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Structural Dynamics Dr.-Ing. Volkmar Zabel Bauhaus-Universit¨at Weimar, Fakult¨at Bauingenieurwesen Institut f¨ur Strukturmechanik Marienstraße 15, 99421 Weimar email: [email protected] EXERCISES

1

Bauhaus-Universit¨at Weimar Institut f¨ur Strukturmechanik Dr.-Ing. Volkmar Zabel

Structural Dynamics Exercises Page 2

Given:

• m, k

Task:

• Moment of inertia with respect to the center of rotation • Equation of motion neglecting gravity • Angular natural frequency ω0 depending on m and k

3. Aufgabe (30) Ein System aus zwei starren Körpern mit den folgenden Parametern ist gegeben: m = 12 kg, k = 3400 kN/m. Die Länge ist als bekannte Größe zu behandeln. Bestimmen Sie die Eigenfrequenz des Systems in Hz!

m k L

b m

L

a

L

L

The following parameters of a system consisting of two rigid bodies are given: m = 12 kg, k = 3400 kN/m. Assume, that the length L is given. Determine the natural frequency of the system, f0 in Hz!

2L

4. Aufgabe (30) Bestimmen Sie die Verdrehung des Trägerquerschnitts ϕ(x) entlang der Stabachse. Stellen Sie den Verlauf von ϕ(x) graphisch dar!

m = 2 kg, c = 0.25 Nms , k = 54 N , ϕ0 = 0.1 rad m

kA−A

2p

5

3p

5

k

p 20

5

L/2

15

10

[m]

m x y

z

ϕ 50cm

25cm 25cm

25cm

p = 8 kN/m G = 6, 25 · 104 kN/m2

L

2,25m

2

1,375m

Schnitt A − A

L/2

a

• Calculate the free vibration response assuming small amplitudes with the initial conditions ϕ(t = 0) = ϕ0 , ϕ(t ˙ = 0) = 0! Neglect gravity! • Determine the time, when the amplitudes have are decayed to 20 % of the initial displacement!

c

• Sketch the diagram of the free vibration response as a function of time!

Bauhaus-Universit¨at Weimar Institut f¨ur Strukturmechanik Dr.-Ing. Volkmar Zabel

F0 sin ωt a

m

m = 10 kg c = 80 Nms k = 2400 N m

g L

2m k

Structural Dynamics Exercises Page 3

x

L = 0.5 m, F0 = 12 N

b c L

L

• Calculate the dynamic amplification factor V1 of the horizontal displacement x at support b at the excitation frequencies ω = 6, 9, 12, 15 and 18 s−1 ! Sketch the diagram of V1 for the frequency range 0 ≤ ω ≤ 18 s−1 ! • Compute the maximum horizontal displacement x at support b due to a harmonic excitation with a frequency ω = 9 s−1 !

The following diagram represents measured response data that was acquired in a free vibration test after the release of an initial displacement.

17.0927

1.3333

0.0833

1.62005

Determine the natural frequency f0 [Hz] and the damping coefficient ζ!

Bauhaus-Universit¨at Weimar Institut f¨ur Strukturmechanik Dr.-Ing. Volkmar Zabel

m

Structural Dynamics Exercises Page 4

x2

L

k

2m

x1

L

k

xb = xb,0 sin ωt

L

• Develop the equations of motion for the system shown in the figure! • Calculate the vibration amplitudes of x1 and x2 relative to the amplitude of the base excitation xb,0 ! for the frequency Assume ω 2 = 4k m of the harmonic base excitation! • Sketch the displacement shape!

F (t) E, I, m(x), ζm kd

cd md

x L

z

E = 2.1 × 1011 N m 4 I = 300000 cm L = 25m m = 1000 kg m ζm = 0.005

• Design for the given bridge system the parameters kd , cd and md of a tuned mass damper (TMD) according to the optimal criteria by Den Hartog for reducing the vibrations of the first modes of the bridge! • Determine the natural frequencies of the equivalent 2-dof system! • Calculate the maximal dynamic amplitude of steady state vibration zˆ at mid-span to a dynamic excitation F (t) = Fˆ sin 4πt; Fˆ = 2 kN for the system with and without TMD, neglecting damping! • Sketch a qualitative diagram of the dynamic amplification functions for the system with and without TMD!

Bauhaus-Universit¨at Weimar Institut f¨ur Strukturmechanik Dr.-Ing. Volkmar Zabel

Structural Dynamics Exercises Page 5

b

t

h

a

b 0.5

A framed machine support carries a machine that generates dynamic forces in x-direction only. The dynamic force results of an unbalanced mass of 0.5 kg that rotates with 900 rpm and a radius of 2.5 m. The respective parameters are as follows

0.5

a

0.4

y

kg E = 3.2 × 1010 N ρ = 2500 m 3 m a = 4.0 m h = 3.0 m b = 5.0 m t = 0.35 m mmachine = 10000 kg • Derive the mass and stiffness matrices!

0.4

x

• Set up the system of equations of motion for free vibration of the undamped system! • Calculate the natural frequencies (in Hz) and mode shapes of the undamped system for movements of the slab in the xy-plane! Neglect the inertia of the columns! • Sketch the mode shapes! • Calculate the maximal dynamic response in x-direction xˆ of the deck due to the excitation by the rotating unbalanced mass assuming a modal damping coefficient of ζi = 0.02!

Bauhaus-Universit¨at Weimar Institut f¨ur Strukturmechanik Dr.-Ing. Volkmar Zabel

Structural Dynamics Exercises Page 6

Suppose, you are requested to perform a numerical dynamic analysis for a two storey structure. Machinery that runs at 1400 r.p.m. is supposed to be installed in both storeys. Your analyses require the use of a complete damping matrix since you want to take into account the discrete damping properties of the machine supports. From prelimenary analyses you know the following 8 mode shapes, natural frequencies and modal damping ratios. Use the model of Rayleigh describe the damping of the structure! • Determine the weighting parameters for the mass and stiffness matrices α and β! Explain how you determined α and β and why you did it the way you have chosen! • Determine the modal damping ratios ζi for all eight modes that are represented by the Rayleigh damping model with your chosen α and β! Compare them with the damping ratios that are given in the table! f1 = 3.45 Hz ζ1 = 0.015

Z YX

f5 = 11.16 Hz ζ5 = 0.025

Z XY

f2 = 3.63 Hz ζ2 = 0.015

Z XY

f3 = 5.15 Hz ζ3 = 0.020

f4 = 10.94 Hz ζ1 = 0.025

ZY X

Z YX

f6 = 16.52 Hz ζ6 = 0.020

f7 = 22.12 Hz ζ7 = 0.022

f8 = 24.98 Hz ζ8 = 0.025

Z YX

Z YX

Z YX

Bauhaus-Universit¨at Weimar Institut f¨ur Strukturmechanik Dr.-Ing. Volkmar Zabel 2m

Structural Dynamics Exercises Page 7

x2

The following parameters are given: H

EI m

2EI H

x1

EI

• E = 3 × 1010

N m2

• I = 6 × 10−4 m4 • H = 3m • m = 18000 kg

• Develop the mass matrix [M ] and the stiffness matrix [K] for the system shown in the figure! • Develop the equations of motion for free vibration of the system shown in the figure! • Calculate the natural frequencies f1 and f2 in Hz! Results: f1 = 1.917 Hz, f2 = 5.088 Hz A machine on an existing foundation as shown in the figure is going to be replaced. About the new machine the following parameters are known:

mmach Fdyn z k

mf ound

k

• dominant frequency: 1200 rpm (rpm min−1 )

=

• expected force amplitude: Fˆdyn = 5000 N • mass of the machine: mmach = 1000 kg Furthermore the following parameters of the foundation are given: mf ound = 5000 kg, k = , allowable total load for the two springs together: max. Ftot = 120 kN , damping ratio 5 × 107 N m ζ=2% The maximal dynamic displacement amplitude of the machine foundation has to be limited to 0.1 mm. • Determine the maximal dynamic displacement amplitude of the machine foundation if its parameters were not changed! • Suggest one option to modify the existing machine foundation such that the requirements are satisfied? Show that all requirements and restraint conditions are satisfied with the changes you suggest! Results: f0 = 2.055 Hz, max. V1 = 2, for ζ = 0.02 → max. Ftot. = 106 kN

Bauhaus-Universit¨at Weimar Institut f¨ur Strukturmechanik Dr.-Ing. Volkmar Zabel

x2 EI

2m

x1 2EI

H

2EI

H

EI

xb = xb,0 sin ω t

Result: x1 = 0.714 xb,0 , x2 = 1.286 xb,0

Assume that the system shown in the figure is excited at the base by a harmonic displacement with q amplitude xb,0 and frequency

ω = m6 EI . H3 Find the amplitude of x1 and x2 relative to xb,0 in the stationary state of vibration! Damping is neglected. The stiffness matrix is given " # as: 3 −1 24 EI [K] = 3 H −1 1

2m

2m

x2

m

x1

E = 21×1010

N m2

I = 5 × 10−4 m4

EI

H = 4.5 m m = 5000 kg

H

The following parameters are given: m

EI

3 EI

H

m

Structural Dynamics Exercises Page 8

Tasks: • Develop the mass matrix [M ] and the stiffness matrix [K] for the system shown in the figure! • Develop the equations of motion for free vibration of the system shown in the figure! • Calculate the natural frequencies f1 and f2 in Hz! Results: f1 = 3.70 Hz, f2 = 13.38 Hz

Bauhaus-Universit¨at Weimar Institut f¨ur Strukturmechanik Dr.-Ing. Volkmar Zabel

Structural Dynamics Exercises Page 9

1. You are given the arrays x = [1 4 8], y = [2 1 5] and A = [3 1 6 ; 5 2 7] in MATLAB. Which of the following statements will correctly execute? Provide the results. If the command will not correctly execute, state why it will not. (a) x + y (b) x + A (c) A - [x’ y’] (d) [x ; y’] (e) [x ; y] (f) A .* 3 2. Evaluate h using the following MATLAB code snippet for a) T = 50, b) T = 7, c) T = 0 . if T < 30 h = 2 * T + 1 elseif T < 10 h = T - 2 else h = 0 end 3. Given are x = [ 4.5 ] and y = [ 5 ]. Provide the MATLAB output for the commands given below. (a) x < y (b) x == y (c) x y) & (y < x) 4. What is the major difference between a for-loop and a while-loop?

Structural Dynamics Exercises Page 10

For the system in the sketch the following parameters are given:

m EI

E = 1011 mN2

2.1 ×

H = 3m

I = −4 10 m4

1.8 ×

m = 2000 kg

Fdyn EI

H

Bauhaus-Universit¨at Weimar Institut f¨ur Strukturmechanik Dr.-Ing. Volkmar Zabel

F [kN ] 1

Tasks: • Calculate the natural frequency f0 in Hz!

0

• Derive the impulse response function h(t− τ )!

1 3

2 3

1

t[s]

• Calculate the system’s response to the dynamic force given in the diagram at time steps t = 31 , 23 , 1 s! Results: x1 = −1.50 · 10−4 mm, x2 = −6.65 · 10−4 mm, x3 = −1.55 · 10−3 mm The following parameters are given:

EI m

2 EI

x2 EI

m EI

H

2 EI

m

m EI

2 EI

• H = 6m • m = 20000 kg

x1 H

m

• for all columns: E = 3.2 × 1010 mN2 I = 2.5 × 10−3 m4

a) Develop the the mass matrix [M ] and the stiffness matrix [K] for the system shown in the figure! b) Develop the equations of motion for free vibration of the system shown in the figure! c) Calculate the natural frequencies f1 and f2 in Hz! Results: f1 = 1.74 Hz, f2 = 3.72 Hz

Bauhaus-Universit¨at Weimar Institut f¨ur Strukturmechanik Dr.-Ing. Volkmar Zabel

kd

For a tower with a TMD the following parameters are given: md

cd H = 60 m

Structural Dynamics Exercises Page 11

E, A, I, ρ

• E = 2.1 × 1011

N m2

• A = 0.2 m2 • I = 0.2 m4 kg • ρ = 7850 m 3

• md = 1600 kg a) Calculate the first natural frequency f1 of the tower (without TMD) in Hz! b) Determine the spring stiffness kd and the damping parameter cd of the TMD according to the optimal criteria developed by Den Hartog! s , cD = 2.2 kN Results: kD = 36.16 kN m m

Bauhaus-Universit¨at Weimar Institut f¨ur Strukturmechanik Dr.-Ing. Volkmar Zabel

Structural Dynamics Exercises Page 12

Consider the illustrated single storey frame structure which carries machinery that generates a harmonic dynamic force in horizontal direction. The following parameters are given: • E = 7 × 1010 Fdyn m2

• I = 3 × 10−4 m4

x

• H = 4.0 m

c

EI

H

m1 EI

N m2

• m1 = 4000 kg • m2 = 500 kg • c = 15000 Nms • Fdyn = 10 kN × sin Ω t

a) Calculate the natural frequency f0 of the system in Hz! b) Determine the maximal dynamic displacements xˆ by means of the concept of dynamic amplification for different working frequencies (or rotational speeds) of the machine. Neglect damping! Complete the following table!

rot. speed [r.p.m.]

180

240

420

600

1500

fmach. [Hz]

η

V1

xˆ [mm]

c) Give a short comment of your results with respect to the operation conditions of the machinery! Results: f0 = 7.29 Hz, xstat = 1.6 mm

Bauhaus-Universit¨at Weimar Institut f¨ur Strukturmechanik Dr.-Ing. Volkmar Zabel

Structural Dynamics Exercises Page 13

In the following lines a apart of a MATLAB file for the computation of the structural response of a multi-degree-of-freedom system by means of the Newmark-β = 1/6 method is given. Read carefully all comments that describe the variables which are given! Complete the missing command lines in the frames!