same in all data sets, then this label is used. If not, a default label will be assigned to it. Note: You can use the general 2D data display options with this display.
The Coherence Tab (Compare Processed Reference Data Window) This display presents the coherence of the estimated spectral density function between two measured degrees of freedom (DOF's) of all signal processed and enabled data sets. In the case of the HCT building in Vancouver, Canada example (Examples Folder: HCT Building\hct.axp) the window look as below:
The two DOF's being displayed are presented in the title in terms of the labels of the reference transducers, in this case Ref. Transducer 2 and Ref. Transducer 1. This entry in the spectral density matrices has been selected using the 2D-Displays tab of the Control Panel. The color coding of the four enabled and signal processed data sets are documented in the legend to the right. In the present case the displayed is a cross-spectral density function. If it were the same transducer, e.g. a display of Ref. Transducer 1 and Ref. Transducer 1, then it would be the auto-spectral density function which is also the power spectral density function having coheremce 1. If the label of a specific reference node is the same in all data sets, then this label is used. If not, a default label will be assigned to it. Note: You can use the general 2D data display options with this display.
The Average Diagonal Tab (Compare Processed Reference Data Window) This display presents the average of the diagonal elements of each of the matrices of the estimated spectral density function. This is presented for all signal processed and enabled data sets. In the case of the HCT building in Vancouver, Canada example (Examples Folder: HCT Building\hct.axp) the window look as below:
This average line provides a fast indication of where the most dominating modes are located and what the average energy level is. The color coding of the four enabled and signal processed data sets are documented in the legend to the right. Note: The above plot uses a dB scale but can also use a linear scale. You can use the general 2D data display options with this display.
The Average All Tab (Compare Processed Reference Data Window) This display presents the average of all elements of each of the matrices of the estimated spectral density function. This is presented for all signal processed and enabled data sets. In the case of the HCT building in Vancouver, Canada example (Examples Folder: HCT Building\hct.axp) the window look as below:
This average line provides a fast indication of where the most dominating modes are located and what the average energy level is. The color coding of the four enabled and signal processed data sets are documented in the legend to the right. Note: The above plot uses a dB scale but can also use a linear scale. You can use the general 2D data display options with this display.
The Correlation Tab (Compare Processed Reference Data Window) This display presents the correlation function between two measured degrees of freedom (DOF's) of all signal processed and enabled data sets. The presented correlation function have been normalized so that the auto-correlation function starts in 1 or -1. In the case of the HCT building in Vancouver, Canada example (Examples Folder: HCT Building\hct.axp) the window look as below:
The two DOF's being displayed are presented in the title in terms of the labels of the reference transducers, in this case Ref. Transducer 2 and Ref. Transducer 1. This entry in the correlation function has been selected using the 2D-Displays tab of the Control Panel. The color coding of the four enabled and signal processed data sets are documented in the legend to the right. In the present case the displayed is a cross-correlation function. If it were the same transducer, e.g. a display of Ref. Transducer 1 and Ref. Transducer 1, then it would be the auto-correlation function. If the label of a specific reference node is the same in all data sets, then this label is used. If not, a default label will be assigned to it. Note: You can use the general 2D data display options with this display.
The FDD Peak Picking Editor The idea of the Frequency Domain Decomposition (FDD) technique is to perfom an approximate decomposition of the system response into a set of independent single degree of freedom (SDOF) systems, one for each mode. The theory is described in R. Brincker, L. Zhang and P. Andersen: Modal Identification from Ambient Responses using Frequency domain Decomposition. Proc. of the 18th International Modal Analysis conference (IMAC), San Antonio, Texas, 2000. The decomposition is performed simply by decomposing each of the estimated spectral density matrices. In the above reference it is shown that the singular values are estimates of the auto spectral density of the SDOF systems, and the singular vectors are estimates of the mode shapes. The FDD technique involves the main steps listed below: 1. Estimate spectral density matrices from the raw time series data. 2. Perform singular value decomposition of the spectral density matrices. 3. If multiple data sets are available, then average the first singular value of all data sets and average the second etc. 4. Peak pick on the average singular values. For well-separated modes always pick on the first singular value. In case of close or repeated modes, pick on the second singular value, the third singular value etc. as well. 5. Optionally, if multiple data sets are available, inspect the singular values of each data set and edit the peak picking position if necessary. The first three steps are performed automatically when the data is processed. The last steps require your input and are done using the FDD Peak Picking editor. The technique is a completely non-parametric technique where the modes are estimated purely by signal processing. In the case of the building model example the FDD Peak Picking editor looks as below when started right after the signal processing of the data is complete:
The editor consists of three windows which are: The Peak Pick Tab. The Animate Tab. The Mode List Tab. The peak picking is performed in the editor of the Peak Pick tab window and the result can be verified immediately after in the Animate tab window. The estimated modes are listed in the mode list window. In the above case this list is empty and the window is gray. Below is the same example presented. However, now there is five modes estimated:
The Peak Pick Tab (FDD Modal Identification) In the case of the building model example the Peak Pick tab window of the Frequency Domain Decomposition (FDD) technique looks as below when started right after the signal processing of the data is complete:
The left-most window is the Peak Picking editor where you pick the modes. To the right you can get information about the currently selected frequency and singular value. In addition, there is a legend explaining that an estimated mode is marked using a brown box and an estimated and selected mode is marked by a blue box. The Peak Picking editor displays singular values of the spectral density matrices. These singular values have been normalized with respect to the area under the first singular value curve (the top curve). If multiple data sets are present the normalized singular values calculated for each data set have been averaged to obtain the displayed curves. In the present example there are two data sets each with four transducers, and in order to obtain the presented four curves the following operations have been performed: 1. The 4x4 dimensional spectral density matrices of data sets 1 and 2 have been estimated. 2. For both data sets all the spectral density matrices have been decomposed using the singular value decomposition. The result is 4 singular values and 4 singular vector for each of the spectral density matrices. The singular values and the singular vectors are ordered in singular value descending order for each of the spectral density matrices, i.e. the first singular value is the largest.
3. For each data set the singular values are normalized. The normalization factor corresponds to the area under the first singular value curve. This normalization prevent that week modes only appearing in one or few data sets disappear. 4. Finally, the first singular value curve of both data sets are averaged frequency by frequency. This operation is repeated for the second, third, fourth etc. singular value curves. By using these averaged singular value curves all modal information can be presented in one display no matter how many transducers and data sets there are. Since this normalized and averaged curve is constructed from several transducers and several data sets the dB reference value of this display is always chosen as 1. If you want to zoom to a specific frequency range you can do this from the 2D-Displays tab of the Control Panel. Note: If you have multiple data sets it is possible to display the singular values of the currently selected data set instead.
Estimating a New Well-separated Mode The way to estimate a mode in this Peak Picking editor is illustrated below:
You can either press the New Mode button in the Modal Toolbar as shown in the lower left corner, or you can use the context-sensitive menu and select the New Mode menu item. In both cases a blue vertical cursor will appear together with a blue cross that snaps to the nearest point of the singular value curves as shown below:
The cursor line show the frequency position of the mouse pointer, whereas the cross show the peak nearest to the mouse cursor. In the right-hand side information window the location of the cross is presented in terms of the frequency (Frequency) and singular value curve (SVD Line) number. To estimate the mode at the peak where the cross is located simply double-click on the lefthand side mouse button. To abort the estimation press the key instead. When the estimation is finished the vertical line lock itself on the frequency at the picked mode. Further, a blue box will appear instead of the cross to indicate which singular value the mode has been picked on. Below the complete editor is shown after the estimation:
As seen the mode is now drawn in blue and listed in the Mode List below. The mode remains drawn in blue as long as it is selected. You can now immediately press the Animate tab to inspect the estimated mode shape.
Estimating Repeated Modes
If you have a system as the building model example with repeated modes you might need to pick the modes on several singular value curves at the same frequency. This means that if you have two repeated modes you will have to pick them on the first and the second singular value curve. It also means that the maximum number of repeated modes that can be estimated is equal to the minimum number of transducers in the data sets. To estimate the two repeated modes at 55 Hz in the building model example it is recommendable to zoom to a small frequency range around the repeated modes:
It is now clear that the two largest singular value curves peaks at the same frequency. Therefore, you will need to pick the first mode on the first singular value curve and the second mode on the second singular value curve as shown below:
Since the modes are estimated on the basis of the singular value decomposition the two estimates are only valid if the two modes indeed are orthogonal.
Estimating Closely Spaced Modes There might also be situations where the modes are closely spaced as it is the case of the modes around 20 Hz in the building model example. In this case it might still be possible to pick the modes on the first singular values. However, the difference between being closely spaced and being repeated depends both on the actual system but also on the frequency resolution of the spectral density estimation. Anyway, in the building model example the two
close modes around 20 Hz can be picked on the first singular value curve as shown below:
When you pick the modes you should try to visualize that you are picking the individual modes on the top of their correponding single degree of freedom (SDOF) spectral density bell function. If you can visualize that the bell of e.g. two modes have peaks on different frequencies then they are not repeated and can both be picked on the first singular value curve. If they have peaks at the same frequency you will have to pick them at different singular value curves instead. This is what the Enhanced Peak Picking take advantage of. It simply identifies these SDOF bell functions constructed from singular values.
Switching Between Average and Individual Singular Value Lines
Until now all operations relating to estimation of a mode has been performed using the averaged singular value display in the peak picking editor. As long as you use this display the same frequency and singular value will be used when the estimator picks out the singular vectors in the individual data sets. This way of estimating the modes are the easiest one and therefore also the most recommendable and it will work in most cases. However, sometimes you are forced to change either frequency and / or singular value in one or more data sets. This can happen if the frequency of the mode drifts a bit or if another mode suddenly become much more energetic. For this reason you can disable the average display and display the individual data set instead. The switch between the two displays can be performed by selecting the menu item Project, Frequency Domain Decomposition, Show Averaged SVD Lines or in the context-sensitive menu:
You can also use the Modal Toolbar and press button . When you are displaying the averaged singular value lines the button will appear pressed. When you choose to display the singular value lines of the individual data set it is always the lines of the currently selected data set that are displayed.
Editing a Mode Already Estimated
You can always edit the currently selected (blue) mode. You might want to adjust the frequency you are picking the mode on or selecting another singular value curve. The latter is applicable when dealing with closely spaced modes. To activate the edit mode, start by selecting the mode you want to edit. Then click the edit mode button on the Modal Toolbar. You can also activate the edit mode from the context-sensitive menu by selecting the menu item Edit Mode as shown below:
You have then activated the edit mode which will stay active until you either press the button / menu item agin or press the key, or the key or the Animate tab in the editor. When you press the left-hand mouse button down and keep it down you will see a blue cross with a gray cursor image behind as shown bellow:
The gray cursor image show which mode you are editing since the gray lines are drawn from the original position of the mode. The blue cross shows you the currently snapped position. If you release the mouse button the mode will move to this position and be reestimated using the new frequency and singular value position. Note: While you are displaying the average singular value lines the changes are reflected in all data sets. However, if you are editing while displaying the singular value lines of an individual data set only the frequency and singular value of this specific data set is changed.
Deleting a Mode You can always delete the currently selected (blue) mode either by pressing the or keys, or from the context-sensitive menu by selecting the Delete Mode menu item as shown below:
For every mode you are deleting you are asked to confirm the deletion. This is the only place where you can delete modes estimated by the FDD Peak Picking editor.
The Animate Tab (FDD Modal Identification) When you have selected a mode either from the Peak Pick editor window or from the Mode List you will immediately be able to see its mode shape animated by pressing the Animate tab. In case of the building model example a mode being animated is show below:
Note: The same window is used for mode shape validation in the Enhanced Frequency Domain Decomposition (EFDD) technique. The animation window supports the general 3D display options. While the window is active it is not possible to create new mode nor edit modes nor delete modes. Further, it is only possible to copy and print the window when the animation is stopped.
The Mode List (FDD Modal Identification) The mode list is continuously updated with the modes estimated by the Frequency Domain Decomposition technique. You can use the list to select the mode to animate the mode shape of in the animate tab window or the mode to edit in the Peak Pick tab window.
You can also use the list to select the modes you want to export out of the application as shown in the example below:
First you select the modes to export, then you select the export format from the File, Export, Modes menu item or from the context-sensitive menu as shown above. Two different export formats are available, the UFF Format (ASCII) or the SVS Format (ASCII). The mode currently being animated can be saved in an AVI Movie file. This list supports the four different form of table views and all other general table options. If you select View Details you are able to enter comments specific to the individual modes. The comments are entered by double-clicking with the left-hand side mouse button on the white fields as shown below:
The comments are presented in printouts or if you copy the list to the clipboard and paste it in e.g. Microsoft Word.
The Properties Dialog (FDD Modal Identification) You can change the appearance of the singular value curves of the Peak Picking editor window by selecting the Properties... menu item either from the View menu or from the context-sensitive menu as shown below:
In either case the following dialog with five tabs appear:
The General tab enables you to change the 2D-display options. From the SVD tab you can select how many singular value curves you want to display:
Since you are always picking the modes from the largest singular value, selecting a number less than the actual number of curves will hide the curves of the smallest singular values. The Geometry tab belong to the Animate tab window of the Peak Picking editor.
Here you can change the colors of the background, coordinate system arrows, undeformed and deformed geometry. You change the colors by pressing the appropiate color-well button. In the Export tab you can select the export format to use when saving modes in ASCII files.
Due to the difference in usage of escape characters when writing ASCII files in a PC and a UNIX environment, it is necessary to specify to what environment files are exported. By default the PC environment is selected. In addition, it is sometimes preferred only to export nodes and trace lines in which case the export of surfaces can be disabled by unchecking the last box. Finally, in the last tab you can set up the AVI Movie recorder. In this dialog the default settings of the mode animation AVI movie recorder can be modified. When the application is started the computer is searched for applicable AVI compression drivers. These are displayed in the drop-down list called Compression Mode. It is also possible to select the uncompressed file format. However, this is not recommendable due to the unnecessary size of the generated AVI movie files.
There are two different modes for recording an AVI movie file of an animation. If an infinite cyclic animation is desired the radio button Cyclic show (one cycle) should be selected. The result will be a single cycle recording that can be repeated infinitely in e.g. the Microsoft Windows Media Player. If a specific number of seconds of recording is desired the Time capture radio button should be selected. This will result in an AVI recording of a specific number of seconds of the current animation.
The EFDD Peak Picking Editor (Pro and Handy versions only) The EFDD Peak Picking editor adds a modal estimation layer to the FDD Peak Picking editor. The modal estimation is therefore divided into two steps. The first step is to perform the FDD Peak Picking, and the second step is to use the FDD identified mode shapes to identify the Single-Degree-Of-Freedom (SDOF) Spectral Bell functions and from these SDOF Spectral Bells estimate all modal parameters. The SDOF Bell Identification The identification of the SDOF Spectral Bell is performed using the FDD identified mode shape as reference vector in a correlation analysis based on the Modal Assurance Criterion (MAC). On both sides of the FDD picked frequency a MAC vector between the reference vector and the singular vectors corresponding to a certain frequency is calculated. If the largest MAC value of this vector is above a user-specified MAC Rejection Level the corresponding singular value is included in the description of the SDOF Spectral Bell. The search on both sides of the reference frequency is continued until no MAC values are above the rejection level. Outside the search range the values of the SDOF Spectral Bell is set to zero. This means that the lower MAC Rejection Level the more singular values are included in the SDOF Spectral Bell. However, at the same time the lower MAC Rejection Level the deviation from the reference vector is allowed. Therefore, a good compromise is to use an initial MAC Rejection Level at 0.8. The Modal Parameter Estimation Besides storing the singular values that describe the SDOF Spectral Bell, the corresponding singular vectors are averaged together to obtain an improved estimate of the mode shape. The average is being weighted by multiplying the singular vectors with their corresponding singular values. This means that the closer the singular vectors is to the peak of the SDOF Spectral Bell the more weight it has on the mode shape estimate. The natural frequency and the damping ratio of the mode is estimated by transforming the SDOF Spectral Bell to time domain. What we then obtain is a SDOF Correlation Function, and by simple regression analysis we obtain the estimates of both the natural frequency as well as the damping ratio. The estimation of the damping ratio is performed by identification of the positive and negative extremes of the correlation function. Taking the logarithm of this decaying curve will for viscous damped linear systems result in a straight line on which the damping ratio can be estimated by linear regression. However, due to broad-banded noise and / or nonlinearities the beginning and end of the curve might not be straight. Such non-straight parts should not be included in the regression.
The estimation of the natural frequency is performed by a linear regression on the straight line that describes the correlation crossing times. However, again the beginning and end of the curve might not be straight and should not be included in the regression. These modal estimates will be good if the correlation function decays to a sufficiently small level of correlation. This can be accomplished by having sufficient frequency resolution. In this case the bias of the natural frequency and the damping ratio will be small. The EFDD Modal Parameter Estimation at a Glance In the case of the building model example the EFDD Peak Picking editor looks as below when started right after the signal processing of the data is complete:
The editor consists of four windows which are: The Peak Pick Tab. The Modal Estimation Tab. The Animate Tab. The Mode List.
The peak picking is performed in the editor of the Peak Pick tab window and the result can be verified immediately after in the Animate tab window. The estimated modes can be validate and fine-tuned in the Modal Estimation tab window and they are listed in the mode list window.
References
1. R. Brincker, L. Zhang & P. Andersen: Modal Identification from Ambient Responses using Frequency Domain Decomposition. Proceedings of the 18th International Modal Analysis Conference (IMAC), San Antonio, Texas, 2000. 2. R. Brincker, J. Frandsen & P. Andersen: Ambient Response Analysis of the Great Belt Bridge. Proceedings of the 18th International Modal Analysis Conference (IMAC), San Antonio, Texas, 2000. 3. R. Brincker & P. Andersen: Ambient Response Analysis of the Heritage Court Tower Building Structure. Proceedings of the 18th International Modal Analysis Conference (IMAC), San Antonio, Texas, 2000. 4. R. Brincker, P. Andersen & Nis Møller: Output-Only Modal Testing of a Car Body Bubject to Engine Excitation. Proceedings of the 18th International Modal Analysis Conference (IMAC), San Antonio, Texas, 2000. 5. N. Møller, R. Brincker & P. Andersen: Modal Extraction on a Diesel Engine in Operation. Proceedings of the 18th International Modal Analysis Conference (IMAC), San Antonio, Texas, 2000. 6. R. Brincker, L. Zhang & P. Andersen: Output-Only Modal Analysis by Frequency Domain Decomposition. Proceedings of the 25th International Seminar on Modal Analysis (ISMA), Leuven, 2000. 7. R. Brincker, C.E. Ventura & P. Andersen: Damping Estimation by Frequency Domain Decomposition. Proceedings of the 19th International Modal Analysis Conference (IMAC), Kissimmee, Florida, 2001.
The Peak Pick Tab (EFDD Modal Identification) (Pro and Handy versions only) In the case of the building model example the Peak Pick tab window of the Enhanced Frequency Domain Decomposition (EFDD) technique looks as below when started right after the signal processing of the data is complete:
As seen it looks exactly like the FDD Peak Picking Editor window, and the operation of is also exactly the same. The only difference from FDD Peak Picking is that the EFDD Peak Picking is used only to obtain a reference frequency and mode shape to be used in the EFDD Modal Estimation. When you estimate these reference parameters you will usually get the best result by picking on top of the peaks. Here the reference mode shape estimate will most accurate in case of well-separated modes. If the modes are repeated or closely spaced you should follow the guidelines listed in the description of the FDD Peak Picking Editor window, for estimation of these special cases. When you have picking a reference mode shape and frequency, the modal parameters estimation is performed immidiately with the default setting of it. The default modal estimation settings are editeed in the Preference dialog. To adjust and / or validate the estimation of the natural frequency and the damping ratio, you click on the Modal Estimation Tab window. To validate the estimated mode shape by animation you click on the Animate Tab window.
The Modal Estimation Main Tab (EFDD Modal Identification) (Pro and Handy versions only) When you have estimated a reference frequency and mode shape using the EFDD Peak Picking Editor, you can go to the Modal Estimation Tab window where you will find the following windows The Frequency Domain Tab. The Time Domain Tab. The Validate Damping Tab. The Validate Frequency Tab. for editing and validation of the damping ratio and natural frequency estimates. As an example the Building Model (Example\Building Model) is used to illustrate the use of this tab window. Before you step into this part you should have completed a peak picking in the Peak Picking editor as shown below:
Optionally, you should step through all the individual data sets to verify, and if necessary correct, that you have picked the peak of the mode you are estimating. By then you are ready to verify, and if necessary correct, the identification of the SDOF Spectral Bell. This is what is done in the Frequency Domain tab of the Modal Estimation tab window.
The Frequency Domain Tab (Modal Estimation Main Tab EFDD) (Pro and Handy versions only) This tab window is where you verify the identification of the SDOF Spectral Bell. In the case of the Building Model example (Examples\Building Model) the SDOF Spectral Bell, identified using a default MAC Rejection Level of 0.8, is shown below:
What you need to verify here is that you have a good representation of the SDOF Spectral Bell around the peak. In this case the representation is good because the peak has been identified completely down to where the noise begin to affect it. If we decrease the MAC Rejection Level using the Modal Estimation Control Panel we get the following result:
where we can see that the SDOF Spectral Bell identification becomes inaccurate in the tails. There is no reason to include all this noise in the modal estimation. Therefore, do not go to low in MAC Rejection Level, but only as low as required to get a good identification of the peak of the bell. In addition to the visual impression that using 0.6 as MAC Rejection Level is bad, you can also see that the estimated standard deviations of both natural frequency and damping ratio increases when this value is used. This is an additional indication of the improved quality of the SDOF Spectral Bell identified using a MAC Rejection Level equal to 0.8. So we will use this value when we proceed to validate the SDOF Correlation Function obtained from the SDOF Spectral Bell.
The Time Domain Tab (Modal Estimation Main Tab - EFDD) (Pro and Handy versions only) The SDOF Spectral Bell is transformed using an algorithm based on the Fast Fourier Transform to obtain the SDOF Correlation Function that is normalized so it always start with the correlation 1. This function is what is presented in the Time Domain tab of the Modal Estimation Tab window. In the case of the Building Model example (Examples\Building Model) the SDOF Correlation Function, identified using a default MAC Rejection Level of 0.8, is shown below:
From time lags 0 to around 5m s the function is seen to decay exponentially. After time lag 5m s noise start to interfere significantly. This part should of course be avoided in the estimation of the natural frequency and the damping ratio. The scattered region indicates the part of the correlation function that by default is used by the estimation algorithm. The choice of the maximum and minimum correlation limits to include is perfomed in the Modal Estimation Control Panel below. As default it uses a maximum equal to 0.95. Now since the maximum correlation is 1 this means that the first values of the correlation function by default are excluded. This is because this part of the
correlation function sometimes is polluted by a broad-banded noise. At the same time the minimum correlation limit is by default set at 0.3 because large time lags typically are estimated with increasing uncertainty. In the example, however, it seems that the maximum correlation limit can be 1 and the minimum limit a value that allows us to use the correlation function up to a time lag around 5m s. Below the adjusted limits are presented:
We can immediately see how the estimated uncertainties of both the natural frequency and the damping ratio dropped by the inclusion of more regression points. What remian to be done is to validate the damping ratio estimate and the natural frequecy estimate.
The Validate Damping Tab (Modal Estimation Main Tab EFDD) (Pro and Handy versions only) Based on the SDOF Correlation function presented in the Time Domain Tab window as well as the maximum and minmum correlation limits selected in the Modal Estimation Control Panel this windows presented the result of the regression leading to the damping ratio estimate. In the case of the Building Model example (Examples\Building Model) the regression result is shown below:
The green curve presents the logarithm of the absolute value of all the positive and negative extremes. The red straight line is the result of the linear regression problem. The damping ratio can be found directly from the slope of this straight line. The standard deviation of the damping ratio is estimated by a linear transformation of the estimation error of the estimated slope. Finally, what remains is to validate the natural frequency estimation.
The Validate Frequency Tab (Modal Estimation Main Tab EFDD) (Pro and Handy versions only) Based on the SDOF Correlation function presented in the Time Domain Tab window as well as the maximum and minmum correlation limits selected in the Modal Estimation Control Panel this windows presented the result of the regression leading to the natural frequency estimate. In the case of the Building Model example (Examples\Building Model) the regression result is shown below:
This window is typically not important to check. The reason is that the zero-crossing times can be determined extremely accurately from the correlation function. In the above window they are presented as the green line. The result of the regression problem is presented as the red straight line. This regression result is based on the part of the correlation function that is inside the correlation limits selected in the Modal Estimation Control Panel. The natural frequency estimate is obtained from the slope of this estimated straight line as well as the estimated damping ratio.
Note: Because of the coupling of the regression problems the estimated standard deviation of the natural frequency depends on the estimation error of the slope of both regression problems.
The Modal Estimation Control Panel (EFDD Modal Identification) (Pro and Handy versions only) There are three values that together with the reference mode shape and frequency uniquely controls the modal estimation using the EFDD Peak Picking Editor. The reference mode shape and frequency are determined by peak picking using the Peak Picking Tab window, whereas the remaining three values are set in the Modal Estimation Control Panel shown below:
The three values are: The MAC Rejection Level. The Maximum Correlation Limit. The Minimum Correlation Limit. The MAC Rejection Level controls how many singular values that are included in the identified SDOF Spectral Bell displayed in the Frequency Domain Tab window located under the Modal Estimation Tab. See the paragraph The SDOF Bell Identification in the description of the EFDD Peak Picking Editor. The Maximum and Minimum Correlation Limits controls how much of the SDOF Correlation Function, obtained by transformation of the SDOF Spectral Bell, to include in the regression problems that leads to the estimates of the natural frequency and damping ratio. See the paragraph The Modal Parameter Estimation in the description of the EFDD Peak Picking Editor.
The Mode List (EFDD Modal Identification) (Pro and Handy versions only) The mode list is continuously updated with the modes estimated by the Frequency Domain Decomposition technique. You can use the list to select the mode to animate the mode shape of in the animate tab window or the mode to edit in the Peak Pick tab window.
This list supports the four different form of table views and all other general table options. If you select View Details you are able to enter comments specific to the individual modes. The comments are entered by double-clicking with the left-hand side mouse button on the white fields as shown below:
The comments are presented in printouts or if you copy the list to the clipboard and paste it
in e.g. Microsoft Word. Specifically, the list presents the following information about an estimated mode: Mode. A short information string presenting the natural frequency and the estimator. Natural frequency (Frequency [Hz]). In case of multiple data sets, the presented value is the average of the natural frequency estimates of the individual data sets. In this case the standard deviation of the resulting natural frequency is also presented (Std. Frequency [Hz]). Damping ratio (Damping Ratio [%]). In case of multiple data sets, the presented value is the average of the damping ratio estimates of the individual data sets. In this case the standard deviation of the resulting damping ratio is also presented (Std. Damping Ratio [%]). Comment. A user-specified comment about the mode. Creation Date & Time. The time and date this mode were created. You can also use the list to select the modes you want to export out of the application as shown in the example below:
First you select the modes to export, then you select the export format from the File, Export, Modes menu item or from the context-sensitive menu as shown above. Two different export formats are available, the UFF Format (ASCII) or the SVS Format (ASCII).
The mode currently being animated can be saved in an AVI Movie file.
The Properties Dialog (EFDD Modal Identification) (Pro and Handy versions only) The properties dialog of the EFDD Peak Picking Editor is the same as explained in the FDD Peak Picking Editor.
Stochastic Subspace Identification (SSI) (Pro version only) In the Stochastic Subspace Identification (SSI) techniques a parametric model is fitted directly to the raw times series data returned by the transducers. A parametric model is a mathematical model with some parameters that can be adjusted to change the way the model fits to the data. In general we are looking for a set of parameters that will minimise the deviation between the predicted system response (predicted transducer signal) of the model and measured system response (transducer signal). This process is often called model calibration. See the following picture
All known time domain modal identification techniques can be formulated in a generalised form as an innovation state space formulation
where the A-matrix contains the physical information, the C-matrix extracts the information that can be observed in the system response and the K-matrix contains the statistical information. The statistical information allows for a covariance equivalent modelling, so that the model can have the correct correlation function and thus also the correct spectral density function. The number of parameters in the model is essential. If this number is to small, then the dynamical- and statistical behaviour cannot be modeled correctly. On the other hand, if the number is too high, then the model becomes over-specified resulting in unnecesary high statistical uncertainties of the model parameters. So the art of parametric model estimation is to determine a model with a reasonable number of parameters. This means that what you must do when you are estimating state space models is to choose the model order also known as the state space dimension,
which is the dimension of the A-matrix. In ARTeMIS Extractor there are three different implementations of the Stochastic Subspace Identification technique. These are: Unweighted Principal Component Principal Component Canonical Variate Analysis However, even though they estimate the state space models in different ways, you will not feel any difference when operating them. See the Technical Paper on the Stochastic Subspace Identification Techniques for a more comprehensive description about how the Stochastic Subspace Identification techniques works and what the mathematical difference between the three implementations are.
Extracting Modal Parameters from the State Space System
When the stochastic state space system is being estimated using e.g. the Stochastic Subspace Identification techniques we obtain what is called a realization of the true but unknown system. So the paramters of the state space system
is only estimates of the true system. You will never be able to estimate the 100% correct parameters but you can indeed estimate very accurate parameters by not using a too large state space dimension. The above system is shown in time domain but can of course also be represented in frequency domain by its transfer function H(z) as below
where z is a frequency dependent complex number. By a complex transformation of this transfer function using the eigenvectors of A the modal decomposed transfer function appear as
This representation of the transfer function expose all the modal parameters. From the eigenvalues µj defined as the diagonal elements of the matrix
the natural frequencies
and damping ratios
are extracted using the following definition
In this equation T is the sampling interval. The mode shape that are associated with the jth mode is given by the jth column of the matrix . The last matrix that completes the modal decomposition contains a set of row vectors. The jth row vector corresponds to the jth mode. This vector distributes the white noise excitation et in modal domain to all the degrees of freedom. So the amplitude values of the degrees of freedom depends on this vector as well as the eigenvalue and the mode shape.
At the initial time step the state vector
is zero. This imply that the contribution of e0 to
from a specific
mode solely is given by the row vector of that corresponds to that mode. For this reason this vector is called the initial modal amplitude. Since this vector describes how the white noise is distributes in modal domain, this vector describes the statistical part of the modal decomposition. All the other modal parameters relates to the dynamic system and are therefore deterministic parameters that should not change if the excitation changes.
Estimating Models (SSI) (Pro version only) In the Stochastic Subspace Identification techniques the order of the model is characterized by the state space dimension. Since structural modes typically are lightly damped they are described by two eigenvalues. Thus for every time you increase the state space dimension by two you actually add the extra parameters that are necessary in order to fit one extra mode. To choose the right state space dimension is essential in the Stochastic Subspace Identification techniques. If the dimension is to small, then the dynamics cannot be modeled correctly. On the other hand, if the dimension is too high, then the estimated state space model becomes over-specified, and as a results, the statistical uncertainty on the estimated parameters increases unnecessarily. This dilemma is illustrated in the following figure:
When you are estimating try to do is the following: Try to over-specify the state space model a little. It is better to over-specify than to under-specify. When you have over-specified the model, you will have some computational modes or noise modes that you want to get rid of before you present your modal results. First of all, get rid of the modes that does not comply with you expectations. You might have some apriori knowledge about the expected range of the modal parameters. For instance, you might know that structural modes must have a damping ratio smaller than 2 % otherwise they are not structural - so you exclude all modes with a damping larger than 2 %. Secondly, verify which modes repeats themselves when you estimate model with different state space dimension. This is done by creating a stabilization diagram. Creating this diagram you can specify the maximum allowed deviations on natural
frequency, damping ratio, mode shape and initial modal amplitude. Increase the stabilization demands until you have only one or a few models left in the stabilization diagram. Make your choice of model taking into account the fitting quality of the different models and that a model with a smaller state space dimension is better than one with a large dimension.
The Stochastic Subspace Identification Editor (SSI) (Pro version only) No matter which of the three Stochastic Subspace Identification techniques you use the editor to use looks the same. You start the editor of one of the techniques either from the Project, Stochastic Subspace Identification, Data Driven Estimation menu item or from the SSI pane of the Task Bar. Below the editor window is shown in case of the building model example:
This editor consist of several windows that relate to the estimation of an optimal state space realization. The upper part of the editor is vertically divided into two. The upper left part contains four tab windows which are: Estimate Tab. Validate Response Predictions Tab. Validate Prediction Errors Tab. The Estimate tab is where the actual estimation take place. The two remaining tabs are
windows used to validate the selection of the optimal model. The upper right part consist of two tab windows. The SVD window presents the singular values of the weighted Common SSI input matrix. The weighting of this matrix is what makes the available techniques different. There are three types of weighting: Unweighted Principal Component, Principal Component and Canonical Variate Analysis. The FPE window display the socalled Final Prediction Error (FPE) criterion, which is a bar diagram helping you to select the optimal model. The lower part of the editor consist of three tab windows which are: Modal Indicators Tab. Estimation Status Tab. Modal Results Tabs. When estimation is in progress the Estimation Status tab presents information about the currently estimated state space realization. In the Modal Results tab you can get a list of the modal parameters of the currently selected state space realization. Finally, the Modal Indicators tab documents the modal indicators used as they have been entered in the Modal Indicators tab of the Control Panel.
The Estimate Tab (SSI) (Pro version only) The window that is found under the Estimate tab of the Stochastic Subspace Identification editor is presented below in the case of the Building Model example:
In the example, it is the Canonical Variate Analysis (CVA) technique that has been chosen. However, it could just as well be the Unweighted Principal Component (UPC) algorithm or the Principal Component (PC) algorithm. The window is divided vertically in two. To the left a stabilization diagram is shown and to the right some information relating to the stabilization diagram is presented.
Stabilization Diagram
The validation of the time domain estimation of the state space models is actually performed in frequency domain. The reason for this is that it is very easy in frequency domain to see the repeated trend of structural modes when estimating multiple state space models. But you should remember that the actual estimation behind the screen is performed
on the raw time series data of the currently selected data set in time domain. The stabilization diagram presents the natural frequencies of all the estimated eigenvalues as well as a background wall-paper of the Singular Value Decomposition of the spectral density matrices of the currently selected data set. This wall-paper has nothing direct to do with the estimation. However, it is a valuable help in the search of structural modes since these will be located at the spectral density peaks. The horizontal axis is a frequency axis ranging from zero to the Nyquist frequency. The vertical axis list the dimensions of the available state space models. This range goes from 1 to the maximum state space dimension specified in the Signal Processing Configuration Selected for Analysis.
The Concept of Stabilization
The problem of parametric model estimation is that you do not know the true model order. In our case it means that that the exact state space dimension is unknown. The way to overcome this is to estimate a range of candidate state space models. The important issue here is that the information of the structural (physical) system will be contained in all the estimated models if the state space dimension is high enough. This is revealed as a repeated trend across the state space models of some of the estimated eigenvalues (the vertical line of red crosses in the above figure). If such a repeated trend is located at a resonance frequency it is a strong indication that a structural modes has been estimated. The problem is that there can be a lot of candidate models and in order to minimize this number you should make use of the so-called modal indicators. You set up a series of requirements that the repeated modes must fullfill in order to be called stable. A stable mode of a model is one that compared with one of the estimated modes of the previous model fullfill the union of the following requirements: 1. A user-specified maximum allowed deviation of the natural frequency of the mode when compared with one of the modes of the previous model. 2. A user-specified maximum allowed deviation of the damping ratio of the mode when compared with one of the modes of the previous model. 3. A user-specified maximum allowed deviation of the Modal Assurance Criterion of the mode shape vector of the mode when compared with one of the modes of the previous model. 4. A user-specified maximum allowed deviation of the Modal Assurance Criterion of the
initial modal amplitude vector of the mode when compared with one of the modes of the previous model. If the deviations of all the above requirements of a mode is less than the specified maximum allowed deviations then a mode is characterized as stable and will be marked with a red cross. If one or more of the above requirements are not fullfilled then the mode is characterized as unstable and will be marked with a green diagonal cross. So by decreasing the maimum allowed deviation the number of candidate models will also decrease. The setting of the requirements are performed in the Modal Indicator tab of the Control Panel. The modal indicators used are documented in the Modal Indicators tab of the Stochastic Subspace Identification editor.
Eliminating Noise or Computational Modes When a state space model is estimated from measured data not only structural modes will be estimated but also socalled noise (computational) modes. These noise modes are used by the algorithms to account for non-fullfilled assumptions. Noise modes appear e.g. in the following situations: Limited data records (Accounts for the violation of the infinite amount of data assumption). Non-linearities (Accounts for the violation of the linear system assumption). Colored noise caused e.g. by non-white excitation and / or filtering (Accounts for the violation of the assumption of having white noise disturbance only). Typically, noise modes are spread in a non-systematic and non-repeated way. Because they usually are very heavily damped and because structural modes usually are lightly damped we can exclude a large parts of the noise modes from the analysis by looking at the damping of a mode. If the damping ratio is larger than some predefined value, say e.g. 5%, then the mode is problably a noise mode. The setting of the range of the damping ratio of a structural mode is performed in the Modal Indicator tab of the Control Panel. Modes having damping ratios outside this range are characterized as noise modes and marked in the stabilization diagram with yellow diagonal crosses. Besides being marked they are also excluded completely from the evaluation of stable/unstable modes.
Estimating a Range of State Space Models
When you start the Stochastic Subspace Identification editor the first time after having processed the loaded data you will see a window like below:
As seen no state space models have been estimated yet. Now in order to find a sensible range of models to estimate you should firstly inspect the singular values of the weighted common SSI input matrix. These value can be inspected in the window located right-most in the editor. For the above example the window is shown below:
The singular values (the yellow horizontal bars) indicate the rank of the weighted common SSI input matrix. What you are doing when you estimate a state space model is to specify what subspace of singular values of this matrix to include in the estimation. This subspace should at least include all singular values significantly different from zero. In order to make the inspection easier we have normalized the singular values so that they are located between 0 and 1. In the figure we have as illustration selected a model with state space dimension 11. In this case the blue horizontal bars indicate the subspace being included in the estimation of this model. However, in the present case a good starting point is to estimate models with dimension around 40 (shown at the vertical axis). So let us estimate models with state space dimensions from 30 to 50. We select the desired range by pointing at the dimension 50 with the mouse and pressing the left mouse button. Keeping the mouse button down drag the mouse to the dimension 30 as shown below:
In the information window to the right we see that we have selected 21 models. To estimate these you can either press the key, press the Estimate Model(s) button on the Modal Toolbar or select the menu item Estimate Selected Models from the context sensitive menu or the Project, Stochastic Subspace Identification, Data Driven Estimation menu. Note: You can also select the range using the keys or . The state space models are then estimated as shown below:
When finished estimating a result as below appear:
During the estimation information is written to the Estimation Status window. Here you get information about the condition of the state space models that also can assist you in judging whether you should increase the estimated range of models or not. The next task is then to verify if one of the estimated state space models is adequate. First you narrow range by decreasing the maximum allowed deviation of the modal indicators. Then you inspect the modal result of the candidate models by pointing the blue cursor rectangle at the models and looking in the Modal Results window and the three validation windows. If you are unsure about the result try to extent the range of estimated models. If a state space model seems to be apropiate you select it for Mode Selection and Linkage (see below). If you have more than one data set in the project you open the Project Control window and changes to the next data set. You can then go back to the Stochastic Subspace Identification editor and estimate an adequate state space model for this data set. Note: If the modes does not seem to stabilize properly, or if a presumably adequate model is close to the maximum state space dimension, or if the singular values of the weighted common SSI matrix does not drop close to zero at say half of the maximum state space dimension, you should definitely try to increase the maximum state space dimension in the Signal Processing Configuration Selected for Analysis. This is done in the Signal Processing Configuration wizard. An example of a bad situation is shown below for the same building model data set and with the same choice of modal indicators. The problem is that the maximum state space dimension only is 16 instead 80 as before.
Find the Optimal of State Space Model When you have estimated a range of models that seem to stabilize properly the next task is to find the optimal model among the estimated ones. There might not be one, but a tool that can help you figure this out is the socalled Final Prediction Error (FPE) criterion. In the figure below a range of models have been estimated.
To the right the Final Prediction Error criterion is presented for all the estimated models. The absolute value of this criterion is unimportant which is why it has been normalized with the maximum value to yield values between 0 and 1. The idea of the criterion is the following: When the model order increase more and more details of the measured response are modelled resulting in a smaller error between the measured response and what can be predicted by the model. So tha variance of the socalled prediction error descrease as the model order increase. On the other hand, when the model order increase more and more parameters are used resulting in an increase of the estimation uncertainty of the individual parameters. So as the model order increase so do the parameter uncertainties. This seemingly paradox is what the FPE criterion is trying to visualize, and is why there is a minimum around model order 40 in the above FPE diagram. For the small model order the criterion is descreasing because the prediction error variance is decreasing. For large model orders the criterion is increasing because the number of parameters is becoming unnecessary large. The criterion is actually constructed by merging the two lines in the figure below:
The model order (state space dimension) where lines are crossing is where the FPE criterion will be minimum. At minimum there is a good balance between the use of parameters and the accuracy of the model. This minimum is a good initial place to start your model validation.
Selecting a Candiate Model for Mode Selection and Linkage
When you have found an adequate state space model you must select it for mode selection and linkage. The mode selection and linkage is performed in the associated Select and Link Modes editor, and it means that you select which of the modes of the state space model that are structural and you link the mode shapes of these modes together across the data sets. First you move the blue cursor rectangle to the adequate model, then secondly you select the menu item Select Cursor Model for Mode Selection and Linkage from the context sensitive menu as shown below:
You are then informed that any previous selected and linked modes now will be overwritten by this action, and when you remove the cursor rectangle from the model you will then see a red rectangle that is locked to this model to indicate the selection. You can always go to the associated Select and Link Modes editor and inspect how the stable modes of this state space model compares to the stable modes of the other data sets.
Change the Appearance of the Stabilization Diagram
You can modify the appearance of the wall-paper with the Singular Value Decomposition of the spectral density matrices as well as the color of the eigenvalue markers. This is done by activating the Properties dialog from either the context-sensitive menu or the View menu. In both cases, select the menu item Properties ....
Zooming on the Axes
As explained above, the horizontal axis is a frequency axis ranging from zero to the Nyquist frequency. This frequency range can be adjusted using the 2D-Displays tab of the Control Panel. Use the sliders of the group shown below:
The vertical axis list the dimensions of the available state space models, ranging from 1 to the maximum state space dimension specified in the Signal Processing Configuration Selected for Analysis. To select another range of models to present you should do the following: Use the mouse to select the range of models you want to have presented. Click with the left mouse button on the state space dimension of the first model to present. Keep the mouse button down and drag the mouse to the state space dimension of the last model to present.
1.
Click on the right mouse button to activate the context-sensitive menu. Select the menu item Show Models in Selected Range as shown below.
2.
To return to the full range that presents all available models just activate the contextsensitive menu and select the menu item Show all Models.
Exporting Estimated State Space Models You can export the estimated state space models to a file in ASCII format. To export either the Cursor Model of the Model Selected for Mode Selection and Linkage, right-click the mouse to activate the context-sensitive menu. From the Export menu item select the model you want to export. Note: You can use the general 2D data display options with this display.
The Properties Dialog (SSI) (Pro version only) You can modify the appearance of the wall-paper of the stabilization diagram with the Singular Value Decomposition of the spectral density matrices as well as the color of the eigenvalue markers. This is done by activating the Properties dialog from either the contextsensitive menu or the View menu. In both cases, select the menu item Properties .... The General tab enables you to change the 2D-display options.
Changing the color of the eigenvalue markers is performed in the property page called Mode Markers shown below:
Just press the color-well button of the markers you want to change the color of. Select one of the predefined colors or press the Other button to define any color you like from the standard Windows color dialog. When you change the colors the changes are also made in the Select and Link Modes editor. Changing the wall-paper with the Singular Value Decomposition of the spectral density matrices is performed in the property page called SVD shown below:
If there are many transducer in the data set it might be too much to present all the singular value lines. Typically you can see all the modal content of the data using only a few of the dominating singular value lines. You can change the number of singular values presented in
the edit box.
Validate Response Predictions (SSI) (Pro version only) One way to validate an estimated model is to compare the response that can be synthezised using the model with the actual measurements. This is what can be done when activating the Validate Response Predictions. The are three ways of doing this: Compare Spectral Density Magnitudes Compare Spectral Density Phase Angles Compare Correlation Functions
The Validate Response Tabs - Spectral Density Magnitude (SSI) (Pro version only) To assist in the selection of an adequate state space model you can display the magnitude of the spectral densities of the data together with synthesized spectral densities of the selected model and the model selected for mode selection and linkage. You simply select the state space models in the stabilization diagram and press the Spectral Density Magnitude tab under the Validate Response Predictions main tab in the Stochastic Subspace Identification editor. If the dynamics as well as the statistical properties are correctly estimated the synthesized spectral densities should be comparable with the spectral densities of the data. Below the magnitude of the spectral densities of the data of the building model example are displayed together with two different state space models estimated using the Canonical Variate Analysis algorithm:
The green curve is the magnitude of the spectral density of the measured data. The blue curve is the magnitude of the synthesized spectral density of the cursor model, i.e. the state space model selected using the blue cursor rectangle in the stabilization diagram. Finally, the red curve is the magnitude of the synthesized spectral density of the model selected for mode selection and linkage, i.e. the state space model selected using the red cursor rectangle in the stabilization diagram.
Note: You can use the general 2D data display options with this display.
The Validate Response Tabs - Spectral Density Phase (SSI) (Pro version only) To assist in the selection of an adequate state space model you can display the phase angle of the spectral densities of the data together with synthesized spectral densities of the selected model and the model selected for mode selection and linkage. You simply select the state space models in the stabilization diagram and press the Spectral Density Phase tab under the Validate Response Predictions main tab in the Stochastic Subspace Identification editor. If the dynamics as well as the statistical properties are correctly estimated the synthesized spectral densities should be comparable with the spectral densities of the data. Below the phase angle of the spectral densities of the data of the building model example are displayed together with two different state space models estimated using the Canonical Variate Analysis algorithm:
The green curve is the phase angle of the spectral density of the measured data. The blue curve is the phase angle of the synthesized spectral density of the cursor model, i.e. the state space model selected using the blue cursor rectangle in the stabilization diagram. Finally, the red curve is the phase angle of the synthesized spectral density of the model selected for mode selection and linkage, i.e. the state space model selected using the red cursor rectangle in the stabilization diagram.
Note: You can use the general 2D data display options with this display.
The Validate Response Tabs - (SSI) (Pro version only) To assist in the selection of an adequate state space model you can display the correlation function of the data together with synthesized correlation function of the selected model and the model selected for mode selection and linkage. You simply select the state space models in the stabilization diagram and press the Correlation Function tab under the Validate Response Predictions main tab in the Stochastic Subspace Identification editor. The presented correlation function have been normalized so that the auto-correlation function starts in 1. For this reason it is called a normalized correlation function. If the dynamics as well as the statistical properties are correctly estimated the synthesized normalized correlation function should be comparable with the normalized correlation function of the data. Below the normalized correlation function of the data of the building model example are displayed together with two different state space models estimated using the Canonical Variate Analysis algorithm:
The green curve is the normalized correlation function of the measured data. The blue curve is the synthesized normalized correlation function of the cursor model, i.e. the state space model selected using the blue cursor rectangle in the stabilization diagram. Finally, the red curve is the synthesized normalized correlation function of the model selected for mode selection and linkage, i.e. the state space model selected using the red cursor rectangle in
the stabilization diagram. Note: Correlation functions can only be calculated for stable state space models. Whether a model is stable or not can be seen in the Estimate tab window. You can use the general 2D data display options with this display.
Validate Prediction Errors (SSI) (Pro version only) Another way to validate an estimated model is to compare the prediction errors between the response that can be synthezised using the model with the actual measurements. This is what can be done when activating the Validate Prediction Errors. The are four ways of doing this: Compare Spectral Density Magnitudes Compare Spectral Density Phase Angles Compare Correlation Functions
The Validate Prediction Errors Tabs - Spectral Density Magnitude (SSI) (Pro version only) To assist in the selection of an adequate state space model you can display the magnitude of the spectral densities of the prediction errors between the response, that can be synthezised using the model, with the actual measurements. These results can be displayed for the selected model and the model selected for mode selection and linkage. You simply select the state space models in the stabilization diagram and press the Spectral Density Magnitude tab under the Validate Prediction Errors main tab in the Stochastic Subspace Identification editor. If the dynamics as well as the statistical properties are correctly estimated the spectral densities of the prediction errors should be a flat spectrum, indicating that the prediction errors are a realization of a white noise stochastic process. In practice the spectrum can only be flat in the parts where no deterministic components are present. This means e.g. below the cut-off frequency of the anti-alias filter. If the measurements contain information about e.g. rotating machinery there will also be errors around the harmonics presented. However, prediction errors can efficiently reveal the good and bad areas of the model fit. Below the magnitude of the spectral densities of the data of the building model example are displayed together with two different state space models estimated using the Canonical Variate Analysis algorithm:
The blue curve is the magnitude of the spectral density of the prediction errors of the cursor
model, i.e. the state space model selected using the blue cursor rectangle in the stabilization diagram. Finally, the red curve is the magnitude of the spectral density of the prediction errors of the model selected for mode selection and linkage, i.e. the state space model selected using the red cursor rectangle in the stabilization diagram. It is seen that the red curve is completely flat from 0 to the cut-off frequency, indicating that all significant information is present in the model. The blue curve on the other hand reveal that some information in the data is not present in the model. So the red curve model is in the case the most optimal one. Note: Prediction errors of a model can only be calculated if response predictor of it is stable. This can be verified in the Estimate tab window.You can use the general 2D data display options with this display.
The Validate Prediction Errors Tabs - Spectral Density Phase (SSI) (Pro version only) To assist in the selection of an adequate state space model you can display the phase of the spectral densities of the prediction errors between the response, that can be synthezised using the model, with the actual measurements. These results can be displayed for the selected model and the model selected for mode selection and linkage. You simply select the state space models in the stabilization diagram and press the Spectral Density Phase tab under the Validate Prediction Errors main tab in the Stochastic Subspace Identification editor. If the dynamics as well as the statistical properties are correctly estimated the spectral densities of the prediction errors should be a flat spectrum with random phase angles for all off-diagonal elements of the spectral density matrix, indicating that the prediction errors are a realization of a white noise stochastic process. In practice this can only be accomplished in the parts where no deterministic components are present. This means e.g. below the cutoff frequency of the anti-alias filter. If the measurements contain information about e.g. rotating machinery there will also be deterministic phase angles around the harmonics presented. However, prediction errors can efficiently reveal the good and bad areas of the model fit. Below the phase angles of the spectral densities of the data of the building model example are displayed together with two different state space models estimated using the Canonical Variate Analysis algorithm:
The blue curve is the phase angles of the spectral density of the prediction errors of the cursor model, i.e. the state space model selected using the blue cursor rectangle in the stabilization diagram. Finally, the red curve is the phase angles of the spectral density of the prediction errors of the model selected for mode selection and linkage, i.e. the state space model selected using the red cursor rectangle in the stabilization diagram. It is seen that the red curve is switching randomly between +180 degrees and -180 degrees, indicating that all significant information is present in the model. The blue curve on the other hand reveal that some information in the data is not present in the model. So the red curve model is in the case the most optimal one. Note: Prediction errors of a model can only be calculated if response predictor of it is stable. This can be verified in the Estimate tab window.You can use the general 2D data display options with this display.
The Validate Prediction Errors Tabs - Correlation Function (SSI) (Pro version only) To assist in the selection of an adequate state space model you can display the correlation functions of the prediction errors between the response, that can be synthezised using the model, with the actual measurements. These results can be displayed for the selected model and the model selected for mode selection and linkage. You simply select the state space models in the stabilization diagram and press the Correlation Function tab under the Validate Prediction Errors main tab in the Stochastic Subspace Identification editor. If the dynamics as well as the statistical properties are correctly estimated the correlation function of the prediction errors should be nearly zero for all time lags greater that zero, indicating that the prediction errors are a realization of a white noise stochastic process. In practice this can only be accomplished if no deterministic components are present. However, there will almost always be some influence from the anti-alias filter. This is a highfrequent contamination of the data which will turn up as contamination of the beginning of the correlation function. Compared to the spectral density inspections of the prediction errors, the correlation functions are a more global measure of the quality of the model fit. If the is only a slight contamination in the beginning it is an indication that overall the model is good. Below a prediction error correlation function of the data of the building model example is displayed together with two different state space models estimated using the Canonical Variate Analysis algorithm:
The blue curve is the correlation function of the prediction errors of the cursor model, i.e. the state space model selected using the blue cursor rectangle in the stabilization diagram. Finally, the red curve is the correlation function of the prediction errors of the model selected for mode selection and linkage, i.e. the state space model selected using the red cursor rectangle in the stabilization diagram. It is seen that the red curve is overall close to zero correlation for the non-zero time lags, indicating that all significant information is present in the model. The blue curve on the other hand reveal that some information in the data is not present in the model. So the red curve model is in the case the most optimal one. The display also present the 95% confidence limits of white noise. If all correlation function estimates available, for all non-zero time lags, are lying inside these limits, it can cannot with more that 5% confidence be rejected that the prediction errors are not a realization of a white noise process. Note: Prediction errors of a model can only be calculated if response predictor of it is stable. This can be verified in the Estimate tab window. You can use the general 2D data display options with this display.
The Modal Indicators Tab (Pro version only) You use the modal indicators tab to set the modal indicators used in the Stochastic Subspace Identification editor.
To enable the use of different symbols for stable/unstable modes as well as noise noise in the Stabilization Diagram of the Stochastic Subspace Identification editor you should check the box labeled Enable Mode Indication on Stabilization Diagram. If the deviations of the Natural Frequency, the Damping Ratio, the Mode Shape MAC and of the Initial Modal Amplitude MAC all are less than the maximum allowed deviations specified in the Control Panel, then a mode is charaterized as stable. You set these maximum allowed deviations in the four edit boxes of the control group called Deviations for Stable Modes between Consequtive Models. Deviations for one model are determined by comparing with the model one model order lower. Typically, structural modes are lightly damped. This means that many modes can be excluded from the search for stable modes. The exclusion can be obtained by setting the range of the damping ratio of stable structural modes. You set this range using the two edit boxes of the control group called Expected Range of Modal Parameters. This range is characterized by a minimum damping ratio and a maximum damping ratio both specified in per cent. When you have set the values you want to change you simply press the key.
The Estimation Status Tab (SSI) (Pro version only) The Estimation Status window present you with information about the progress and condition of the state space realization estimation. In general there are three types of information presented for each of the estimated model as shown in the example below:
In this a state space realization with a dimension of 17 has been estimated. The information you get is: Numerical performance of the estimation algorithm. The following messages are possible events:
Estimation: OK
The estimation were successfully completed without numerical problems.
Estimation: Skipped - Already estimated
The estimation were skipped since the state space model already is estimated.
Estimation: Warning - Internal estimation failure
An unexpected internal error occurred during estimation.
Estimation: Warning - Statistical part of model may be inaccurate
The part of the state space model that relate to the statistical modelling of the data could not be completed successfully.
1.
Model stability. The estimated state space model is of the following form (see the Technical Note on the Stochastic Subspace Identification Techniques for more information)
2.
As you can see the upper part of the state space system is a recursive formula. The recursion will only be stable if all the eigenvalue of the state matrix A are located inside the complex unit circle. In this case the model is called stable. If one or more eigenvalues are outside the complex unit circle this recursion breaks down due to floating-point overflow. In this case the model is called unstable. Therefore, unstability is an indicator for that at least some parts of the system dynamics are porely estimated. The following messages are possible events: Model: Stable
All eigenvalues of the state matrix A are located inside the complex unit circle.
Model: Unstable
One or more eigenvalues of the state matrix A are located outside the complex unit circle.
Response predictor stability. From the estimated state space model a socalled steady-state Kalman filter can be established as shown below (see the Technical Note on the Stochastic Subspace Identification Techniques for more information)
With this filter you can make predictions of the future system response based on your measured system response. You can also use the filter the calculate the socalled
prediction error which is actually a very powerfull validation tool.
3.
As you can see the upper part of the filter is a recursive formula. The recursion will only be stable if all the eigenvalue of the matrix A-KC are located inside the complex unit circle. In this case the filter is called stable. If one or more eigenvalues are outside the complex unit circle this recursion breaks down due to floating-point overflow. In this case the filter is called unstable. Therefore, unstability is an indicator for that at least some parts of the system dynamics and / or the statistical modelling are porely estimated. The following messages are possible events: Response Predictor: All eigenvalues of the state matrix A-KC are located inside Stable the complex unit circle. Response Predictor: One or more eigenvalues of the state matrix A-KC are Unstable located outside the complex unit circle.
You can use the following quadrature as a rough guideline when evaluating the health of an estimated state space model. However, always use all the available validation techniques of the Stochastic Subspace Identification editor. There are not two identifications that are the same.
Model: Unstable / Response Predictor: Stable Model: Stable / Response Predictor: Stable This is the most desirable condition for the estimate. Both dynamics as well as the statistical modeling seems to be good. If the model is large enough the modal parameters as well as synthesized spectral densities and correlation functions will probably be of good quality.
In this case the overall dynamics will probably be estimated more or less correctly. However, the modal parameter estimates may be inaccurate since parts of the dynamics are wrongly estimated. Since the response predictor is stable it is an indication that the unstable part of the model only contribute a little to the system response. If the model is large enough the modal
parameters as well as synthesized spectral densities and correlation functions could be of fairly good quality.
Model: Stable / Response Predictor: Unstable In this case the dynamics will probably be estimated more or less correctly. However, the modal parameter estimates may be inaccurate since the statistical modeling of the state space model is wrongly estimated. If the model is large enough the modal parameters could be of fairly good quality. However, the synthesized spectral densities and correlation functions will probably be of bad quality.
Model: Unstable / Response Predictor: Unstable This is the most undesirable condition for the estimate. Both dynamics as well as the statistical modeling are wrongly estimated. You should avoid using a model like this.
The Modal Results - Cursor Model Tab (Pro version only) The Modal Results - Cursor Model window present the natural frequencies and damping ratios of all the modes of the currently selected state space model, i.e. the model selected by the blue cursor rectangle. In the case of the building model example the window could look like below:
Here the result of a 21 dimensional state space model is presented. What is presented from left to right in the stabilization diagram is presented from top to bottom in the Modal Results window. For each mode the table present the natural frequency, the damping ratio and a remark. If the mode is visible in the stabilization diagram the remark indicate literally and with a color if the mode is stable, unstable or just a noise mode. If the natural frequncy is outside the current frequency axis or larger than the Nyquist frequency the remark is Not Plotted. If the natural frequency becomes negative or the damping ratio is lying outside the range from 0 to 100% the remark will say NaN (Not a Number). This can happen for computational (noise) modes.
As shown above, it is sometimes difficult to see very close modes in the stabilization diagram. However, this is of course not a problem in the table of the Modal Results window.
The Modal Results - Model Selected for Select & Link Tab (Pro version only) The Modal Results - Model Selected for Select & Link window present the natural frequencies and damping ratios of all the modes of the mode selected for Select & Link, i.e. the model selected by the red cursor rectangle. In the case of the building model example the window could look like below:
Here the result of a 30 dimensional state space model is presented. What is presented from left to right in the stabilization diagram is presented from top to bottom in the Modal Results window. For each mode the table present the natural frequency, the damping ratio and a remark. If the mode is visible in the stabilization diagram the remark indicate literally and with a color if the mode is stable, unstable or just a noise mode. If the natural frequncy is outside the current frequency axis or larger than the Nyquist frequency the remark is Not Plotted. If the natural frequency becomes negative or the damping ratio is lying outside the range from 0 to 100% the remark will say NaN (Not a Number). This can happen for computational (noise) modes.
As shown above, it is sometimes difficult to see very close modes in the stabilization diagram. However, this is of course not a problem in the table of the Modal Results window.
Unweighted Principal Component (SSI) (Pro version only) The Stochastic Subspace Identification techniques all uses the same estimation engine for estimation of state space realizations (models). In general, the input to this engine is a weighted version of the so-called Common SSI Input matrix that consist of compressed time series data. The difference between the three Stochastic Subspace Identification techniques is how this matrix is weighted. The Unweighted Principal Component algorithm is the most simple because no weighting is performed at all. So the input to the estimation engine is the Common SSI Input matrix itself. This algorithm works best with data having modes with comparable energy level. In such cases it will produce good results using resonable small state space dimensions. The Stochastic Subspace Identification editor initialized for state space estimation using the Unweighted Principal Component algorithm can be launched either from the SSI Pane of the Task Bar by pressing the button called UPC Estimation or from the Project, Stochastic Subspace Identification, Data Driven, Unweighted Principal Component menu item. See the Technical Paper on the Stochastic Subspace Identification Techniques for a more comprehensive description about how the Stochastic Subspace Identification techniques works and the specific mathematical formulation of the Unweighted Principal Component algorithm.
Principal Component (SSI) (Pro version only) The Stochastic Subspace Identification techniques all uses the same estimation engine for estimation of state space realizations (models). In general, the input to this engine is a weighted version of the so-called Common SSI Input matrix that consist of compressed time series data. The difference between the three Stochastic Subspace Identification techniques is how this matrix is weighted. The Stochastic Subspace Identification editor initialized for state space estimation using the Principal Component algorithm can be launched either from the SSI Pane of the Task Bar by pressing the button called PC Estimation or from the Project, Stochastic Subspace Identification, Data Driven, Principal Component menu item. This algorithm works best with data having modes with comparable energy level. In such cases it will produce good results using resonable small state space dimensions. See the Technical Paper on the Stochastic Subspace Identification Techniques for a more comprehensive description about how the Stochastic Subspace Identification techniques works and the specific mathematical formulation of the Principal Component algorithm.
Canonical Variate Analysis (SSI) (Pro version only) The Stochastic Subspace Identification techniques all uses the same estimation engine for estimation of state space realizations (models). In general, the input to this engine is a weighted version of the so-called Common SSI Input matrix that consist of compressed time series data. The difference between the three Stochastic Subspace Identification techniques is how this matrix is weighted. The Stochastic Subspace Identification editor initialized for state space estimation using the Canonical Variate Analysis algorithm either from the SSI Pane of the Task Bar by pressing the button called CVA Estimation or from the Project, Stochastic Subspace Identification, Data Driven, Canonical Variate Analysis menu item. This algorithm typically forces the use of a larger state space dimension that the two other available algorithms. The reason is its ability to estimate modes with a large difference in energy level. In order to see low excited modes among well-excited modes, it is necessary to force a large state space dimension. If you have data only with well-excited modes use e.g. the Unweighted Principal Component algorithm instead as your first choice. See the Technical Paper on the Stochastic Subspace Identification Techniques for a more comprehensive description about how the Stochastic Subspace Identification techniques works and the specific mathematical formulation of the Canonical Variate Analysis algorithm.
The Select and Link Modes Editor (Pro version only) When you have completed the estimation of state space models for all data sets using a specific Stochastic Subspace Identification technique the next step is to launch the Select and Link Modes editor associated with the specific estimation technique. The Select and Link Modes editor can be initialized for use with the three different Stochastic Subspace Identification techniques: Unweighted Principal Component Principal Component Canonical Variate Analysis However, no matter which technique you have used to estimate the state space models for each of the data sets of the project the way to operate the Select and Link Modes editor is the same. The Select and Link Modes editor have three windows which are: The Select & Link window. The Animate window. The Mode List window. The mode selection and linkage is performed in the editor of the Select & Link tab window and the result can be verified immediately after in the Animate tab window. The estimated modes are listed in the mode list window. When no modes have been found this list is empty and the window is gray. In case of the building mode example the editor look as below after selecting and linking the stable modes of the state space models estimated using the Stochastic Subspace Identification editor for both data sets:
The Select & Link Tab (Select and Link Modes) (Pro version only) In the case of the building model example the Select & Link tab window looks as below when started after an adequate state space model has been estimated and selected for each of the two data sets:
The left-most window is the Select & Link editor where you estimate the modes. The horizontal axis is a frequency axis ranging from zero to the Nyquist frequency. The vertical axis list the data sets by their order in the Data Set tree in the Project Control window. Data set number 1 is the first tree item and so on. To the right you can get information about the currently selected frequency and the data set the mouse will snap on. In addition there is a legend explaining that an estimated mode is marked using a brown box and an estimated and selected mode is marked by a blue box. Further, there are three legends explaining how the mode markers of stable, unstable and noise modes appears. As a wall-paper, the Select & Link editor displays singular values of the spectral density matrices that has been normalized with respect to the area under the largest singular value curve. If multiple data sets are present the normalized singular values calculated for each
data set have been averaged to obtain the displayed curves. The absolute scaling of the singular values is arbitrary since their peaks are only used to indicate where structural modes might be. In the present example there are two data sets each with four transducers, and in order to obtain the presented four curves the following operations have been performed: 1. The 4x4 dimensional spectral density matrices of data sets 1 and 2 have been estimated. 2. For both data sets all the spectral density matrices have been decomposed using the singular value decomposition. The result is 4 singular values and 4 singular vector for each of the spectral density matrices. The singular values and the singular vectors are ordered in singular value descending order for each of the spectral density matrices, i.e. the first singular value is the largest. 3. For each data set the singular values are normalized. The normalization factor corresponds to the area under the first singular value curve. This normalization prevent that week modes only appearing in one or few data sets disappear. 4. Finally, the first singular value curve of both data sets are averaged frequency by frequency. This operation is repeated for the second, third and fourth singular value curves. By using these averaged singular value curves all modal information can be presented in one display no matter how many transducers and data sets there are. Since this normalized and averaged curve is constructed from several transducers and several data sets the dB reference value of this display is always chosen as 1. If you want to zoom to a specific frequency range you can do this from the 2D-Displays tab of the Control Panel.
Estimating a New Well-separated Mode
The way to estimate a mode in this Select & Link editor is illustrated below:
You can either press the New Mode button in the Modal Toolbar as shown in the lower left corner, or you can use the context-sensitive menu and select the New Mode menu item. In both cases a blue vertical cursor will appear together with blue diamonds that snaps to the nearest local mode of each data set as shown below:
The cursor line shows the frequency position of the mouse pointer. In the right-hand side information window the location of the mouse pointer is presented in terms of the frequency (Frequency) and nearest data set (Data Set) number. To select and link all the snapped local modes, represented by the crosses selected by the diamonds, simply double-click on the left-hand side mouse button. To abort the estimation press the key instead. When the estimation is done the vertical line link the crosses that were used to create the global mode. In the top and bottom the vertical line lock itself on the average frequency of the estimated global mode. Further, a blue box will appear instead of the diamond to indicate the crosses that were used to create the mode. Below the editor is shown after the estimation:
As seen the global mode is now drawn in blue and in the Mode List below it is presented as an icon. The mode remains drawn in blue as long as it is selected. You can now immediately press the Animate tab to inspect the estimated mode shape. A valuable hint is that you should normally make sure that the local modes you are snapping on are stable modes (having red crosses). In this case you can inform the snap function to disregard all local modes except the stable ones. This is accomplished by selecting the menu item Snap on stabilized modes in the context-sensitive menu as shown below:
When enabled the menu item is replace by the Snap on all modes as shown below:
which will bring the snap function back to the default which is snapping on all the available local modes.
Estimating Repeated Modes
If you have a system as the building model example with repeated modes things are a little more complicated than if all modes are well-separated as above. To estimate the two repeated modes at 56 Hz in the building model example it is recommendable to zoom to a small frequency range around the repeated modes:
It is now clear that there are two almost repeated local modes in both data sets and it is no longer a difficult task to estimate both of them as shown below:
Editing a Mode Already Estimated
You can always edit the currently selected (blue) mode. In this case you edit the existing local modes data set per data set. This feature is very applicable in cases with close modes as is the case in the example below:
It might be that the local mode seems wrong for e.g. data set number 2. To activate the edit mode, start by selecting the mode you want to edit. Then click the edit mode button on the Modal Toolbar. You can also activate the edit mode from the context-sensitive menu by selecting the menu item Edit Mode as shown below:
You have then activated the edit mode which will stay active until you either press the button / menu item agin or press the key, or the key or the Animate tab in the editor. When you press the left-hand mouse button down and keep it down you will see a blue diamond drawn around the cross of the local mode nearest to the mouse pointer with a gray cursor image behind as shown bellow:
The gray cursor image show which local mode/data set you are editing since the gray lines are drawn from the original position of the local modes of the surrounding data sets. The blue diamond shows you the currently snapped local mode. If you release the mouse button the new local mode will be used instead of the old and the global mode is be re-estimated as shown below:
Removing a the Local Mode of a Data Set
If the project has nultiple data sets and you want to exclude the influence of some of these data sets on a mode you can remove the data set while the edit mode is turned on. Point at the local mode/data set you want to exclude an select the menu item Remove Data Set from Mode from the context menu as shown below:
There will then no longer be a blue box surrounding the local mode. To include the data set again simply point and click on a local mode of the data set.
Deleting a Mode
You can always delete the currently selected (blue) mode either by pressing the or keys, or from the context-sensitive menu by selecting the Delete Mode menu item as shown below:
The Animate Tab (Select and Link Modes) (Pro version only) When you have selected a mode either from the Select & Link editor window or from the Mode List you will immediately be able to see its mode shape animated by pressing the Animate tab. In case of the building model example a mode being animated is show below:
The animation window support the general 3D display options. While the window is active it is not possible to create new mode nor edit modes nor delete modes. Further, it is only possible to copy and print the window when the animation is stopped.
The Mode List (Select and Link Modes) (Pro version only) The mode list is continuously updated with the modes estimated by the Select & Link editor. You can use the list to select the mode to animate the mode shape of in the animate tab window or the mode to edit in the Select & Link tab window.
This list supports the four different form of table views and all other general table options. If you select View Details you are able to enter comments specific to the individual modes. The comments are entered by double-clicking with the left-hand side mouse button on the white fields as shown below:
The comments are presented in printouts or if you copy the list to the clipboard and paste it in e.g. Microsoft Word. Specifically, the list presents the following information about an estimated mode:
Mode. A short information string presenting the natural frequency and the estimator. Natural frequency (Frequency [Hz]). In case of multiple data sets, the presented value is the average of the natural frequency estimates of the individual data sets. In this case the standard deviation of the resulting natural frequency is also presented (Std. Frequency [Hz]). Damping ratio (Damping Ratio [%]). In case of multiple data sets, the presented value is the average of the damping ratio estimates of the individual data sets. In this case the standard deviation of the resulting damping ratio is also presented (Std. Damping Ratio [%]). Comment. A user-specified comment about the mode. Creation Date & Time. The time and date this mode were created. You can also use the list to select the modes you want to export out of the application as shown in the example below:
First you select the modes to export, then you select the export format from the File, Export, Modes menu item or from the context-sensitive menu as shown above. Two different export formats are available, the ASCII Universal File Format or the SVS ASCII
format.
The Properties Dialog (Pro version only) You can change the appearance of the singular value curves of the Select & Link editor window by selecting the Properties... menu item either from the View menu or from the context-sensitive menu as shown below:
In either case the following dialog with five tabs appear:
The General tab enables you to change the 2D-display options. From the SVD tab you can select how many singular value curves you want to display:
Since you are always picking the modes from the largest singular value, selecting a number less than the actual number of curves will hide the curves of the smallest singular values. The Geometry tab belongs to the Animate tab window of the Select and Link Modes editor.
Here you can change the colors of the background, coordinate system arrows, undeformed and deformed geometry. You change the colors by pressing the appropiate color well button.
In the Export tab you can select the export format to use when saving modes in ASCII files.
Due to the difference in usage of escape characters when writing ASCII files in a PC and a UNIX environment, it is necessary to specify to what environment files are exported. By default the PC environment is selected. In addition, it is sometimes preferred only to export nodes and trace lines in which case the export of surfaces can be disabled by unchecking the last box. Finally, in the last tab you can set up the AVI Movie recorder. In this dialog the default settings of the mode animation AVI movie recorder can be modified. When the application is started the computer is searched for applicable AVI compression drivers. These are displayed in the drop-down list called Compression Mode. It is also possible to select the uncompressed file format. However, this is not recommendable due to the unnecessary size of the generated AVI movie files.
There are two different modes for recording an AVI movie file of an animation. If an infinite cyclic animation is desired the radio button Cyclic show (one cycle) should be selected. The result will be a single cycle recording that can be repeated infinitely in e.g. the Microsoft Windows Media Player. If a specific number of seconds of recording is desired the Time capture radio button should be selected. This will result in an AVI recording of a specific number of seconds of the current animation.
Mode Comparison Window The philosophy of ARTeMIS Extractor is that you use different modal identification techniques to actually identify the same modes. In this way you cross-validate the results and thereby make a quality control of the modal results. It is pretty straight forward to compare natural frequencies and damping ratios of the estimates modes. However, to compare the mode shapes in case of complex structure ARTeMIS Extractor has a visual tool for this purpose. This tool is called the Mode Comparison window. In the Mode Comparison window you can compare the modes from different techniques of the same project but also across projects as long as the geometries of the two projects are exactly the same. Below you can see a typical comparison situation:
The Mode Comparison window consist of the following windows: The Project Modes Trees. The Information Tab. The Overlaid Animation Tab.
The Error Animation Tab. The Side by Side Animation Tab. The MAC Tab. The idea is to compare only two project modes A and B at the same time. You select the modes to compare from the Project Modes trees. From the Information tab window you can get some information about the selected projects and modes as well as the nodal differences between the modes. The Overlaid Animation tab window is shown above and is used to animate the mode shapes together. In the Error Animation tab window the two mode shapes are subtracted and the result is animated. In the Side by Side Animation the two modes are animated next to each other. Finally, the Modal Assurance Criterion (MAC) matrix between all the modes of the two selected modal identification techniques is shown in the MAC tab window.
The Project Modes Tree (Mode Comparison Window) When you start the Mode Comparison window the two Project Modes trees will look like the example below:
Both of them list the project loaded into ARTeMIS Extractor. In this case we have the building model example (mes32set.axp) and another project (AnotherProject.axp) loaded. If you expand one of the projects you will get a list of the modal identification techniques available:
You can see the four available techniques: the Frequency Domain Decomposition (FDD) Peak Picking technique, the data driven Unweighted Principal Component (UPC) technique, the data driven Principal Component (PC) technique and the data driven Canonical Variate Analysis (CVA) technique. The techniques that have been used in the modal identification are expandable, and as leaves the estimated modes of that specific technique are list:
For each of the modes the natural frequency (f) is presented as well as the damping ratio (z).
Comparison Within the Same Project
The idea is that you select two modes A and B from the two threes which are then compared in various ways in the other windows. Below we compare the third FDD mode (Project Modes - A) with the third UPC mode (Project Modes - B):
Comparison Across Two Projects
You can also compare modes from different projects as long as the geometry of both projects are equal. The geometries are equal if the node number, the node coordinates and trace lines are the same. Such a case is shown below:
Comparison of Project Modes and Imported Modes
You can compare the modes of a project with modes created by modal software from third party software vendors. The third party software could be other modal analysis software packages or Finite Element Method (FEM) software. In this way you can make verification of results obtained in different ways using the Mode Comparison Window. Importing a Mode First you have to select the project you want to import a mode to. You do this by clicking anywere on the Project Modes Tree of that specific project:
Then right-click the mouse button to activate the context-sensitive menu and select the menu item Import Modes, UFF ASCII Format:
An Open dialog will appear and you will be asked to enter the name and location of the file that contains the ASCII Universal File Format representation of the mode you want to import. The ASCII Universal File Format representation of the mode must fullfill the same requirements as modes being exported in ASCII Universal File Format. Note: The imported mode must contain the same node numbers and node coordinates as the project modes. The imported modes will then be located in the Imported Modes folder as shown below:
The modes can now be accessed on the same terms as any other modes of that specific project. Delete an Imported Mode To delete an imported mode from a project you simply select the imported mode and rightclick the mouse to activate the context-sensitive menu. From this menu you select the menu item Delete Imported Mode:
You can also use the key to delete the selected mode.
The Information Tab (Mode Comparison Window) When you start the Mode Comparison window the first tab window to appear is the Information window. In the example below two modes from different modal identification techniques but from the same project have been selected using the Project Modes trees:
This window is divided horizontally into two. In the lower window the errors between the mode shapes of the two modes in each of the principal X, Y and Z directions of each node are presented in a table. The table consists of the following columns: 1. DOFs - The node number. 2. Direction - The principal direction in the global X, Y and Z coordinate system. 3. Magnitude(A-B) - The absolute value of the complex valued error. 4. Phase(A-B) - The phase angle of the complex valued error. 5. Real(A-B) - The real part of the complex valued error. 6. Imag(A-B) - The imaginary part of the complex valued error.
Each of the columns are sortable in both ascending and descending order which makes it easy to inspect where the largest and smallest error are. Note: The errors presented are the resulting errors of all nodes after slave node equations have been applied. In the upper part, textural information about the two project modes are presented. For each of the two project its title is shown and which mode is selected and from what modal identification technique. Additionally, the Modal Assurance Criterion between these two modes is presented. Finally, information relating to the selected node in the lower window is presented.
The Overlaid Animation Tab (Mode Comparison Window) When you start the Mode Comparison window the second tab window to appear is the Overlaid Animation window. In the example below two different estimates of the same mode but from the same project have been selected using the Project Modes trees:
The blue deformed geometry relates to the selection in the Project Modes - A tree, whereas the red deformed geometry relates to the selection in the Project Modes - B tree. The yellow undeformed geometry can be removed from the 3D-Displays tab of the Control Panel. Note: The two mode shapes are automatically phase corrected for synchronized animation. The window support the general 3D display options. While the window is animating it is not possible to copy and print the window.
The Error Animation Tab (Mode Comparison Window) When you start the Mode Comparison window the third tab window to appear is the Error Animation window. In the example below two different estimations of the same mode but from the same project have been selected using the Project Modes trees:
The blue slightly deformed geometry is the animation of the error added to the undeformed geometry. The yellow undeformed geometry can be removed from the 3D-Displays tab of the Control Panel. The scale of the animated error is the same as the scale of the Overlaid Animation. Note: The two mode shapes are automatically phase corrected before the errors are calculated. The window support the general 3D display options. While the window is animating it is not possible to copy and print the window.
Side by Side Animation Tab (Mode Comparison Window) When you start the Mode Comparison window the fourth tab window to appear is the Side by Side Animation window. In the example below two different estimates of the same mode but from the same project have been selected using the Project Modes trees:
The blue deformed geometry relates to the selection in the Project Modes - A tree, whereas the red deformed geometry relates to the selection in the Project Modes - B tree. The yellow undeformed geometry can be removed from the 3D-Displays tab of the Control Panel. Note: The two mode shapes are automatically phase corrected for synchronized animation. The window support the general 3D display options. While the window is animating it is not possible to copy and print the window.
The MAC Tab (Mode Comparison Window) When you start the Mode Comparison window the fourth and last tab window to appear is the MAC window. In the example below two identical modes from different modal identification techniques but from the same project have been selected using the Project Modes trees:
The window presents the full Modal Assurance Criterion (MAC) matrix of all the modes of the two selected modal identification techniques. As tic marks of the axes of this matrix has the natural frequencies of the modes. As an additional label the project name shown together with the name of the modal identification technique. The natural frequency of the two selected modes are written in yellow. If you want to change the three fundamental colors in the contour legend this is accomplished by activating the context-sensitive Properties dialog as shown below:
The upper limit corresponds to a MAC value of 1, the middle to 0.5 and the lower to 0. In this dialog you can also change the background color. By selecting the Matrix Viewpoint in the context-sensitive menu as shown below:
the matrix is turned into a contour diagram as shown below:
Note: The window support the general 3D display options for static windows. You can also get a tabular presentation of the MAC matrix by pressing the Table MAC View tab. In this case the following Excel compatible table appear:
The rows correspond to the Project A modes and the columns to the Project B modes. This is also marked on the project tree labels to the right. The maximum MAC number per row or column is marked with a light red color. The MAC value between the selected modes are marked with yellow. If you like to have the color codes from the 3D representation of the matrix you simply activate the context-sensitive Properties dialog shown below:
and check the box called Syncronize color for table and press . Then the table will looks as shown below:
The Properties Dialog The Properties Dialog - Overlaid Animation and Side by Side Animation Tabs You can change the colors of the Overlaid Animation or Side by Side Animation Tab windows from the Properties... menu item of the context sensitive menu or from the View menu. Selecting this item start the Properties dialog:
Here you can change the colors of the background, coordinate system arrows, undeformed geometry and deformed geometries of the two modes. In either case the color is change using the color well button. The last tab you can set up the AVI Movie recorder. In this dialog the default settings of the mode animation AVI movie recorder can be modified. When the application is started the computer is searched for applicable AVI compression drivers. These are displayed in the drop-down list called Compression Mode. It is also possible to select the uncompressed file format. However, this is not recommendable due to the unnecessary size of the generated AVI movie files.
There are two different modes for recording an AVI movie file of an animation. If an infinite cyclic animation is desired the radio button Cyclic show (one cycle) should be selected. The result will be a single cycle recording that can be repeated infinitely in e.g. the Microsoft Windows Media Player. If a specific number of seconds of recording is desired the Time capture radio button should be selected. This will result in an AVI recording of a specific number of seconds of the current animation. The Properties Dialog - Error Animation Tab You can change the colors of the Error Animation Tabwindow from the Properties... menu item of the context sensitive menu or from the View menu. Selecting this item start the Properties dialog:
Here you can change the colors of the background, coordinate system arrows, undeformed geometry and deformed geometry error. In either case the color is change using the color
well button. The last tab you can set up the AVI Movie recorder. In this dialog the default settings of the mode animation AVI movie recorder can be modified. When the application is started the computer is searched for applicable AVI compression drivers. These are displayed in the drop-down list called Compression Mode. It is also possible to select the uncompressed file format. However, this is not recommendable due to the unnecessary size of the generated AVI movie files.
There are two different modes for recording an AVI movie file of an animation. If an infinite cyclic animation is desired the radio button Cyclic show (one cycle) should be selected. The result will be a single cycle recording that can be repeated infinitely in e.g. the Microsoft Windows Media Player. If a specific number of seconds of recording is desired the Time capture radio button should be selected. This will result in an AVI recording of a specific number of seconds of the current animation. The Properties Dialog - Project Modes Tree Due to the difference in usage of escape characters when writing ASCII files in a PC and a UNIX environment, it is necessary to specify to what environment files are exported. In the Export tab shown below you can select the export format to use when saving modes in ASCII files.
SVS ASCII Format You can export a list of estimated modes in the socalled SVS Format. This can be done from the mode list of the FDD Peak Picking editor, the mode list of the Select and Link Modes editor and from the Project Modes trees of the Mode Comparison window. When exporting an ASCII file with extension .svs is created. The idea of this format is to use a number of key word followed by values specific to that block. The philosophy is the same as for the SVS Configuration File. Since there can be information about multiple modes saved in the same file each mode definition start with the keyword BEGIN MODE DEFINITION and ends with the keyword END MODE DEFINITION. Note: This format is still subject to changes. Below an example (mes32set.axp) of such a file containing the information for one mode is shown:
BEGIN MODE DEFINITION
ESTIMATOR FDD
CREATION: DATE / TIME 27-11-2000 11:21:31
FREQUENCY [HZ]: MEAN / SDEV 3.368018e+001 0.000000e+000
DAMPING [%]: MEAN / SDEV 0.000000e+000 0.000000e+000
MODE SHAPE: NODE / X-ABS / X-ANG / Y-ABS / Y-ANG / Z-ABS / Z-ANG 1 0.000000e+000 0.000000e+000 0.000000e+000 0.000000e+000 0.000000e+000 0.000000e+000 2 6.623489e-001 -3.066718e+000 6.516432e-001 1.566424e-001 0.000000e+000 0.000000e+000 3 8.881654e-001 2.810976e+000 9.163309e-001 -2.093329e-001 0.000000e+000 0.000000e+000 4 0.000000e+000 0.000000e+000 0.000000e+000 0.000000e+000 0.000000e+000 0.000000e+000 5 6.623489e-001 -3.066718e+000 5.945545e-001 -2.872727e+000 0.000000e+000 0.000000e+000 6 8.881654e-001 2.810976e+000 9.488018e-001 3.038225e+000 0.000000e+000 0.000000e+000 7 0.000000e+000 0.000000e+000 0.000000e+000 0.000000e+000 0.000000e+000 0.000000e+000 8 5.946954e-001 3.609918e-001 6.516432e-001 1.566424e-001 0.000000e+000 0.000000e+000 9 1.000000e+000 0.000000e+000 9.163309e-001 -2.093329e-001 0.000000e+000 0.000000e+000 10 0.000000e+000 0.000000e+000 0.000000e+000 0.000000e+000 0.000000e+000 0.000000e+000 11 5.946954e-001 3.609918e-001 5.945545e-001 -2.872727e+000 0.000000e+000 0.000000e+000 12 1.000000e+000 0.000000e+000 9.488018e-001 3.038225e+000 0.000000e+000 0.000000e+000
END MODE DEFINITION
UFF ASCII Format You can export a list of estimated modes in the ASCII Universal File Format (UFF). See e.g WEB-site of the Structural Dynamics Research Laboratory at University of Cincinnati, Ohio. This can be done from the mode list of the FDD Peak Picking editor, the mode list of the Select and Link Modes editor and from the Project Modes trees of the Mode Comparison window. When exporting an ASCII file with extension .uff is created. This file will contain geometry information in terms of UFF data set number 15 (nodes), UFF data set number 82 (trace lines) and as many UFF data sets number 55 (modal results) as there are modes to be exported. Note: Only one global cartesian coordinate system is exported from the software. This restriction is also imposed on any modes imported into the software through UFF ASCII format. Below an example (mes32set.axp) of such a file containing the information for one mode is shown: -1 15 1 0 0 0 0.00000E+00 0.00000E+00 0.00000E+00 2 0 0 0 0.00000E+00 0.00000E+00 1.50000E+01 3 0 0 0 0.00000E+00 0.00000E+00 3.00000E+01 4 0 0 0 1.50000E+01 0.00000E+00 0.00000E+00 5 0 0 0 1.50000E+01 0.00000E+00 1.50000E+01 6 0 0 0 1.50000E+01 0.00000E+00 3.00000E+01 7 0 0 0 0.00000E+00 1.50000E+01 0.00000E+00 8 0 0 0 0.00000E+00 1.50000E+01 1.50000E+01 9 0 0 0 0.00000E+00 1.50000E+01 3.00000E+01 10 0 0 0 1.50000E+01 1.50000E+01 0.00000E+00 11 0 0 0 1.50000E+01 1.50000E+01 1.50000E+01 12 0 0 0 1.50000E+01 1.50000E+01 3.00000E+01 -1
-1 82 1 60 0 NONE 0 1 2 0 2 3 0 3 6 0 6 5 0 5 4 0 2 5 0 7 8 0 8 9 0 10 11 0 11 12 0 8 11 0 9 12 0 5 11 0 2 8 0 6 12 0 3 9 0 1 4 0 4 10 0 10 7 0 7 1 -1 -1 55 SVS - ARTeMIS Extractor Version Estimator: FDD Torsional Mode 27-11-2000 11:21:31 Project: mes32set.axp 1 3 2 12 5 3 2 6 0 3 0.00000E+00 2.11619E+02 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 1 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 2 -6.60493E-01 -4.95467E-02 6.43665E-01 1.01658E-01 0.00000E+00 0.00000E+00 3
-8.40065E-01 2.88321E-01 8.96327E-01 -1.90420E-01 0.00000E+00 0.00000E+00 4 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 5 -6.60493E-01 -4.95467E-02 -5.73194E-01 -1.57936E-01 0.00000E+00 0.00000E+00 6 -8.40065E-01 2.88321E-01 -9.43737E-01 9.79012E-02 0.00000E+00 0.00000E+00 7 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 8 5.56365E-01 2.10048E-01 6.43665E-01 1.01658E-01 0.00000E+00 0.00000E+00 9 1.00000E+00 0.00000E+00 8.96327E-01 -1.90420E-01 0.00000E+00 0.00000E+00 10 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 11 5.56365E-01 2.10048E-01 -5.73194E-01 -1.57936E-01 0.00000E+00 0.00000E+00 12 1.00000E+00 0.00000E+00 -9.43737E-01 9.79012E-02 0.00000E+00 0.00000E+00 -1
AVI Movie File All animations of modes can be saved in a movie file format called AVI (Audio Video Interlaced). To save an animation you should do the following: 1. Select the mode you want to animate. 2. Start the animation in the 3D-Displays tab window in the Control Panel. 3. Arrange the animation window and zoom level as you want it to look in the movie. 4. Select a compression mode and set up the screen capture in the Properties dialog of the animation window. 5. Start the animation from the File, Export, Modes, AVI Movie File... or from the pop-up menu of the animation window. A save As dialog appear to let you specify the AVI Movie file name. During the recording the animation speed will drop, but the final movie will be played back with a rate equal to the speed prior to the recording.
Below there is some internet links specified to more information about the AVI Movie format and its usage. Windows Media Player Download Site: http://windowsmedia.com/download/download.asp The AVI Movie format: http://www.digicelinc.com/avi_format.htm and http://web.cs.mun.ca/k12media/resources.formats.video.avi.html
OLE Programming Below is the type library for ARTeMIS Extractor presented in the Object Description Language. The compiled type library is located in the file artx.tlb in the Bin folder. The application interface is called IARTX and the interface for a ARTeMIS Extractor project is called IARTX.Document. // ARTX.odl : type library source for ARTX.exe // This file will be processed by the MIDL compiler to produce the // type library (ARTX.tlb). [ uuid(09F7AE37-2B05-11D4-99C5-0090272EF27C), helpstring("ARTeMIS Extractor Application Type Library Version 1.2"), version(1.2) ] library ARTX {
importlib("stdole32.tlb"); importlib("stdole2.tlb"); [ uuid(09F7AE38-2B05-11D4-99C5-0090272EF27C) ] dispinterface IARTX {
properties:
[id(1), propget, helpstring("String containing a description of the last occurred error.") ] BSTR LastError; [id(2),
propput, helpstring("Set boolean to true if safearrays of type long is passed as safearrays of type double. (Used e.g. in MATLAB).") ] boolean DoubleAsLong; [id(3), propput, helpstring("Set boolean to true if comments entered in SetProjectInfo and SetDataSet are in the Rich Text Format.") ] boolean RTFComments;
methods:
[id(4), helpstring("Update all project windows.") ] boolean RefreshWindow(); [id(5), helpstring("Show the project control window.") ] boolean ShowWindow(); [id(6), helpstring("Select which data set to work on.") ] boolean SetCurrentDataSet(long nCurrentSetup); [id(7), helpstring("Set general project information (Should called first when creating a new project).") ] boolean SetProjectInfo(BSTR Title, BSTR CommentsRTF, long nSetups, double T); [id(8), helpstring("Update the project information (Should be the final call when creating a new project.") ] boolean UpdateProject();
[id(9), helpstring("Insert a data set (setup).") ] boolean SetDataSet(long index, BSTR Label, BSTR CommentsRTF, VARIANT TransducerNodesArray, VARIANT TransducerDirectionsArray, VARIANT TransducerReferencesArray, VARIANT TransducerUnitsArray, VARIANT TransducerQuantitiesArray, VARIANT TransducerLabelsArray, VARIANT TransducerDataArray); [id(10), helpstring("Set the geometry related to the new project.") ] boolean SetGeometry(VARIANT NodeNumbersArray, VARIANT NodeDirectionsArray, VARIANT TraceLinesArray); [id(11), helpstring("Set the equations for description of movements of nodes not having a transducer mounted.") ] boolean SetEquations(VARIANT EquationsArray); [id(12), helpstring("Close the project and its all windows. Optionally, save the project before closing.") ] boolean CloseProject(BOOL bSaveOnClose); [id(13), helpstring("Set the geometry including surfaces related to the new project. (Version 1.1 feature)") ] boolean SetGeometry2(VARIANT NodeNumbersArray, VARIANT NodeDirectionsArray, VARIANT TraceLinesArray, VARIANT SurfacesArray); [id(14), helpstring("Set a default project title. (Version 1.1 feature)") ] boolean SetDefaultProjectTitle(BSTR Title); [id(15), helpstring("Check if signal processing is running. (Version 1.1 feature)") ] boolean IsSignalProcessingRunning(); [id(16), helpstring("Upload SPC file with signal processing configuration and start signal processing of the data. (Version 1.1 feature)") ] boolean StartSignalProcessing(BSTR SPCFileName,BOOL bProcessAllSetups,BOOL
bProcessPSD,BOOL bProcessCorFnc,BOOL bProcessOPHankel); [id(17), helpstring("Insert a data set (Version 1.2 feature") ] boolean SetDataSet2(LONG nDataSet, LONG nTransducers, LONG nMeasurements, BSTR Label, BSTR CommentsRTF, VARIANT TransducerNodesArray, VARIANT TransducerDirectionsArray, VARIANT TransducerReferencesArray, VARIANT TransducerUnitsArray, VARIANT TransducerQuantitiesArray, VARIANT TransducerLabelsArray); [id(18), helpstring("Insert transducer data for the current data set (Version 1.2 feature") ] boolean SetDataForDataSet2(LONG nTransducer, VARIANT TransducerDataArray);
}; [ uuid(09F7AE36-2B05-11D4-99C5-0090272EF27C) ] coclass Document {
[default] dispinterface IARTX;
};
};
An example showing how to use the interface is provided in the MATLAB m-file artx.m located in Examples\MATLAB. If you have any questions concerning the use of the interface in your own custom applications please contact: Structural Vibrations Solutions ApS NOVI Science Park Niels Jernes Vej 10 DK-9220 Aalborg East
Denmark. [email protected]
Below are some good references on OLE automation listed.
References 1. Automation - Programmer's Reference. Microsoft Press 1997. ISBN 1-57231-584-9. 2. Programming Microsoft Visual C++. 5th Edition. D.J. Kruglinski, G. Shepherd & S. Wingo. Microsoft Press 1998. ISBN 1-57231-857-0. 3. Understanding ActiveX and OLE. D. Chappell. Microsoft Press 1996. ISBN 1-57231216-5. 4. Inside COM. D. Rogerson. Microsoft Press 1997. ISBN 1-57231-349-8.
Technical Paper on the Stochastic Subspace Identification Techniques By Palle Andersen - Structural Vibration Solutions ApS.
1 Introduction In the traditional input-output modal analysis the estimation of modal parameters have been performed using a somewhat deterministic mathematical framework. One of the major hurdles for people of this traditional modal community to overcome, when turning to outputonly modal analysis, is the switch of the mathematically framework. In output-only modal analysis the mathematically framework involves the use of statistics and introduction of concepts such as optimal prediction, linear system theory and stochastic processes. The two general assumptions made in output-only modal analysis are that the underlying physical system behaves linearly and time-invariant. The linearity imply that if an input with a certain amplitude generates an output with a certain amplitude, then an input with twice the amplitude will generate an output with twice the amplitude as well. The time-invariance implies that the underlying physical system does not change in time. One of the typical parametric model structures to use in output-only modal analysis of linear and time-invariant physical systems is the stochastic state space system. (1)
The first part of this model structure is called the state equation and models the dynamic behavior of the physical system. The second equation is called the observation or output equation, since this equation controls which part of the dynamic system that can be observed in the output of the model. In this model of the physical system, the measured system response yt is generated by two stochastic processes w t and vt. These are called the process noise and the measurement noise. The process noise is the input that drives the system dynamics whereas the measurement noise is the direct disturbance of the system response. The philosophy is that the dynamics of the physical system is modeled by the n´n state matrix A. Given an n´1 input vector w t, this matrix transforms the state of the system, described by the n´1 state vector xt, to a new state xt+1. The dimension n of the state
vector xt is called the state space dimension. The observable part of the system dynamics is extracted from the state vector by forward multiplication of the p´n observation matrix C. The p´1 system response vector yt is a mixture of the observable part of the state and some noise modeled by the measurement noise vt.
2 The Statistical Framework 2.1 Properties of stochastic state space systems The state space model (1) is only applicable for linear systems that do not have timevarying changes of its characteristics. However, this is not the only restriction. The only way to obtain an optimal estimate of a state space model on the basis of measured system response, is to require that the system response is a realization of a Gaussian distributed stochastic process that has zero mean. In other words, in the applied stochastic framework the system response is modelled by a stochastic process yt defined as (2)
and the principal assumption is that the measured system response is a realization of this process. It is seen that this process is completely described by its covariance function L i. This means that if we can estimate a state space model having the correct covariance function this model will completely describe the statistically properties of the system response. An estimated model fulfilling this is called covariance equivalent. The estimator that can produce such model is called an optimal estimator. Since the system response of the linear state space model is a Gaussian stochastic process it implies that xt, w t and vt all are Gaussian stochastic processes as well. Since the input processes w t and vt are unknown we make the simplest possible assumption about their statistical properties, which is to assume that they are two correlated zero-mean Gaussian white noise processes, defined by their covariance matrices as (3)
The Gaussian stochastic process describing the state xt is also zero-mean and completely described by its covariance function
(4)
Using (1) to (4) the following relations can be established
(5)
The matrix G is the covariance between system response yt and the updated state vector xt+1. The covariance function of yt can also be expressed in terms of the system matrices as (6)
There are two additional system matrices turns out to play an important role
(7)
These are the extended observability matrix G i and the reversed extended stochastic controllability matrix D i.
2.2 Optimal prediction One of the most important parts of all estimation is the ability to predict the measurements optimally. In output only modal analysis this means to be able to predict the measured system response optimally. An optimal predictor is defined as a predictor that results in a minimum error between the predicted and measured system response. If the system response can be predicted optimally it implies that a model can be estimated in an optimal sense. Recall that the state vector xt completely describes the system dynamics at time t. In order to predict the system response yt optimally it is necessary to start by defining an optimal predictor of xt. Now assume that we have measurements yk available from some initial time k = 0 to k = t-1. Collect these measurements in a vector
(8)
In the Gaussian case the optimal predictor of xt is then given by the conditional mean value
(9)
So, the optimal predictor of xt is defined as the mean value of xt given all measured system response yk from k = 0 to k = t-1. The difference between prediction error and is defined as
and xt is called the state
(10)
This error is the part of xt that cannot be predicted by
.
In order to predict the system response a similar conditional mean can be formulated for yt
(11)
The last part of this equation is obtained by inserting (1) and assuming that vt and yk from k = 0 to k = t-1 are uncorrelated.
2.3 The Kalman filter. The two predictors (9) and (11) are related through the so-called Kalman filter for linear and time-invariant systems, see e.g. Goodwin et al. [6] (12)
The matrix Kt is called the non-steady state Kalman gain and et is called the innovation and is a zero-mean Gaussian white noise process. Defining the non-steady-state covariance matrix of the predicted state vector
as Pt the Kalman gain Kt is calculated from (13)
The last of these equations is called the Ricatti equation. The Kalman filter predicts the state on the basis of the previous predicted state and the measurement yt. The covariance Q of the innovations et can be determined from the last equation in (12) as (14)
Given that the initial state prediction is and the initial state prediction covariance matrix P0 = 0 and assume that we have measurements yk available from k = 0 to k = t-1, then this filter is an optimal predictor for the state space system (1) when the measurements yt are Gaussian distributed.
2.4 The innovation state space system. At start up the Kalman filter (12) will experience a transient phase where the prediction of the state will be non-steady. However, if the state matrix A is stable the filter will enter a steady state as time approach infinity. When this steady state is reached the covariance matrix of the predicted state vector becomes constant, i.e. Pt = P, which imply that the Kalman gain becomes constant as well, i.e. Kt = K. The Kalman filter is now operating in steady state and is defined as (15)
The steady state Kalman gain is now calculated from
(16)
The last equation is now called an algebraic Ricatti equation. Assuming all matrices but P is known this equation can be solved using eigenvalue decomposition, see Aoki [2] and Overschee et al [1]. If the last equation in (15) is rearranged the following state space system is obtained (17)
This system is called the innovation state space system. The major difference between this system and (1) is that the state vector has been substituted with its prediction, and that the two input processes of (1) have been converted into one input process – the innovations. This state space system is widely used as model structure in output only modal analysis, see e.g. Ljung [3] and Söderström et al. [4].
3 The Stochastic Subspace Identification Framework The Kalman filter defined in the last section turns out to be the key element in the group of estimation techniques known as the stochastic subspace techniques. From (17) it is seen that if sufficiently many states of (1), let’s say j states, can be predicted, i.e. and , then the A and C matrices can be estimated from the following least regression problem (18)
This is a valid approach since the innovations are assumed to be Gaussian white noise. Since A and C are assumed to be time-invariant this regression approach will be valid even though the predicted state and originates from a non-steady state Kalman filter. So the fundamental problem to solve in the stochastic subspace identification technique is to extract the predicted states from the measured data. To show how this is performed, consider the state space system in (1) and take the conditional expectation on both sides of both equations to yield
(19)
Now assume that a recursion is started at time step q. Inserting the first equation in (19) recursively into itself i times and finally inserting the result the last of the equations in (19)
leads to the following formulation
(20)
This equation shows the relation between the initial predicted state
and the prediction of
the free (noise free) response of the system . By stacking i equations on top of each other the following set of equations are obtained
(21)
By introducing the vector oq as the left-hand side and noticing that the first part of the righthand side is equal to the extended observability matrix G i we actually obtain the following expression for the predicted states
(22)
The matrix G i-1 is actually the pseudo-inverse of G i. This equation shows that if we can estimate G i and oq for several values of q, we can in fact estimate the predicted states for several values of q as well.
3.1 Estimation of free system response. In this section we will focus on the estimation of the predicted free response . We will estimate a set of vectors Ot and gather them column by column in a matrix O. In order to predict the system response a conditional mean similar to (11) can be formulated.
(23)
This conditional mean is the prediction of the future system response yi+q given the past system response from time t = i+q-1 down to t = q. This conditional expectation is only an approximation of (11) since the conditioning vector stops a time t = q and not t = 0. The approximation is only good if i is sufficiently high. For zero-mean Gaussian stochastic processes this conditional mean can be calculated by, see e.g. Melsa et al. [5].
(24)
Since the error
is zero-mean and uncorrelated and is independent of the
conditional mean
and the conditioning vector yqi+q-1 the conditional mean (24) is also
called the orthogonal projection of the vector yi+q onto the vector yqi+q-1. In order to estimate all elements of oq we need to extend (24) to allow estimation of
to
in one operation. This is done by using (8) to extend the conditional mean in (24) to the following
. This results in the following
equation for oq
(25)
In the last equation a new ip´ip matrix Lk is introduced for simplicity. This matrix is defined as
(26)
Incidentally, the matrix Li is also equal to
(27)
As seen in (18) we need a bank of predicted state estimates for q = i to q = i+j-1 for a sufficiently large value of j. To estimate these state in one operation based on the approach in (25) we need to define the following two matrices Oi and Y p as
(28)
(29)
The index p in (29) signifies that the matrix contains system response of the past compared to the system response we are predicting. Since we assume that the system response is stationary, i.e. that (28) and (29) to yield
, equation (25) can easily be extended using
(30)
With this equation the first of the two major tasks in the stochastic subspace identification technique has been fulfilled. If the extension in (30) is carried on to (22) we obtain the following relation
(31)
The matrix
is a bank of predicted states and is defined as
(32)
As seen the matrix Oi only depends on system response and system response covariance, and can therefore be estimated directly from the measured system response. In Overschee et al. [1] a method based on the QR decomposition is presented (For more on the QR decomposition, see e.g. Golub et al. [7]). This method estimates Oi directly from the measurements without explicit need of the covariance estimates. By using that method the stochastic subspace identification techniques can surely be called data driven identification techniques. Since the matrix Oi is the same no matter which data driven identification technique that is used this matrix is also referred to as the Common SSI Input Matrix.
3.2 Estimation of the extended observability matrix. In order to estimate A and C in (18) what remain is to estimate the extended observability matrix G i as shown in (22). It is actually the estimation of this matrix that can be done in different ways and results in that several stochastic subspace identification techniques exist. In this section we will treat the matter in a generalized way by introducing two so-called weight matrices that takes care of the differences between the techniques. In chapter 4 we will show how to choose these weight matrices in order to arrive at different techniques. The only input we have for the estimation is still only the matrix Oi, i.e. only information related to the system response. The underlying system that has generated the measured
response is unknown, which means that we do not know the state space dimension of underlying system. What this means can be seen from equation (31) that defines the matrix Oi as the product . The outer dimension of Oi and therefore also of is ip´ j. However, the question is what the inner dimension of this product is. The inner dimension is exactly the state space dimension of the underlying system. So to find G i the first task is to determine this dimension. We determine this dimension from Oi by using the Singular Value Decomposition or SVD, see e.g. Golub et al. However, before taking the SVD we pre- and postmultiply Oi with the before mentioned weight matrices W1 and W2 which are user-defined. Now taking the SVD of the resulting product yields
(33)
Assuming that W1 has full rank and that the rank of W2 is equal to the rank of Y pW2, the dimension of the inner product is equal to the number of non-zero singular values, i.e. number of diagonal elements of S1. From the last two equations of (33) we see that G i is given by
(34)
The non-singular n´ n matrix T represents an arbitrary similarity transform. This means that we have determined the extended observability matrix except for an arbitrary similarity transformation, which merely means that we have no control over the exact inner rotation of the state space system. As seen the state space dimension is determined as the number of diagnonal elements of S1, and G i is found on the basis the reduced subspace of W1OiW2. For these reasons it is no wonder why the estimation techniques are called subspace identification techniques.
3.3 A general estimation algorithm.
Independently of the choice of weight matrices W1 and W2 the estimation of the system matrices can be done in the general way presented in this section. This approach presented here is not the only one, but in the current context properly the most obvious choice. In Overschee et al. [1] two other approaches are also described. The estimation can be divided into three parts.
3.3.1 Data compression. Assuming that N samples of measured system response are available the user needs to specify how many block rows i the matrix Oi should have. As seen from (33) the maximum state space dimension depends on the number of block rows and will be ip, where p is the dimension of the measured system response vector yt. It should be remembered that the maximum state space dimension corresponds to the maximum number of eigenvalues that can be identified. It should also be remembered that i is the prediction horizon and as such depends on the correlation length of the lowest mode to be identified. Oi are the estimated using (30). However, in order to estimate the matrix we also need to estimate the matrix Oi-1 since
(35)
This can be proven by proper substitutions in the above equations, see also Overschee et al. [1]. Oi-1is estimated by deleting the first p rows of Oi.
3.3.2 Subspace identification. Pre- and post multiply the matrices W1 and W2 which are dependent upon the actually identification algorithm. Determine the SVD (33) of W1OiW2, and calculate the extended observability matrix G i. G i-1 is obtained from G i by deleting the last p rows.
3.3.3 Estimation of system matrices. Now we have all the information available that is needed in order to estimate a realization of the innovation state space system defined in (17). Estimate the predicted states
and
using (31) and (35), and set up the following matrix of measured system response
(36)
Solve the least squares problem
(37)
where is the pseudo inverse of . The steady state Kalman gain K is estimated by the following relations. First estimate the reversed extended stochastic controllability matrix D i from (27) (38) The covariance matrix G can then be extracted from the last p columns of sample covariance matrix L 0 rom e.g.
D i.
Estimate the
(39)
Estimate the Kalman gain K in (16) by solving the algebraic Ricatti equation in (16) first. Finally, estimate the covariance matrix Q of the innovations using (14).
4 Some Stochastic Subspace Identification Algorithms As mentioned in section 3.2, the only significant difference between the different stochastic subspace algorithms is the choice of the weight matrices W1 and W2. In this chapter we will focus on three algorithms, the Unweighted Principal Component algorithm, the Principal Component algorithm and the Canonical Variate Analysis algorithm.
4.1 The Unweighted Principal Component algorithm. The Unweighted Principal Component algorithm is the most simple algorithm to incorporate into the stochastic subspace frame work. As the name says it is an unweighted approach which means that both weight matrices equals the unity matrix, see Overschee et al. [1]
(40)
The reason is that this algorithm determines the system order from the left singular vectors U1 of the SVD of the following matrix
(41)
Since we have chosen the weight to be unity the covariance of W1OiW2 equals
(42)
This show that covariance (42) is equal to (41) which means that (41) and (42) has the same left singular vectors. From (34) we see that the extended observability matrix is determined as
(43)
This algorithm is also known under the name N4SID.
4.2 The Principal Component algorithm. The Principal Component algorithm determines the system matrices from the singular values and the left singular vectors of the matrix Li. This means that the singular values and left singular vectors of W1OiW2 must equal the singular values and left singular vectors of Li. To accomplish this the weight matrices are chosen as
(44)
The covariance of W1OiW2 now equals
(45)
This shows that the singular values and the left singular vectors of (45) and Li are equal. From (34) we see that the extended observability matrix is determined as
(46)
Even though it appears that (46) is equal to (43) they are not. We must remember that the SVD has been taken on different matrices due to the different weights.
4.2 The Canonical Variate Analysis algorithm. The Canonical Variate Analysis algorithm, see Akaike 74,75, computes the principal angles and directions between the row spaces of the matrix of past outputs Y p and the matrix of future outputs Y f. The matrix of past outputs Y p is defined in (29), and the matrix Y f is defined in a similar manner
(47)
The principal angles and directions between the row spaces of Y p and Y f are determined from the SVD of the matrix, see Overschee et al. [1]
(48)
Comparing with (30) we obtain the same covariance matrix of (47) and W1OiW2, and therefore also the same principal angles and directions between the row spaces of Y p and Y f , if we choose the following weights
(49)
The covariance of W1OiW2 now equals
(50)
The covariance of (47) is given by
(51)
We see that the covariance matrices of W1OiW2 and (48) are equal. In the above it has been used that determined as
. Finally, we see that the extended observability matrix is
(52)
Since L0 is a byproduct of the determination of Oi this estimator is very easy to implement into the common framework.
References 1. Overschee, P. van & B. De Moor: Subspace identification for linear systems – Theory, Implementation, Applications. Kluwer academic Publishers, ISBN 0-7923-9717-7, 1996.
2. Aoki, M.: State Space Modeling of Time Series. Springer-Verlag, ISBN 0-387-528695, 1990. 3. Ljung, L.: System Identification – Theory for the user. Prentice-Hall, ISBN 0-13881640-9, 1987. 4. Söderström, T. & P. Stoica: System Identification. Prentice-Hall, ISBN 0-13-127606-9, 1989. 5. Melsa, J. L. & A.P. Sage: An Introduction to Probability and Stochastic Processes. Prentice-Hall, ISBN 0-13-034850-3, 1973. 6. Goodwin, G.C. & K.S. Sin: Adaptive Filtering, Prediction and Control. Prentice-Hall, ISBN 0-13-004069-X, 1984. 7. Golub, G.H. & C.F. Van Loan: Matrix Computations. 2nd Ed. The John Hopkins University Press, ISBN 0-8018-3772-3, 1989.
The Modal Results Tab (SSI) (Pro version only) In the stabilization diagram of the Stochastic Subspace Identification editor you can see the results of the individual modes of a model presented as crosses lying on a horizontal line. You see the location of the modes as a function of the natural frequency of the modes. This is only a visualization tool and if you need to look at the specific values of a mode, i.e. the natural frequency and the damping ratio, you need to go to the Modal Results window tabs. There are two window tabs: one for the blue cursor model and one for the red model selected for Select & Link.
Technical Paper on the Stochastic Subspace Identification Techniques By Palle Andersen - Structural Vibration Solutions ApS.
1 Introduction In the traditional input-output modal analysis the estimation of modal parameters have been performed using a somewhat deterministic mathematical framework. One of the major hurdles for people of this traditional modal community to overcome, when turning to outputonly modal analysis, is the switch of the mathematically framework. In output-only modal analysis the mathematically framework involves the use of statistics and introduction of concepts such as optimal prediction, linear system theory and stochastic processes. The two general assumptions made in output-only modal analysis are that the underlying physical system behaves linearly and time-invariant. The linearity imply that if an input with a certain amplitude generates an output with a certain amplitude, then an input with twice the amplitude will generate an output with twice the amplitude as well. The time-invariance implies that the underlying physical system does not change in time. One of the typical parametric model structures to use in output-only modal analysis of linear and time-invariant physical systems is the stochastic state space system. (1)
The first part of this model structure is called the state equation and models the dynamic behavior of the physical system. The second equation is called the observation or output equation, since this equation controls which part of the dynamic system that can be observed in the output of the model. In this model of the physical system, the measured system response yt is generated by two stochastic processes w t and vt. These are called the process noise and the measurement noise. The process noise is the input that drives the system dynamics whereas the measurement noise is the direct disturbance of the system response. The philosophy is that the dynamics of the physical system is modeled by the n´n state matrix A. Given an n´1 input vector w t, this matrix transforms the state of the system, described by the n´1 state vector xt, to a new state xt+1. The dimension n of the state
vector xt is called the state space dimension. The observable part of the system dynamics is extracted from the state vector by forward multiplication of the p´n observation matrix C. The p´1 system response vector yt is a mixture of the observable part of the state and some noise modeled by the measurement noise vt.
2 The Statistical Framework 2.1 Properties of stochastic state space systems The state space model (1) is only applicable for linear systems that do not have timevarying changes of its characteristics. However, this is not the only restriction. The only way to obtain an optimal estimate of a state space model on the basis of measured system response, is to require that the system response is a realization of a Gaussian distributed stochastic process that has zero mean. In other words, in the applied stochastic framework the system response is modelled by a stochastic process yt defined as (2)
and the principal assumption is that the measured system response is a realization of this process. It is seen that this process is completely described by its covariance function L i. This means that if we can estimate a state space model having the correct covariance function this model will completely describe the statistically properties of the system response. An estimated model fulfilling this is called covariance equivalent. The estimator that can produce such model is called an optimal estimator. Since the system response of the linear state space model is a Gaussian stochastic process it implies that xt, w t and vt all are Gaussian stochastic processes as well. Since the input processes w t and vt are unknown we make the simplest possible assumption about their statistical properties, which is to assume that they are two correlated zero-mean Gaussian white noise processes, defined by their covariance matrices as (3)
The Gaussian stochastic process describing the state xt is also zero-mean and completely described by its covariance function
(4)
Using (1) to (4) the following relations can be established
(5)
The matrix G is the covariance between system response yt and the updated state vector xt+1. The covariance function of yt can also be expressed in terms of the system matrices as (6)
There are two additional system matrices turns out to play an important role
(7)
These are the extended observability matrix G i and the reversed extended stochastic controllability matrix D i.
2.2 Optimal prediction One of the most important parts of all estimation is the ability to predict the measurements optimally. In output only modal analysis this means to be able to predict the measured system response optimally. An optimal predictor is defined as a predictor that results in a minimum error between the predicted and measured system response. If the system response can be predicted optimally it implies that a model can be estimated in an optimal sense. Recall that the state vector xt completely describes the system dynamics at time t. In order to predict the system response yt optimally it is necessary to start by defining an optimal predictor of xt. Now assume that we have measurements yk available from some initial time k = 0 to k = t-1. Collect these measurements in a vector
(8)
In the Gaussian case the optimal predictor of xt is then given by the conditional mean value
(9)
So, the optimal predictor of xt is defined as the mean value of xt given all measured system response yk from k = 0 to k = t-1. The difference between prediction error and is defined as
and xt is called the state
(10)
This error is the part of xt that cannot be predicted by
.
In order to predict the system response a similar conditional mean can be formulated for yt
(11)
The last part of this equation is obtained by inserting (1) and assuming that vt and yk from k = 0 to k = t-1 are uncorrelated.
2.3 The Kalman filter. The two predictors (9) and (11) are related through the so-called Kalman filter for linear and time-invariant systems, see e.g. Goodwin et al. [6] (12)
The matrix Kt is called the non-steady state Kalman gain and et is called the innovation and is a zero-mean Gaussian white noise process. Defining the non-steady-state covariance matrix of the predicted state vector
as Pt the Kalman gain Kt is calculated from (13)
The last of these equations is called the Ricatti equation. The Kalman filter predicts the state on the basis of the previous predicted state and the measurement yt. The covariance Q of the innovations et can be determined from the last equation in (12) as (14)
Given that the initial state prediction is and the initial state prediction covariance matrix P0 = 0 and assume that we have measurements yk available from k = 0 to k = t-1, then this filter is an optimal predictor for the state space system (1) when the measurements yt are Gaussian distributed.
2.4 The innovation state space system. At start up the Kalman filter (12) will experience a transient phase where the prediction of the state will be non-steady. However, if the state matrix A is stable the filter will enter a steady state as time approach infinity. When this steady state is reached the covariance matrix of the predicted state vector becomes constant, i.e. Pt = P, which imply that the Kalman gain becomes constant as well, i.e. Kt = K. The Kalman filter is now operating in steady state and is defined as (15)
The steady state Kalman gain is now calculated from
(16)
The last equation is now called an algebraic Ricatti equation. Assuming all matrices but P is known this equation can be solved using eigenvalue decomposition, see Aoki [2] and Overschee et al [1]. If the last equation in (15) is rearranged the following state space system is obtained (17)
This system is called the innovation state space system. The major difference between this system and (1) is that the state vector has been substituted with its prediction, and that the two input processes of (1) have been converted into one input process – the innovations. This state space system is widely used as model structure in output only modal analysis, see e.g. Ljung [3] and Söderström et al. [4].
3 The Stochastic Subspace Identification Framework The Kalman filter defined in the last section turns out to be the key element in the group of estimation techniques known as the stochastic subspace techniques. From (17) it is seen that if sufficiently many states of (1), let’s say j states, can be predicted, i.e. and , then the A and C matrices can be estimated from the following least regression problem (18)
This is a valid approach since the innovations are assumed to be Gaussian white noise. Since A and C are assumed to be time-invariant this regression approach will be valid even though the predicted state and originates from a non-steady state Kalman filter. So the fundamental problem to solve in the stochastic subspace identification technique is to extract the predicted states from the measured data. To show how this is performed, consider the state space system in (1) and take the conditional expectation on both sides of both equations to yield
(19)
Now assume that a recursion is started at time step q. Inserting the first equation in (19) recursively into itself i times and finally inserting the result the last of the equations in (19)
leads to the following formulation
(20)
This equation shows the relation between the initial predicted state
and the prediction of
the free (noise free) response of the system . By stacking i equations on top of each other the following set of equations are obtained
(21)
By introducing the vector oq as the left-hand side and noticing that the first part of the righthand side is equal to the extended observability matrix G i we actually obtain the following expression for the predicted states
(22)
The matrix G i-1 is actually the pseudo-inverse of G i. This equation shows that if we can estimate G i and oq for several values of q, we can in fact estimate the predicted states for several values of q as well.
3.1 Estimation of free system response. In this section we will focus on the estimation of the predicted free response . We will estimate a set of vectors Ot and gather them column by column in a matrix O. In order to predict the system response a conditional mean similar to (11) can be formulated.
(23)
This conditional mean is the prediction of the future system response yi+q given the past system response from time t = i+q-1 down to t = q. This conditional expectation is only an approximation of (11) since the conditioning vector stops a time t = q and not t = 0. The approximation is only good if i is sufficiently high. For zero-mean Gaussian stochastic processes this conditional mean can be calculated by, see e.g. Melsa et al. [5].
(24)
Since the error
is zero-mean and uncorrelated and is independent of the
conditional mean
and the conditioning vector yqi+q-1 the conditional mean (24) is also
called the orthogonal projection of the vector yi+q onto the vector yqi+q-1. In order to estimate all elements of oq we need to extend (24) to allow estimation of
to
in one operation. This is done by using (8) to extend the conditional mean in (24) to the following
. This results in the following
equation for oq
(25)
In the last equation a new ip´ip matrix Lk is introduced for simplicity. This matrix is defined as
(26)
Incidentally, the matrix Li is also equal to
(27)
As seen in (18) we need a bank of predicted state estimates for q = i to q = i+j-1 for a sufficiently large value of j. To estimate these state in one operation based on the approach in (25) we need to define the following two matrices Oi and Y p as
(28)
(29)
The index p in (29) signifies that the matrix contains system response of the past compared to the system response we are predicting. Since we assume that the system response is stationary, i.e. that (28) and (29) to yield
, equation (25) can easily be extended using
(30)
With this equation the first of the two major tasks in the stochastic subspace identification technique has been fulfilled. If the extension in (30) is carried on to (22) we obtain the following relation
(31)
The matrix
is a bank of predicted states and is defined as
(32)
As seen the matrix Oi only depends on system response and system response covariance, and can therefore be estimated directly from the measured system response. In Overschee et al. [1] a method based on the QR decomposition is presented (For more on the QR decomposition, see e.g. Golub et al. [7]). This method estimates Oi directly from the measurements without explicit need of the covariance estimates. By using that method the stochastic subspace identification techniques can surely be called data driven identification techniques. Since the matrix Oi is the same no matter which data driven identification technique that is used this matrix is also referred to as the Common SSI Input Matrix.
3.2 Estimation of the extended observability matrix. In order to estimate A and C in (18) what remain is to estimate the extended observability matrix G i as shown in (22). It is actually the estimation of this matrix that can be done in different ways and results in that several stochastic subspace identification techniques exist. In this section we will treat the matter in a generalized way by introducing two so-called weight matrices that takes care of the differences between the techniques. In chapter 4 we will show how to choose these weight matrices in order to arrive at different techniques. The only input we have for the estimation is still only the matrix Oi, i.e. only information related to the system response. The underlying system that has generated the measured
response is unknown, which means that we do not know the state space dimension of underlying system. What this means can be seen from equation (31) that defines the matrix Oi as the product . The outer dimension of Oi and therefore also of is ip´ j. However, the question is what the inner dimension of this product is. The inner dimension is exactly the state space dimension of the underlying system. So to find G i the first task is to determine this dimension. We determine this dimension from Oi by using the Singular Value Decomposition or SVD, see e.g. Golub et al. However, before taking the SVD we pre- and postmultiply Oi with the before mentioned weight matrices W1 and W2 which are user-defined. Now taking the SVD of the resulting product yields
(33)
Assuming that W1 has full rank and that the rank of W2 is equal to the rank of Y pW2, the dimension of the inner product is equal to the number of non-zero singular values, i.e. number of diagonal elements of S1. From the last two equations of (33) we see that G i is given by
(34)
The non-singular n´ n matrix T represents an arbitrary similarity transform. This means that we have determined the extended observability matrix except for an arbitrary similarity transformation, which merely means that we have no control over the exact inner rotation of the state space system. As seen the state space dimension is determined as the number of diagnonal elements of S1, and G i is found on the basis the reduced subspace of W1OiW2. For these reasons it is no wonder why the estimation techniques are called subspace identification techniques.
3.3 A general estimation algorithm.
Independently of the choice of weight matrices W1 and W2 the estimation of the system matrices can be done in the general way presented in this section. This approach presented here is not the only one, but in the current context properly the most obvious choice. In Overschee et al. [1] two other approaches are also described. The estimation can be divided into three parts.
3.3.1 Data compression. Assuming that N samples of measured system response are available the user needs to specify how many block rows i the matrix Oi should have. As seen from (33) the maximum state space dimension depends on the number of block rows and will be ip, where p is the dimension of the measured system response vector yt. It should be remembered that the maximum state space dimension corresponds to the maximum number of eigenvalues that can be identified. It should also be remembered that i is the prediction horizon and as such depends on the correlation length of the lowest mode to be identified. Oi are the estimated using (30). However, in order to estimate the matrix we also need to estimate the matrix Oi-1 since
(35)
This can be proven by proper substitutions in the above equations, see also Overschee et al. [1]. Oi-1is estimated by deleting the first p rows of Oi.
3.3.2 Subspace identification. Pre- and post multiply the matrices W1 and W2 which are dependent upon the actually identification algorithm. Determine the SVD (33) of W1OiW2, and calculate the extended observability matrix G i. G i-1 is obtained from G i by deleting the last p rows.
3.3.3 Estimation of system matrices. Now we have all the information available that is needed in order to estimate a realization of the innovation state space system defined in (17). Estimate the predicted states
and
using (31) and (35), and set up the following matrix of measured system response
(36)
Solve the least squares problem
(37)
where is the pseudo inverse of . The steady state Kalman gain K is estimated by the following relations. First estimate the reversed extended stochastic controllability matrix D i from (27) (38) The covariance matrix G can then be extracted from the last p columns of sample covariance matrix L 0 rom e.g.
D i.
Estimate the
(39)
Estimate the Kalman gain K in (16) by solving the algebraic Ricatti equation in (16) first. Finally, estimate the covariance matrix Q of the innovations using (14).
4 Some Stochastic Subspace Identification Algorithms As mentioned in section 3.2, the only significant difference between the different stochastic subspace algorithms is the choice of the weight matrices W1 and W2. In this chapter we will focus on three algorithms, the Unweighted Principal Component algorithm, the Principal Component algorithm and the Canonical Variate Analysis algorithm.
4.1 The Unweighted Principal Component algorithm. The Unweighted Principal Component algorithm is the most simple algorithm to incorporate into the stochastic subspace frame work. As the name says it is an unweighted approach which means that both weight matrices equals the unity matrix, see Overschee et al. [1]
(40)
The reason is that this algorithm determines the system order from the left singular vectors U1 of the SVD of the following matrix
(41)
Since we have chosen the weight to be unity the covariance of W1OiW2 equals
(42)
This show that covariance (42) is equal to (41) which means that (41) and (42) has the same left singular vectors. From (34) we see that the extended observability matrix is determined as
(43)
This algorithm is also known under the name N4SID.
4.2 The Principal Component algorithm. The Principal Component algorithm determines the system matrices from the singular values and the left singular vectors of the matrix Li. This means that the singular values and left singular vectors of W1OiW2 must equal the singular values and left singular vectors of Li. To accomplish this the weight matrices are chosen as
(44)
The covariance of W1OiW2 now equals
(45)
This shows that the singular values and the left singular vectors of (45) and Li are equal. From (34) we see that the extended observability matrix is determined as
(46)
Even though it appears that (46) is equal to (43) they are not. We must remember that the SVD has been taken on different matrices due to the different weights.
4.2 The Canonical Variate Analysis algorithm. The Canonical Variate Analysis algorithm, see Akaike 74,75, computes the principal angles and directions between the row spaces of the matrix of past outputs Y p and the matrix of future outputs Y f. The matrix of past outputs Y p is defined in (29), and the matrix Y f is defined in a similar manner
(47)
The principal angles and directions between the row spaces of Y p and Y f are determined from the SVD of the matrix, see Overschee et al. [1]
(48)
Comparing with (30) we obtain the same covariance matrix of (47) and W1OiW2, and therefore also the same principal angles and directions between the row spaces of Y p and Y f , if we choose the following weights
(49)
The covariance of W1OiW2 now equals
(50)
The covariance of (47) is given by
(51)
We see that the covariance matrices of W1OiW2 and (48) are equal. In the above it has been used that determined as
. Finally, we see that the extended observability matrix is
(52)
Since L0 is a byproduct of the determination of Oi this estimator is very easy to implement into the common framework.
References 1. Overschee, P. van & B. De Moor: Subspace identification for linear systems – Theory, Implementation, Applications. Kluwer academic Publishers, ISBN 0-7923-9717-7, 1996.
2. Aoki, M.: State Space Modeling of Time Series. Springer-Verlag, ISBN 0-387-528695, 1990. 3. Ljung, L.: System Identification – Theory for the user. Prentice-Hall, ISBN 0-13881640-9, 1987. 4. Söderström, T. & P. Stoica: System Identification. Prentice-Hall, ISBN 0-13-127606-9, 1989. 5. Melsa, J. L. & A.P. Sage: An Introduction to Probability and Stochastic Processes. Prentice-Hall, ISBN 0-13-034850-3, 1973. 6. Goodwin, G.C. & K.S. Sin: Adaptive Filtering, Prediction and Control. Prentice-Hall, ISBN 0-13-004069-X, 1984. 7. Golub, G.H. & C.F. Van Loan: Matrix Computations. 2nd Ed. The John Hopkins University Press, ISBN 0-8018-3772-3, 1989.