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SYNTHESIS MECHANISMS Editor-in-Chief N.I. LEVITSKII

Published for the National Aeronautics and Space Administration and the National Science Foundation, Washington, D.C. byAmerind Publishing Co. Pvt. Ltd., New Delhi

This collection of articles deals with the analysis and synthesis of plane, and spatial mechanisms. Various theorteical and practical problems associated with the design of plane mechanisms as well as of adjustable lever and cam-lever mechanisms are considered. A major part of the book is devoted to articles in which the problems of kinematic and dynamic analysis of cam mechanisms as well as problems of selecting optimum laws of movement of driven links are solved. Mechanisms of certain machines and instruments, e.g.,radialpiston multipath hydromotors, molding presses, pneumatic-hydraulic drives for stopcocks and electrical switches are also described in this collection. The material presented in this book is intended for scientists and engineer-technicians engaged in the computation and design of mechanisms.

TT 71-58004 NASA TT F-680

AKADEMIYA NAUK SSSR Otdelenie mekhaniki i protsessov upravleniya Gosudarstvennyi Nauchno-issledovatel'skii Institut Mashinovedeniya ACADEMY OF SCIENCES OF THE USSR Department of Mechanics and Control1 Processes State Scientific-Research Institute of Machine Design

ANALYSIS AND SYNTHESIS OF MECHANISMS ' (Analiz i Sintez Mekhanizmov)

Editor-in-Chief N. I. Levitskii

Nauka Publishers Moscow 1970

Translated from Russian

Published for the National Aeronautics and Space Administration and the National Science Foundation, Washington, D. C. by Amerind Publishing Co. Pvt. Ltd., New Delhi 1975

Editorial Board: Academician I. I. Artobolevskii (Chairman), G. G. Baranov, A. P. Bessonov, V. A. Gavrilenko, E. V. Gerts, V. A. Zinov'ev, A. E. Kobrinskii, L. V. Petrokas, N. P. Raevskii, M. A. Skuridin, A. V. Shlyakhtin

1975 Amerind Publishing Co. Pot. Ltd., New Delhi

Translated and Published for the National Aeronautics and Space Administration, pursuant to an agreement with the National Science Foundation, Washington, D. C. by Amerind Publishing Co. Pvl. Ltd., 66 Janpath, New Delhi 110001

Translators: S. K. Gael & K. L. Awasthy General Editor: Dr. V. S. Kothekar

Available from the U. S. Department of Commerce National Technical Information Service Springfeld, Virginia 22161

Printed at Prem Printing Press, Lucknow, India Production: M. L. Gidwani

UDC 62.23

The manual deals with the analysis and synthesis of plane, spatial and spherical mechanisms, problems of selection of optimal laws of movement of links of cam mechanisms. • ' • Material presented in the manual is intendedfor scientists and engineer-technicians engaged in the computation and design of mechanisms.

FOREWORD This collection of articles deals with the analysis and synthesis of plane and spatial mechanisms. Various theoretical and practical problems, associated with the design of plane mechanisms as well as of adjustable lever and cam-lever mechanisms, are considered in the articles of N. I. Aleksishvili, Yu. M. Zingerman, P. G. Mudrov and V. I. Kulyugin and others. A major part of the book is devoted to articles (K. V. Tir and D. N, Senik, E. N. Dokuchaeva, Yu. V. Epshtein and V. A. Novgorodtsev and others) in which the problems of kinematic and dynamic analysis of cam mechanisms as well as problems of selecting optimum laws of movement of driven links are solved. Synthesis of automats using photoelectronic devices is described in the articles of B. N. Sklyadnev et al. The articles by L. B. Maisyuk, A. E. Kropp and V. S. Karelin describe the synthesis of complex cam-planetary-connecting rod and hingedtoothed mechanisms. Nomographic methods of synthesis of hinged lever and cam mechanisms are covered in the articles of L. P. Storozhev and M. M. Gernet. Mechanisms of certain machines and instruments, e.g. radial-piston multipath hydrometers, molding presses, pneumatic-hydraulic drives for stopcocks and electrical switches, are described in articles by A. S. Gel'man et al., D. M. Lukichev et al., M. S. Rozovskii et al. and A. V. Sinev et al.

CONTENTS Foreword .. .. .. v Kinematic Analysis of the Steering Linkage of an Automobile— JV. I. Aleksishvili .. .. .. .. 1 New Methods of Synthesis of Mechanisms for Reproducing and Enveloping Curves—E. G. Berger .. .. ' .. 10 Determination of Momentum in Three-Link Mechanisms with a Higher Pair of the 4th Class—L. JV. Borisenko .. .. 21 A Method of Calculating the Change in Primary Dimensions of Mechanisms Due to Wear for the Purpose of Estimating Their Reliability—E. L Vorob'ev .. .. .. . . 3 1 Calculation of Kinematic Parameters of Radial-Piston Multipass Hydraulic Motors with Guide Block Profile Consisting of Arcs of Circles—A. S. Gel'man, Tu. A. Danilov, L. V. Krymova and A.-M.Makeev . . .. .. .. . . 4 1 Nomographs for Selecting Optimal Laws of Motion of Driven Links of Cam Mechanisms—M. M. Gernet .. .. 52 Design of a Four-Link Mechanisms for Reproducing a Given Motion—JV. M. Guseinov and S. I. Gamrekeli .. 60 Synthesis of Cam Lever Mechanisms for Different Types of Motions of Driven Links—R. P. Dzhavakhyan .. .. . . 7 1 Effects of Errors in the Working Profile of the Cam on Velocity and Acceleration of the Follower—E. JV. Dokuchaeva - .. 88 Kinematic Study of Spatial Mechanisms by the Technique of Vector Analysis—Tu. M. ^ingerman .. .. 102 A Method of Analytical Synthesis of Plane Toothed Lever Mechanisms— V. S. Karelin .. .. .. ..112 Two Simple Methods of Regulating Motion of the Driven Link in Three-Dimensional Single Contour Mechanisms—A. A. Kasamanyan .. .. .. .. ..132 Synthesis of a Toothed Lever Transmission Mechanism—A. E. Kropp 143 Design of Three Dimensional Four-Link Mechanisms Conforming to the Travel of the Driven Link and Coefficient of Increase in Velocity of the Reverse Stroke—V. I. Kulyugin .. ..152 Classification of Slotted Bar and Lever Mechanisms of Discontinuous Motion—P. G. Kukharenko .. .. 163

viii

Contents

Peculiarities of the Lagrangian Method in the Synthesis of Mechanisms—N. I. Levitskii and Yu. L. Sarkisyan .. 180 Design of Cam'Elements of Electrical Switches—D. M. Lukichev, V. A. Nikonorov and £. S. Gazizova .. .. ..188 Synthesis of Cam-Planetary-Connecting Rod Mechanism with Reverse Stroke and Stop—L. B. Maisyuk .. ..196 Determination of the Zones of Existence of Slotted Lever Mechanisms— V. A. Mamedov .. .. .. .. 207 Three-Dimensional Five-Link Hinged Mechanisms—P. G. Mudrov .. 213 Kinematic Study of Three-Dimensional Three-Link Lever Mechanisms Using Analytical Methods—£. S. Natsvlishvili.. .. 220 An Approximate Method for Synthesizing a Hinged Lever Amplifying Mechanisms of a Molding Press—M. S. Rozovskii, E. N. Dokuchaeva and V. G. Lapteva .. .. ..233 Dynamic Force Analysis of Plane-Mechanisms in Three-Dimensional Space—K. F. Sasskii .. .. .. 246 Analysis of the Structural and Technical Errors of Six-Link Mechanisms in the Dwell Zone—V. I. Sergeev, S. A. Cherkudinov and /. G. Oleinik .. .. .. .. ..262 Fundamentals of the Theory of Dynamic Synthesis of Cam Mechanisms—P. V. Sergeev .. .. .. .. 277 Kinematic and Force Analysis of Spatial Mechanism of Pneumatic and Hydraulic Drive for Stopcocks of Gas Mains of Type DU 1000—A. V. Sinev, I. £. Brodotskaya and I. S. Charnaya . . 284 Design of Automats with Photoelectronic Devices for the Control and Measurement of Linear Dimensions and Areas—B. N. Sklyadnev, B. N. Yurukhin, Yu. I. Evteev and E. I. Astakhov .. 296 A Nomographical Method for the Synthesis of Multiple Contour Plane Hinged-Lever Mechanisms—L. P. Storozhev . . .. 309 Techniques for Synthesizing Adjustable Lever Mechanisms— B. S. Sunkuev .. .. .. .. ..323 An Approximate Technique for the Synthesis of Disk Cam Mechanisms—A". V. Tir and D. N. Senik ,. .. . . 334 The Problem of Existence of the Crank in the Spatial Four-Link" Mechanism with Two Turning and Two Spherical • Pairs —K. A. Tonoyan .. .. .. .. 349 Functions with Minimum Deviations from Zero in Problems of Synthesis of Cam Mechanisms—Yu. V. Epshtein and -V. A. Novgorodtsev .. .. .. .• • • 366 Abstracts .. ' .. .• •• .. 378

JV. /. Aleksishvili KINEMATIC ANALYSIS OF THE STEERING LINKAGE OF AN AUTOMOBILE

The steering linkage installed on all existing models of automobiles with one-piece front axles is a spatial four-link mechanism [1]. Conforming to the generally accepted classification [2], such a mechanism has two rotating pairs of the 5th class (forked turning cams, connected to the shaft of the front axle with the help of steel cotter pins) and two spherical joints of the 3rd class. The latter are ball joints at the junction of turning cam levers with the transversal rod of the steering linkage. There are no common connections affecting the motion of links of such a mechanism and therefore, it belongs to the category of mechanisms of the zeroth family. The degree of mobility of a mechanism of the zeroth family is in general expressed by the following equation:

where n is the number of mobile links and pt is the number of kinematic pairs whose class is indicated in the form of index i'= 1, 2, 3, 4 and 5. Using this equation, the degree of mobility of the steering linkage of an automobile is given by : (0=6x3-5x2-4x0-3x2-2x0-0=2, i.e., there is a surplus degree of mobility due to the possibility of rotation of the transversal link AC of the linkage about its longitudinal axis (Fig. 1). . This motion however does not take part in the functioning of the linkage as a whole, i.e., actually the mechanism has one degree of mobility.

JV. I. Aleksishvili

The presence of ball joints gives -rise to a number of difficulties in the manufacturing process as well as during the operation of the steering linkage. Firstly, fabrication of spherical surfaces requires special tools and fixtures. Secondly, inevitable deviations of the ball joint from the drawing dimensions give rise to intensive wear and tear and an increase in the play between the mating surfaces. Attempts to compensate for this wear and tear with the help Fig. 1. of adjustable spring inserts (automobiles MAZ-200, ZIL-164, UAZ-69, etc.), or the insertion of hemispherical pins (GAZ-51A, URAL355M, etc.) complicate the design and operation of the ball joint. The above problems initiated research into the possibility of replacing ball joints by spherical ones (for example, in automobiles GAZ-63 and others). However, the necessary condition—intersection of the axes of all the pairs of steering linkages at one point—was not fulfilled for cylindrical pairs in any of the proposed designs of such joints. As a result, there was additional wear and tear of mating surfaces and other undesirable effects which forced designers to pursue a more complicated and expensive method of installing spherical joints. It should be stated at this point that the condition for intersection of all cylindrical pairs at one point is a purely theoretical requirement. In practice, this point occupies a volume constrained by conical surfaces. These surfaces are obtained as a result of possible angular displacements of the axes of cylindrical pairs in clearances permissible between the mutually rotating elements of these kinematic pairs—sleeves and pins. Therefore, the farther the theoretical point of intersection of axes and the larger the permissible clearance, the greater will be the value of the above mentioned volume as well as the limits of possible deviations from drawing dimensions of the links of the steering linkage. Let us examine closely die condition of substitution of spherical pairs of the steering linkage of an automobile by cylindrical (turning) pairs. The steering linkage of an automobile is represented in Fig. 1 in the form of a four-link mechanism ACO'O. The following symbols refer to Fig. 1:

Kinemetic Analysis of Steering Linkage

3

S — length of the transversal rod of the steering linkage, AC=S; Ri=R T =R — length of the lever of the rotating cam along a straight line perpendicular to the axis of the pin, OA=R\; 0'C=Rr; d — distance between the projections of the. ends of levers of turning cams on the pin axis, O0'=d; 0*— point of intersection of the pin axis inclined at an angle of2p; 00 — angle between the coordinate axis OX and lever of turning cam OA in the case of straight motion of the automobile; 61 — angle of rotation of the lever of turning cam,'facing the internal side of the trajectory of the automobile on a turn; Be — the same angle but toward the outer side of the automobile on a turn (in the case of straight motion of the automobile 0 e =0 1= 0); (J — angle of transversal inclination of the pin of the turning cam. Cartesian coordinates XY£ are selected in such a manner that the axis 0£ is directed along the pin axis, center 0 lies at the intersection of straight lever of the linkage with the axis of the corresponding pin and the axis OT lies in the plane, passing through points 0 and 0' perpendicular to the plane of linkage in the case of neutral (straight) position of the wheels of the automobile. Let us examine some of the kinematic problems of this mechanism. In A^00 2 and ACO'02, the sides CO'=RT and AO=R\ are equal in accordance with the requirement of similar kinematics of turning of the automobile toward left as well as toward right." The sides 00Z=0'0Z, since &00'02 is an isosceles triangle in view of the equality of angles , at the base Thus, in /^AOO^ and C0'02, two sides and the angle between them (Z.CO'02=,/^002=const) are equal, i.e. &A002= ACO'02. The last equality leads to the conclusion that 0ZA=0ZC. Since this expression is valid for an arbitrary position -of the links of the steering linkage of an automobile, by taking point 02 as the center of sphere of radius r= OtA=02C, it can be shown that points A and C remain on the s'urface of this sphere in the case of motion -of the elements of this fourlink mechanism in any direction. Thus, the steering linkage of an automobile is a spherical mechanism [3]. It also follows from the equality 02C=OaA that /\0 2 AC is an isosceles triangle, i.e. angles at the base, Z_0ZAC and /_02CA are equal. Since this equality is valid for all link positions of the linkage, the spherical pairs A and C can be replaced by cylindrical ones with axes intersecting at point, 02. The position of the axes of rotating pairs with centers , at; points A and C is denned by the angles X and A (see Fig. 1). J.

JV. /. Aleksishuili From the right-angled ^AOOZ, tan f.= 002IR, while in right-angled Substituting the last equality 2, hypotenuse 0O 2 =rf/(2 sin p). in the expression for X, we get the following value for this angle: X=arctan d/(2 sin p), where d=d/R Angle A is determined from the

(1)

right-angled

cos *=AD/A02 , where AD=S/2; A02=R/cos X, the final expression for A reduces to: \=arccos

S cos

(2)

where S=SJR. Because of symmetry of the steering linkage of an automobile, equations (1) and (2) are also valid for the rotating pair at point C. It should be noted that the parameters d and S are mutually related by the requirement of optimum kinematics of the steering linkage of an automobile [1] and consequently, the angles X and A in the general case are functions of {J, d (or S) and P/L where P is the distance between the points of intersection of the axis of pins of turning cams with the plane of the track and L is the base of the automobile. The relationship between 90°—X .. and the parameters d and [J are 90'-X^,deg\. graphically shown in Fig. 2. As is obvious from these graphs, the angle X deviates considerably from 90° as angle p increases and the deviation is significant at small values of d. At p=0°, angle x=A=90°, i.e., the steering linkage functions as a twodimensional four-link mechanism. The functional relation of angle 90°—A with parameters d and (J is shown in Fig. 3. Here, parameter d 4° is related to S by the requirement of minimum deviation of the dependence of the actual angles of turning of the steering (front) wheels of the automo^ * bile [1] from the equation: Fig. 2.

Kinematic Analysis of Steering Linkage

5

cot 6e=cot Fig. 3 shows that the angle A tends to 90° as decreases and PjL increases. This rule holds strictly for decreasing 0',

0;

0; yo'—y0; Jo-yc yo'-y^,

J>A—J>C;

0; 0;

0;

s \.

0

^

yA-yo'; 0 0; yc-yo>,

= (0),

"9

(5)

8

JV. /. Aleksishvili

where (0) is the null matrix: « =

Similarly for the second system of equations: ZA—ZC; ZC—Z Q ; Z O —ZA; 0; X O —X A ; ZA—Zo", ZQ'—z0; 0; z 0 —ZA; x0—XA ; 0; Zo'—z0; z0—Zc ; 0; ; Zc—ZQ",

xc—x0 ;

0;

0; xo'—x 0 ; * 0 —x c ; x 0 '—A: O ;

0; 0;

\

0; XA—XO'; 0 0; xc—xo'j

= (0),

(5a)

\ where (0) is the null matrix:

4*0 • 4*0';

»]0=

As the rank of a matrix composed of the coefficients of the system of equation (5) [similarly for system (5a)] is less than the number of unknown variables, equations (5) and (5a) have several basic systems of solutions. However, if the direction of axes of any two kinematic pairs of the linkage is specified, equations (5) and (5a) make it possible to determine the only one possible direction of the axes of all the four pairs at which this mechanism can function. Generally this solution does not give values of angles X and A which coincide with theoretical values calculated from equations (1) and (2). Varying the positions of the two given axes, it is easy to achieve minimum deviation of these angles from their theoretical values. The direction cosines are related to each other by the equation:

(6)

Kinemetic Analysis of Steering Linkage

9

The clearance A restricts the possible change of only one of the three directional cosines while the values of the remaining two should be selected in such a manner that the two selected axes intersect in space. The permissible clearances, at which there-will be minimum deviation of the values of angles X and A from theoretical values, will be the minimum possible required to retain the efficiency of the automobile steering linkage considered as a four-link spatial mechanism. Since numerous calculations are possible, they should be done on highspeed electronic computers using standard programs for solving determinants. In this case, the results obtained serve as reference material for designers and can be drawn up in the form of tables or graphs.

REFERENCES 1. DVALI, JR.. R. and N. I. ALEKSISHVILI. Kinematika rulevoi trapetsii avtomobilya (Kinematics of the steering linkage of an automobile). Soobshch. AN GruzSSR, XLIV, No. 3, 1966. 2. ARTOBOLEVSKII, I. I. Teoriya mashin i mekhanizmov (Theory of Machines and Mechanisms). Moscow—Leningrad, GITTL, 1952. 3. TAVKHEUDZE, D. S. K voprosu sinteza i kinematiki prostranstvennykh chetyrekhzvennykh mekhanizmov (Problems of Synthesis and Kinematics of Spatial Four-Link Mechanisms). Tbilisi, Izd-vo Tsodna, 1960.

E. G. Berger NEW METHODS OF SYNTHESIS OF MECHANISMS FOR REPRODUCING AND ENVELOPING CURVES

Modern methods of synthesis of mechanisms, designed for reproducing and enveloping plane curves are based either on different geometrical transformations (projections, cissoidal, [m-n] marking, inversion, etc.) or on preliminary determination of the properties of curves leading to their construction [1-3]. However, the problem of determining the properties of a curve from its equations or of finding out transformations from which the curves can be obtained without using the general methods of solving the equation, is extremely time consuming and complicated. The methods which follow are helpful in designing mechanisms for constructing curves directly from their equations without taking into account the properties of the given curves. Mechanisms used in reproducing and enveloping curves are commonly found in subassemblies of various automatic machines and in precision instruments and calculators. Mechanisms whose rectilinear links are used in1 enveloping particular curves are of special importance as their working component, which can take the form of a rack, hobbing cutter or a wide cutter, facilitates machining of profile surfaces by the method of generation [4, 5]. 1. Reproduction of Curves

1. T h e o r y o f t h e m e t h o d : The method is based on the theory of plotting curves of the type: F( X ,y)^Q

(1.1)

as the geometrical location of intersection of corresponding curves belonging

New Methods of Synthesis of Mechanisms

11

to the design' groups : - A (*,y, *)=0,. . ., (a) ' and/ 2 (*, y, A)=0,. . ., (b).

[6]

A technique of synthesizing mechanisms directly from the equations of the curves without going into the laborious examination of their properties is the main feature of this article. The equivalence between the curves of groups (a) and (b) is determined in the following manner. Initially the equation of one of the groups, for example (a), is selected arbitrarily. Then, the equation of the second group (b) is found out by simultaneously solving the equations (1.1) and (a). Eliminating parameter A from equations (a) 'and (b) results in the initial equation (1.1). Consequently, the points of intersection of curves (a) and (b) at A=Ai, A2, A 3 ,- • • , lie on the given curve (1.1). Selecting different variables such as an angle or its trigonometric function, a segment of a straight line, etc. as A, we define the method of plotting the' curves (a) and (b) respectively. Sets of straight lines and circles can be taken as the design groups for curves of all orders [7] and in many cases this 'simplifies the process of plotting these curves. Considering the structure of these graphical representations as the abstract geometrical drawing of the mechanism to be designed, we fabricate it by replacing the points by rollers or slide blocks and straight lines by rocker arms, linear guides or by cranks. 2. Let t h e . c o n i c a l s e c t i o n s be g i v e n by the equation (l-sV.

(1-2)

To synthesize a conicograph, let us arbitrarily select the following group of straight lines passing through the origin as one of the design groups : y=Xx.

(1.3)

On simultaneously solving the expression (1.2) and (1.3) we get the equation of the second group : (*+0/(*+0 = 0'-*/A)/(-//A),

(1.4)

where /=2//e2; k=2p/(\- s z ). Equation (1.4) represents a group of straight lines passing through two points with coordinates (k, 0) and (— /, //A). On the basis, let us determine the method of plotting the groups of straight lines (1.3) and (1.4) at variable A=tan tp (Fig. 1, a). The straight lines AN, passing through a fixed point A (k, O) and point N (—1, //A) of intersection of straight line BL(x=—l) with the line ON A. OM, corresponds

12

E. G. Burger

to the arbitrary ray OM of group (1.3). The points of intersection of straight lines OM and AM lie on the conic sections (1.2). A universal conicograph can be fabricated from this drawing. For plotting ellipses, the slide block A of the rocker arm AJV should be fixed on the right of the origin 0 at a distance k, since when sQ. For drawing hyperbolas (e>l, £