This is an authorized facsimile and was produced by microfilm-xerography in 1969 by University Microfilms, A Xerox Company, Ann Arbor, Michigan, U.S.A.

This dissertation has been microfilmed exactly as received 67-17,846 TIMM, Robert Frederick, 1931SIMULATION OF PLANAR MECHANICAL LINKAGE SYSTEMS. The University of Michigan, Ph.D., 1967 Engineering, mechanical University Microfilms, Inc., Ann Arbor, Michigan

SIMULATION OF PLANAR MECHANICAL LINKAGE SYSTEMS Robert Frederick Timm A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the University of Michigan 1967 Doctoral Committee: Professor Joseph E. Shigley, Chairman Associate Professor Francis E. Fisher Professor Donald T. Greenwood Assistant Professor Alexander Henkin Professor J. Raymond Pearson

SIMULATION OF PLANAR MECHANICAL LINKAGE SYSTEMS Robert F. Timm ABSTRACT As a simulator, the modern electronic analog computer can be a powerful and highly flexible tool in studies involving mechanical engineering design; however, potential applications in many areas of mechanical engineering remain unexplored. It has been the purpose of this study to broaden the scope of applications by developing techniques for simulating mechanical linkage systems. The analog simulation yields all kinematic and dynamic information for a linkage system with either a prescribed motion input, as with a mechanism; or with a prescribed force or torque input, as with a machine. Parameters may be easily varied, and variables may be recorded or displayed with the investigator in immediate and direct control of optimization. Generalized procedures are described for devising computer circuits on a modular basis for linkage systems with rigid elements. Examples are given with derivations of the mathematical models and the computer circuits along with samples of computer records. The four-bar crank and rocker linkage is simulated as a mechanism with a prescribed motion input. The kinematic simulation provides the investigator with a visual display of the mechanism, with the ability to make displacement records of the motion of any point on the mechanism (the coupler curves being of prime interest), and to simultaneously record the angular velocities, and accelerations of all links. In addition, the absolute velocity or acceleration of any point may be recorded.

The simulation for dynamic studies yields records of forces at the pin connections due to inertial and external loading. The four-bar linkage is simulated for dynamic analysis with the crank driven at a constant angular velocity. The slider-crank is simulated as part of a single cylinder engine operating on a two-stroke cycle. All kinematic and dynamic information is provided as the engine is brought up to speed. Because links possessing finite elasticity may be treated with varying degrees of approximation, a generalized procedure is not described. The case of a four-bar linkage (driven at a constant angular velocity) with an elastic coupler, and rigid crank and rocker links, is simulated with elastic motions derived as perturbations from the rigid-body case.

ACKNOWLEDGEMENTS The author is very grateful for the interest and counsel of his thesis committee, and is especially grateful to the chairman, Professor Joseph E. Shigley. The helpful suggestions of Mr. Theodore Oliver are also appreciated. This work was made possible with the support of the National Science Foundation and the Ford Foundation, and sincere thanks is extended to these organizations.

TABLE OF CONTErS ACKNOLEDGEMENTS................................................ LIST OF FIGURES.................................................. Chapter 1 THE STUDY OF MECHANICAL LINKAGE SYSTES.................. 1.1 Problems in the Study of Linkage Systems............ 1.1.1 Mechanisms................................... 1.1.2 Linkage Systems as Machines.................. 1.2 Historical Survey of the Role of Models in Mechanical Analysis and Synthesis................... 1.3 Current Techniques in the Study of Mechanical Systems............................................. 2 OUTLINE OF TECHNIQJES FOR SIMULATIG PLATAR LINKAGES ON THE ELECTRONIC DIFFERENTIAL ANALYZER..................... 2.1 Modular Construction of Analog Computer Circuits for the Simulation of Linkages...................... 2.2 Trigonometric Resolution Using Electronic Multipliers......................................... 2.3 Scaling............................................ 3 THE SIMULATION OF PLANAR RIGID LIE= MECHANISMS FOR KINEMATIC ANALYSIS...................................... 3.1 The Four-Bar Linkage................................ 3.1.1 Simulation for Velocities.................... 3.1.2 Displacements and Display Techniques......... 3.1.3 Accelerations................................ 3.2 The Slider-Crank................................... 3.2.1 Velocities.................................. 3.2.2 Displacements and Display of the SliderCrank..................................... 3.2.3 Accelerations............................... Page ii v 1 1 2 6 8 12 18 21 25 27 27 27 39 46 53 53 55 61 iii

TABLE OF CONTENTS (CONT'D) 3.3 Scaling.................................... 63 Chapter 4 SIMULATION FOR DYNAMIC ANALYSIS OF LINKAGES WITH RIGID ELEMENTS......................................... 66 4.1 Four-Bar Linkage Dynamics.......................... 66 4.2 Simulation of a Single. Cylinder Engine............. 78 5 THE SIMULATION OF LINKAGE SYSTEMS WITH ELASTIC LINES.... 98 5.1 An Elastic Link With A Lumped Mass................. 98 5.2 Distributed Mass Linkage Analysis................. 103 5.2.1. Simulation of the Elastic Coupler of a Four-Bar Linkage............................ 114 BIBLIOGRAPHY.................................. 129 APPENDIX 1 DIGITAL COMPUTER SOLUTIONS............................. 132 2 CALCULATIONS OF FOURIER COEFFICIENTS................... 135 iv

LIST OF FIGURES Figure Page 1.1 Schematic of the Four-Bar Linkage................... 3 1.2 Schematic of the Slider-Crank............................ 7 2.1 Flow Diagram Showing the Modular Construction of Computer Circuits......................... 20 2.2 Rate Resolver Circuit................................. 20 2.3 Rate Resolver Circuit With Error Generation and 20 Corrective Feedback............................... 3.1 Four-Bar Linkage Showing Length and Angle Parameters for Kinematic Analysis............................ 28 3.2 Velocity Polygon for the Four-Bar Linkage............... 31 3.3 Initial Configuration of the Four-Bar Linkage............ 32 3.4 Circuit for the Four-Bar Linkage Velocities.............. 34 3.5 Records of 4 and...................................... 35 3.6 Circuit for Deriving Polar Plots of the Absolute Velocities of Points A, B, and C on the Four-Bar Linkage................................................. 37 3.7 Polar Plots of Velocities of Points A, B, and C on the Four-Bar Linkage..................................... 38 3.8 Circuit for Displacements of the Four-Bar Linkage........ 40 3.9 Schematics Showing Points Which Generate Coupler Curves Associated With the Movement of Link f2.......... 42 3.10 Four-Bar Linkage. Circuits for Coupler Curve Generation................. 43 3.11 Four-Bar Linkage Coupler Curves. a) Point C. b) Point D................................. 44 3.12 Sweep Frequency and Gate Signal Generators With a Gate Circuit For the Oscilloscope Display of the Four-Bar Linkage...................................... 47 v

LIST OF FIGURES (CONT'D) Figure Page 3.13 Circuit and the Oscilloscope Display of the Four-Bar Linkage.......,........................................ 48 3.14 Acceleration Polygon for the Four-Bar Linkage............ 50 3.15 Circuit for the Accelerations of the Four-Bar Linkage................................................. 51 *a ~** G2 2 3.16 Records of (, A, a and p of the Four-Bar Linkage......... 52 5.17 Schematic of the Slider-Crank........................... 54 3.18 Circuit for the Velocities of the Slider-Crank........... 56 ~ ~ 3.19 Records of YB and p For a Constant Angular Velocity of the Crank of the Slider-Crank Mechanism,............. 57 3.20 Circuit for Generating Rectangular Components of Points on the Connecting Link of the Slider-Crank............ 58 3.21 Coupler Curves of the Slider-Crank, a) Point C. b) Point D............................................ 59-60 3.22 Circuit for the Oscilloscope Display of the SliderCrank................................................ 62 3.23 Circuit for the Accelerations of the Slider-Crank........ 65 3.24 Records of B and p of the Slider-Crank with a =.628 radians/second.................................. 65 4.1 Schematic of the Four-Bar Linkage Showing Parameters for a Dynamic Analysis.......................... 67 4.2 Four-Bar Linkage Free-Body Diagram................. 68 4.3 Four-Bar Linkage. Link 12 Showing the Relative Acceleration Components of m2......I..*.................. 70 4.4 Circuit for the Dynamics of the Four-Bar Linkage......... 72 4.5 Polar Plots of the Forces at Pins A and B of the Four-Bar Linkage....................................... 74 vi

LIST OF FIGURES (CONT'D) Figure Page 4.6 Computer Record of the Torque at the Crank of the Four-Bar Linkage.................................... 75 4.7 Circuit for the Pin Reactions at the Ground Link of the Four-Bar Linkage....................... 76 4.8 Polar Plots of the Forces at Pins 01 and 03 of the Four-Bar Linkage.......................................... 77 4.9 Single Cylinder Engine Schematic.......................... 79 4.10 Slider-Crank Free-Body Diagram............................ 80 4.11 Circuit for the Generation of Gas Force Fe.......... 83 4.12 Circuit for the Rectangular Components of Acceleration of m2 on the Connecting Rod.............................. 84 4.13 Circuit for Forces FAx, FAy, FBX, and FBy............. 85 4.14 Circuit for Ts, a, and a of the Single Cylinder Engine.... 86 4.15aRecords of Fe, Ts, a, a, A, A, YB, and ay as the Engine Comes Up to Speed.............................. 87 4.15bRecords of Fe, Ts,, &, a,, B and Yg at Full Speed..................................................... 88 4.16 Polar Plots of Forces at Point B a) First Cycle b) All Cycles as the Engine Comes Up to Speed c) One Cycle at Full Operating Speed................... 89 4.17 Polar Plots of Forces at Point A a) First Two Cycles (Inner Loops) and a High Speed Cycle (Outer Loop) b) Forces at Point A as the Engine Comes Up to Speed....... 90 4.18 Oscilloscope Photographs of Bearing Forces............... 92 4.19 Engine Mounting................ e... 95 4.20 Circuits for the x', y', and ( Motions of the Crankcase........,,...................0.......... 96 4.21 Records of x', y', and 4 as the Engine Comes Up to Speed..................................................... 97 vii

LIST OF FIGURES (CONT'D) Figure Page 5.1 Elastic Links With Bending Loads.......................... 99 5.2 Elastic Link With Beam-Column Loading.................... 103 5.3 Differential Element of a Link With Beam-Column Loading................................................... 103 5.4 Rigid Link Suspended as a Pendulum........................ 106 5.5 Four-Bar Linkage With Force Components at Pin B........... 115 5.6 Acceleration Components at Point A of the Four-Bar Linkage................................................. 118 5.7 Circuit for the Perturbed Motions of the Elastic Coupler of a Four-Bar Linkage..................................... 121 5.8 Photographs of Individual Modes and Combined Modes of Elastic Link Vibrations................................. 126 5.9 Records of T1, T2 and T3.............................. 128 viii

CHAPTER 1 THE STUDY OF MECHANICAL LINKAGE SYSTEMS The study of mechanical linkage systems entails many facets of investigation including both analysis and synthesis, with the mechanical elements considered either rigid or elastic. The investigation may be concerned with purely kinematic considerations, it may entail questions of the dynamic aspects of system performance, or it may encompass both of these problems. To some degree, mechanical systems have been categorized according to the type of system or to the type of information which is desired about the system. In attempting to use the electronic analog computer to aid in the study of linkage systems it is first necessary to recognize the categories and to define the problems. It is also helpful to examine, from an historical viewpoint, the methods that have been developed to solve these problems. 1.1. Problems in the Study of Linkage Systems A linkage system is defined as an assemblage of mechanical elements, or links, which are connected so as to produce a desirable effect through constrained motion. One link is fixed and, in the analysis, constitutes the inertial frame of reference. Two broad categories of mechanical systems which can be distinguished are those of the "mechanism" and the "machine". Linkage systems are ordinarily thought of as mechanisms; however, this need not always be the case. When the input to a mechanical system is a prescribed motion of some -1

-2 element, which, in turn, sets the entire system in motion, the system is referred to as a mechanism. When the motivating input is a prescribed force or torque, the system is called a machine. Problems in the study of linkage systems are described below, with those problems generally associated with mechanism study distinguished from those of machines. As will be seen, some overlap in desired information can occur. 1.1.1. Mechanisms The study of mechanisms can be divided into three areas of investigation; kinematic analysis, kinematic synthesis, and dynamic analysis. The general planar four-bar crank and rocker linkage is used here to describe typical problems. Length and angle parameters are illustrated in the schematic shown in Figure l.l1The linkage consists of a crank 1, a coupler l2, a follower or rocker L3, and a fixed frame with pin connections at 01 and 03 separated by the distance I. 0 Kinematic Analysis A typical kinematic analysis problem is one in which the dimensions of the links are given together with the position, angular velocity, and angular acceleration of the driving crank. The links are assumed to be rigid. The kinematic problems to be solved are as follows. 1. Determine the angles 4 and 3 as functions of time, and the path of any point attached to link 12, These paths are called coupler point curves.

y I I! x 0 / Figure 1.1. Schematic of the Four-Bar Linkage.

2. Determine the angular velocities of links 12 and 13 and the absolute velocity of any point on the linkage. 3. Determine the angular accelerations of 12 and l3 and the absolute acceleration of any point on the linkage. Absolute accelerations are the accelerations referred to the fixed frame and are ordinarily determined in terms of components which are normal and tangential to the path traversed by the point. Kinematic Synthesis A more difficult problem is that of the kinematic synthesis of:mechanisms. One aspect of this problem deals primarily with displacement considerations, although efforts have been reported which establish limited velocity or acceleration criteria.l Typical displacement problems are: 1. Determine the link lengths which will produce a prescribed functional relationship between a and 4 2. Determine the link lengths which will cause a point, ordinarily on the coupler link, to generate a prescribed path. A great variety of other synthesis problems are encountered involving forces in the links, transmission angles, the creation of dwells, etc. 1C.W. McLarnan, Analytical Synthesis of Function Generators Using the Slider-Crank Inversion, Presented at the Winter Annual Meeting of the American.Society of Mechanical Engineers, New York, November 29 to December 3, 1964. ASME Paper No. 64-WA/MD-13.

-5 Dynamic Analysis In mechanism studies, the emphasis is placed upon kinematic considerations; indeed, the science of kinematics evolved through efforts to derive certain information about a system without involving the complicating effects of dynamics. Nevertheless, in the practical problems of mechanical design and fabrication, a dynamic analysis, which includes the effects of external loading as well as inertial reactions, yields information as to the magnitude of bearing loads at the pin connections, and the nature of the forces transmitted to the ground &ink. In such a problem the link lengths along with their gravity centers, masses, and radii of gyration are specified. The input motion at the driving crank is prescribed and the following problems are to be solved. 1. Determine the magnitudes and directions of forces at the pin connections. 2. For the prescribed motion input, determine the torque reaction encountered at the input crank. In the problems discussed above, there has been traditionally implicit in the analysis or synthesis, the assumption of the rigidity of the links. With the raising of performance standards and increasing speeds of operation, the effects of finite elasticity of the links is becoming more important. If a linkage is synthesized to provide a specified coupler point path, finite elasticity could lead to high frequency deviations, which under extreme operating conditions would render the design inadequate. Another possibility is that of operating under conditions of elastic instability which could lead to total collapse or breakup of the linkage.

1.1.2. Linkage Systems as Machines The slider-crank linkage of Figure 2 comprises the basic system of reciprocating machinery. For purposes here it is regarded as an integral part of a single cylinder internal combustion engine. The system consists of a frame of mass mo, a piston of mass m3, a connecting rod of length 12 and mass m2, and a crank of length 11 and mass mi. Radii of gyration are specified for m, mI, and m2. Some form of engine mounting is specified along with the input force Fe, and a load torque TL applied at the flywheel. The problems of kinematic analysis, kinematic synthesis, and dynamic analysis, as with mechanisms, also apply here with, however, a shift in emphasis. The problems to be solved are as follows. 1. Determine the kinematic variables for each element of the system. 2. Determine the force reactions at the pin connections. 3. Evaluate the vibration response of mo and the mounting system as the engine is operated. A systematic study of the aforementioned problems began in the eighteenth century, and historically, in the analysis of mechanical systems, and particularly in the synthesis of linkage systems, the investigator has relied heavily upon some form of model to enhance his analytic and perceptive powers. In a modern interpretation of the term "model", a computer simulation of a mechanical system constitutes a model of that system. A consideration of early modeling methods and associated analytical and graphical techniques provides insight into the various approaches currently employed in dealing with the problems of mechanical analysis and synthesis.

mO FLYWHEEL Figure 1.2. Schematic of the Slider-Crank.

-8 1.2. Historical Survey of the Role of Models in Mechanical Analysis and Synthesis In modern technology, the term "model" is interpreted in many ways, and the model itself can assume many forms ranging from a physical miniature to an abstract mathematical formulation. Its purpose too, may be diverse; it may complement conceptual powers in design, provide engineering data, help to establish the validity or the invalidity of a hypothesis, or it may simply express an idea in a way which enhances one's ability to communicate the idea. The role of models in the engineering of mechanical systems and how man's peculiar needs led to their subsequent development, is tied in with the evolution of the sciences of mathematics, mechanics, and, in particular, kinematics. Three types of early models are distinguished; the hardware model, the mathematical model which is purely analytic in nature, and the graphic model which portrays information pictorially, often with the support of mathematical concepts. The work of Euler gave impetus to what were to later evolve as models.of the mathematic and graphic types.2 Of great significance is Euler's observation (1775): The investigation of the motion of a rigid body may be conveniently separated into two parts, the one geometrical, the other mechanical. In the first part, the transference of the body from a given position to any other position must be investigated without respect to the cause of motion, and must be represented by analytical formulae, which will define the position of each point of the body after the transference with respect to its initial place. This investigation will therefore be referable solely to geometry, or rather to sterotomy. 2 Richard S. Hartenberg and Jacques Denavit, Kinematic Synthesis of Linkages, McGraw-Hill, Inc., New York, N. Y., 1964, p. 7.

-9 It is clear that by the separation of this part of the question from the other, which belongs properly to Mechanics, the determination of the motion from dynamical principles will be made much easier than if the two parts were undertaken conjointly.3 Thus thedistinction between dynamics and kinematics, and the origin of the notion of a massless, rigid body.4 Euler's contemporaries, D'Alembert and Kant, also treated motion in a purely geometric manner, but little significant activity took place until after the time of Ampere. In 1834, in his "Essay sur la philosophie des sciences", Ampere coined the word "cinematique", derived from the Greek word for motion.5 With Ampere's work, Euler's idea received reiteration and identity, kinematics was recognized as a science, and there followed considerable activity by many investigators.6 One particularly significant event took place with the development of a "graphic method" presented in a paper before the Royal Society of Edinburg in 1885 by Professor R. H. Smith.7 This method, the method of acceleration and velocity polygons, and the concept of "images", enabled study by relatively simple graphical techniques. 3 Novi commentarii Academiae Petrop., Vol. XX, 1775. Translation by Robert Willis, Principles of Mechanism, Longmans, Green and Co., London, 1870, p. viii. A perhaps archaic but precise designation is that of dividing the general problem of dynamics into kinetics and kinematics. Aubrey F. Burstall, A History of Mechanical Engineering, The M.I.T. Press, Cambridge, Mass., 1965, p. 236. Hartenberg and Denavit, o. cit., pp. 7-22. The authors outline developments by many contributors including those of Monge, Hachette, DeCoriolis, Charles, Poinsot, Bernoulli, Reuleaux, Burmester, Mohr and others 7 Ibid., p. 20.

-10 The significance of this contribution lies in the subsequent widespread use of graphical methods which took place, and which are used extensively even today. Because the graphic methods became so popular also pointed up the unwieldy character of purely analytical methods. The graphical methods were a compromise which approached the advantage of the information yield of purely analytical methods and at the same time, enabled the investigator to more readily relate his analysis with the hardware counterpart. Though the method serves the analyst well, it is not without shortcomings. The graphical techniques supported by vector algebra, follow the example of the geometer. The graphics of the geometer are models enabling one to perceive the relations between points, lines, angles, and, to some degree, surfaces. But kinematics is geometry in motion, and displacement, the first aspect of motion, requires a movable geometric figure capable of relating continuous changes in angles and line lengths as well as the curves traced by points. Furthermore, with graphical techniques, subsequent velocity and acceleration analysis requires the construction of polygons in a discrete fashion for every linkage configuration of interest. To help alleviate the problem, particularly in synthesis, hardware models were constructed and their use and development paralleled the progress in analytical and graphical methods. The use of hardware models which yield kinematic information 8 can be traced to the ancients but systematic efforts were not made until the nineteenth century. A record of the early use of hardware 8 Ibid., p. 70. Archimedes is known to have made a planetarium in which the motion of.the sun, moon and planets were displayed.

-11models is described by Willis, and indicates an exhibition of many of the important "machines" of the period by Rev. William Farish in a "Course of Lectures on Arts and Manufactures" at Cambridge, England.9 Willis succeeded Farish to the office of Jacksonian professor of natural and experimental philosophy in 1813 and expanded Farish's modeling idea by devising a mechanisms kit. The kit has a multifold purpose ranging in its use as a teahing aid to "the trial of new combinations and original research". One device which was essentially a drafting machine, is described by Willis...a machine to describe the curves that belong to parallel motions under various proportions of the rods and [connecting] link...a pencil is attached to this link and when the rods are made to turn about their fixed centres of motion, the pencil describes a curve,....a part of its length is so nearly rectilinear, that in practice it may be employed as if it were a true straight line.10 This is a description of a model of a four-bar linkage and the "curves" are the coupler curves familiar to the current technology. The four-bar linkage is the same type of assemblage which enabled Watt to devise the double acting steam engine a century earlier.ll One hundred and ten years later, in 1951, Hrones and Nelson published an immense compilation of coupler curves drawn by a device 12 similar to Willis' drafting machine.2 This tremendous volume containing 730 linkage configurations and displaying over 4000 coupler curves 9 Robert Willis, A System of Apparatus for the Use of Lecturers in Mechanical Philosophy, Especially in those Branches Which are Connected with Mechanism, John Weale, London, 1851. 10 Hartenberg; and Denavit, op.cit., p. 74. Friedrich Klemm, A History of Western Technology, The M.I.T. Press, Cambridge, Mass., 1964, pp. 256-261. 12 John A Hrones and George L. Nelson, Analysis of the Four-Bar Linkage, The M.I.T. Press, Cambridge and John Wiley & Sons, Inc., New York, 1951.

-12 is today, probably the publication most often referred to by engineers engaged in four-bar linkage synthesis. One other early modeling activity is worthy of note, and that is the effort of Reuleaux in the latter half of the nineteenth century.13 Reuleaux's models were, as contrasted with the kits of Farish and Willis, permanent miniatures constructed of iron and brass, each showing a unique kinematic charateristic. A collection of several hundred of Reuleaux's models was destroyed in Berlin in World War II. Smaller collections of Reuleaux's models still exist; the principal one, numbering 266 items, is at Cornell University in Ithaca, New York. In the discussion thus far, attention has been focused primarily on the kinematic aspects of linkage study. It should be noted that it is also possible to make a dynamic analysis of linkages using graphic polygon methods as a step subsequent to the velocity and acceleration analysis. Mass is attributed to each of the links and through the application of D'Alembert's principle, force polygons can be constructed. However, there still exists the tedium associated with these techniques. With more recent developments in technology, much of the tedium in both analysis and synthesis is relieved. 1.3. Current Techniques in the Study of Mechanical Systems With the relatively recent introduction of high speed digital computation and progress in the development of electronic analbg computers, mathematical formulations have been undertaken which have heretofore been regarded as too involved for hand computation. 13 and Denavit, op. cit.. 75. Hartenberg and Denavit, op. cit., p. 75.

-13 The vectorial method, that used by Newton, continues to be well suited to the analysis of rigid body systems. Analytical vector techniques which are easily programmed for the digital computer have 14 been successfully applied to both kinematic and dynamic analysis.l Techniques for employing the digital computer for synthesis of mechanisms have also been reported. The most general ones are those of Freudenstein and McLarnan for the synthesis of function generators and the Sandor-Freudenstein program for the synthesis of path-generating mechanism. 15 In 1964, Mann discussed the use of the M.I.T. Sketch Pad in mechanism study.6 With this technique, an image of the linkage appears on the screen of a cathode ray tube and is caused to execute constrained motion by subroutines programmed into a digital computer.17 With continued progress in hardware design, and further developments in software, 14 Milton Chace, "Development and Application of Vector Mathematics for Kinematic Analysis of Three-Dimensional Mechanisms", Ph.D. Thesis, University of Michigan, Ann Arbor, Mich., 1964; T. W. DeVries, "Generalized Linkage Analyzer", Dept. of Mechanical Engineering, Purdue University, West Lafayette, Ind., 1961; A. S. Hall and F. E. Hahn, "Four-bar Mechanism Analysis", Dept. of Mechanical Engineering, Purdue University, West Lafayette, Ind., (no date); Joseph E. Shigley, Kinematic Analysis of Mechanisms, McGraw-Hill Book Co., Inc;, New York, 1959. 15 C. W. McLarnan, "Program for Designing Four-bar Linkages with Five Precision Points", Dept. of Mechanical Engineering, The Ohio State University, Columbus, Ohio, (no date); G. N. Sandor and F. Freudenstein, "Kinematic Synthesis of Path-Generating Mechanisms", IBM 650 program library file no. 9.5.003, IBM data proc. div., New York, 1959. 16 Robert W. Mann, Computer Aided Design, Presented at the ASME Mechanisms Conference, Purdue University, October 19 to 21, 1964. ASME Paper No. 64-Mech-36. 17 There are currently available, commercial consoles which include a small digital computer coupled with a graphic display unit. Such units can be wired into large time sharing digital systems.

-14 the digital computer combined with the graphic terminal will provide a powerful design tool. This combination will provide the user with the flexibility and accuracy of the digital computer along with a convenient means of affording two-way communication. This latter point, according to Adams, et. al., in a study reassessing the application of digital computers in mechanism studies, has been one important factor which has precluded more extensive use of this machine in this area.l8 This same factor is one of the inherent attributes of the analog computer. The first uses of analog computers in the study of linkages were reported in 1962. These first applications were concerned primarily with displaying displacement relationships of planar, rigid link mechanisms, and of particular interest was the display of coupler curves. A method described by Lenk is directed primarily at coupler curve generation.19 The method entails the writing of algebraic equations which describe, in rectangular components, the links of the system. These equations are implemented using conventional analog circuitry. Keller applied feedback control techniques. 20 Noting that points on a rigid link mechanism are constrained to move on prescribed 18 D. P. Adams et. al., "Kinematic Aid From Graphical Computer-Output", Transactions of the Seventh Conference on Mechanisms, Purdue University, 1962, p. 22. 9E. W. Lenk, "Instrumentelle und elektrische Verfahrung zur Erzeugung und Aufzeichnung von koppelkurven", Konstruction, November 1962, PP. 393-396. 20 Robert E. Keller, "Application of the Analog Computer to the Study of Kinematics", Ph.D. thesis, Stanford University, Stanford, Calif., 1962.

-15 paths, he developed a circuit which he describes as a "link simulator". If the extremity of a radial link, for instance, deviates from its circular path, an error signal is generated and corrective feedback is incorporated to maintain the fixed link length. His work also describes useful oscilloscope display techniques and he reports to have successfully differentiated the displacement variables to derive velocities and accelerations. The techniques offer economy from the standpoint of equipment utilization, but do require the construction of special purpose circuitry with. some critical elements. Crossley describes another approach using the. four-bar linkage 21 as a model.21 In this method, a prescribed mass is attributed to the follower link (Figure 1, link 13 ) and the coupler link 12, undergoes linear elastic deflections along the axis of the link. By sunming torques on link I3, a nonlinear ordinary differential equation is derived and implemented on the analog computer. Letting the elasticity of the coupler link approach infinity (with the use of a high gain amplifer) the rigid link case is approximated. Some velocity information is inherent in the implementation, but acceleration data was not found to be reliable. It should be noted that Crossley was not interested in investigating the elastic system, nor was the method directed at deriving velocities or accelerations. The method was intended only to provide an economical means of solving the position and coupler point problem. 21 F. R. E. Crossley, "Die Nachbildung eines mechanischen kurbelgetriebes mittels eines elektronischen Analog-rechnersi', Feinwerktechnik, Vol. 67, June, 1963.

The methods of Keller, Lenk and Crossley are described by Shigley, and their techniques are enlarged upon and extended to a 22 variety of mechanical problems.22 Also included in this work are many original contributions in the simulation of a variety of other mechanical devices, including the treatment of elements such as cams and gears. In the reported uses of analog computers in linkage studies, the full potential of modern machines has not been exploited. The efforts have been directed at deriving an image of the linkage with particular emphasis on the drawing or displaying of coupler curves. With the increasing availability of large modern computer installations, more elaborate and complete simulations of systems in all areas of technology have been undertaken.23 It is the purpose of this study to describe general procedures for simulating planar linkage systems on the electronic analog computer. The resulting simulation is intended to provide the investigator with a model which is capable of generating a graphic display of the system, and which will enable important parameters to be varied. The computer model also simultaneously provides any other desired kinematic or dynamic information. The simulation thus constitutes a flexible design tool, capable of yielding an abundance of information, which the author hopes will contribute significantly in mechanical engineering design. 22 2Joseph E. Shigley, Simulation of Mechanical Systems, Mc-Graw-Hill Book Co., New York, 1967. 23 To enhance progress in this -area of technology, the technical society, "Simulation Councils, Inc." was formed with the express purpose of advancing the design and.use of computers and similar devices employing mathematical or physical analogies and to widen their application in all fields.

-17In addition to presenting a generalized approach for simulating rigid link mechanisms, methods for dealing with elastic links are also described.

CHAPTER 2 OUTLINE OF TECHNIQJES FOR SIMULATING PLANAR LINKAGES ON THE ELECTRONIC DIFFERENTIAL ANALYZER In this study, the equations which describe the kinematic and dynamic variables of planar mechanisms are derived using the vector approach of classical Newtonian mechanics. In the case of rigid link systems, the equations are exact insofar as representing the assumed system, and a generalized procedure can be described for their derivation and computer implementation. When elastic effects are included, varying degrees of approximating a solution are plausible and it is more difficult to. generalize. Comments on techniques for handling elastic links are deferred to Chapter 5. The generalized procedure for constructing the analog model for rigid link systems is presented here. 2.1. Modular Construction of Analog Computer Circuits for the Simulation of Linkages In simulating a physical system on the analog computer, mathematical expressions describing the system are first derived which, in turn, purvey the logic for constructing the circuit. The implementation of the equations for linkage simulation, as presented in this study, does not, however, follow a conventional pattern. A system of ordinary differential equations can usually be implemented by first assuming the highest ordered derivative and then performing successive integrations to generate the lower ordered derivatives. The lower ordered derivatives are then fed back and appropriate coupling connections are made, as dictated by the equations. This procedure can be followed for the case of linkages, however there

-19 are two major disadvantages. If, for instance, in a kinematic analysis, the expression for an acceleration is implemented in the customary manner, initial conditions would be required on both the velocities and the displacements. Furthermore, if information regarding the accelerations is not required, superfluous circuitry is being used. By analytically deriving expressions for displacements, velocities, and accelerations, the circuits can be constructed on a modular basis. If accelerations are not required, fewer circuit elements are required. As will be seei, the modular construction also precludes the necessity for the a priori determination of initial velocities. The circuit construction follows the pattern indicated in Figure 2.1. Explicit expressions for angular velocities are first derived. If a link moves in rectilinear translation, the expression for its translational velocity is derived. The circuit for the velocity equations provides terms which are required to satisfy the equations for displacements and accelerations. Circuits for displacements or for accelerations may be constructed independent of one another. The three circuits taken together constitute a complete kinematic model of a linkage. Examples of circuit derivations are presented in Chapter 3. A dynamic model is fully dependent upon the.kinematic model. For the case when the linkage is operated as a machine, kinematic variables which become available in the solution of the dynamic equations must be fed back to drive the velocity and acceleration circuits. For the case of a mechanism, the velocity and acceleration circuits are driven directly with the prescribed motion inputs. If only displacement information is required, the method of Lenk discussed in Chapter 1 offers the most economical recourse.

o o 1 N I a I N w Pj R b nr a.. 8 &{2 9 q J rN q R S I 91 11 no 0:3 i 0 0 It Fi

The symbols used in the analog computer circuits are given in Table 2.1.2 2.2. Trigonometric Resolution Using Electronic Multipliers An important subcircuit which is essential to techniques presented here, involves trigonometric resolution employing conventional electronic quarter-square multipliers. The circuit performs simultaneously the operations indicated in Equations (2.1) and (2.2). The circuit, hereafter referred to as a rate resolver, f8 sin e dt= - cos e (2.1) f8 cos dt = sin e (2.2) appears in Figure 2.2. The rate resolver is driven with the velocity o0 0o e and generates the terms sine, cose, esine and ecose. This method offers the advantage of allowing continuous angular travel through an indefinite number of revolutions, and is particularly convenient for linkage simulation. A complete evaluation of this circuit, with error analysis and circuit improvements is given by Howe and Gilbert.3 Drift or decay problems may be encountered, depending upon the quality of the capacitors and the accuracy of the multipliers. In the event these problems are encountered, remedial steps can be taken. 2The symbols indicated in Table 2.1 adhere to the standards recommended by Simulation Councils Inc., Vol. 6, No. 3, March 1966, pp. 137-140. R.M. Howe and E.G. Gilbert, "Trigonometric Resolution in Analog Computers by means of Multiplier Elements", IRE Transactions on Electronic Computers, Vol. EG-6, no. 2, June 1957, pp. 86-91.

TABLE 2.1 SYMBOLS AND NOTATION FOR ANALOG COMU]TER CIRCUITS Element - Symbol Function Notes Signal low Connection -T- t No Connection -4- _ Amplifiers Summer 10 b N C d Integrator a b c d a - _x x * -(l0 + 5b + c + pd) t x a -a - / (lOb+5c+d)dt o Shape clearly indicates direction of signal flow. No gain label on unity gain input. "N" indicates the identification number of the amplifier and is used only if the amplifier is referred to in the text. High gain inputs, as with input d, are fed directly to the summing Junction of the amplifier. The gain ~ is a property of the particular amplifier. Small symbols are used for inverting amplifiers. x " -a

TABE 2.1 (coUT'D) Element Sybol Function Notes Function The circuit used to perform the divide operatica Generators Divider as indicated by the symbol is given belaw. (cont'd) a x. - b Vref a b -' a j a ma -b GC-ate-ax ~ -a g > qThe gate circuit is described in detail in | 1- ~G at-x l x O i<lChapter 3, Section 3.1.2. x w 0, g<1 I! wO I2 Rate Resolver ec -- in!-9 cos 8 -- cos sin es The rate resolver circuit is described in detail in Chapter 2, Section 2.2. ec and es represent the initial condition voltagesfor the cosine and sine of- 0.

TABLE 2.1 (CONT'D) A tnt cmSiNbolo Fnptinn Notes Potentiometers Two Terminal (bottom terminal The functional relationships assme ideal grounded) conditions with neglible loading effects. a -x x k a 0 k < 1 Ok<l Three Terminal a x- k(a-b) + b I ro!P Function Generators Arbitrary Dual Squarer bQ& X2 Multiplier b x a f(a) x1 - 2/Vrer x2 b2/ref A vertical line within the symbol indicates that an amplifier is attached to the passive nonlinear element. Absence of the line signifies the passive element only. Vref is the reference voltage of the computer. Commercial standards are 10 and 100 volts. Computers used in this study use the 100 volt system. -ab x = Vref.

-25 One way of compensating-for decay is to incorporate regenerative feedback in the circuit. This can be done by connecting a feedback loop from the output of amplifier 2 to the input of amplifier 1 in Figure 2.2. The feedback loop must contain a potentiometer to enable adjustment for proper neutrally stable performance. A more precise, but more costly method, as suggested by Howe and Gilbert, involves the generation of an error signal, as indicated in Equation (2.3), with corrective feedback. The circuit is shown in Figure 2.3. = 1 - (cos2e + sin2e) (2.3) No difficulties were encountered in employing the circuit of Figure 2.2 in this study. For best results, the following operational techniques were observed. (1) To minimize static multiplier errors, scale the circuit at 100 volts equal.to unity. (2) The circuit should be time scaled to avoid dynamic errors due to bandwidth limitations, but yet be sufficiently fast to minimize errors due to drift or decay. Since the overall circuitry will vary, trial and error is probably the most expedient recourse to proper time scaling. 2.3. Scaling The method used to scale an analog computer circuit is arbitrary, and of the many techniques, each has some particular advantage. The method of machine scaling, employed throughout this study, is described by Shigley, and it offers the advantage of enabling one to interpret.the computer output directly, without going through the intermediary of machine variables

-26 or machine equations.' All amplifier outputs in the circuits are labeled as physical variables rather than machine variables; hence, a single amplitude scale factor applies throughout the circuit. The relationship given in Equation 2.4, relates the voltage at an amplifier output to the physical variable which it represents. a is the amplitude scale factor. volts (a)(physical variable) (2.4) Equation 2.5 relates computer time (T) to real time (t), with a time scale factor. = t (2-5) Time and amplitude scale factors for the circuit of Figure 2.2 are interdependent. It is convenient to use multiples of ten in scaling, and, for a 100 volt reference system, if = 10n, then a = 10(2-2n) The amplifier outputs as labeled in Figure 2.2 correspond to one hundred volts being equal to unity, or a = 100. Thus n = 0 and = 1. If circuitry and records are to be interpreted in terms of different scale factors, the amplifier outputs, in terms of physical variables, would be labeled as indicated below. Amplifier 2 p2 cos e Amplifier 3 p e cos e Amplifier 4 p2 sin e Amplifier 5 p e sin e Input p e 4 Shigley, Joseph E., Simulation of Mechanical Systems, McGraw-Hill, Inc., New York, 1967, p. 73.

CHAPTER 3 THE SIMULATION OF PLANAR RIGID LINK MECHANISMS FOR KINEMATIC ANALYSIS Analog Computer circuits are developed for- the general fourbar linkage and for the slider-crank. Each of these linkages constitutes a basic building block for more ccmplicated systems, with the four-bar example demonstrating the handling of turning pairs, and the slidercrank showing the method for dealing with sliding pairs. The outputs of the kinematic circuits are required to drive the circuits for a dynamic analysis. The four-bar linkage is simulated for the case of a specified motion input and records of the kinematic variables are included. The slider-crank circuitry is developed for the case of a force input with records of some of the variables included in the dynamic analysis of the single cylinder engine and flywheel of Chapter 4. 3.1. The Four-Bar Linkage A schematic of the four-bar linkage is shown in Figure 3.1. Link l1 is the driving crank with its motion specified by a and a. Circuits are derived for velocities, displacements and display, and accelerations. 3.1.1. Simulation for Velocities Explicit expressions for the angular velocities, 3 and, must be derived in terms of the linkage parameters. Expressions for o o v and. may be-derived either analytically or graphically. Both methods are described in the velocity and acceleration analysis of the four-bar linkage, with subsequent derivations using either one method or the other. -27

Figure 3.1. Four-Bar Linkage Showing Length and Angle Parameters for Kinematic Analysis.

-29 o o The expressions for 3 and ( can be derived by writing an equation in polar vector form which describes the linkage. eja + I2 eJP = o + 13 ejl (3-1) Equation 3.1 is differentiated and transformed into the two algebraic equations, (3.3) and (3.4). j & 11 e + j P 2 ej = j ( ej 3 (3.2) o o s a 11cos a+ 1 2 cos =Q 13 cos Q (3.3) a l1 sin a+ P l2 sin P =Q 43 sin Q (3.4) Equation (3.3) is multiplied by -sine, Equation (3.4) by cosp, and the equations are added to form Equation (3.5). 0 o 11 a sin(a-p) 0 O=- (3.5) 13 sin(O-P) By multiplying (3.3) by -sin( and (3.4) by cos( and adding, Equation (3.6) may be derived. o a sin(a-) (3 = =1 _ {3.6) 12 sin(O-p) A graphical method, which is particularly convenient for deriving the equations in the required form, involves inspecting a Francis H. Raven, "Velocity and Acceleration Analysis of Plane and Space Mechanisms by Means of Independent Position Equations", J. App1. Mechanics, Trans. ASME, Vol. 80, pp. 1-6, 1958.

-30 sketch of the velocity polygon as shown in Figure 3.2. Equations (3.5) and (3.6) can be derived directly by applying the law of sines to the polygon as shown below. i1o a L a, 3 4= _ 13 = = sin(O-P) sin(x+3-a) sin(a-p) Hence, o 1, a sin(a-p) -...'.........- (3.5) ~3 sin(4-p) Similarly, " o:i a _ 12 2'sin(~-p) sin(a-4) Hence, o X1'i sin(a-4) (3.6) 12 sin('4-) Three rate resolver circuits are required for the solution of these equations, each of which requires initial conditions on the integrators. The required initial conditions are those of the inital values of the sines and cosines of the angle differences, a -, a - *, and ( -. A general initial linkage configuration is shown in Figure 3.3, with the initial value of a equal to zero. The initial values of the cosines are derived by applying the law of cosines to the triangle described by fg, 13 and o0 - 1l. The sines are then determined from the trigonometric identity relating the sine and cosine. f2,2 _ ). \2 Cos((4-) = 2 + ~ (Lo - )11 (3.7) 2 12 13

-51 y x Figure 3.2. Velocity Polygon for the Four-Bar Linkage.

B /, Figure 3.3. Initial Configuration of the Four-Bar Linkage.

-332 + (1 - 2 -2 2 0 1( 3 2 12(1o- 11) cos(a-p) = cos(P) = *2 _ 12 _(10 - 1f)2 cos(a-() = cos = (2 3 - 2 13(1- 1l) sin(4-p) = 1- - cos2 (4sin(a-P) = - 1 - cos2? sin(a-4) - 1 - cos24 (3.8) (3.9) (3.10) (3.11) (3.12) The following parameters are assumed with an initial configuration as designated by the angles. o ='3 inches 1 = 1 inch 12 = 2 s/inches 13 = 2 inches a =.628 radians/second, constant a(o) = o P(0) = 45 degrees 4(0) = 90 degrees For these specified parameters, the initial values of the sines and cosines are as follows: sin(a-4) = - 1 cos(a-() = 0 sin(a-.) = -.707 cos(a-p) =.707 sin(4-$) =.707 cos(4-P) =.707 The circuit for Equations (3.5) and (3.6) is shown in Figure 3.4. The circuit is scaled in real time with 100 volts equal to unity. Records of the angular velocities appear in Figure 3.5.

-34 Ov. Figure 3.4. Circuit for the Four-Bar Linkage Velocities. [Equations (3.5) and (3.6), a = 100, A = 1

radians/second 2 volts/line radians/second 2 volts/line - / i, "!___ __i-!B~ _!J_ __ ___ __,__ i o: ii;'? i~!i9' 10'I _ I ne t _ _ _!o _ __-6 — _ — — ~!' ___ T^ —-— t --- - T=='1 1 1 11 <I Figure 3.5. Records of') and B. =.628 radians/second.

The absolute velocities of points on the linkage are vectors which lie in the plane of the linkage. Since both magnitudes and directions vary, vector variables are best recorded as polar plots. The absolute velocities of points. on links 11 and 13 are directly proportional to the angular velocities of the links. To make polar plots, the rectangular components of the velocities are generated and used to drive an XY plotter. Equations (3.13) and (3.14), written in unit vector form, represent the velocities of points A and B, respectively. VA. vAx i + vAy = -1 & sin. i + 1 & cos a j (3.13) VB " VBx i + vBy j = -3 4 sin 4 i + 13 i cos { j (3.14) The velocity of point C on the coupler is given in Equation (3.15). VC = Vcx i + VC, = (-'1 a sin a -C f sin i) i + (11 a cos a + Ic p cos a) j (3.15) The circuit for Equations (3.13) through (3.15) appears in Figure 3.6 with the polar plots shown in Figure 3.7. The derivation of the sines and cosines for the circuit of Figure 3.6 is given in Section 3.1.2.

-37 -sina - sin 8 -B I cos B cos a - cos# V'sx sin # Figure 3.6. Circuit for Deriving Polar Plots of the Absolute Velocities of Points A, B, and C on the Four-Bar Linkage. [Equations (3.13), (3.14), and (3.15). a = 10, = 1.]

0 8 2 3 7 8 2 I t co I 3 3 7 5 4 5 0.5 Scale inches/second 1.0 5 Figure 3.7. Polar Plots of Velocities of Points A, B, and C on the Four-Bar Linkage. [Time Marks Correspond to One Second Intervals]

-39 3.1.2. Displacements and Display Techniques The equations for the sines and cosines of the angles of the links, derived from the linkage geometry, are given in Equations (3.16) through (3.21). The circuits are given in Figure 3.8. sin a = I [i2sin(a-.) -13sin(a-_)] (3.16) 0 cos a =1 [ l 1j2 cos(a -))-]3cos(a-4)J (3.17) lo sin p =! [13sin(q_-P)-llsin(a-_)] (3.18) cos f =-l [L1cos(a-)-13cos(4-) + 12] (3.19) 1 sin' = [(12sin(4 —p)-1sin(a-4)] (3.20) so cos [ = [llcos(a-q)+l2cs(O-P)-l3] (3.21) Note that terms necessary to drive the circuits of Figure 3.8 are taken from the velocity circuit of Figure 3.4. Through the use of XY indicating or recording devices, the terms available in Figure 3.8 enable a visual display of the displacement characteristics. To obtain plots of the coupler curves, the rectangular components of points A and B are first derived. These are described in Equations (3.22) through (3.25). XA = 11 cos a (3.22) YA = 1 sin a (3.23)

sin (a-+) -sin (<-/) sln(a-/.) — o00 v\ -cos(a-9) _\cosa cos(a-1 -coovs-) -cos(Oa-/) - cos(+-b) 4?W - / sln(a-. ) Figure 3.8. Circuit for Displacements of the Four-Bar Linkage. [Equations (3.16) Through (3.21).] [o = 100, i = 1]

-41 xB = I + 3cos Q (3.24) B = 13 sin (3.25) The rectangular components of a point C lying a distance IC along 12, as shown on the schematic of Figure 3.1, can be conveniently generated using three terminal potentiometers to satisfy Equations (3.26) and (3.27). C = (gB - xA) +A (3.26) ~2 YC - A(Y ) + YA (37) 2 All possible cases for points associated with the coupler movement are covered by the eight configurations shown in Figure 3.9, and with the information thus far developed, coupler curves for points anywhere in the plane of the linkage can be generated. The circuits for coupler curve generation appear in Figure 3.10 with examples of curves drawn with an XY plotter given in Figure 3.11. Keller describes a technique for providing a visual display of the linkage on an oscilloscope screen which involves the construction 2 of special purpose circuitry. An alternative method, using conventional analog computer elements, is described here using a technique suggested to the author by F.R.E. Crossley of the Georgia Institute of Technology. 2 R.E. Keller, "Mechanism Design by Electronic Analog Computer," Trans. of the 7th Conf. on Mechanisms, Oct., 1962, pp. 11-21.

-42 Figure 3.9. Schematics Showing Points Which Generate Coupler Curves -Associated With the Movement of Link 12 ~

4 Q - (11 I ~\ Q3 A "o o0 0 H 0 U) g x 0<

0, (a) Figure 3.11. Four-Bar Linkage Coupler Curves. a) Point C. IC' 2/4p 2/2, 312/4

Di A (b) Figure 3.11. Four-Bar Linkage Coupler Curves. b) Point D. Se -9 2/4k l2/2 and 312/4 with d - 1 0,

-46 Let a sinusoid of unity amplitude and of a frequency which is on the order of magnitude of one hundred times that of the linkage frequency be designated as n. This sinusoid is rectified in a manner which makes both the positive and the negative half waves available. The latter are designated 11+ and i-, respectively. If, for example, the products, (T+)llsina and (n+)llcosa are applied to the x and y inputs of a dual beam oscilloscope, an image of the link is scintillated on the screen. The scintillation occurs at the frequency of I and the link is seen to rotate at the angular velocity a. To display the four-bar linkage, two high frequency sinusoids, r and X are generated. The frequency of n is twice that of X. Equations describing the x and y components are given in (3.28) and (3.29). The. circuits are shown in Figures 3.12 and 3.13a. x = [(n-)(llcosa)+(I+ )(l2cos) + (flcosa)] gx+ + [((lcos) + (X-)(i3cosO) + lo] gX- (3.28) y = [(n-)(1lsina)+(n+)(2sinP) + (Ilsina)] ga+ (3.29) + [(A-)(Z3sin4) + (z3sinV)] gXExamples of the oscilloscope display are shown in Figures 3.13b and 3.13c. Figure 3.13b shows an instantaneous position and 3.13c shows a time exposure as the crank completes one revolution. 3.1.3. Accelerations 0 oo Expressions for C and A can be derived by either of the methods used to derive. 0 and'. Equation (3.2), repeated below,

-47-.1 A+ *sine wave generator +100v 5.1 Circuit for the Generation of the Scintillation Signals. x +LOOv.05 - +9 dx I ^A^- * +gX- -ge+gk +._ I" oGate Signal Generator -gX t~ ~ with gX+ as the gate signal y =x, x > 0 -100v y = 0, < 0 x - Gate circuit with gX- as the gate signal y = 0, > 0 y =x, k<0 x: Gate circuit symbol Figure 3.12. Scintillation Frequency and Gate Signal Generators with a Gate Circuit for the Oscilloscope Display of the Four-Bar Linkage..1 I+

-48 Isin a sin a. -sin' - ( I ~-Vp --- _~ ITo oscilloscope vertical amplifier — cos a A _ To oscilloscope horizontal amplifier -100v. A Figure 3.13a. Circuit for the Oscilloscope Display of the Four-Bar Linkage. (Equations (3.28) and (3.29) o = 10, a = 1 a i~' Y`"-"'-"t 1 i-. ~-? -~ i it -- T- — - L-~-~~-L-L- -- —._-~ i i-. 1 i t_.:,..._.?.. r - -J —;- ~- - - - --— L. —- —: — -—'-:i t i ri Figure 3.13b. Photograph Showing an Instantaneous Position of the Four-Bar Linkage Display. Figure 13.3c. Time Exposure of the Four-Bar Linkage as the Crank Completes One Revolution.

is differentiated to give Equation (3.30). J a I1 e J+ j 2 ejl = J 3 eJl (3.2) j 1a eJa- a 11 eJ + j e e (330)'1eoa+J 2e. 82 2 21(3.30) j I00 eJ? - 02 = 1J 13 e J - 2 1 ej This vector equation can be transformed to the scalar expressions given in (3.31) and (3.32). 02 00 0 0oo 1a sin(a-)-iSllcos(ca-_) + 2i_ 02 3os(2 -3) 13 sin(Q-p) o0o 11 sin(a-d)+iLap2cos(a-t)+1gf2cos( )- 422 2 sin(4O-) The equations can also be derived by summing vector components of the acceleration polygon. A sketch of the polygon is shown in Figure 3.14. Equation (3.31) is derived by summing vector compooo nents normal to the 12 p vector; Equation (3.32), by summing vector 00 components normal to the L3z vector. The circuit for Equations (3.31) and (3.32) appears in Figure o 3.15 with a constant. Some terms for this circuit must be taken from the velocity circuit of Figure 3.4. The important terms available from the circuit of Figure 3.15 are 02 and F which are related to the normal components of accelera00 00 tion, and the terms 4 and p which enable derivation of the tangential

*3 to34 \r e A Figue 3 A P\ ag tP2 Figure 3.14. Acceleration Polygon for the Four-Bar Linkage.

a cos (a - P) -COB (t - a) cos (a ( f) cEquations (3.31) and (3.32), a = 100, = 1] [Equations (3.31) atld (3.32)~ a = 100, ~J = l

-52 r *d rad/sec2 00 rad/sec2 *2 rad/sec2 * 2 rad/sec2,,- I. __ _ __ - _ _ — t --- - - --—.~ Li L X -L-: i L- - - I t i i i -- - r - _ i —-—! I ~' t... I, S r..- -I- - - -I. f t-r-:1F 1 I- " [ - --- + i _ I_ I -' i - -._, i L I I I, 4 i I. Figure 3.16. Records of p, p, 2 and 2 of the Four-Bar Linkage. a =.628 radians/second.

-53 02 02 00oo components of acceleration. Records of,,, and p, from the circuit of Figure 3.15, are shown in Figure 3.16. The absolute acceleration of any point can be plotted in polar form as was done for velocities in Section 3.1.2. Since the method for deriving polar plots has been described, these plots are not repeated for the accelerations. Rectangular components of the accelerations of gravity centers on the linkage are derived and applied in the dynamic analysis of Chapter 4. 3.2. The Slider-Crank The equations and circuits for a kinematic study of the slidercrank mechanism are derived here. These will be used in the dynamic analysis of an engine and flywheel in Chapter 4. A schematic of the slider-crank mechanism is shown in Figure 3.17. Parameters are assumed as indicated below with an initial configuration as designated by the angles a(o) and (0). 11 = 1 inch a(0) = 90 degrees L2 = 3.4 inches A(0) = 90 degrees 3.2.1. Velocities The expressions for the angular velocities are derived by first differentiating the polar vector Equation (3.33) which describe the linkage. i1 eja + 2 ej = (333) J a eJ ea.= (.4) J a l1 +' j "~ (2 eJA'= J YB (3.34)

-54 y 12 I. I __,/ Figure 3.17. Schematic of the Slider Crank.

-55 Equation (3.34) is resolved into the scalar Equations, (3.35) and (3.36). Two rate resolver circuits are required o - a llsin a =- (3.35) 12 sin A YB = al.cos a coss p (3.36) in the circuit for these equations, with initial conditions applied to the integrators. From the given initial configuration, the initial values for the. sines and cosines are given below. The circuit for Equations (3.35) and (3.36) appears in Figure 3.18. Records of 3 and yg for a constant are shown in Figure 3.19. sin(O) = si 1 sin(O) = sing = 1 cosa(O) = cos- = 0 coso(O) = cos = o 3.2.2. Displacements and Display of the Slider-Crank The sines and cosines of the angles of the links are available from the circuit of Figure 3.18. The x and y components of points on the linkage are derived in the same manner as for the fourbar linkage in Section 3.1.2. Equations describing the x and y components of points A, B, C, and D of Figure 3.17 are given below. The circuit for these equations appears in Figure 3.20. and examples of coupler curves are shown in Figure 3.21. xA = 11 cos a (33) YA = 11 sin a (3.38)

100V. Figure 3.18. Circuit for the Velocities of the Slider-Crank. Equations (3.35) and (3.36). [ = 100, = = 1. ]

P -b ii!ni! 4* — -- rad/sec _ 1 T X..1~~~~~~~~~~~~~~~~~~1 l'"i,'I l z~'U'I:'.' in/sec te il ~,i: -g7 I:... t [ t. i _ 1 -.' i! Figure 3.19. Records of and f for a Constant Angular Velocity of the Crank of the Slider-Crank Mechanism. of the Crank of the Slider-Crank Mechanism.

-cos a''..... — I -sin YA 10 Figure 3.20. Circuit for Generating Rectangular Components of Points on the Connecting Link of the Slider-Crank. [Equations (3.37) through (3.44). a = 10, i = 1]

(a) Figure 3.21. Coupler Curves of the Slider-Crank. a) Point C. [IC= X2/4, a2/2, and 32/4. ]

B A (b) Figure 3.21. Coupler Curves of the Slider-Crank. b) Point D. [XC = 1a2/4' I2/2, and 352/4 with d = 1.

-61 XB = 0 (3.39) YB = tl sin a + 12 sin P (3.40) 12 - (C 12 Y = (YB - YA) + YA (3.42) 2 XD= XC + d sin P (3.43) D - YC + d cos P (3.44) An oscilloscope display of the crank and connecting rod in motion can be generated using the circuit of Figure 3.22. Using a dual beam oscilloscope, the x component is applied to the time sweep amplifier, and each of the y components are applied to the vertical amplifiers. 3.2.3. Accelerations 00 00 Expressions for the accelerations 1 and YB are derived by differentiating vector Equation (3.34) to give Equation (3.45). Equation (3.45) is transformed to the equivalent scalar expressions given by (3.46) and (3.47). -, eja + ja 0' 2 e j ~00.0 e3j + - 12e + j 12 = J Y (.45) 1O Oo 2 oo - o11cos a - -a flsin a - 22cos (3.46) 12 sin f

-62 Y, X - isman — i y" l Figure 3.22. Circuit for the Oscilloscope Display of the Slider-Crank,

c.) 2. I a Coa s yB - - lsina +c llcosa - li2sinf + i 122s (3.47) The circuit for Equations (3.46) and (3.47) is shown in Figure 3.23. Plots of and YB for the case of a constant angular velocity of the crank are shown in Figure 3.24. Additional plots of the kinematic oo variables with ac finite are presented in Chapter 4. 3.3 Scaling All circuits are scaled in real time with the rate resolvers time scaled for a maximum frequency of one radian per second. This low frequency was chosen to accomodate an XY plotter. A constant angular drive frequency of a =.628 radians/second was chosen to simplify the interpretation of strip chart records. The circuit can be interpreted in terms of any desired drive frequency by applying the relations, T = i t and volts = a (physical variable) which were discussed in Section 2.3. If, for example, it is desired to quantitatively interpret the velocity 4 of Figure 3.5, one first notes that a = 100 and J = 1. Also, there is a voltage scale of 2 volts/line and a chart speed of 10 millimeters/second (2 vertical divisions/second). Thus at t = O, there is a voltage of approximately -31 volts. Hence, volts = a (physical variable), -31 = 10oo. (), 4 = -.31 radians/second. Similarly, at t = 2 seconds, which corresponds to a crank angle of a = 72 degrees, the voltage is approximately 25 volts.

Hence, volts = a (physical variable), 25 = 100 (), 0 4 =.25 radians/second. If the system is to be interpreted in terms of another drive frequency, a different time scale factor is assumed. For a crank frequency of a =628 radians/second (6000 revolutions per minute), the time scale factor would be p = 1000 and the amplitude scale factor would be a = 104. Thus in Figure 3.5, at t = O, with -31 volts, the interpre0 tation of 4 would be as follows. volts = a (physical variable) -31 = 104 (103 4) 4 = -310 radians/second Digital computer solutions for verification of the kinematic records of both the four-bar and the slider-crank mechanisms are given in Appendix 1. These solutions are for a constant drive frequency of a =.628 radians/second.

-65 d sina -a cos a cosOa d cos a sin a A cosa *Y YB sin Figure 3.23. Circuit for the Accelerations of the SliderCrank. [Equations (3.45) and (3.46) a = 100, p = 1.] I 6- r *f - - -4 - -t — - - -- ~e p. rad/sec2 YB in/sec2 *tt-t -titt-t -.-Ti- 1,,t-P — — t0 —--- t - t. 2 - 0_ - --. 2 7I 1 It I I[ _,i' 1 i I T -I:-t T-i;#,GEl, ^_!: - 1:! i I t f, _!.!..N 4::.., i_ -- t t..! I 1 -. I I i i-i -rFj i Figure 3.24. Records of y and i of the Slider-Crank with &-.628 radians/second. Two Cycles.

CHAPTER 4 SIMJLATION FOR DYNAMIC ANALYSIS OF LINKAGES WITH RIGID ELEMENTS A dynamic analysis, through analog computer simulation for both the cases of a mechanism and-a machine are described in this chapter. The four-bar linkage is analyzed for the case of a motion input, and the slider-crank, as part of a single cylinder engine and flywheel, is analyzed for the case of a prescribed force input. 4.1. Four-bar Linkage Dynamics The mechanism for this analysis is shown in Figure 4.1. The masses, ml, m2, and m3 are specified for each link, and gravity centers are located by the distances pi, P2, p3. In addition, each link is to have a radius of gyration designated by the letters rl, r2, and r3 A free-body breakdown of the linkage is shown in Figure 4.2. The system of forces and torques includes an arbitrary load torque, TL, acting upon link i3. The crank is driven at a constant angular velocity. The torques and forces are summed in Equations (4.1) through (4.5). The notation E TT1 indicates the summation of torques acting upon link 1 about a moment center taken at 01. Similarly, E F12 indicates the summation of forces acting upon I2 which are in the x direction. E TO1; Ti + FAyllcos a - FAX1 sina = 0 (4.1) -66

p y A I' x Figure 4.1. Schematic of the Four-Bar Linkage Showing Parameters for a Dynamic Analysis..

F1B B FAX A FAy FBy f F ey iy FBX FAX _... _ TL co I 0 a Ti Forx-a 01 Fo0x FO3y Figure 4.2. Four-Bar Linkage Free-Body Diagram.

-69 E T; FBx n2sin1 - FBy n2cos$ + FAy P2CosP 20. - FA P2 sin = m2r2 (4.2) F2; - F - FBx = m2 x (4.3) E F12y; - FAy - FBy = m2Am2y (4.4) ~ T; FBy 13 cos 4 - Fx 13sin Q - T m3r3 + p) O0 (4.5) Equations (4.1) through (4.5) enable a solution for the five unknowns, FAx FA FBx, FBy, and Ti Ti is the torque at the crank which is required to maintain the constant angular velocity of the crank. These equations are rewritten as explicit expressions of the five unknowns in Equations (4.6) through (4.10). FY = - sin 4 - M sin p (4.6) B~y sin(4-1) Bx - N cos ( - M cos (47) sin(4-p) FAy =-2 AY -FBy (4.8) x Bx (49) Ti = FAx L1 sin a - FAy 11 cos a (4.10)

p. pp/3 0 am2 ama Q^ Omty B. 18 1 A,~~z P A. Figure 4.3. Four-Bar Linkage. Link l2 Showing the Relative Acceleration Components of m2.

-71 Expressions for N and M are given in (4.11) and (4.12). m2 2 r2 - 2 pI &2sin(a-. ) (4.11) m2 mr0 2 1 M = 3(p + r) 0 + L (4.12) 3 3 In equations (4.8) and (4.9), Am2x and Am2y are the rectangular components of the absolute acceleration of mass m2. The rectangular components of the acceleration of m2 relative to point A are designated as am2x and am2. These components are shown on the schematic of link 12 in Figure 4.3, and the equations are given in 02 00 (4.13) and (4.14). am2x = - p2 cosp - p2D sinp (4.13) a *2 cy.~,~~.i.,~,~.~.B (4.14) y = - P21 sinp + p2p cosp (414) Ax = p22cos p - sincosa (4.15) mc:= - P2g sinp + P2 P cosp - l lsina (4.16) The circuits for the equations derived above require terms from the circuitry for the kinematic analysis which was derived in Section 3.1. The length parameters for this analysis are prescribed as those used in the kinematic analysis. All of the parameters are specified below and the circuits appear in Figure 4.4.

cooa OF AJ3g2 0, ~_O —.^ _0 v - s ine-2 x V sin(O -0) U ] pi -s _0 __2 Ile xV:= I A. t s-i e) -i,' - -cosO /j( s FA/,2 ot - Figure 4.4. Circuit for the Dynamics of the Four-Bar Linkage. [Equations (4.6) through (4.10). a' 10 4, =1.]

-73 fo = 3 inches f1 = 1 inch 2 = 2 N2 inches f = 2 inches TL = 0 in-lb P1 =.5 inches P2 = J2 inches p3 = 1 inch r2 =.8 inches r3 = 1 inch ml = 3xl0-4 lb-sec2/inch m2 = 6.5x10-4 lb-sec2/inch m = 5.0xlO 4 lb-sec2/inch 0 a =628 ra0/sec a =0 Polar plots of the forces at pin connections A and B are shown in Figure 4.5. A record of Ti appears in Figure 4.6. The force reactions at pins 01 and 03, called "shaking forces", can be derived by a force summation on links l1 and 13. The rectangular components of these forces are given in Equations (4.17) through (4.20). Z F; Fx = FAx mlPlsina "Ffly; 1= Fy - mlPl2COoSa z F3x; xO F3x - m3P3(2coso + 4 sink) (4.17) (4.18) (4.19) ~ 00 Z F3y; F = FBy - m3P(42sin4 - 4 cos4) (4.20) The circuit for these equations is shown in Figure 4.7 and polar pl6ts of the shaking forces are shown in Figure 4.8. The amplitude scale factor for all circuits of Section 4.1 is -4 a = 10 and the time scale factor is = 103. Circuits of Section 5.1 which are used in the dynamic analysis must be interpreted in terms of these scale factors.

0 1 2 3 9 S,5 4 I/ / 0 100 200 pounds scale / / 7 F. Figure 4.5. Polar Plots of the Forces at Pins A and B of the Four-Bar Linkage, Time Marks Correspond to One Second Intervals.

I':5d,~~ t e I l~ ~ ~I-~ Il~~'~l'llll f~ lI~ I t~ [ I-iI... iI - r *I Ji j I - _ -.; -i! L~. - - _,,_L 2.,l,,,,3_ 4 $ 6 ^5 7 | 1,! 93: 1 - l I Figure 4.6. Computer Record of the Torque at the Crank of the Four-Bar Linkage.

.2167 - FJ -PF./m2 sina - FAy/m cosoa.1972 ^ cos4,. 4 sin4, — Fox/lO,"l > —-- Fly/10l1 F03x/m / 3 - FBy/m2 -V14 I.13 cos4 -~1 Figure 4.7. Circuit for the Pin Reactions at the Ground Link of the Four-Bar Linkage. [Equations (4.17) through (4.20). a - 10-, l = 103. 3

For 0 ok r Fos 0, 0 SO 100 I I 00 1000 force scales, pounds I I5 3 2 3 4 2 5 I I 7 0 Figure 4.8. Polar Plots of the Forces at Pins 01 and 0- of the Four-Bar Linkage. Time Marks Correspond to One Second Intervals.

-78 4.2. Simulation of a Single Cylinder Engine A schematic of the slider-crank, for which a kinematic analysis was made in Section 3.2, is shown in Figure 4.9. In addition to length and angle parameters, masses ml, m2, and m, and radii of gyration r1, r2, and r, are specified. The crank is connected to a flywheel through a crankshaft which undergoes angular elastic deflections. The system is driven by a force, Fe, which acts upon the piston. The force equations listed below are derived by taking torque and force summations on the links shown in the free-body schematic of Figure 4.10. E F -mF y0+F (4.21) ~ my -'FBy YB + Fe (421) zF r2; - FAX FBX 2 x (4.22) F2 Fy; -FAy+F3y m2 A4my (4.23) s 2y; - By + y ~ T; Fi n2 cosp - FBx n2sip FX sin - F P sn + Fd P2 cosS = m2r2~ (4.24) ~ T1; Ts + FAY 11 cosa - FAX 1 sina = mr2 a (4.25) 0 - T o00 ET01; - Ts.- TL =I e (4.26) Ts = k(G-cr) (.27l)

IF 8 a Y8 Figure 4.9. Single Cylinder Engine and Flywheel.

-80 I" Fe s M,3.,Fox m, FAX FAY FAX Figure 4.10. Slider-Crank Free-Body Diagram.

-81 The six Equations, (4.21) through (4.26), can be solved for O the six unknowns, FAx FAy, FBx FB, a and T s. Ts is the torque in the crankshaft and is given by Equation (4.27). The equations are rearranged below as explicit expressions of the unknowns. F - Y F. (4.28) FBy 115 Fe F -F - m2 A (4.29) A By - (4.29) a = m —r (Ts + FAy 1l cosa - FAx1 sina) (4.32) - T = Ie+ TL (4.33) Ts = k(e-a) (4.34) Expressions for the rectangular components of the acceleration of m2, 4x and Amy are given in Equations (4.35) and (4.36). 00 02 oo 0 2 Amx = - p2 sin2 - p2 cosP - 1asina - 11 cosa (4.35) O= o 2 + o 2 AY = " cosP - P2P sinp + 1cx cosa - ip. sina (4.36)

-82 Parameters as listed below are assumed. t1 = 1 inch 2. = 3.4 inches P2 = 1 inch r1 = 1 inch r2 = 1io inches ml =.01 lb-sec2/inch m2 =.0015 lb-sec2/inch m3 =.001 lb-sec2/inch I =.02 in-lb-sec2 Femax = 1000 lb c1 =.008 in-lb-sec C2 = 200 in-lb-sec k = 106 in-lb/radian a(o) P(o) = i/2 = i/2 TL = [-cl /1/ -c2(8-) ] in-lb &(0) = -(0) = 200 rad/sec For the gas force in the cylinder, empirically derived pressure characteristics might be simulated with a variable diode function generator. Since an actual characteristic would be pertinent only to the study of a particular engine, the gas force here is approximated for a two-stroke cycle by a truncated exponential. Equations (4.37) and (4.38) describe the force. Fe = F.9 0 < t < tl (4.37) _ (t-tl) Fe = F e T max, < t < t2 (4.38) The time interval, 0 < t < t2, represents the duration of the force during the power stroke. The circuit of Figure 4.11, which satisfies Equation (4.39), is used to simulate the gas force. 1 deo -RC1 dei 1 dt + b dt (4.39) The letter b represents the setting of potentiometer P1, and truncation occurs through saturation of amplifier 2. Potentiometer P1 adjusts the dwell period of Fm, and potentiometer P2 adjusts the

- -S e I R, C Fir eo 0.1 Figure 4.11. Circuit for the Generation of;as Force Fe. Fe

-84 sin/-.2 -cos —o2 -a sin a - *o j9 sinj6 00 -a cosa o2 o2 a sin a Am2x Am2y Figure 4.12. Circuit for the Rectangular Components of Acceleration of m2 on the Connecting Rod. [Equations (4.35) and (4.36) a = 100, p = 1.]

-85 00 Fey *-.3 FAy m3 F^ 2 667 1 "2 ms 1-.294 COS., 8 -- 0. - sin i + m28 FBx sinB m2 mAx Figure 4.13. Circuit for Forces F,F F and Fy [Equations (4.28) through (i31 ). a = 10Ip = 103.]

0I -oa.o r FA, -COS al/ FAx xn e V// ^.15 sin a.15 -- Figure 4.14. Circuit for TS, a, and a of the Single Cylinder Engine. [Equations (4.32) and (4.33)]

y. pounds TI 0 inch-pou d 2.400 -600 -800 800 a 600 radians/econd 400 200 0 300,000 200,000 rad/ue2 100,000 0 200 ~ 100 redains/second 0 100,000 50,000 rsd/sec2 0 1000 iB 500 in/sec 0 00,000 ^i/.ec2 200,000 0 hr -.e 4.15a. Records of Fe, To, &, a, &' 3, a', YB d YB as. the Engine Comes Up to Speed.

,poods TIroa 4v 9A ptm 694 ag el fR C1p de 32t Jo aplowas qc~q ff euhou 0 ooo'oo " oooos O B 0001 oor DHf oOLs Qe 0 0o 000'05 0 oa w o 008 00 o oo'ooZ / ooo'ooC D 000 009 00o9 CoB 0 l OO -98

v Force scale, pounds I S5N -.... V Power Y Is1 cycle (Inner loop) and 2nd cycle Power I I rc) (a) (b) Figure 4.16. Polar Plots of Forces at Point B. a) First Cycle, b) All Cycles as the Engine Comes Up to Speed. c) One Cycle at Full Operating Speed.

Force scale, pounds * Sw t.... __ / / \ 2" cycl Recorder transients BDC (a) Figure 4.17. Polar Plots of Forces at Point A. a) First two Cycles (Inner Loops) and a High Speed Cycle (Outer Loop).

V Force scale, pounds S S5., A a I - I i(b) Figure 4.17. Polar Plots of Forces at Point A. b) Forces at Point A as the Engine Comne Up to Speed.

-920 v.....:.:.:......:::.:4 ^- - ^ 4.........I.^ c i,;::*:;*::..;: l'':'/'^' C 4)::* ^::;:,;.:-;.'ITO^^ ^^ <

-93 magnitude. The time constant T can be changed by changing R1. The force is applied throughout the downward travel of the piston and is triggered as y becomes negative. Circuits for Equations (4.29) through (4.36) are shown in Figures (4.12), (4.13), and (4.14). A strip chart record of Fe, Ts 0 00 0 00 0 00 a, a, a YB', and YB is shown in Figure 4.15. These variables are plotted as functions of time as the engine comes up to speed from a specified initial velocity of the crankshaft. A polar plot of forces at the wrist pin (point B) is shown in Figure 4.16, and at the crank pin (point. A) in Figures 17a and 17b. Piston positions which correspond to top dead center (TDC) and bottom dead center (BDC) are marked in Figures 4.16 and 4.17. In Figures 4.16 and 4.17, the transients marked"Recorder Transients" are not part of the solution.1 Oscilloscope photographs which correspond to Figures 4.16b and 4.17b are shown in Figure 4.18. These photographs, although having poorer resolution than the XY plotter records, show that the transients are nonexistant in the solution. A hypothetical engine mounting is shown in Figure 4.19. Vibrations of the engine in the x' y' coordinate system, due to the load torque fluctuations and to the internal "shaking" forces, are analyzed with the assumption that the vibratory motions of the engine have a negligible effect upon the previously determined internal reactions. The equations for vibrations with assumed small motions of the engine casing are given below. The mass of the engine is mo The transients in Figures 4.16 and 4.17 are attributed to the XY plotter which was used to draw the figures, and which did not have an adjustable damping feature. The plots are included here because of their high resolution.

the radius of gyration is assumed constant and is given by ro, and the gravity center is assumed fixed at a distance b along the vertical centerline. (Fe + FAy) - 2 ky' - 2 Cy = my (4.40) (FAx - F) 2 k2x - 2 C2x = mx - b (4.41) * 00 F 2 k 2 4 - 2clo = m (r2+ b2) -mob (4.42) a jB l 0 00 Assumed parameters are given below and the computer circuit appears in Figure (4.20). Records of vibrations in x, y, and, as the engine comes up to speed, are shown in Figure 4.21. mo =.1 lb-sec2/in k1 = 10,000 lb/in rO = 20 inches k2 = 5,000 lb/in b = 0 inches C1 = C2 = 40 lb-sec/in t = 5 inches All circuits of Section 4.2 are scaled with an amplitude scale factor of a = 104 and a time scale factor of 1 = 103. The kinematic circuits of Section 3.2, which are required in the dynamic analysis, are to be interpreted accordingly.

ye Figure 4.19. Engine Mounting.

~~~~6~.Z -10 sina JLi 6 toI~~~~~ 0 F2L L m(Fe + FAy) i ------.~~( -^ -IO' I. F' _ - F - -,o 7L _ m2r turbn ~' i^ Figure 4.20. Circuits for the x', y', and q Motions of the Crankcase. [Equations (4.40), (4.41) and (4.42)]. [a = 104, p 103.

-97-.10 i —t'.e_ —_'t 1 i _'eL i inches 0 ""':^sl' L5 """ - V I ~' - p LP.'00 4 * I - -t.: y'.05 _ IIi i t *p, - _! ~I 1 i 1T 1 Ut-t-t-, it b]-~ —: —. 4 - l ——;~ 4 -r — —: -L Figure 11.21. Records of x', y' and <' as the Engine Comes Up to Speed. * 7'. 0 0 2, - - I I - Aradians 0 A' A ~.; I \' It i.... ~ L - r Fi gure..:21. Becords of x'I yI and f as the Engine Comes Up to Speed.

CHAPTER 5 THE SIMULATION OF LINKAGE SYSTEMS WITH ELASTIC LINKS Interest in elastic effects arises when high operating speeds and high performance standards are required in the design of linkage systems. Two cases are considered in this chapter for which a simulation is described that takes into account the elasticity of a particular link. In both cases, elastic deflections are determined as perturbations from the rigid body motions. 5.1 An Elastic Link With A Lumped Mass For some linkage systems, the configuration of a particular link may be such that it can be assumed that only bending loads result from the link accelerations. Shigley first described this situation with the example of an inversion of the slider-crank mechanism. Figure 5.1a shows a schematic of the slider-crank inversion and also included in the figure are examples of other linkages to which this analysis can apply. A mass m, which is large relative to the rest of the link, and which is located at the extremity of the link, will result in a single predominate mode of vibration. If the distributed mass of the rest of the link is assumed to be negligible, the perturbed motions of the point mass can be described with the solution of an ordinary differential equation. The equation can be derived by summing forces on the mass in the direction of A Joseph E. Shigley,'Simulation of Mechanical Systems, The University of Michigan, Engineering Summer Conferences, Ann Arbor, Michigan, August 8-19, 1966, p. 338. -98

-99 (a) (b) 4 A4 ^^ ^^ _ ^^. (c) Figure 5.1. Elastic Links With Bending Loads.

-100At is the absolute tangential component of the acceleration of point m on the rigid link. Examples which describe techniques for deriving the equations, and computer circuits for rigid body accelerations are given in Chapter 3. m(+ At) =. - k A (5.1) The coefficient k characterizes the elasticity of the link and is derived by assuming that the elastic strains due to dynamic loading are the same as those for a statically loaded link. Thus k is given by Equation (5.2). k = 3EI (5.2) Im(Im-p)2 For the schematics in Figures 5.lb and 5.1c, k is a constant; for the slider-crank inversion of Figure 5.1a, the length Ip varies and k is a function of time. In any case, the circuit for the equation of the perturbed motions is easily implemented once the terms required from the rigid linkage simulation are derived. If the axial component of force at a pin connection is appreciable, that portion of the link between pins undergoes column loading; and if, in addition, mass in the portion between pins is accelerated, the link has time variant loading as a beam-column. With this type of loading, there exists the possibility for elastic instability.2 Therefore, an analysis based upon static conditions is not valid. A method for simulating this case is described in Section 5.2. Stephen P. Timoshenko and James M. Gere, Theory of Elastic Stability, 2nd ed., Mcgraw-Hill, Inc., 1961, p. 160. Also, Heinz Houben, "Untersuchungen uber die stabilitat elastisher Bewegungen in der Koppel eines Viergelenkgetriebes', Ph.D. thesis, Rheinisch-Westfalischen Technischen Hochschule, Aachen, 1965.

-101 TABLE 5.1 NOTATION WHICH APPLIES TO THE VARIABLES AND THE PARAMETERS FOR THE ANALYSIS OF SECTION 5.2. A point A on the linkage AX translational acceleration of point A in the x direction A AY translational acceleration of point A in the y direction A Ax translational acceleration of point P in the x direction P p AY translational acceleration of point P in the y direction ax = AX/ an alternate form of the acceleration of point A in the x direction ay = Af/1 an alternate form of the acceleration of point A in the y direction B point B on the linkage b width of the rectangular cross-section of the elastic link Ci integration constant E modulous of elasticity F axial component of force acting on the elastic link FH component of force which acts at point B of the four-bar linkage and which is normal to the follower link g gravitational constant h height (or depth) dimension of the rectangular cross-sectional area of the elastic link I3 moment of inertia of link ~3 about its center of rotation Izz second moment of the cross-sectional area of the elastic link about the neutral bending axis EIzz k = Z stiffness parameter of the elastic link wl4 k' = n4k stiffness parameter of the elastic link I link length; with subscript, designates a particular link M bending moment

-102 TABLE 5.1 (CONT'D) O origin of the inertial reference frame with coordinates XY P point P on the linkage Ti the time variant portion of the product solution to the partial differential equation for the ith mode of vibration t independent variable, real time V shear force w mass per unit length of the elastic link X abscissa of the inertial reference frame x abscissa of the noninertial reference frame Y ordinate of the inertial reference frame y ordinate of the noninertial reference frame a angular position of the crank, link 11 of the four-bar linkage angular position of the noninertial reference system 7 the angle of the slope of the elastic link deflection curve = y/I dimensionless form of the perturbed deflection of the elastic link generalized eigenfunction O = x/i dimensionless.form of the length dimension of the elastic link = Ci/12 an alternate form of the integration constant IP proportionality constant relating real time t to computer time T. T = it v a parameter inversely proportional to the elastic link's ability to resist buckling p density per unit volume a proportionality constant relating a computer voltage to the physical variable which it represents; volts = a (physical variable)

-103 TABLE 5.1 (CONT'D) T computer time 4 angular position of the follower, link 13 of the four-bar linkage o r= 2 4k angular frequency of the fundamental mode of vibration of the elastic link 5.2 Distributed Mass'Linkage Analysis The link shown in Figure 5.2 is accelerated in the XY plane by forces applied at pins A and B. The pins are assumed frictionless and the link is assumed to have a uniformly distributed mass of w units per unit length. As the link is accelerated, it undergoes loading as a beamcolumn and it will deflect elastically with potentially unstable vibratory modes, depending upon the link parameters, the external loading at the pins, and the excitation frequency. It is the objective of this analysis to derive a mathematical model which is suitable for analog computer implementation, and which will enable the investigator to analyze vibratory motions. In addition, the model is to enable the variation of important parameters and thus provide a convenient means of evaluating elastic stability. The procedure is to derive elastic deflections as perturbations from the rigid body motions. These elastic deflections are determined

-104 y Y P.. O. X Figure 5.2. Elastic Link With Beam-Column Loading. a A aA.+ x -,A O Y Pdt M+SM F+SF A i A O A. x Figure 5.3. Differential Element of a Link With Beam-Column Loading.,y at

-105 with respect to a noninertial reference frame xy, the motion of which is assumed known with respect to the-inertial coordinates XY In the analysis which follows, it is assumed that the material is homogeneous, that it is isotropic, and that the link cross-sections remain plane, and normal to the neutral bending axis. The equations of motion are derived by taking moment and force summations on the differential element shown in Figure 5.3. zMp; M -(M+8M) + (V+5V)6x = 2 () (5xa3 _P 12 C~xb12t2 EF; Vcos7 - (V+5V)cos(7+87) - F siny + (F+bF)sin(y7+8) = w A 8x (5.4) EFx; -Fcos7 - Vsin7 + (F+5F)cos(7+57) +(V+8V)sin(7+8y) = w A 8x (5.5) The rotary inertia of the differential element involves two acceleration terms. These terms, which are included in Equation (5.3), are the angular acceleration o3y/ xat2 which is due to the perturbed *0 motion, and the angular acceleration of an otherwise rigid link,. 3 2 Timoshenko shows that including the term, a y/axat produces a negligible effect when the cross-sectional dimensions of the link are small compared to-the link length.3 The effect of the rotary inertia of the differential element which is due to the rigid body acceleration term can be assessed by considering the importance of this term as it affects the free oscillations 3 S. Timoshenko, Vibration Problems in Engineering, 3rd. ed., D. Van Nostrand Co., Inc., Princeton, N.J., 1965, pp. 331-335.

of the rigid link when suspended as a pendulum. This situation is depicted in Figure 5.4, and the equation of motion, given in (5.6), is derived by summing torques about point A. 1 O h2 1 w 1(1 + 4) + 2 w g I2 sinp = 0 (5.6) 2 wh o Including the term, -- p dx, and integrating over the link length, results in a correction to the inertia torque given by the expression (1 + h?); thus, if h, the link depth dimension, is small compared 4l2 to the link length I, the rotary inertia term due to finite link depth may be neglected. The analysis proceeds with the condition that the link depth dimension h is to be small compared to the link length I. Equation (5.3) thus reduces, in the limit, to Equation (5.7). aM V(5.7) Expanding terms in Equations (5.4) and (5.5), and neglecting terms involving differentials of second order, results in the expressions given in (5.8) and (5.9). w A 5x + B(Vcosy)-5(Fsiny) = 0 (5.8) w; 6x - (Fcosy)-5(Vsiny) = 0 (5.9) For small deflections, the angle y remains small and cosy _ 1, sin7 ~ tan7 = ~y/ox. Consistent with the condition of small perturbed deflections, the differential equation of the deflection curve is given

-107 A. I wxjdx wgdx Figure 5.4. Rigid Link Suspended as a Pendulum.

-108 by Equation (5.10), and the shear force is described by (5.11). 2 EI x2 = M (5.10) *! a EI -) (5-11) Using these relations in Equations (5.8) and (5.9), they become, in the limit, the equations given in (5.12) and (5.13). 2 EI -t (F i) + w(A + x -y + ) = (5.12) d-)x4 Xe x A dt_F = w(A~ - yp3 - x~- 2p ) (5-13) vP ~-xl?-2 A ) (5.13) Cx -A a A final simplifying assumption is possible by noting that the terms 21 3y/ot and yF3, which appear in Equation (5.13), may be omitted. Omission of these terms suggests that the variation of the axial force along the link is not appreciably affected by the perturbed motions, and is due primarily to loading at the pins which is manifested by the rigid body accelerations. Justification for omitting these terms stems from the condition that the perturbed motions y(x,t) are to be small. Thus, Equation (5.13) may be integrated and substituted into (5.12). The integrated form appears in Equation (5'.14). F = w[A x - 2x2 + Ci(t)] (5.14) Had the omitted terms been included, it is observed that, upon substitution into (5.12), the perturbed variable would appear in terms of second order or higher, which would further diminish their significance in the equation.

-109 - The variable Ci in Equation (5.14) is a constant of integration in x.which is time variant and must be determined at a pin connection. With the assumptions and conditions thus imposed, the equation of motion for the perturbed deflections is given in (5.15). The boundery conditions are as stipulated below. c4 xa A'2 + wA + x + y2] = 0 (5.15) A. t2 y(O,t) = 0 (5.16). y(l,t) = O (5.17) (o,t) = o (5.18) (i,t) = O (5.19) Initial displacements and velocities of the perturbed variable may be arbitrarily prescribed. For this analysis they are to be zero and their specification is given in (5.20) and (5.21). y(x,O) = o (5.20) (x,0) = o (5.21) dat The following parameters and variables are designated which enable a representation of the equation and the boundary conditions in an alternate form. EIzz A = y/ ay' = AY/t Al e = x/l X = CO 22

-110 With the substitution of these parameters and variables, Equation (5.15) and the boundary and initial conditions appear as given below. 2k 4A -a [(2aXe-22+2kX)] + 2(aY +' + A^ 32) = 0 (5.22) ot2 A(0,t) = 0 (5.23) A(l,t) - 0 (5.24) (o,t) = 0 (5.25) o92 a)A g2 (lt) = o (5.26) A(,0O) = 0 (5.27) 9 (@,.) = 0 (5.28) A method for simulating this link is suggested by a technique first described by Oliver for the simulation of a cantilever beam with a distributed mass undergoing free vibrations. The technique, which involves the derivation of modal solutions, requires the assumption of a solution in product form that enables representation, for n modes of vibration, of the differential equation in the form given in (5.29). n ~ fi(t) i(Q) = 0 (5.29) i-1 T.. A. Oliver, "Cantilever Beam Simulation on the Analog Computer", Unpublished paper presented at the Engineering Summer Conference Course, Simulation of Mechanical Systems, University of Michigan, Ann Arbor, Michigan, August, 1965.

-111 Each time variant function fi(t) is implemented on the analog computer to determine a solution Ti. The eigenfunctions ~i(@) are chosen to.satisfy the boundary conditions, and the independent space variable of the eigenvalues is represented by the independent variable "time" on the computer, but at a frequency which is much higher than the frequency of fi(t). The two domains of time are possible on the computer by allowing the integrating amplifiers associated with the generation of the ti(q) to operate in the repetitive mode. The generation of the eigenfunctions is simplified if they are trigonometric. Individual modes or their sums in the proper proportion, as given in Equation (5.30), are applied to the vertical amplifier of an oscilloscope. a= Z Ti Bi(O) (5.30) i=l The sweep amplifier is synchronized with the repetitive mode operate period, and the period of each mode is adjusted so that the horizontal display corresponds to the link length. An image of the elastic link is thus displayed as it vibrates. Parameters can then be varied and the elastic response and stability can be evaluated. This procedure is applied to Equations. (5.22.) through (5.28) by assuming a solution of the form given in (5.31). n A = Z Ti sin(i It ) (5.31) i=l The terms required in Equation (5.22) are determined by differentiating (5.31) with respect to the appropriate independent variable. 2 = z - sin(i i e) (5.32) at2 i=l t2

-112n.4= Z i A cos(i A O) (5-33) aQ i=lJ^vt =' i 4 Ti sin(i 9) (5.34) Substituting Equations (5.32) through (5.34) into Equation (5.22) results in Equation (5.35). 2k ~ i 44 TTisin(i A o) i=l n - [(2aX - 2g2 + 2X)( Z itTi cos(ing))] 09ae~ ~ i=l n 62Ti + 2[ay + GP + i l2 —sin(it9) n - 2 Z Ti sin(ir)] = 0 (5.35) i=l Since Equation (5.35) is not yet in the form of (5.29), and hence is not separable, separation must be forced with Fourier series approximations to the terms aY, 9, 9cos(itQ), and @2cos(ir). The terms aY and' are expanded in a sine series, and, since Gcos(irQ) and 92cos(irs) are to be differentiated with respect to 9, these terms are expanded in a cosine series. The derivation of the Fourier coefficients is given in Appendix 2. Including terms for the first three modes of vibration, and using the first three terms of the Fourier series, Equation (5.35), upon expansion, appears as given in (5.36).

-113 2k(~4TL sin7r + 244T2 sin 2Q + 34x4T3 sin 3ie +...) + (2aXlTl(2 sinvi -Q ~sin2S t...) 2 9r2 12 52*3-r sin3iQ+...) + 2ax. 2tT2(- 201 sin -9 + 2 sin 2i9 - 52'3 sin39 +...) 9 A2 2 25n2 + 2aX3tT3 (- 52'2n sin2it@ + 3J sin 3:9 +...) 3 25n2 2 - T 2( 6. siniQ - 9 sin 2,9 + 1 sin 3 + * *) - 2T2^2(- 20 sin si8ni +3 sin2n9 - 523 sin37n +...) 9K 12xt 25xr'*~ 2 3-10 52-2 18.A+3 - 3-AT 32(e-3 singc - -2 isin2nQ + 1'3 sin3tQ -...) -3 (16ic 25x 18i + 2r2X(Tl sintQ + 4T2 sin2ne + 9T3 sin3ng)] + 2 (4 ay sinsQ + - aY sin3xO + + sin 2 - 2 ~. 2- p sin2?t + 2'f sin3 -... + T1 sinnQ + T2 sin2tQ + T3 sin3jt +... - 2 T1 sinnr. 3 - 2 si2 - Tsin3rn -...) = (536) Terms which are coefficients of each of the eigenfunctions are collected, and, because of the orthogonality of the eigenfunction set, the collected terms may be equated to zero. This results, for the first three modes of Vibration, in the system of three ordinary differential equations given below. T1Ekl +aXoI 2 +22+3 ~2+ 22 23T T1.+ [k4 + ax. 2232 + 2] T 2 12 + [209 _ 4~ a] T2 - S 2 T3 4^- 2 ( ) - Jr- - p (5 37) V it

T2 + [24 t4 k + F - 2 + 12 - 2] T2 - [.5 ax 3.52;2] T3 [40 ax 20 2] T.=.! -a B - 2]0 T1_ (5.38) 9 9 IT T3 + [3 A k + 92 ax _ 18+ 3* f2 + 9a- - 2 12 T3 + [3 22 62 x] T2 225 25" 15 2 T1 4 y 2*.- T -_ aY- - (5.39) 16 3 i 37t To this point, the formulation is general and can apply to any link in plane motion which satisfies the conditions of the analysis. The link would ordinarily be a coupler link, such as the connecting rod in the slider-crank) or the coupler of a four-bar linkage. Specifying a particular linkage system enables the derivation of specific expressions for the integration constant X, and the rigid body acceleration terms. This, in turn, enables the implementation of the perturbation equations and the derivation of a solution. Since it is difficult to make a meaningful generalization beyond this point, an example is described for the case of a four-bar linkage with an elastic coupler link. 5.2.1 Simulation of the Elastic Coupler of a Four-Bar Linkage The four-bar linkage with parameters as specified for the kinematic analysis of Chapter 3, Section 3.1, is used here. The rigid crank is driven at a prescribed velocity, and, through the elastic coupler, drives the follower link which has a finite moment of inertia 13 about its center of rotation.

-115 Derivation of the Integration Conttant The coupler of length 12 and the follower 13 are shown in the schematic of Figure 5.5. FH, the component of the force reaction at pin B which is normal to the follower link, is defined, in Equation (5.40), in terms of the axial force FB and the shear force VB which act on link 12 FB and VB are given in Equations (5.41) and (5.42) FH = Fg sin(4-p) + VB cos(O-0) (5.40) FB = w I [ax _ 1 2 + A] (5.41)'2 o2 ~3 VB EIzz_ (lt) 1 EIx [T - 8T2 + 27T3] (5.42) 12 By summing moments about 03 on link 13 FH is derived in terms of the torque parameters of 13 @0 FH - 1 3 (5.43) 13 To facilitate scaling, FHmax is defined in terms of the maximum angular acceleration of I3 which can be determined from the previously derived kinematic analysis. (See Figure 3.16, page 52.) 1 00 FHma= I3 max (5.44) 3 To characterize the constants of link 13 in terms of the parameters of L2, the term indicated in (5.45) is defined. h. =' IZ (5.45) 3 max 2

FS /2 B F/ I H I A /3,/l Figure 5.5. The Four-Bar Linkage With Force Components at Pin B.

-117 In Equation (5.45), 13 represents the load which the- coupler link must drive, and, through the acceleration of 13, a reaction at pin B is generated which results in the column loading of the elastic link. In Equation (5.45), the inertia of 13 is defined in terms of the Euler critical load for a column with pinned ends. This applies to link 20 12 and is given by the expression 2EIzz/l2. Thus, for a given -max' the term v, which is proportional to 13,. can be interpreted as a parameter which is inversely proportional to the elastic link's ability to resist buckling. Substituting Equation (5.45) into (5.43) results in a description of FH given by (5.46). n2EIz FH v zz (5.46) 12ax By substituting the expressions of (5.41) and (5.42) into Equation (5.40) and equating (5.40) to (5.46), an explicit expression for X can be derived as given in (5.47). vA2k i- - k(Tl-8T2 + 27T3)cos(I4-0) Tmax sin(4-1) + a 2 - a (5.47) 2 Note that substituting the above expression into Equations (5.37) through (5.39) makes the latter equations nonlinear. Expressions for the rigid link acceleration terms remain to be determined. Rigid Link Accelerations The terms,, d may be taken direct The terms l,,c and may be taken directly from th e circuit for the rigid link acceleration analysis which appears in Figure 3.15,

page 51. The terms ax and ay can be derived from a consideration of the rigid link accelerations at point A as shown in Figure 5.6. The accelerations are given in Equations (5.48) and (5.49). ax = 1 Ax = 1 [-~1_Ocos(a-p) 12 A 12 -. sin(a-) ] (5.48) aY = 1 Ay= 1- [-lc2sin(a-P) 2 A I 2 2 + I1 a cos(a-P) (5.49) The parameters can now be specified and the equations can be scaled for computer implementation. Computer Implementation Parameters as listed below are assumed for this example. to = 3 inches E = 30xlO61b/in2 11 = 1 inch p =.281b/in3 2 = 2 2 inches a = 628 rad./sec., constant 13 = 2 inches a(0) = 0 k' = i4 k, variable v, variable The equations which are to be implemented are listed below. The circuits appear in Figure 5.7 with scale factors of a = 10-4 and = 103. lOT1 + 10[k' + 9.86X + 4.93ax - 2.89P2] T 22.2[2ax - p2] T2 -9.8332T3 6.37(2J+13) (5.50)

\ A Y a Aa:, A. a t x Figure 5.6. Acceleration Components at Point A of the Four-Bar Linkage.

-120 100'2 T2 + 1OO[4k' + 9.86 + 4.93aX - 1.97 IT2 - 1562a - 216 T3 - 55.6[2aX - 2]T1 = 7.973 (5.51) l T3 + 1000[9k' + 9.86X + 4.93aX - 1.782F32 T3 - 104i2 T 692a - 9[2x 2] T2 =- 23.6[2ay + (5.52) Ti(O) = 0, T1(O) = 0 (5.53) T2(0) = 0, 2( = = 0 (5.54) T3(0) = 0, T3(0) = 0 (5.55) k' [- -.- (T1-8T2+ 27T3)cos(4-p) ] h^ ^ t max __ sin(4-p) + 2 (5.56) aX =.1 a sin(a-p) (5.57) 12 ay- a cos(a-p) (5.58) 12 = T1 sinnt (5.59) 2 = T2 sin 2Q9 (5.60) A3 = T3 sin 3i~ (5.61) n A(t) = ZA, i = 1,2,3 (5.62) i~l

.1000. 1000013g X07T1.10 25(2.542).55_-108l' loro25. 1o /os (*) X 1 - l-l lOaOOT2. lOjT 2 - I n drts -bOOOT) — 2.OBwo.10 Tj(2 —-- -109T > Vln jutplr r eur 1 h 10 nI-.) OOT~ 10 -. vk* \^/lO 10 l10T0 0___ l ( - IOOT2 \-.986 1In addition to the circuits giveM above, 2. P^ \ 1S^ following terms: IOOT20 -100T20 IQW^. b2 1i T Figure 5.7a. Circuit for the Perturbed Motions of the Elastic Coupler of a Four-Bar Linkage. a) [Equations (5.50) through (5.58) a. 1,-4, - 103].

10 10.15 Figure 5.7b. Circuit for the Perturbed Motions of the Elastic Coupler of a Four-Bar Linkage. b) [Equations (5.59) through (5.62). a = 10'4, 4 = 103].

-123 The terms which must be taken from the kinematic circuitry of Section 3.1 are listed below. These terms are listed with the coefficients which correspond to the given scale factors.'4 106 cos(a-0) p 10 sin(a-p) 2 106 cos(4-0) 106 sin(4-. ) With these circuits, the parameters v and k' can be varied and records can be derived. The parameter k' = 4k characterizes the natural frequency of the fundamental mode of vibration. Let this frequency be designated as c, and, as seen from Equation (5.36), it is given by (5.63). X, _ i2' k (5.65) The circuits are scaled so that this frequency may be varied in the neighborhood of resonant operating conditions. Varying this frequency requires the resetting of fewer potentiometers than if the drive frequency were to be varied. Noting, from the kinematic records of Figure 3.16 in Chapter'3, that significant second harmonics of the drive frequency are present in the accelerations, it can be assumed that appreciable excitation of the elastic link due to inertial reactions will occur at a frequency which is twice the crank drive frequency. In order to operate below resonance, the parameter k' must be selected so that the frequency of the fundamental mode is somewhat" greater than twice the crank drive frequency. From the definition of k, repeated below, the depth dimension, for a given material and link length, can be derived once k (or k') is specified.

-124 k- EIzz E h2 (5.64) wE4 12pLf 2 2 In this equation, Izz is the second moment of the cross-sectional area, and, for a rectangular link of width b and depth h, is given by Equation (5.65). Izz = b h (5.65) 12 Hence, with the specified parameters, h, in terms of k, is given in (5.66). h = 1.36x10-4 k inches (5.66) Examples of computer records are shown in Figures 5.8 and 5.9 with k' equal to 2.5x106 sec2, and v =.4 6 Figures 5.8a, 5.8b, and 5.8c show the individual modes, and Figure 5.8d shows the link vibrations with all three modes in their proper proportions. All figures show configurations at random intervals of time for a stable mode of operation. Figure 5.9 shows records of T1, T2, and T3 as the linkage is set in motion. The method for deriving the oscilloscope display of the link as it vibrates was described, in general, earlier, and is reviewed here with reference to amplifiers in the computer circuits. The first step is that of synchronizing the sweep frequency of the oscilloscope with the eigenfunctions so that the horizontal display corresponds to the link length. The harmonic loops used to generate the 6h With k' = k4 2.5x106, the depth dimension, as given by Equation (5.66), is.022 inches. Hence, the depth to length ratio for this very slender link is.008. This value is quite within the requirements set forth for neglecting the rotary inertia terms.

-125 eigenfunctions (see Figure 5.7b) are operated in the repetitive mode, and the outputs of each of the amplifiers numbered 3, 6, and 9 are applied to the oscilloscope vertical amplifier. The repetitive mode trigger signal is used to trigger the sweep amplifier. For each eigenfunction, fine adjustments for the period of the sinusoids are made with potentiometers PI, P2, and P4. The outputs of amplifiers 10, 11, 12, or 13 in Figure 5.7b can then be applied to the oscilloscope vertical amplifier to observe the individual or combined modes of vibration. In all of the circuits, for both the rigid and elastic linkage simulations, parameters are varied by resetting potentiometers. This requires, *in most instances, considerable a priori hand calculation. The relatively recent development of hybrid computers should overcome this difficulty. For a hybrid computer, equipped with servo-set potentiometers, digital software development, as an adjunct to the analog circuits developed here, will enable the rapid and automatic resetting of the many dependent potentiometers, once key parameters are set on their corresponding independent potentiometers. This ultimate development will provide a final facility in computer operation, which should enhance those efforts of the investigator that require many trials and combinations in analysis and synthesis.

N 03 F;:E?: e * c* * cni*: u~ ~ - $1 It HO C' O tt 0 N) Sg: ^: <:H'' S-^ 0 "~~~~~N * 20 o N)'0' C> Sfr 0'r 03 ~ t 0Z NN) a I 8 8 8 8 8 111111 8':? pS H rt~~~~~~~~~~~~~~~~~a:a: - o S I~~ I I 1. *... 000.4004X2440000::.. I:f,.:;E 1. l.E:i r ~ ~ ~ ~ i E i ifi iiE.:-i ~E.i:::-:... i: ddif; 11<&1*1 ^- |EI^ I I 1 * O: i ff'..ir:':-::.:!? f i1::::::::, 1::,:::; U * *::::E:::E: i8 1:: L;:::;l::::::::: ~ p:;; E:'1:.:.:*1:. <: *:':::''": -:i: 1: >:::::: I^^: *. i i. di f f010S;70l 0id:ij0i~ i: i!40.:X::j: 0:it::: X: 0:: -:40040ii;A::0jS:Xjtg >1:I:::::K~~~~~~~~~~~~~~~~~~~~~~~~~~: K (% ~ ~ ~ ~ ~ ~~~~:::::':::::::::::::: —: N). I t N) 0N t 0b

ro II t o-_?U t si -:.......,..-....... 4 - 1 1:.: ----:- - 0 4 - So ~I l I I.1 14I t,: O -0 H t,):::::::::::::: t':': _:::;:;. ":;'::":~:::i:':.::: *,sos~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ l::::;,::: x \J1 S 0 0;:;:x,...i.... k * 1 c o -;:' *1: 1'1':-:.: 1:::: *: * 00;:**:; E il D_: i _ _:::~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~il:-: 1- o: -- II I1' I

Il i J #%;. l I A t il IIf I i t.1i I t- - I i it I! -.1.1 I'i i R fi, A i - _. _ 1 [7'I.-7.i: -J-1 -1 - i I- A -. IV.. If I l A iN -r i I Al I II I/ I\ A It Il I!\ ift i I:-it I W I tI 11\ v N T. I il I,\ I / 1It 11 I. \I l - - - - --- -...1 - I. -I 1, - 1: I. L - I- I - i. -. -,j.. I L -1 Lj LJtt. i!.1 I I - I I i. -1.,. J:A -.I: i 1 41-t. -I 3 1, I1a"xea.04.06 t i' i |' I:. 08;.09.10. T1 t z r.....~ -"} 1.F!11 A JA P1 J. rP pA I tI - - t —/Mat L - I. tW iWl — I i I/ I L i t I UW,'i I iWIL ilI 1.WLJjA I LU. —.... I l - t A / Il iA, If I CO.I~ I -i.. I- l —-t.t:. tV I vu I L1I.' tvu I1 I, I I: X:'F-' i —:-: I -'- f I - i i.: -W- i - i - t I v 4 f- -- f i 4...:.. -., -.'.7 -; i` I, I - i'.': - - -'" _ _. 4.. t- -- t I! __t "4 —-.: 4- - i 1 l. i!: i': * i.; I' t' I1.!!-! 1".1 ~~~~~~~~~~~~,.'' ~ i...!.i -:i'! t I I__ -Y.I q -: f t -C 5'4 J i X'\C8W!'tf,,," ^i 2^,,'i,^:./1 A,, I 1 i' I i.'.. -iI: -zr~ziizzizzi~z~., T-T —,[FT'T.PT iTTT,_+_. i i I t i I.. 4- t. I I I I I 11! I I! I.1 1 i I I -: i _; I I II I I t Figure 5.9. Records of T1, T2, and T as the Linkage Is Set In Motion With the Elastic Link in an Initially Unperturbed State.

BIBLIOGRAPHY Adams, D. P., et al., "Kinematic Aid From Graphical Computer-Output", Transactions of the Seventh Conference on Mechanisms, Purdue University, November, 1962. Burstall, Aubrey F., A-History of Mechanical Engineering, M. I. T. Press, Cambridge, Mass., 1965. Chace, Milton, "Development and Application of Vector Mathematics for Kinematic Analysis of Three-Dimensional Mechanisms", Ph.D. thesis, University of Michigan, Ann Arbor, Mich., 1964. Churchill, Ruel V., Fourier Series and Boundary Value Problems, McGraw-Hill, Inc., New York, 1941. Clymer, Ben A., Methods of Simulating Structural Dynamics, Presented at the Midwestern Simulation Council meeting, Columbus, Ohio, March 21, 1966. Crossley, F. R. E., "Die Nachbildung eines mechanischen kurbelgetriebes mittels eines elektronischen Analogrechners", Feinwerktechnik, vol. 67, June, 1963. Cyr, L. Jr., "Kinematic and Dynamic Analysis of the Crank-and-Rocker Mechanism", Department of Mechanical Engineering, University of Michigan, Ann Arbor, Michigan, 1961. DeVries, T., W., "Generalized Linkage Analyzer", Department of Mechanical Engineering, Purdue University, West LaFayette, Ind., 1961. Fifer, Stanley, Analogue Computation, Vols. I-IV, McGraw-Hill, Inc., New York, 1961., Freudenstein, F., Robbin, J., and Chinlund, T., "Five-point Four-Bar Function Generators", Department of Mechanical Engineering, Columbia University, New York, 1958. Hall, Allen S., Kinematics and Linkage Design, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1961. Hall, Allen S., and Hahn, F. E., "Four-Bar Mechanism Analysis" Department of Mechanical Engineering, Purdue University, West LaFayette, Ind., (no date). Harrisberger, Lee, Mechanization of Motion, John Wiley and Sons, Inc., New York, 1961. -129

-130 Hartenberg, Richard S. and Denavit, Jacques, Kinematic Synthesis of Linkages, McGraw-Hill, Inc., New York, 1964. Hirschhorn, Jeremy, Kinematics and Dynamics of Plane Mechanisms, McGraw-Hill, Inc., New York, 1962. Houben, Heinz, "Untersuchungen uber die stabilitat elastisher Bewegungen in der Kippel eines Viergelenkgetriebes", Ph.D. thesis, RheinischWestfalische Technische Hochschule, Aachen, 1965. Howe, R. M. and Gilbert, E. G., "Trigonometric Resolution in Analog Computers by means of Multiplier Elements", IRE Transactions on Electronic Computers, vol. EC-6, no. 2, June 1957, pp. 86-91. Hrones, John A. and Nelson, George L., Analysis of Four-Bar Linkage, M. I. T. Press, Cambridge, and John Wiley and Sons, Inc., New York, 1951. Kaplan, Wilfred., Advanced Calculus, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1952., Ordinary Differential Equations, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1958. Keller, Robert E., The Accuracy of One Method for Mechanism Simulation by Electronic Analog Computer, Delivered at the ASME meeting November 29 to December 4, 1964. ASME Paper No. 64-WA/MD/2., "Application of the Analog Computer to the Study of Kinematics", Ph.D. thesis, Stanford University, Stanford, California, 1962., "Mechanism Design by Electronic Analog Computer", Trans. of the 7th Conference on Mechanisms, October, 1962, pp. 11-21. Klemm, Friedrich, A History of Western Technology, M. I. T. Press, Cambridge, Mass., 1964. Korn, G. A. and Korn, T. M., Electronic Analog Computers, 2d ed., McGraw-Hill, Inc., New York, 1956. Lenk, E. W., "Instrumentelle und elektrische Verfahrung zur Erzeugung und Aufzeichnung von koppelkurven", Konstruction, November, 1962, pp. 393-396. McLarnan, C. W., Analytical Synthesis of Function Generators Using the Slider-Crank Inversion, Presented at the Winter Annual Meeting of the American Society of Mechanical Engineers, New York, November 29 to December 3, 1964. ASME Paper No. 64-WA/MD-13., "Program for Designing Four-Bar Linkages with Five Precision Points", Department of Mechanical Engineering, The Ohio State University, Columbus, Ohio, (no date).

-131Mann,. Robert W., Computer Aided Design, Presented at the ASME Mechanisms Conference, Purdue University, October 19 to 21, 1964. ASME Paper No. 64-Mech-36. Oliver, T. A., "Cantilever Beam Simulation on the Analog Computer," Unpublished paper presented at the Engineering Summer Conference Course, Simulation of Mechanical Systems, University of Michigan, Ann Arbor, Michigan, August, 1965. Raven, Francis H., "Velocity and Acceleration Analysis of Plane and Space Mechanisms by Means of Independent Position Equations", J. Appl. Mechanics, Trans. ASME, Vol. 80, 1959, pp. 1-6. Sandor, G. N. and Freudenstein, F., "Kinematic Synthesis of Path Generating Mechanisms", IBM 650 program library file no. 9.5-003, IBM data proc. div., New York, 1959. Shigley, Joseph E., Simulation of Mechanical Systems, The University of Michigan, Engineering Summer Conferences, Ann Arbor, Michigan, August 8-19, 1966.._, Kinematic Analysis of Mechanisms, McGraw-Hill, Inc., New York, 1959.,Theory of Machines, McGraw-Hill, Inc., New York, 1961. _, Simulation of Mechanical Systems, McGraw-Hill, Inc., New York, 1967. Timoshenko, Stephen P., Strength of Materials, 2nd ed., vol. II, Van Nostrand, New York, 1941. _____, Vibration Problems In Engineering, D. Van Nostrand Co., Princeton, N. J., 1965. Timoshenko, Stephen P. and Gere, James M., Theory of Elastic Stability, 2nd ed., McGraw-Hill, Inc., 1961. Willis, Robert, A System of Apparatus for the Use of Lecturers in Mechanical Philosophy, Especially in those Branches Which are Connected with Mechanism, John Weale, London, 1951.

APPENDIX 1 In this appendix, digital computer solutions are presented which correspond to the strip chart records of Chapter 3. (See Figures (3.5), (3.16), (3.19), and (3.24) of the text). The digital computer solutions are given for both the four-bar linkage and for the slidercrank. Numerical values are given for each parameter for each fifth of the drive crank rotation. This corresponds, for a drive crank frequency of.628 radians/second, to time increments of two seconds. These solutions show excellent agreement with the analog computer records, and verify the proper operation of the rate resolver circuits which are the only critical elements in the analog circuitry. -132

-133FOUR-BAR LINKAGE DIGITAL COMPUTER SOLUTION LINK LENGTHS CORRESPOND TO THOSE OF THE ANALOG COMPUTER PROBLEM AND ARE AS FOLLOWS: LI(CRANK) = 1, L2(COUPLER) = 2.828 L3(ROCKER) = 2, LO(GROUJD LINK) = 3. ALPHA DOT = 0.62B32 RADIANS/SECOND. ALPHA = 0 RADIANS T = 0 SECONDS BETA DOT =-.314162303E+O PHI DOT =-. BETA DOT SQ =.986979557E-1 PHI DOT SQ BETA DBL DOT =-.872445133E-4 PHI DBL D01 ALPHA = 1.2566 RADIANS T = 2 SECONDS BETA DOT =-.822586978E-1 PHI DOT =. BETA DOT SQ.67664933E-2 PHI DOT SQ BETA DBL DOT =.921146177E-1 PHI DBL DOI ALPHA = 2.5132 RADIANS T = 4 SECONDS BETA DOT =.39859156E-1 PHI DOT. BETA DOT SQ.158875231 E-2 PHI DOT SQ BETA DBL DOT =.879233823E-1 PHI DBL DOT ALPHA = 3.7698 RADIANS T = 6 SECONDS BETA DOT =.220355814E+0 PHI DOT BETA DOT SQ =.485566546E-I PHI DOT SQ BETA DBL DOT =.170084756E-1 PHI DBL DOT ALPHA = 5.0264 RADIANS T = 8 SECONDS BETA DOT =.644278918E-1 PHI DOT =-. BETA DOT SQ =.415095312E-2 PHI DOT SQ BETA DBL DOT =-.201928524E+0 PHI DBL DOT ALPHA = 6.283 RADIANS T = 10 SECONDS BETA DOT S-.314162266E+0 PHI DOT =-. BETA DOT SQ =.986979303E-1 PHI DOT SQ BETA DBL DOT =-.196992196E-3 PHI DBL DOT.314162307E+0 =.9697952 E-1 r =.29600301F+0,255644736 E+O =.65354231 -1 r =.121323027E+0 285724001 E+O =.81638205E-1 =-.S&5124269E-1 214946155E-1 =.462018494E-3 =-.128710645E+0 263957332E+O =.69673475E-1 =-. 169391452E+O 314249 589E+O =.987528054E-1 =.295893246E+O

-134SLIDER-CRANK DIGITAL COMPUTER SOLUTION LINK LENGTHS CORRESPOND TO THOSE OF THE ANALOG COMPUTER PROBLEM AND ARE AS FOLLOWS: LI(CRANK) =1.0, L2(CONNECTING ROD) = 3.4 ALPHA DOT = 0,628 RADIANS/SECOND ALPHA = 1.5708 RADIANS T = 0 SECONDS BETA DOT' =-.184794118E+0 YB DOT =-.2 BETA DBL DOT =.389581495E-6 YB DBL DC ALPHA 2 2.8274 RADIANS T = 2 SECONDS 3ETA DOT =-.594849232E-1 YB DOT =-.6 BETA DBL DOT =.113982511E-0 YB DBL DC ALPHA = 4.084 RADIANS T:= 4 SECONDS BETA DOT =.151779686E+0 YB DOT =-.2 BETA DBL DOT =.6525178E-1 YB DBL DOT = ALPHA - 5.3406 RADIANS T = 6 SECONDS BETA DOT =.151798094E+0 YB DOT =.2 BETA DBL DOT =-.652343562E-l YB DBL DO ALPHA = 6.5972 RADIANS T = 8 SECONDS BETA DOT =-.594525914E-1 YB DOT =.6 BETA DBL DOT =-.113990702 E0 YB DBL-DO ALPHA = 7.8538 RADIANS T = 10 SECONDS BETA DOT =-.184794115E+0 YB DOT =.1 BETA DBL DOT =-.193551617E-4 YB DBL DO 298666763E-5 )T =-.510867034E+O;54 l1 5236 E+O )T =-.251482295E-1:80118772E+0.: 280563192 E+O:80040695E+0 IT =.280564937E+O;54121 787E+0 IT —.250557442E-1 46018377E-3 iT =:-.510367021E+O

APPENDIX 2 CALCULATIONS OF THE FOURIER COEFFICIENTS FOR EQJATION (5.21) ay-1 1 = ei sin(ine) ai 2 f sin(iOt)de 0 li o -.- cos(ie) - ai 4 4 4 e 0.-= a, sin(ixe) ai= 2 1 e sin(i8x0)de 0 2 = 2;, [sin(it0) - ine cos(iAO)] i2x2 o X 2 2 2. - 2 ai X.P 2n' 3x' o- -* a cos ite ecos(mne)= ai cos(ine) a'Li = 2 f/ e cos(mx0) cos(i0O)de a = 2 f cos2(ine)d i= m 0 f i f 0: 0 ai f- F1f [sin(m-f ) sin(m+i)rede o (m-i)n (+i i)r ai t [ i1 +.... (m + i) odd (m-i)7i2+ (m+i)2(2 ai m O (m + i) even

-136 m=iO: bai= 2 f1 2 cos2(aie)de 0 = Ile[1 + cos(2ien)]de o j1 ode 0 2 2 ecos ire 02cos(mg0) = acos(ie0) ~ai = 2 f e2 cos(me0)cos(i<O)de o m i ai = 2 1 02 cos2 (i(e)de i = o1 + 1 m i / O: ai=+2 )22 (m+ i)2 + for (m+i) even - for (m+i) odd m iO aI= + 21. i [2i22+ 3 3 2i2x2 6i2n2