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## Material Information- Title:
- Dynamic response analysis of complex mechanisms with multiple inputs
- Added title page title:
- Complex mechanisms with multiple inputs, Dynamic response analysis of
- Creator:
- Benedict, Charles Edward, 1939- (
*Dissertant*) Tesar, D. (*Thesis advisor*) Oliver, C. C. (*Thesis advisor*) Boykin, W. H. (*Reviewer*) Bullock, T. E. (*Reviewer*) Nahig, Joseph (*Reviewer*) Vance, J. M. (*Reviewer*) - Place of Publication:
- Gainesville, Fla.
- Publisher:
- University of Florida
- Publication Date:
- 1971
- Copyright Date:
- 1971
- Language:
- English
- Physical Description:
- xii, 85 leaves. : illus. ; 28 cm.
## Subjects- Subjects / Keywords:
- Coordinate systems ( jstor )
Degrees of freedom ( jstor ) Dynamic response ( jstor ) Inertia ( jstor ) Kinematics ( jstor ) Lagrangian function ( jstor ) Mechanical springs ( jstor ) Mechanical systems ( jstor ) Torque ( jstor ) Velocity ( jstor ) Dissertations, Academic -- Mechanical Engineering -- UF Dynamics ( lcsh ) Mechanical Engineering thesis Ph. D Mechanics, Applied ( lcsh ) - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Abstract:
- The holonomic constraints associated with complex, multiple input linkage systems complicate the procedures and methods used in determining their dynamic response. Large systems of nonlinear, second-order differential equations, requiring additional algebraic equations of constraint, occur as a result of these constraints. Double iteration algorithms, which are both time-consuming and subject to error, are necessary to integrate numerically these differential equations of motion. In this dissertation the concepts of kinematic influence coefficients of complex, planar, rigid link mechanisms with multiple inputs are developed and utilized to eliminate the holonomic constraints associated with such systems. Kinematic influence coefficients associated with series and parallel linkage combinations are developed, based on the addition of Assur groups (dyads, tetrads and more complex groups) to the basic system group. These complex, multiple input linkage systems are then reduced to coupled equivalent mass systems acted upon by variable rate springs, variable coefficient viscous dampers, and equivalent external forces and torques. The holonomic constraints associated with the original system are eliminated, thus leaving the equivalent mass system free of all such constraints. The number of generalized coordinates required to describe the motion of the equivalent system now equals the number of independent system inputs. The differential equations of motion describing the system's dynamical behavior can then be determined by established methods and put in a suitable form for numerical integration.
- Thesis:
- Thesis--University of Florida, 1971.
- Bibliography:
- Bibliography: leaves 82-84.
- General Note:
- Manuscript copy.
- General Note:
- Vita.
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Resource Identifier:
- 000953376 ( AlephBibNum )
16919120 ( OCLC ) AER5832 ( NOTIS )
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DYNAMIC RESPONSE ANALYSIS OF COMPLEX :.MEClH S.IS WITH MULTIPLE INPUTS By CHARLES EDWARD BENEDICT A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA To Patricia ACKNOWLEDGMENTS The author expresses his appreciation to Dr. Delbert Tesar for his interest, encouragement, and assistance in all phases of his doctoral program as chairman of the supervisory committee, particu- larly in the preparation of this dissertation. Through Dr. Tesar's efforts and support the author was afforded an opportunity to pre- sent a major part of this research to the leading researchers in this area at the NSF Advanced Training Workshop in Mechanisms held at Oklahoma State University. Sincere appreciation is expressed to Dr. Calvin C. Oliver for his assistance as co-chairman of the supervisory committee. Appreciation is also expressed to the other committee members for their guidance and support: Dr. W. H1. Boykin, Jr., Dr. T. E. Bullock, Dr. J. Mahig, Dr. J. M. Vance. The author is indebted to the Graduate Faculty for their support in making it possible for the author to receive financial assistance from a NDEA Title IV Fellowship. Deep gratitude goes to the author's wife, Patricia, and daughter, Sharla, for their patience and understanding. TABLE OF CONTENTS ACNOWLEDGMENTS . . . . . . . LIST OF TABLES . . . . . . . LIST OF FIGURES . . . . . . . NOMENCLATURE . . . . . . . . ABSTRACT . . . . . . . . CHAPTER I LITERATURE SURVEY . . . . . . Dynamic State Analysis . . . Displacement Analysis . . . . Dynamic Response Analysis . . . II CINEMATIC INFLUENCE COEFFICIENTS OF CONSTRUCTED FROM ASSUR GROUPS . . Kinematic Influence Coefficients of Input Poles . . . . . . Velocity Influence Coefficients COMPLEX MECHANISM Primary . . . .. Is Acceleration Influence Coefficients . . . . . Kinematic Influence Coefficients of Intermediate Input Poles . . . . . . . . . . . . Velocity Influence Coefficients . . . . . . Acceleration Influence Coefficients . . . . . Total Input-Output Kinematic Influence Coefficients . . Velocity Influence Coefficients . . . . . . Acceleration Influence Coefficients . . . . . III EQUIVALENT SYSTEM FORMULATION IN TERMS OF KINEMATIC INFLUENCE COEFFICIENTS . . . . . . . . . Equivalent System Torques . . . . . . . . Page 111 . . . vi . . . . vii . . . viii xi 1 2 3 TABLE OF CONTENTS (Continued) Chapter Page III (Continued) External Forces and Torques . . . . ... 30 Internal Springs . . . . . . . ... 32 Internal Viscous Dampers . . . . . . .. 35 Total Equivalent Torque . . . . . ... 38 Equivalent System Inertias . . . . . ... 38 Two Degrees of Freedom Example . . . . ... 41 Equivalent System Inertia Power . . . . ... 45 Two Input Example . . . . . . . ... 48 IV TIME RESPONSE OF EQUIVALENT MASS SYSTEMS . . .. 51 Lagrange's Method . . . . . . . . ... 52 Example . . . . . . . . ... . . 53 Hamilton's Principle . . . . . . ... 56 Example . . . . . . . .... . .. 57 V SUMMARY AND CONCLUSIONS . . . . . . . .. 60 APPENDICES A DIRECT DERIVATION OF FIRST-ORDER DIFFERENCE EQUATIONS FOR DYNAMICAL SYSTEMS . . . . .. 69 Derivation of Method . . . . . . . ... 69 B NUMERICAL SOLUTION TO A TWO DEGREES OF FREEDOM EXAMPLE . . . . . . . ... 74 BIBLIOGRAPHY . . . . . . . . . . . . . 82 BIOGRAPHICAL SKETCH . . . . . . . . ... . . 85 LIST OF TABLES Table Page 2-1 Influence Coefficients . . . . . . . ... 27 3-1 Equivalent System Forces and Torques . . . . .. 37 3-2 Equivalent System Formulation . . . . . . .. 50 B-1 Five-Bar Parameters . . . . . . . ... 80 LIST OF FIGURES Figure Page 2-1 Binary Groups . . . . . . . . . . 7 2-2 Assur Groups . . . . . . . . ... . . 9 2-3 General System Point . . . . . . . . .. 10 2-4 Point Paths of Assur Groups . . . . . . ... .14 2-5 Seven-Link System Group . . . . . . . ... .20 2-6 Sliding Pair Constraint . . . . . . . ... .28 3-1 Equivalent System Elements . . . . . . ... 31 3-2 Complex Multiple Input Linkage System . . . . .. 39 3-3 Differential Gear System . . . . . . . .. 42 3-4 Angular Relationship to Input #1 . . . . . .. 43 3-5 Angular Relationship to Input #2 . . . . . . 43 3-6 Complex Two Input Linkage System . . . . . .. .49 4-1 Two Input System . . . . . . . . ... . 54 5-1 Complex Multiple Input Mechanism and Its Coupled Equivalent Mass System . . . . . . . ... 61 5-2 Linkage System with Elastic Coupler Link . . . .. .64 5-3 Linkage Models with Deformable Bearings . . . .. .65 5-4 Optimal Open Loop Control Example . . . . . .. 67 B-l Two Degrees of Freedom Five-Bar Example . . . ... .76 B-2 Kinematic Position Equations . . . . . . ... .77 B-3 Polar Phase Plane: 1 vs pl .. . . . ...... 78 B-4 Polar Phase Plane: p2 vs P 2 . . .... 79 B-5 Equivalent Inertias vs Time . . . . . . ... .81 vii NOMENCLATURE B General linkage pin joint or points in links m,n m,n C Viscous damping coefficient of dashpot attached to link i and ground * C.. Equivalent viscous damping coefficient associated with the 13 th th i input link due to a unit angular velocity of j input E Coordinate point denoting center of gravity of system link F General output coordinate point F External force acting through general system point E e * th F Equivalent force acting on ith system input due to a unit e/1 external force at point E g.i Velocity influence coefficient of link I with respect to input link i Gci Row vector of velocity influence coefficients h.ij Acceleration influence coefficient of link a with respect S .th to the ith and jth input links Hij Square matrix of acceleration influence coefficients i Denotes input link or generalized coordinate I Effective moment of inertia of link I taken about its center of gravity I.. Equivalent moment of inertia term 13 viii j Denotes input link counter k Corresponding time counter k Corresponding position counter K Effective spring constant associated with a spring attached between link a and ground K Equivalent spring constant of K with respect to system input i 2 Denotes general system link L Lagrangian of the equivalent mass system m Total number of general system outputs M, Effective mass of link i at center of gravity n Denotes total number of system inputs N Denotes total number of system links P Total system inertia power th ..P ij equivalent inertia power coefficient with respect ij3 r th to the r input q Denotes input link counter r Denotes input link counter S1 Summation representing the time integral of the Lagrangian S2 Summation representing the time integral of the virtual work of the nonconservative forces T General system external torque acting on link i * th T,e,d,s/i Equivalent torque acting at i input link resulting from unit torques on link I, unit forces through system point E, unit velocities on equivalent viscous damper, and unit displacements on equivalent springs ix T Total equivalent torques acting on ith system input link 1 v Linear velocity of general system point E e WV Weight of link A acting through center of gravity x Denotes x-coordinate of system input X Denotes x-coordinate of system output y Denotes y-coordinate of system input Y Denotes y-coordinate of system output a. Angular acceleration of link i 1 6 Variation of some parameter A Finite increment 1 Denotes angles \ Undetermined Lagrangian multiplier Y. Angle of ith system input link Angle of 2th system output link W. Angular velocity of link i Denotes column vector Denotes differentiation with respect to time Denotes transpose of matrix S Denotes partial differentiation Sign Convention Right-hand Cartesian coordinate system Angles measured positive ccw from positive x-axis Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy DYNAMIC RESPONSE ANALYSIS OF COMPLEX MECHANISMS WITH MULTIPLE INPUTS By Charles Edward Benedict December, 1971 Chairman: Dr. Delbert Tesar Co-Chairman: Dr. Calvin C. Oliver Major Department: Mechanical Engineering The holonomic constraints associated with complex, multiple input linkage systems complicate the procedures and methods used in determining their dynamic response. Large systems of nonlinear, second-order differential equations, requiring additional algebraic equations of constraint, occur as a result of these constraints. Double iteration algorithms, which are both time-consuming and subject to error, are necessary to integrate numerically these differential equations of motion. In this dissertation the concepts of kinematic influence coef- ficients of complex, planar, rigid link mechanisms with multiple inputs are developed and utilized to eliminate the holonomic constraints asso- ciated with such systems. Kinematic influence coefficients associated with series and parallel linkage combinations are developed, based on the addition of Assur groups dyadss, tetrads and more complex groups) to the basic system group. These complex, multiple input linkage systems are then reduced to coupled equivalent mass systems acted upon by variable rate springs, variable coefficient viscous dampers, and equivalent external forces and torques. The holonomic constraints associated with the original system are eliminated, thus leaving the equivalent mass system free of all such constraints. The number of generalized coordinates required to describe the motion of the equivalent system now equals the number of independent system inputs. The differential equations of motion describing the system's dynamical behavior can then be determined by established methods and put in a suitable form for numerical integration. CHAPTER I LITERATURE SURVEY A substantial survey of the literature representing the state of the art in dynamic response analysis of constrained mechanisms appears in Il]. Virtually all systems treated in the literature possess a single degree of freedom. Furthermore, these systems are analyzed for their output dynamic state. Few researchers have attempted to deter- mine the input dynamic state of complex mechanisms with multiple inputs (degrees of freedom) to forces and torques. Those who have, invariably give examples which are constrained four-link mechanisms, thus leaving their methods essentially untested. This emphasizes the need for a formalized procedure to model complex systems mathematically, utilizing direct methods for determining their dynamic response to known forces, torques and energy crossing the system boundaries. This will eliminate iterative procedures necessary in solving the differential equations of motion. Dynamic State Analysis As stated in [1] "dynamic state" implies that the velocity and acceleration of each point of every link of the system are piecewise continuous, differentiable functions of the input characteristics. In 1957 Modrey [2) developed a graphical method whereby the velocities and accelerations of the links of a system of higher-order-complexity (one which cannot be analyzed as a system of four-link mechanisms con- nected in series) could be determined, using velocity and acceleration influence coefficients obtained through a "zero-relax" procedure. In a discussion of this paper, T. P. Goodman showed the analytical equiv- alent to Modrey's method and proved the linearity property necessary for superposition with respect to single degree of freedom systems. Other graphical methods existed for solving the dynamic state question of complex mechanisms; namely, Hall and Alt's method, Carter's method and the method of normal accelerations. All of these methods assume a known geometric configuration. That is, the point path is known, and the dynamic state is determined based on this known configuration. Displacement Analysis Various methods have been developed for the displacement analysis of mechanisms containing more than one vector loop. Hain [3] shows how to determine the input-output position information of six- and eight-bar mechanisms which do not contain a basic four-bar at ground but do contain an internal four-link loop. This is accomplished by inverting the mechan- ism, solving the position information and then reinverting. However, most methods rely on harmonic analysis, including those by Meyer zur Capellen [4], Denavit and Hasson [5], Flory and Wolford [6], Markus and Tomas [7], and the Romanian school [8,9]. They obtain the harmonics by approximate numerical means within a prescribed error criterion. Crossley and Seshachar [10] analyze the displacement of planar Assur groups by an iterative algebraic method which allows half the unknowns from the matrix of complex numbers to be found first with the other half found later. All of these methods rely on iteration or approx- imations because of the complex loop equations resulting from the geometry. However, they yield results which can be made as accurate as desired. Digital computation capability increases the desirability of these methods. Dynamic Response Analysis Time response of mechanical systems, spatial as well as planar, by use of Lagrangian mechanics has been advanced during the most recent years. Chace [11] uses relative coordinates in determining the dynamic response of multiple degree of freedom spatial mechanisms. He utilizes Lagrangian multipliers to account for the physical (geometric) con- straints. Smith [12] employs this same technique in analyzing the reaction forces in generalized machine systems. In both [11] and [12] the examples are planar single degree of freedom four-link mechanisms, leaving the general nature of the method untested. Uicker [13] and Carson and Trummel [14] apply matrix notation and methods to the kinematics problem. By applying Lagrangian techniques, they develop the system differential equations based on the vector loop displacement equations. This method has been the most useful and power- ful for analyzing the dynamic response of complex multiple degree of freedom systems to date. As with other methods, it has its weaknesses. A dual iteration algorithm results, which is ideal for digital computa- tion but is time-consuming. The works of Wittenbauer [15], Federhofer [16], and Beyer [17] are significant in the development of equivalent mass systems and appear to be the most direct antecedents of the concepts developed in this dissertation. Wittenbauer and Federhofer introduce the concepts of reduced mass in terms of general mass content and velocity ratios. Beyer effectively summarized this work for the modern reader in terms of the time response problem. The authors in references [11-14] ignore this property, preferring to treat these systems as ones with large numbers of generalized coordinates, coupled by algebraic equations of constraint. This property necessitates the use of the Lagrangian X-method in treating the holonomic auxiliary conditions, replacing the kinematical constraints by the forces necessary in maintaining them. The X's, then, indicate the degree to which the constraints are violated. The need for a more concise system formulation, allowing one to determine the dynamic response in a more precise and direct manner, is paramount. This will be accomplished in the following chapters by developing the concepts of kinematic influence coefficients of complex, planar, rigid-link mechanisms with multiple inputs (degrees of freedom). These, in effect, account for the holonomic constraints on the system, resulting in coupled, equivalent mass systems requiring only as many generalized coordinates as its degrees of freedom. These equivalent systems can then be uncoupled by standard methods, allowing their time response to be determined by direct methods such as that developed in [18]. CHAPTER II KINEMATIC INFLUENCE COEFFICIENTS OF COMPLEX MECHANISMS CONSTRUCTED FROM ASSUR GROUPS Pelecudi [19,20] defines kinematic multipoles as kinematic chains interpreted as lines for transmitting motion information (posi- tion, velocity, acceleration) of rigid links from an input to an output or, more generally, from one point in the chain to another. This is accomplished by motion transforms called kinematic ratios, velocity and acceleration ratios, or, more precisely, kinematic influence coefficients. These are well established for the case of four-link mechanisms and cams in [1,21-25]. Kinematic chains receive information from input poles and transmit it to output poles. The generality of this concept enables one to construct highly complex mechanisms consisting of groups of links known as Assur groups. The degree of mobility of the final sys- tem can therefore be controlled or determined by the mobility associated with each group added to the basic kinematic chain. Two types of chains are most important in the construction of complex mechanisms. (1) Structural groups having a degree of mobility (freedom) f 0, forming pa dive multipoles. A group may be overconstrained (f < 0) to form a mechanism with a degree of mobility which is less than the basic chain. (2) Mechanisms with a fixedlink having a degree of freedom f 1, formingactive multipoles. Thus,one is free to construct mechanisms from basic chains whose degree of freedom remains invariant under the addition of active and padLive multipoles. This points out the importance of the struc- tural group in the construction of mechanisms of high complexity. For the purpose of analysis it is desirable to reduce these systems to their basic kinematic groups in order to derive the input- output transformation of position, velocity, and acceleration. Systems possessing these transformations can be reduced to coupled equivalent mass systems. This coupling will be quadratic and will depend on velocity influence coefficients. It is necessary to discuss the types of kinematic groups avail- able for constructing complex mechanisms of desired mobility. Assur groups [26] are defined in terms of kinematic groups which form structures when the outer joints are fixed after being separated from the input links. The binary group is the simplest of the Assur groups. Figure 2-1 illustrates its use in constructing the four-, five-, and six-bar mechanisms. Each binary group forms a struc- ture when it is separated from its primary or intermediate input and has its outer joints fixed. The dyad in the four- and five-bar mechan- isms and the dyad attached to the intermediate input of the six-bar mechanism form dipoles while the dyad in Figure 2-1(c) attached to the primary input forms a triple, capable of transmitting information from the system input through the output pole to the dipole. A B 00 0 1 (a) B A 00i O (b) (c) Figure 2-1. Binary Groups The next Assur group of higher complexity is the tetrad (see Figure 2-2(a)). When the outer joints A, Oc, and E are fixed, the group becomes a structure with f = 0. This group requires two vector loop equations to determine the angular relationships between the links. More complex Assur groups are illustrated in Figures 2-2(b) and (c). These and other groups are classified and analyzed for their displace- ments by Crossley and Seshachar [10]. Markus and Tomas [7] define a system group as a group of links which has zero degrees of freedom when the input links are fixed. The system groups contain Assur groups but are more general, enabling the authors to solve for displacement information of any system group by a single harmonic analysis algorithm. Curtis and Tomas [27] point out the importance of the binary group in determining position information of mechanisms of varying complexity. They utilize the law of cosines in developing a closed form algorithm to obtain the position information of a large class of mechanisms consisting of various binary group con- figurations. Kinematic Influence Coefficients of Primary Input Poles Velocity Influence Coefficients An implicit functional relationship, f(x,y) = 0, describing the path of some general system point E (see Figure 2-3(a)) associated with the outer joint of a binary group, can be written based on known posi- tion information. In general, f(x,y), ge, he, me, and (+P will be known as a function of some system input parameter such as p9. It is 1 (b) (c) Figure 2-2. Assur Groups Y h, e f (x,y) 0 ge 1+ I x (a) y -he E I 'I I jf (x,y)= 0 L_ 9e (b) Figure 2-3. General System Point noted that g and h are defined in [1) as e e dS -e e dcp d2S e2 h- i and m = slope of path tangent of point S e e In order to proceed through the chain and determine the influence coefficients associated with F(X,Y) the values for gx, gy hx, h must be determined. It is noted that (see Figure 2-4(a)) dS ex. x xi =-p. i' (2-1) dS y =y Ci =dp. i 'i and d y f v f S[(f(x,y)j = x- + s v dt x x + y =f af - =T- gx + -3 g = 0 (2-2) This gives (2-3) Sf gy - -x f gx TSy The expressions for me and g are given by S= (2-4) e and -2 .-2 ge = +y (2-5) Since g is known, then e o g = (2-6) and gy = me g (2-7) Acceleration Influence Coefficients Now consider h and h in terms of m g and h . x y e e e It is noted that t h = a x x (2-8) h = a for 9. = 1.0, (p = 0. y y 1 i Define T as follows (see Figure 2-3(b)) S= tan- { } (2-9) gx For f(x,y) = 0, a closed path of point E in a general system link, g is always positive. Then h is positive if directed in the same sense as g, negative if directed in the opposite direction. sense as ge negative if directed in the opposite direction. Therefore, h = h cos (71) , x e h = h sin () . y e (2-10) Now consider a dyad as a dipole or binary group in the Assurian sense with points El and E2 having general point paths defined by implicit functions fl(x1,Y1) = 0 and f2(x2, 2) = 0 (see Figure 2-4(a)). Then, ge gx g = x1 /1+ i ge2 e 2 h = cos (6 , x1 el 1 h =h cos ( 2) , x2 e2 2 --- g el x1 1 1 g = m g Y2 e2 gx2 h = h sin (71 y1 e 1 = h sin (2) Y2 e2 This can be generalized immediately to all classes of Assur groups (see Figures 2-4(a) and (b)) as ge. gx S = 1 i g = m g Yi ei xi (2-13) h = h cos (0 ) h = h sin () . x e i Yi ei and (2-11) (2-12) (x, Y2) (x ,yl) (a) (x2,y2) (XI ,y1) (x3 ,Y) (x,,y,) (x3, y3) ( b) Figure 2-4. Point Paths of Assur Groups Kinematic Influence Coefficients of Intermediate Input Poles Let E1 and E2 be the inputs and F be the output of the binary group in Figure 2-4(a). Point F is an internal joint to links I and S+ 1, Hlowover, the following derivation holds for output polo as general points in either of the links 2 or + 1. The point path of F can be expressed as an implicit function f(X,Y) = 0, (2-14) where X and Y are functions of xl, be expressed by the projections of dyad, E1F + FE2 + E E1 = 0, y1, x2' and y2. Relation (2-14) may the vector loop equation for the (2-15) fx(1, x2,X,ylY2, Y) = 0, fy(X1'2,X,Yy12,Y) = 0. (2-16) Velocity Influence Coefficients Since X,Y are functions of xl, y1, x2, and y2, then they are differentiable with respect to them. The velocity of point F is given as (2-17) VF X Y where 2 2 ax ax VX x. Vy. , S1 1 1 1 1 (2-18) 2 r f Ay ay vV =+V i + T yi VV 1=1 1 1 1 1 The partial derivatives in Equations (2-18) represent the velocity influence coefficients for multiple input linkage systems. They are defined as follows in order to remain consistent with previous literature. Let A ax i TX (2-19) A 3y Yigy = Substituting these into Equations (2-18) gives 2 vx= x xg xi + i vy i i= 1 1 1 1 2 V = v + Yy v } (2-20) i= 1i1 1 1 y Equations (2-20) can be put into compound matrix form as I 1- VX XGx XG i i xI ] [ ] [ Ivy! SImIlrJl\ for angular odULpa information GL a G i G ----I 1i (2-22) v yi where ax x xGx i x2 G --(2-23) Ivxl xl Lx2 Acceleration Influence Coefficients Differentiating Equations (2-18) with respect to time yields 2 2 2 2x ax2 ax x Vxi y i= j=l 1 j 2 2 ax + y fax t ax t 1 -24) Y i yj i1 x + ai (2-24) 1n 1 1 1 1 and 2 2 a2Y + axi +t aY ' + 3-Vy Vyj a a (2-25) y 3 y/ j=1 yi Y I Equations (2-24) and (2-25) can be written in compound matrix form and generalized to n inputs as a = vxI I v + 2v vx x ] Ivy| + |vi' | iY j] i'Viy + I Xi K f y IV y Ii X i -] t 11 i,j = 1,2,...,n, Xhxxx Xxx x Xhx x 1 1 1 2 n ix ] ~~xh2x1 X2x2 tx22n Xhx x X x nx2. Xh x nxn n 1 X x2 X XnXn 82X Xhx x. axx i G g j x x ... X g y1 X n] [- | i 1'^ l-n (2-26) (2-27) (2-28) (2-29) (2-30) where Corresponding expressions relating angular input to angular or linear output can be derived. Since most kinematic chains consist of lower pair connections, the remaining derivations will be based on angular inputs to both linear and angular outputs. Complex kinematic chnins constructed from basic chains with invarinnt mobility are developed by adding Assur groups to output poles of the established chain of pro- determined mobility. II the mobility does not remain invariant, then the basic kinematic chain with its known input-output position relation- ships is destroyed and the point paths are no longer predetermined with respect to the original system group inputs. A new system group must be established and new input-output relationships developed. Total Input-Output Kinematic Influence Coefficients If the inputs to a linkage system group are angular, then the intermediate linear inputs (xi,Yi) and their dynamic states (vxi'vyi) t t and (axi,ayi) can be expressed as functions of the input position param- eters c 's and their dynamic states W 's and a 's. The derivation of q q q the total input-output transformation is now carried out. Velocity Influence Coefficients Consider the system group in Figure 2-5. The expression for the velocity components of point F with respect to points E1 and E2 are G G 1 .2xI ----- ---- --- (2-31) _Y Yx YG x_ i : iy -~ F I (x,y) E2(x2,y2) Figure 2-5. Seven-Link System Group The intermediate input data,in terms of the primary input information, are given by v x i x q -- ---- i,q = 1,2, (2-32) LIV .I- ..Lyi,59q where 8x1 bx1 ax 6 [Xi q] x2= x2 and Substituting the right-hand side of Equation (2-32) into Equation (2-31) yields Vy xi Yi Li c q = ----;---- --- Jwl (2-33) [ Y x Y yi i q_ which reduces to --- ||, q = 1,2. (2-34) v- YGq Notu that when q=l, Equation (2-34) becomes Vx = Xg '1 (2-35) V = Y= I1 ' which are recognized as the velocity influence coefficients developed in reference [1] for single input systems. Acceleration Influence Coefficients The component accelerations of point F (Figure 2-5) in terms of the intermediate inputs are r - 3X]i] ;x = -------0----V - i + .- (2-36) x O x Yx x j 0 HL 2 ------------ IV I +2-36 I Si I i 0 7 The first and third terms of Equation (2-36) are recognized as real quadratic forms while the second is real bilinear. Substitution of Equations (2-26), (2-27), and (2-32) into Equation (2-36) gives (2YX1) (2N X 2n) t -- ---- -- [ 0 t I o -ayj I +2 --- --- 0 |uw| (2n X 2n) Y xy 1i3 $ [_ ----- ------- -- --- --- Y Yi I i _-- \ N,n-= 2. G + I q I - Yi q (2n xn) (n xn) (n xl) XH x x. I ji j-dy ] YRX. X (2-37) The complexity of Equation (2-37) is reduced by collecting like terms and writing it in a more concise and familiar form as aH G i = j| ..----- I + l |a, q,r= 1,2, (2-38) Y q Yr q L where 11wll is a compound diagonal matrix given by u 0 = (2-39) 0 jo| and |)l is the transpose of the column vector IwI. The expressions (2-34) and (2-38) for the velocity and acceler- ation can be generalized to treat systems with n inputs and m outputs. These expressions take the general form X1 XGlpq X X G m m 'q ---.- | q = 1,2,...,n, (2-40) v G 1 1 (q V YG m m q t a X 1 q r X G ~m / Xm qr Xm = ml ----|--- + -- (2-41) tly 1 1 q'r 1 q t a H G- G m Ym q r m q q,r = 1,. ,n. It is clear that expressions similar to Equations (2-40) and (2-41) can be written for the angular properties of the system links. For the purpose of illustration, both forms will be used where appro- priate throughout the remainder of this dissertation. The discrete form [28] of the kinematic influence coefficients of velocity and acceleration are expressed as 1 ik = 1) g i,k+l Ai,k-l (2-42) agik = 2Ap. 2 Lhijk =) ' i,k+l i,k+l i,k-1 i,k-1 Iij k+1 j,k-1 j1,k+1 + z j,k-1 (2-43) AhiJ,k = 4(A )2 where the input type (i.e., angular or linear) is considered understood and its designation is dropped for simplification of notation as follows: Agi,k ( i) (2-44) hij ,k i(i j kjk Table 2-1 is a compact collection of the influence coefficients derived thus far, expressed in discrete form. Each column is headed by a characteristic set of input conditions and each row is designated by a particular system model. Rows and columns intersect in blocks con- taining the influence coefficients for which these properties hold. Blocks 1.1 1.3 and 3.1 3.3 have been derived and are expressed in discrete form. The coefficients in blocks 2.1 2.3 pertain to the relative angular motion between any two links m,n in the system (where m,n are not to be confused with m,n denoting the number of inputs and outputs as in Equation (2-41)). The links are shown with coincident points at B for purposes of simplicity of notation. The influence coefficients in blocks 4.1 4.3 apply to the relative motion between any two points B ,B in different links m,n in the system. The system model has been m n represented as a sliding pair connecting these two points. It may be helpful to consider this pair as massless when it is not required for constraint in the system. An example of when it would be required for constraint is that shown in Figure 2-6. The influence coefficients in Table 2-1 are a precise defin- ition of the system geometry up to the second order. As such they 27 C\?I jr r-) -, 3 fT ~ x? 9 -- N E : 3 -,, g - ;^ '. < Cr 1^_ .<< : j7^ C^ .- ^- -^ ^ ~P I -c^ - ^i [^j cLj ch It I II II (\j ro - 1. C7? - 3 3/) II tI (J* A cW CVJEI -' .- II I OIo Figure 2-6. Sliding Pair Constraint represent a powerful definition of the meaning of multiple input link- age systems. These influence coefficients enable one to determine the dynamic state of every link in a general linkage system, given the dynamic state of the system inputs. They will be used in the follow- ing chapter to reduce complex, multiple input linkage systems acted upon by external forces and torques, internal springs, and viscous dampers, to equivalent mass systems acted upon by equivalent external torques, variable rate springs and variable coefficient viscous dampers. The remaining coefficients will be treated on a discerte time basis, since some of them will depend on their past history. This means that expressions such as that for Sk imply that k S = f(t) (2-45) CHAPTER III EQUIVALENT SYSTEM FORMULATION IN TERMS OF KINEMATIC INFLUENCE COEFFICIENTS The coefficients developed in Chapter II will now be utilized to eliminate the holonomic constraints associated with complex linkage systems, reducing them to coupled, equivalent mass systems. These same coefficients will be used to reduce the generalized internal and external force generators acting on the system to equivalent general- ized torques acting at the system inputs. The Equivalent System Torques The force related influence coefficients are derived by replac- ing the effect of external forces on the system (i.e., TV, Fe) and inter- nal forces generated by system elements, such as springs and viscous dampers, with equivalent torques acting on the n system inputs. External Forces and Torques If a linkage system (see Figure 3-1(a)) is given a virtual displacement by each separate input from some given system position k, th then the virtual work done by the equivalent torque at the i input must be equal to the virtual work done by the system external torque, or (T1/i iAk = l )k (3-1) T41. (a) K~. (b) C .. (c) Figure 3-1. Equivalent System Elements CK C.. solving for Tz/ik and taking the limit as 5 ik 0 gives .A T/ik = ln T )k TLk = ik) Tk (3-2) T/ik i- 1 Similarly, for an external system force Fek T/ik (egik Fek (3-3) The total equivalent torque seen at each input link i, resulting from all system torques T k, is given by Te 1 g 2 1 Ngl G T = 12 2g2 N' 2 (3-4) T* G T n I -1 n 2 n L Ng n k -k N k i = 1,2,...,n, N = number of system links. Equations (3-2) and (3-3) show that Agik and egik are the influence of a unit torque or force applied to system link A or point E on the ith system input link. The gik is the mechanical advantage of link Z with respect to the ith input link and vice versa. Internal Springs Perhaps the most difficult system parameter to develop in terms * of an equivalent coefficient is the equivalent spring constant KA/ik. Suppose a torque TLk is generated by a linear spring K between link I and the ground link. The question arises: What is the equivalent spring constant K /ik of the system spring K,? By giving the system a virtual displacement, the potential energy change in the system spring can be equated with the potential energy in the equivalent spring associated with the ith input link (see Figure 3-2(b)). This yields the following relation 2 (3-5) i2 /ik (ik = (ik (3-5) Solving for K /ik and taking the limit as p ik 0 yields (At 2k (3-6) K/ik = K lim = gk3-6 tipik -0 i k It is noted that S cannot continuously increase with increas- ing pi otherwise the spring would be destroyed. Rather, is cyclic or quasi-cyclic with increasing cp.i Therefore, for part of the cycle A, (i.e., 2gi) must be negative. This implies the need for the signum function a = sgn (,gi) which changes sign whenever the gi.'s go through zero. The torque T/ik is then determined-from its past history in the form a/ik /i,k-1 + K/ik ik (3-7) where 2 2 K* gik-i+ gik (3-8) K/ik ik 2 3-8 The equivalent torques at each system input resulting from all system springs are given in matrix form as * Ts/l T s/2 T s/n * Ts/1 T s/2 Ts s/n N 2=1 K/l &1 K /2 2 K/n Mn 2/n n (3-9) An alternative approach to determine the equivalent torque T2/ik generated by a system spring between link A and ground is to treat the torque acting on link I due to spring Ke as an external torque as in Equation (3-2). The external torque is given by Tk = (K f ( .k)= *k (3-10) where I,, = free length of spring measured from the reference axis, A k = deformation of system spring K from free length. Substituting Equation (3-10) into (3-2) gives T/ik = K (gik) ik (3-11) The equivalent torques at each system input resulting from all system springs K are now given in matrix form as Ts/1 1Gi 1K s/n N N N < -k -k -k (3-12) , i = 1,2,.ber of system lines. N = number of system links. Internal Viscous Dampers Let the torque TIk be generated by a system dashpot C between link ; and ground (see Figure 3-1(c)). The torque generated by the dashpot is proportional to the angular velocity of the link attached to the dashpot, or Using Equation (3-2), TVk is transferred to the i system input as TA/ik (ik) C (gj CW j (3-14) j=l k The total torque acting on the ith input link resulting from all system dashpots is n Td/ik = C (3-15) where N C j,k= gi gJ)k (3-16) The equivalent torques at all system inputs resulting from all system dashpots aregiven in matrix form as I ~ d/l C11 C12 Cln d/2 C21 C22 2n S WI k (3-17) Cnl * T C C ** C d/n nl n2 nn k -'k 1 k Table 3-1 is a concise collection of the preceding equivalent torque concepts. Row 2 corresponds directly with row 1. Those in row 2 are more general, since the generated torques occur between two moving links m and n in the system. The relative influence coefficients brik correspond to those in Table 2-1, i.e., brick = ngik mgik (3-18) The equivalent torques in row 3 refer to a force Fek applied to a particular point or points in the system. Since the direction angle Tek of this force relative to the path tangent is known as a function of the input angles i then the appropriate equivalent torque can be determined. The equivalent torque, T e/ik, is obtained by first taking its component F along the path tangent and transferring this ek th force to the i input link by the influence coefficient egik. Here, the product egik cos ( ek) is the effective influence of a unit force .th applied in the direction F at point E on the i input link. As such, ek it corresponds directly with gik in row 1. Blocks 3.2 and 3.3 follow directly as a result of this correspondence. Here, the system spring K and dashpot C are anchored at some fixed point 0 . e e e Finally, the equivalent torques in row 4 apply to forces in extensible two-force members attached to links m and n at points B and m B n. The relative influence coefficient rk is defined in Table 2-1, n b ik row 4. This implies the treatment of very general force generators in the system. 3z 4:: K) *-o *1I F- ~F U uL4~ - ~ + II I I Total Equivalent Torque The total equivalent torque acting upon the ith input link is given as -19) T = T +T + T +T (3-19) ik A/ik e/ik + Ts/ik +d/ik If one or more of the input elements is actually a sliding pair, then sik represents that input's reference position parameter and Fik would be the equivalent force acting on that system input. Equivalent System Inertias The objective of this section is to transfer the effective mass of link a to each of the n system inputs in order to obtain equiv- alent mass systems for a linkage system such as that shown in Figure 3-2. Consider the kinetic energy of a complex, multiple input system. I2 2 + M 2 } (3-20) (KE)k =k 2 {2 (3-20) where 1) = effective moment of inertia of link I about its center of gravity E, M, = effective mass of link 2 located at its center of gravity E. Writing Equation (3-20) in terms of the system input velocities yields N n 2 n 2 (KE)k = ikk + (eik)Wik =1 i=l 1=1 (3-21) n=3 N=13 Figure 3-2. Complex Multiple Input Linkage System Rewriting Equation (3-21) in a more compact form gives n n N (KE)k Z [ )( ) + M i)( j) wj i=l j=1 2=i (3-22) where it is noted that 1 2=i,j; i=j ( gXi)( g ) = (3-23) 0 o =i,j; i j, a direct consequence of the meaning of independent system inputs. The kinetic energy can be written in matrix form as (K E)k k ij IWk (3-24) (nxn) k where N ij,k = i i(gj) 2+ ei e j(3-25) &=1k Matrix Equation (3-24) is recognized as being real quadratic form since the [I..] matrix is real symmetric. Thus it is seen that any complex multiple input linkage system can be transformed into a system of coupled equivalent masses with n degrees of freedom whose equiva- lent inertias are given by Equation (3-25). Two Degrees of Freedom Example For the purpose of illustration, consider the differential gear (Figure 3-3), where the translating links are constrained to move in the horizontal direction and rotation of link 4 is positive counterclockwise. The linear velocity influence coefficients of 3 are 3X3 d2 63 2 3gl x dl+d2 1 12 (3-26) X3 d1 32 = x2 dl+d2 and 3 = 3 1 v1 + 3 2 v2 (3-27) The angular velocity influence coefficients of 4 are given by (see Figures 3-4 and 3-5) a 4 1 4g 1 dl+d2 (3-28) 4 1 4 2 dx2 + dl+d2 and 4 = 4 1 v1 + 42 v2 (3-29) The kinetic energy of the two input differential gear is 1 2 -2 2 2 KE = 2 Iv + M2v2 + (M3 +M14)v3+ I4w4 (3-30) 1 2v2 ~3 3 44 CL x 0 4-t C _ x- Q. X dx, = (d,+ d2)(-d34) Figure 3-4. Angular Relationship to Input #1 dx2= (d,+ d2)d 4 Figure 3-5. Angular Relationship to Input #2 Substituting Equations (3-27) and (3-29) into (3-30) gives 1 2 2+ 2 KE = (MV1 + M22 + (M3+M4)(3gv1 3g2v2) + 14(4g1 1 + 4g2v2) (3-31) where v3 = v4 Collecting like terms and rewriting Equation (3-31) gives 1 2 + + )2 2 KE = 2 {I + 4(4gi) + (3 + M(31 1] + [M2 + T4(4g)2 + (M13 + M)(3) ]v2 + 2[4I4g ) (4g2) + (M3 + M4) 3g1) (392)]Vlv2 (3-32) or 1 2 2 * KE = IlV + 1222 + 21121v2} (3-33) where 2 -2 I = M11 4 4 1 + (M3 + M4)(31) 2 2 12 = 4(4g)(4 g2) + (M3 + M) (3gl)(3g2) (3-34) 2 2 2 + 4(42)2 + (M3 +M432 For the system in Figure 3-3, Equation (3-24) becomes 11 12 v1 KE=[V1 2] (3-35) 21 122_- - which is recognized as a quadratic form. In the case of a differential gear the equivalent inertias are constant. However, for a general linkage system (Figure 3-2), the I 's are not constant but a function of the system input parameters. Note that whenever v2=0, then 1 2 KE = vI (3-36) and whenever v1 = O, then 1"2 KE = 22 v2 (3-37) which are simply the expressions for the kinetic energy of a single input equivalent system, Inertia coupling terms, I.. i f j, appear for systems with * two or more inputs, indicating a relationship between the I..'s iJ through the velocity product terms w.iW.. Quinn's energy method [29] Ij becomes invalid for these types of systems, since the distribution of kinetic energy between the links of the system is no longer invariant with respect to the input velocities. The equivalent mass system described above remains valid provided certain modifications are made with respect to the solution technique. Equivalent System Inertia Power The final equivalent system parameter to be considered is the equivalent system inertia power. The coefficients derived in this section are necessary when deriving the differential equations of * motion of complex linkage systems. If the I..'s were constant, as in the preceding example, these terms would be zero. Although they appear unwieldy in size and computation for systems with more than two inputs, they readily lend themselves to computer computation. The power necessary to drive a linkage system against its inertias is given as i d P = t (KE) (3-38) where KE = W I*j Il i,j = 1,2,...,n (3-24) Differentiating Equation (3-24) with respect to time gives P i- j* + JI + i I 1. (3-39) 2 '' dtI ij The second product on the right side of Equation (3-39) is self- explanatory. The time derivative of the equivalent inertia matrix, however, is not as trivial. Consider a typical element of [I..] 1J given by N ij = g ) + M(egie j (3-40) Differentiating Equation (3-40) with respect to time gives d n N dij = [(Zg) ( r) + , d k r=1 1 I =g +WNIkt 9 (3-41) + i(ei jr+ (oj)( ei k rk', (-41 where (ii)(hjr) = pr (3-42) Equation (3-41) can be written as the product of a row and column vector as 0W n (3-43) where N ij r T=1 g Zg hj fj (Ah ir + M[ z egi)( ejr)+ (egj)(hir k The subscript notation is defined by 'I ij ij r = ' r Substituting Equation (3-43) into Equation (3-39) gives (IXn) (nxnn) ( I ,7 p* p i w o 11 r 12 r n r ------- i---- ---- - ..I I *1 p- p . p . 21 r 22 r 2 2nr 0 o ^JK ---------- r--- -- I *I P P ... P 0 0 nl r n2 r nn r k+ LI jk I where ijP is a compound nxnn matrix whose submatri row vectors. (3-44) (3-45) nnxn) II I * I -- r-- 0 --1 r--- -I S I k k (3-46) ces are 1 Xn Two Input Example Consider the system in Figure 3-6 whose kinetic energy is given by I (KE)k = 1 [2 k 21 I 12 Wl I* [2 22 k k where ij i k e gi Zgj + i k] Differentiating Equation (3-47) with respect to time gives 1 P P P P r1 11 1 11 2 12 1 12 2 W2 0 i 1 = ------------- -------------- ---- k 2 1 2k k 1 1 0 W 21 1 21 2 22 1 22 2 1 Ik + [Yl 2 k i21 22 where 8 12 1 [I(gi h11+ g2 2h21)+ M(eg1 eh11+ e2 eh21)]k (3-48) W k 2- k (3-49) (3-50) Table 3-2 represents the concepts of equivalent system inertias and equivalent inertia power for complex, multiple input linkage systems. With this formulation the holonomic constraints are eliminated and the equations of motion derived, utilizing only those generalized coordinates associated with the independent inputs. (3-47) Figure 3-6. Complex Two Input Linkage System 50 n < Q -I 0P O ,. ., > Sxrn Z: Fr F- C 0 0 c-I i +.I -. I , --I I.-. '- i lil::^~' '--- t ZII " .-7s + c^ Ir r~ ^^^ \r<\ '.< s . -< \ - ^< O `Q' <-< -f.M - _ L^- -* ^x << >^ ,Xi5~ ? i-- / tt 5< ^ - CHAPTER IV TIME RESPONSE OF EQUIVALENT MASS SYSTEMS Complex, multiple input linkage systems present a unique problem with respect to determining their dynamic response. Methods for deter- mining the time response of single input systems are well established, either in terms of Lagrangian mechanics [11-14] or equivalent mass sys- tems. Each method has its strengths and weaknesses. The method chosen to solve any given problem should be based on need and solution form. It is possible to obtain phase plane solutions for single input systems by using techniques in [1] if the input velocity is never less than zero. With slight modifications in the predictor equation, negative velocities can also be treated. Here the independent variable is the input angle, i rather than time, a result of the cyclic nature of most mechanisms. However, a large class of problems does not meet this requirement. Hence, a more general method must be employed. As pointed out earlier, the energy distribution method is invalid for multiple input systems. Similarly, the kinetic energy and power concepts utilizing the equivalent mass and force system developed in [1] fail because they yield only one second-order differ- ential equation in n independent variables. These failings are a result of the geometric constraints imposed on the system. Two methods will be discussed in this chapter. The differ- ential equations of motion for a two input system will be derived, using both methods. The complete set of first-order difference equa- tions will be derived for only the second method for reasons which will be explained later. The methods used to derive the differential equations and to solve them are not new. The kinematic influence coefficients and coupled equivalent systems (inertias) described in Chapters II and III are new and unique, enabling established methods to be used in deriving and solving the differential equations of motion. Lagrange's Method A system of n second-order differential equations is derived from Lagrange's equations as d (iL aL* * dt Ti ( = 1,2...n) (4-1) where ci generalized coordinates, pi generalized velocities, L Lagrangian of the coupled equivalent mass system, T. equivalent nonpotential forces and torques. 1 The n second-order differential equations resulting from Equation (4-1) are nonlinear, coupled, nonhomogeneous, and contain variable coefficients. It is necessary to reduce this set of equations to 2n first-order differential equations in order to solve them on a digital computer. Although this reduction is entirely possible, it is often tedious, yielding unwieldy equations. The variable coefficients are functions of geometry and are hence implicit functions of time. Tlii characteristic requires the determination of the equivalent inertia power coefficients of the system, terms which are very diffi- cult to obtain (see Chapter III) and only adds to the complexity of the problem. Example The five-bar mechanism in Figure 4-1 is a representative two input system. The generalized coordinates are 91 and 92; a helical spring and viscous damper are attached between links 3 and 4 at pin joint B, while external torques T1 and T2 are applied to links 1 and 2, respectively. The Lagrangian for this system is given as L = (KE PE), (4-2) and the equivalent nonconservative torques are T = -C34 brl [brl 1 + br2 2] + Tl' (4-3) T2 =-34 br 2 brl br2 + T2 where 2 2 KE = j L i j* (4-4) i=l j=1l 1 2 pE -- (34(f ) (4-5) PE 2 34 34f 34 (4 Substituting Equations (4-3), (4-4), and (4-5) into Equation (4-1) and performing the designated differentiation with respect to the generalized coordinates and velocities yields C34 Figure 4-1. Two Input System dt { 1 l + 112 + 34 (34f 34)(brl) =T1 34 [br)l + (brl) br2 (4-6) d - d I12 91 + 22 ~2 + K34 (P34f 34)(br2) 2 -34 (b br2 1 (b 2] (4-7) Completing the differentiation with respect to time gives * .+ .2 - 11 '9 + 1 + 12P) l + 12P 2 11 1 + 12 2 - + K34 34f 34)brl) = 34 r)l + (brl)(br2 2 (4-8) *2 .2 * _ 21P 1 + (21P2 + 22P112 22 2 2 12 1 122 2- + K34(C34f P34)(br2) = 3 (bl) (br2 + br2 (4-9) Equations (4-8) and (4-9) are nonlinear in the velocities and are unwieldy in their present form. In addition, they should be reduced to four first-order equations before they are numerically integrated. The ..P terms as defined by Equations (3-44) and (3-45) are not simple expressions, and require calculation at each integration step. expressions, and require calculation at each integration step. Hamilton's Principle The method used in this section for deriving first-order difference equations by direct application of Hamilton's principle was developed by Vance and Sitchin [18]. The section on the derivation of the method from [18] is included in the Appendix for completeness. The motion of a dynamical system is determined by solving the 3nN equations derived from as,1 N 6f. 3 Z x ij F -Fk At (4-10) ik j=l ik as N 8f.. + .. = 0 (4-11) 2ik j=1J aik fik= -ik ik At = 0, (i= 1,2,...,n; k= 1,2,...,N), (4-12) where X.. it undetermined Lagrangian multiplier at j time 13 interval, th F i generalized equivalent nonpotential force, at ik kth time interval, N and S = Lk(9~Ik' 2k ... k' k' CP2k ... nk) At. (4-13) k=1 The 3nN first-order difference equations resulting from Equa- tions (4-10) (4-12) in conjunction with an appropriate difference expression for p ik are obtained from only one differentiation and are in a form which allows their solution to be marched out with time. Furthermore, the differentiations (aS /Sp ) and (aS l/i.) produce the . .P terms and uncouple the equations in one step. ij r Example In order to compare the two methods, the above procedure will be applied to the system in Figure 4-1. Since the internal spring K34 produces an internal torque which is transferrable to each input as a generalized equivalent torque (see Chapter II), it will be included in the Fik terms, leaving the sum S1 made up of only the kinetic energy of the system. Therefore, 1 1 212, }* Y (4-14) S1 2 + 2112 2 22 2 (4-14) 1 ( 0 + TI, (4-15) S=- C 9 + C12 2 + Ts/lk-l + Kb/ik lk (4-15) S- C21 + C2 2 + T/2,k-1 + Kb/2k 2k +T2, (4-16) ik = ik yik-l' (4-17) where the C and K k are defined in Table 3-1. ij,k b/ik Substitution of Equation (4-17) into Equation (4-12) reduces Equations (4-10) and (4-11) to the following forms i = (iN + (F)k (4-18) ik+l = and as ki,k = (4-19) Let ki,k+1 be given by the finite difference relation i,k+l = ,k+ ,k)/At. (4-20) Then Equation (4-18) becomes sk+1 +.i ^(Fi)kk] At i, k+l = i,k k\~o / Substituting Equations (4-14) (4-17) into Equations (4-12), (4-19), and (4-21), and solving for ci,k+l equations p111 1+1 1 - -2k+1 L- -k 11 2 12 1 12 2 yields the following set of six * 21 1 22 1 0 2 1 2k 21 2 22 2 k 2 k 1 11 C12 1 s/1 Kb/1 1 + + + At, (4-22) 2 k 21 22k 2 s/2] k-1 b/2 N2 = + At , 2k+ 2 k 2 k and -1 *1 11 12 k1 ? k+1 L- -2k+l 2- k+l (4-23) (4-24) (4-21) The initial values of the X's are obtained by solving Equation (4-19), using the initial values for the p's and t's. Hence I 1 11 1 2 I1 (4-25) X2 121 1220 -20 The matrix Equations (4-22) (4-24) require one less numerical step of integration as pointed out in [30]. It is also pointed out in refer- ence [30] that the X's are the moment, usually called pi's, and that Hamilton's canonical equations are obtained directly without formerly deriving the generalized moment. These equations may now be marched out with time to obtain the solution to the dynamical equations of motion. They require one less integration step as opposed to solving Equations (4-8) and (4-9) numerically. The matrix inversion is only necessary once per integration and hence any error is only integrated once as opposed to twice for Lagrange's equations. The significance of this development is the system formulation in terms of kinematic influence coefficients developed in Chapters II and III, allowing complex mechanisms with multiple inputs to be analyzed for their dynamic response by established numerical methods. This for- mulation provides a way to reduce complex, multiple input mechanisms to coupled equivalent mass systems, yielding differential equations of motion possessing variable coefficients. These coefficients are known in terms of the mechanism geometry through the kinematic influence coef- ficients of velocity and acceleration. CHAPTER V SUNVMARY AND CONCLUSIONS Complex multiple input linkage systems have been difficult to analyze for their dynamic response because of their nonlinear geometric character. This characteristic generates holonomic constraints asso- ciated with the generalized coordinates necessary in describing the motion of the linkage system. Large numbers of generalized coordinates (see Figure 5-1(a)) have been required to obtain the systems dynamical equations of motion. Algebraic equations of constraint are required to account for the generalized coordinates other than those associated with independent system inputs. The result is a large number of coupled, nonlinear, second-order differential equations together with a set of algebraic equations in terms of undetermined Lagrangian multipliers which account for the geometrical constraints on the system. The algo- rithm required to integrate numerically and solve this set of equations requires a dual iteration scheme, one to solve the differential equa- tions of motion and one to satisfy the geometric constraints on the linkage system. Methods such as those developed by Chace [11], Uicker [13], and Carson and Trummel [14], utilizing relative coordinates and 4 X 4 matrix coordinate transformations, have been the only tools available to solve the dynamic response question for these systems. The set of N= 8 n =2 8 Generalized Coordinates 6 Algebraic Equations of Constraint 212 n= 2 (b) Figure 5-1. Complex Multiple Input Mechanism and Its Coupled Equivalent Mass System second-order differential equations resulting from their methods has been sufficient, though unwieldy, time-consuming, and subject to error, to describe the dynamical behavior of linkage systems. The goal of this dissertation has been threefold: to develop a systematic method whereby linkage systems of high-order-complexity can be constructed from Assur groups in terms of kinematic influence coef- ficients of velocity and acceleration of the basic system group; to reduce these highly complex linkage systems to coupled, equivalent mass systems acted upon by equivalent variable rate springs, variable coeffi- cient viscous dampers, and equivalent external forces and torques; and, to determine the differential equations of motion for the coupled equiv- alent mass system in terms of the minimum number of generalized coordi- nates (i.e., the number of independent system inputs). The construction of general linkage systems of higher-order- complexity, as discussed in Chapter II, is seen to be expressed in terms of series and parallel link connections. The connection types are defined by multiplication (series) and addition (parallel) of successive velocity influence coefficients. The use of Assur groups to construct mechanisms of higher-order-complexity from existing system groups, with- out modifying the mobility of the basic chains, allows the displacement analysis to be performed by established procedures. This is the basis for eliminating the holonomic constraints on the system, reducing the number of generalized coordinates required to describe the motion from N (number of system links) to n (number of independent system inputs). Elimination of the holonomic constraints subsequently reduces complex, multiple input linkage systems (see Figure 5-1(a)) to coupled, equivalent mass systems (Figure 5-1(b); rcquirinr only n g-nci1i-:eJ coordinates to describe its motion. This reduction tl1iminjt,: ther need for relative coordinates and their matrix transformations required by the existing methods. Rather, the coefficients of the resulting differential equations of motion become known variables of the system's independent input parameters expressed in terms of kinematic influence coefficients. Second-order differential equations describing the dynamical behavior of the equivalent mass system have been derived by the clas- sical Lagrangian method, while first-order difference equations were derived by the direct application of Hamilton's principle. This method [18], yields a set of first-order difference equations derived through only one differentiation. These difference equations can be marched out with time, requiring only one matrix inversion per integration step. The equivalent mass system formulation developed here provides a convenient and unique medium through which many problems concerning the dynamical behavior of linkage systems can be studied and simulated. The influence of elastic deformation of system links on the dynamic response of the linkage system primary input can be studied, based on a hinged beam model such as the one shown in Figure 5-2. The effect of bearing deformation on the input dynamic response could be investigated, based on the system models in Figure 5-3. The equiv- alent mass systems are shown below the system models. The elimination of the holonomic constraints places complex linkage systems in a convenient form for directly applying the prin- ciples of optimal control theory. Optimal open loop control laws may C34 Figure 5-2. Linkage System with Elastic Coupler Link ---` N t I I I CM I I / /\ , 's- e-- H +-4 U) o N SI . be determined which minimize specified performance indexes while satis- fying prescribed constraints on the control itself, the states, or both. Figure 5-4 illustrates this concept as applied to complex linkage systems. The problem could be formulated as follows: Determine the control u which minimizes the variation of the velocity s from the velocity s1 associated with position s1 over the range s 1 s 5 s2, subject to the ..* inequality constraint on the control, u u This in essence places an upper bound on the jerk u,thus giving third-order control for the cam surface designed to produce the required control u. The preceding problems are not intended to be solved here. Rather, they are provided to point out possible research areas which can be pursued, utilizing the system formulation developed in this dissertation. (C -0m APPENDIX APPENDIX A DIRECT DERIVATION OF FIRST-ORDER DIFFERENCE EQUATIONS FOR DYNAMICAL SYSTEMS The derivation of the set of first-order difference equations used in Chapter IV to solve the time response of complex linkage systems is presented in part from the paper [18] "Derivation of First-Order Difference Equations for Dynamical Systems by Direct Application of Hamilton's Principle" by Vance and Sitchin. The purpose for presenting this derivation is for completeness and convenience to the reader of this dissertation, since the method proves to be ideal for treating complex multiple input linkage systems. The nomenclature in this der- ivation does not correspond directly with that in the main text. It is therefore listed separately at the end of this appendix. Derivation of Method Hamilton's principle for nonconservative systems with k degrees of freedom is T T k 61 = 6 S Ldt + Y' ( Fiq.)dt = 0, (A-l) 0 0 i=1 where the integrand of the second integral is the virtual work of the nonconservative forces. This second integral is zero for conservative systems. After partitioning the interval 0 to T into N small increments AT, the two integrals can be approximated by sums and the principle can be rewritten as 6S1 + S2 = 0, (A-2) where the functions S and S2 are sums given by N S1= E n (qln'q2n..A ',qkn'4ln'2n'. qkn)Lt (A-3) n=1 N k S2 = FinqinAt. (A-4) n=l i=l Equation (A-2) requires that the variation of the function S1 equal the negative S2. Since the displacements and veloc- ities are to be related by some finite-difference expression, they are not independent of each other. The problem lends itself to the use of Lagrangian multipliers in order to achieve an independent variation of coordinates. The equations of constraint, defined by the previously mentioned finite- difference relationship, have the general form Aq. in = i (A-5) in At or fin = Aqin int = 0, (A-6) where Aqin is any desired expression for the first-order dif- ference of qin. Taking the variation of the constraint function (A-6) gives N "of. of" ) 6fin = (ln 6q + 6i.)= 0. (A-7) j=1 1j ij1 Equation (A-2) can now be written as N k S + infi + S2 = 0. (A-8) 1 Zn f+ 2= n=l i=l Substitution of Equation (A-7) into Equation (A-8) and rearrangement of terms gives N k as N fa ] N k SS N af.. ++ 6q. = 0. (A-9) +-' ij Eqj a in n=l i=l n j=1 in The kN Xi are chosen so that the kN bracketed expressions ij in the second double summation are all zero. This leaves the 6qin as independent variations. Equation (A-9) can then be satisfied by independently requiring the kN bracketed expres- sions in the first double summation to be zero. Thus the motion of the dynamical system will be such that the following 3kN equations are satisfied (i = 1,2,...,k; n = 1,2,...,N): as1 N 6fij 7 ij + =A F. At, (A-10) in j=l in in S+ kij x 6 = O, (A-ll) in j=l in f. = Aqin in. At = 0. (A-12) When the first-order difference form Aq. is substituted in into Equation (A-12), the summations in Equations (A-10) and (A-ll) are reduced to only a few terms. For example, if qin = qin qin-l 3f.. 6f.. 3- = 0, j n, n+ l; and J = 0, if j j n. (A-13) in in As is characteristic of the Lagrangian multiplier method, the convenience of treating dependent variables as if they were independent has been gained at the expense of an added set of unknowns, the X's. Unlike many applications of Lagrangian multipliers, however, the \'s are not in general constant. In fact the X's represent moment, ., and have the status of independent coordinates [3] . The set of Equations (A-12) may be considered trivial (although necessary) in the sense that they are simply the equations of constraint between the velocities and displace- ments. The X's will be constant only in the case of ignorable coordinates. Goldstein, H., Classical Mechanics, Addison-Wesley Publishing Co., Atlanta, 1965, p. 227. The derivation just presented is not tied to any particular inite-difference form. A determination of the best form to use will depend on the application. The symbols are defined as 6 = "variation of" A = finite increment S= dot appearing directly above a variable designates derivative with respect to time X = undetermined Lagrangian multiplier constant i = the corresponding generalized coordinate n = the corresponding time interval j = the corresponding time interval t = time F. = generalized nonpotential forces in APPENDIX B NUMERICAL SOLUTION TO A TWO DEGREES OF FREEDOM EXAMPLE The purpose of this appendix is to illustrate the actual responses obtained from the dynamical equations of motion derived for the five-bar mechanism in Chapter IV (see Figure (B-l(a))). The procedure for obtaining the variable coefficients to Equations (4-22) (4-25), describing the dynamical behavior of the system in Figure B-l(a), is described below. The necessary coefficients * are K C. Tik I. k and P are K/ik' Cij,k' i,k ij,k' and ij r,k. (1) For the given initial position of the mechanism defined by p1 and (p2' calculate the kinematic position informa- tion pertaining to the other links $3' ,4 etc., by using the equations in Figure B-2. (2) Substitute this position information into the appropriate blocks of Table 2-1 to determine the velocity and acceller- ation influence coefficients igi and hij. For example, 3 l,k+l 3 l,k-l 3gi 2p (B-l) (3) Once the 1g.'s and Lhij's are determined, substitute them into the appropriate blocks of Tables 3-1 and 3-2. This furnishes the following expressions for the coefficients to the differential equations: 2 2 IEgik-I Igik K/ik = a gik K ik-12 ik (B-2) N S(k C ) (B-3) ij,k = k'-' 9 Tik = (gik)T (B-4) I and P are determined from the expressions in Table 3-2, ij,k ij r,k blocks 1.1 and 1.2. The right-hand side of Equations (4-22) (4-25) are now completely known, allowing the k+1st 's, p's and p's, to be deter- mined. This procedure is repeated until the equations of motion are integrated over a predetermined time interval. Equations (4-22) - (4-25) represent the equations of motion for the coupled equivalent mass system shown in Figure (B-l(b)). The solutions to the dynamical equations of motion are shown in Figure (B-3) and (B-4). Polar plots of c. vs p1 and c2 vs c2 are shown for the parameters listed in Table B-l. For added clarity, Figure (B-5) illustrates the equivalent inertias as functions of time. The equivalent inertias show no cyclic phenomenon due to the noncyclic character of the input links. However, the inertia coupling term, I12, illustrates the coupling between the * positive inertia terms Ill and I22. The influence coefficients were calculated by finite differences. The total problem was programmed on the IBM 360-65. The program con- sisted of 270 cards and required 0.38 minutes to execute for an integra- tion step size of At = .000025 second. This step size can be increased considerably without affecting the solution accuracy, thus decreasing the execution time by that factor. B K 34 4 C34 (a) (b) Figure B-1. Two Degrees of Freedom Five-Bar Example 34 - k34 - =- + - 2/3/4,OS34 /324 (05 COSc -ACOS S-r-2COSS2+ (CSIN 5+ NSIN(+- 1SININ0 3 -04 34 C34= C os 2- 2- 4 -4 2-e3 ,^ C<3 SIN -12-SINO5+ C4 SIN I_ r5SIN0B + PSINO, - 134 , SINN, - -34 ,4SIN 2 .+ 4SIN03 S 1E .,r 1 .1034 --SIN-2- +SIN SIN SSINS 1 34 Figure B-2. Kinematic Position Equations O/ O \~ ~ v \ oTOs / cs 14 .8-" 0 '-4 1 bo U, U, * U, '-4 0 C. U, '-4 C3 w 0 00 Ijp 002 (T)4 o o o .9- 4 02 02 No 02 01 02 12 'I //i TABLE B-1 FIVE-BAR PARAMETERS Length (inches) Weight (lbs.) Moment of inertia (in. lb. sec.2) 1 4.0 2.97 .0488 34 = C34= T = 2 = 2 2.0 1.48 1.14 50.0 in. lb./rad. 5.0 in. lb. sec./rad. .05 in. lb. -.02 in. lb. (c1)o (Yp2)o (2 o) (9 o = 00 = 250 rad./sec. = 900 = 0 Solution to differential equations for known forcing functions, constant T1 and T2, are shown on Figures (B-3) and (B-4). "3 8.0 5.52 .1385 4 12.0 4.16 .4062 5 11.18 - 0 - - - 06 4 *a I F Im EC3 cr * w coo o w i 3i 3N I *I I I I * * I ** * r, l *s( tR *'~l3)~ 358 H-~l~~N N~nn3 BIBLIOGRAPHY 1. Benedict, C.E., "Dynamic Response Analysis of Real Mechanical Systems Using Kinematic Influence Coefficients," Master's Thesis, University of Florida, December, 1969. 2. Modrey, J., "Analysis of Complex Kinematic Chains with Influence Coefficients," Journal of Applied Mechanics, Vol. 26, Transactions of the ASME, Vol. 81, June, 1957, pp. 184-188. 3. Hain, K., Applied Kinematics, McGraw-Hill Book Company, Inc., New York, 1967, pp. 53-56. 4. Meyer zur Capellen, W., ". . Harmonic Analysis of Periodic Mechanisms' Proceedings of the International Conference for Teachers of Mechanisms, Shoe String Press, New Haven, 1961, pp. 171-185. 5. Denavit, J., and S. Hasson, "On the Harmonic Analysis of the Four- Bar Linkage," Proceedings of the International Conference for Teachers of Mechanisms, Shoe String Press, New Haven, 1961, pp. 171- 185. 6. Flory, J. F., and J. C. Wolford, "Harmonic Analysis of Kinematic Linkages," ASME Mechanisms Conference, Paper No. 64-Mech-38, October, 1964. 7. Markus, L., and J. Tomas, "Harmonic Analysis of Planar Mechanisms- Kinematics," Journal of Mechanisms, Vol. 5, No. 4-A, 1971, pp. 171- 185. 8. Bogdan, R. C., and T. V. Huncher, "General Systematization and Unified Calculation of Five- and Four-Bar Plane Basic Mechanisms," ASME Mechanisms Conference, Paper No. 66-Mech-ll, October, 1966. 9. Bogdan, R. C., D. Larionescu, and I. Carutasu, "Complex Harmonic Analysis of Plane Mechanisms, Programming on Digital Computers and Experimental Examples," ASME Mechanisms Conference, Paper No. 68-Mech-62, October, 1968. 10. Crossley, F. R. E., and N. Seshachar, "Analysis of the Displace- ment of Planar Assur Groups of Computer," Transactions of the International Federation of Theory of Mechanisms and Machines, September, 1971. 11. Chace, M. A., "Analysis of the Time-Dependence of M.ulti-Freedom Mechanical Systems in Relative Coordinates," ASME lMechlanisms Conference, Paper No. 66-Mech-23, October, 1966. 12. Smith, D. A., "Reaction Force Analysis in Generalized Machine Systems," Ph.D. Dissertation, University of Michigan, 1971. 13. Quicker, J. J., Jr., "Dynamic Behavior of Spatial Linkages: Part 1 Exact Equations of Motion; Part 2 Small Oscillations About Equilibrium," Journal of Engineering for Industry, February, 1969, pp. 251-265. 14. Carson, W. L., and J. M. Trummel, "Time Response of Lower Pair Spatial Mechanisms Subjected to General Forces," ASME Mechanisms Conference, Paper No. 68-Mech-57, October, 1968. 15. Wittenbauer, F., Graphische Dynamik, Springer-Verlag, Berlin, 1923. 16. Federhofer, K., Kinetastatik flachenlaufiger Systeme, S.-B. Akad. Wiss. Wien, Math.-Naturwiss. Kl., Abt. IIa. Jg. 139, 1930. 17. Beyer, R., Kinematisch-Getriebeanalytisches Practikum, Springer- Verlag, Berlin, 1960. 18. Vance, J. M., and A. Sitchin, "Derivation of First-Order Differ- ence Equations for Dynamical Systems by Direct Application of Hamilton's Principle," Journal of Applied Mechanics, Paper No. 70-APM-PP, 'June, 1970. 19. Pelecudi, CHR., "Kinematic Multipoles with Rigid Links," Rev. Roum. Sci. Techn. Mec. Appl., Tome 13, No. 5, pp. 997-1013, Bucarest, 1968. 20. Pelecudi, CHR., "Interpretation of the Dyad as Kinematic Dipole and Quadripole," Rev. Roum. Sci. Techn. Mec. Appl., Tome 13, No. 6, pp. 1225-1237, Bucarest, 1968. 21. Benedict, C. E., and D. Tesar, "Analysis of a Mechanical System Using Kinematic Influence Coefficients," Proceedings of the Applied Mechanisms Conference, Paper No. 37, July, 1969. 22. Benedict, C. E., and D. Tesar, "Optimal Torque Balance for a Complex Stamping and Indexing Mechanism," ASME Mechanisms Con- ference, Paper No. 70-Mech-82, November, 1970. 23. Benedict, C. E., and D. Tesar, "Dynamic Response Analysis of Quasi-Rigid Mechanical Systems Using Kinematic Influence Coef- ficients," accepted for publication in 1971 in the Journal of Mechanisms. 24. Benedict, C. E., and D. Tesar, "Dynamic Response of a Mechanical System Containing a Coulomb Friction Force," Transactions of the International Federation of Theory of Mechanisms and Machines, September, 1971. 25. Benedict, C. E., G. K. Matthew, and D. Tesar, "Torque Balancing of Machines by Sub-Unit Cam Systems," Proceedings of the Applied Mechanisms Conference, Paper No. 15, October, 1971. 26. Assur, L. V., "Issledovanie Ploskikh Sterzhnevykh Mekhanizmov S Nizshimi Parami," Izdat. Akad. Nauk SSSR, Moscow, 1952. 27. Curtis, M. F., and J. Tomas, "Analytical Solution of Planar Binary Groups," Proceedings of the Applied Mechanisms Conference, Paper No. 30, October, 1971. 28. Abramowitz, M., and I. A. Stegun, Handbook of Mathematical Functions, Dover Publications, Inc., New York, 1968, pp. 883-884. 29. Quinn, B. E., "Energy Method for Determining Synamic Character- istics of Mechanisms," Journal of Applied Mechanics, Vol. 16, September, 1949, pp. 283-288. 30. Sitchin, A., "Problems in Attitude Stability of Dual-Spin Space- craft," Ph.D. Dissertation, University of Florida, 1970. BIOGRAPHICAL SKETCH Charles Edward Benedict was born March 21, 1939, at Tallahassee, Florida. He graduated from Leon High School in June, 1957. He began studies at Florida State University in 1958, and graduated with the degree Bachelor of Science with a major in Mathematics in December, 1963. After working for three and one-half years for Florida Gas Trans- mission Company, he began studies at the University of Florida in the field of Mechanical Engineering in April, 1967. He graduated with the degree Bachelor of Science in Engineering with high honors in August, 1968. He received an Engineering College Fellowship and continued his advanced education, receiving a Master of Science in Engineering from the University of Florida in December, 1969. He was awarded a NDEA Title IV Fellowship and continued his studies toward a degree of Doctor of Philosophy. This dissertation completes these studies. Charles Edward Benedict is married to the former Patricia Ann Casey and has one daughter, age seven. He is a member of Kappa Alpha Order, Tau Beta Pi, Pi Tau Sigma, Phi Kappa Phi, Florida Engineering Society, and the American Society of Mechanical Engineers. I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. D. Tesar, Chairman Professor of Mechanical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. C. C. Oliver, Co-Chairman Professor of Mechanical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. W. H. Bykin Jr.- Assistant Professor of Engineering Science & Mechanics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. T. E. Bullock Associate Professor of Electrical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Associ te Professor of MechLnical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. ,K1l. Vance ASsistant Professor of Mechanical Engineering This dissertation was submitted to the Dean of the College of Engineer- ing and to the Graduate Council, and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. December, 1971 Dea College of Engineering Dean, Graduate School |