The vector-network method for the modelling of mechanical systems

Mathematics and Computers North-Holland

in Simulation

31 (1989/90)

THE VECTOR-NETWORK METHOD OF MECHANICAL SYSTEMS

565

565-581

FOR THE MODELLING

M.J. RICHARD Dept. of Mechanical Engineering,

Lava1 University, Quebec, Quebec, Canada GIK 7P4

R.J. ANDERSON Dept. of Mechanical

Engineering,

Queen’s University, Kingston,

Engineering,

University of Waterloo,

Ontario, Canada K7L 3N6

G.C. ANDREWS Dept. of Mechanical

Waterloo, Ontario, Canada N2L 3GI

This paper presents an extension of the vector-network approach to the problem of motion prediction. The entire procedure is a basic application of concepts of graph theory in which laws of vector dynamics have been combined. A comprehensive mathematical model for the systematic formulation of the equations of motion of dynamic three-dimensional constrained multi-body systems is derived. The method embodies simultaneously the three-dimensional inertial equations associated with each rigid body and the kinematic constraints into a symmetrical format yielding the differential equations governing the response of the system. The modelling technique is thoroughly described in this work and its validity is impartially established.

1. Introduction The last two decades have witnessed tremendous growth in the number of computational techniques developed for efficiently simulating mechanical systems. Most of the problem-oriented computer programs for the analysis of dynamic mechanical systems were sponsored by academic institutions. They all possessed basic capabilities for analyzing different classes of mechanical systems. Specific features of general-purpose dynamic programs include the following; simulation of two and three-dimensional multi-degree-of-freedom systems, assumption that mechanical devices and bodies are rigid, library of interconnecting mechanical elements such as springs and dampers, possibility to input non-standard forces via user-written subroutines, constrained or unconstrained motion, library of kinematic joints, kinematic and dynamic/vibration analysis. It should be noted that the “self-formulating” aspect of the program is the prime concern of external users. In addition, good resolution time, generality and a reasonable interaction between analyst and computer must be combined in a simulation program in order to sustain a productive implementation. All general-purpose computer programs must comply with five major steps. The first phase of the dynamic analysis consists in specifying the system to the computer. Then, we must deal with 0378-4754/90/$3.50

0 1990, Elsevier Science Publishers

B.V. (North-Holland)

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the model construction. It identifies all interconnecting points throughout the system. Whether the discretization is carried out by computer or by a set of manual procedures, whether it is intuitive or explicit, a proper discretization requires the application of some topological concepts. Clearly, if this phase is assigned to the computer, the program is called “self-formulating”. The third phase of the study focuses on the creation of the model. Due to the non-linear effects encountered in dynamic systems, all advanced simulation techniques require the use of the computer. The fourth phase is where the theoretical equations of motion are solved. As expected, these differential expressions are complex in nature and must be solved by numerical techniques. Finally, the last step would basically involve the translation of these results into a detailed format issued in an understandable form to the analyst. Although the sequence of execution is by no means complete it seems generally acceptable and represents the main artery of a complex numerical procedure required by all dynamic investigations. The refinement of formulation techniques have spawned several algorithms for the dynamic analysis of mechanical systems [l-6]. All are capable of automatically generating and numerically integrating the differential equations governing the motion of dynamic systems. The major difference between each of the algorithms resides in the second and third step of the general procedure suggested above for simulating the response of mechanical systems. Interrelated advancements in the field of dynamics and computer science have seized the interest of many researchers [7-lo]. They have laid the groundwork for practical applications of mechanical principles to physical systems. Surely, many more computer programs exist and the number is growing. The vector-network formulation originated from the University of Waterloo. In the mid-seventies, Andrews [ll-131 suggested the association of vector algebra with linear graph theory bringing many topological concepts out of the realm of mathematics and into the world of dynamic analysis. His contribution was significant since it enabled, for the first time, many direct applications of graph theory to traditional dynamics. It is proposed to study the relationship between some spatial dynamic concepts and geometric constraints guiding the motion of mechanical systems. Hence, the frame of reference of this research is entirely based on the application of the vector-network algorithm to advanced mechanisms. The objective is to describe a general-purpose computer program based on vector-network techniques [14,15] called RESTRI (RESeau TRIdimensionel) which stands for the French version of three-dimensional network.

2. Vector-network theory This section provides a general description of the vector-network formulation including a brief review of linear graph principles [16] required to formulate the algorithm. Many researchers have studied the theory of graphs [17-191 mainly due to the fact that among all the fields of human interest, there are few where graph theory cannot be applied to the process of analyzing or synthesizing problems [20]. Hence, graphs can be considered as mathematical abstractions of the real world and can be useful in enhancing the understanding of composite systems. Formally, a graph G(v, e) consists of a set of vertices v and a set of edges c such that the edges intersect only at the vertices. Graphs as defined here are not restricted in either Y and <,

H. J. Richard et al. / Modelling of mechanical systems

567

Y

L

REFEREN&

Fig. 1. A simple planar mechanical

system.

but if depicted from dynamic systems, they will always represent a finite topology which implies that this study is completely based on finite graphs. The network of vectors can be generated in three sequential steps: 1. define an inertial reference frame (root), 2. locate the points of interconnection as nodes, and 3. draw the displacement vectors between all nodes which are connected. This diagram idealizes the original system as an oriented linear graph. As an illustration of the procedure, consider a simple planar dynamic system, shown in Fig. I, which consists of a rigid body with non-negligible rotational inertia and center of gravity located at C.G., acted upon by two springs and a dashpot. Assuming negligible weight, the vector-network diagram, including the rigid body traced in dashed lines to make the network easier to identify, is depicted in Fig. 2 and portrays a linear graph with six vertices and eight edges. The vector rr is the inertial displacement vector representing the center of gravity, r, and r3 define dependent drivers and are called “rigid-arm elements”, r4 and rs specify displacement drivers while r, and r7 represent the springs and r, the dashpot. All vectors such as r,, r, and q, the properties of which are established from an inertial frame, must emanate from the single ground node since that node is the only absolute fixed reference in the diagram, as compared to rz and r, which are defined relative to the rigid body. Visibly, vectors r,, r, and r, span their respective two points of interconnection in the system. The formulation tree requires that all branches be associated with inertial elements such as masses, displacement drivers and rigid-arm elements. Therefore, all other elements, each associ-

Fig. 2. Vector-network

diagram

of the planar system in Fig. I.

M.J. Richard et al. / Modelling of mechanical systems

568

ated with a chord, such as linear springs, dampers, geometric constraints, kinematic joints and a variety of dependent drivers are considered part of the cotree. Many interesting graphical properties can be studied using three matrix formulations of importance in vector-networks. Although closely interrelated in their derivation, comparisons among them are not meaningful since each investigates a different aspect of the graph and is generated for different purposes. 2.1. Incidence matrix Given a matrix with v vertices and e edges, the order of interconnection of a system can be summarized in a v by c incidence matrix. The incidence matrix of G(v, C) is denoted by [K] and is defined as follows: i. each of the v rows corresponds to a vertex of G, ii. each of the e columns corresponds to an edge of G, and iii. entries kjj = + 1 (or - 1) if the ith node is the initial (or final) vertex of the j th edge, or k,, = 0 otherwise. The incidence matrix can always be compiled from inspection of the graph. As an illustration, the graph of Fig. 2 has the following incidence matrix: Edge

2

1

Node

3

4 0

5 0

6 0

7 0

8 0

1

0

0

0

0

1

0

0

1

0

0

1

0

1

0

0

1

0

0

0

0

0

0

7

0

0

-1

5

6

All the geometrical of this matrix.

0

-1

information

-1

necessary

-1

-1

(1)

-1

for the solution

of dynamic

systems lies in the creation

2.2. Cutset matrix A cutset is a disconnecting set of edges such that after removal of that set of edges, the graph is divided into two or more components. Since a cutset exists for each tree branch, there will be (v - 1) cutsets in a graph with v vertices. Therefore, all cutsets can be assembled in a (v - 1) by c cutset matrix identifying all cotree terminal graphs acting through each vertex. Following this definition, the cutset matrix can be redefined as [V, A], where [U,] is a (v - 1) by (v - 1) unit sub-matrix associated with tree branches and [A] is a (v - 1) by (e - v + 1) sub-matrix associated with cotree chords. For example, the cutset matrix of the diagram sketched in Fig. 2 can be obtained by simple row operations [16] performed on the (v - 1) rows of the

H.J. Richard et al. / Modelling of mechanical systems

incidence

matrix.

569

In this case, rows (2) and (3) are added to row (1) leading to the cutset matrix: Edge

Branch 1

[t&4]=

12345 -1 0

0

0

0

6 1

7 1

8 1

0

1

0

0

0

0

1

0

:

0

0

1

0

0

1

0

1

4

00010

0

0

5

_o

-1 0 TREE

0

0

1

0

-1

(2)

-l_

COTREE

which has five cutsets. Since a cutset isolates a part of the system, its application to dynamic systems will become obvious when solving the instantaneous relationship among forces or torques which requires the construction of a “free-body diagram”. 2.3. Circuit matrix A circuit is a connected subgraph in which exactly two edges are incident with each vertex. The circuit is defined as a subgraph which contains a single cotree chord and all the other edges, forming the closed chain, are tree branches. Following the definition, each chord of the cotree will form a circuit, thereby producing an independent set of closed chains since at least one edge will not be found in any other circuit. The circuit matrix can be arranged in the form [B UC], where [UC] is a (C - Y + 1) by (6 - v + 1) unit sub-matrix associated with cotree chords and [B] is a (c - v + 1) by (v - 1) sub-matrix associated with tree branches. The circuit matrix can be obtained by applying another basic theorem [16] or graph theory which proves that the cutset and circuit matrices are orthogonal. This orthogonal relationship states that the scalar product of the cutset matrix and circuit matrix vanishes. In matrix notation, the sub-matrices [A] and [B] are related by the negative transpose relation:

[B] = -[AIT

(3)

and

[A] =

-[BIT

where the superscript T denotes matrix transpose. This liaison regulates the entire structure of the vector-network formulation. corroborates the bond between the graph and dynamic theory. We can demonstrate by using the sample graph of Fig. 2. Equation (3) can be used to automatically

(4)

Moreover, it this principle transform the

M.J. Richard et al. / Modelling

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of mechanicalsystems

cutset matrix into the circuit matrix: Edge

Chord

[Bu,]=

1

6

-1

7 8

I -1 -1

-1

2 0

34567 -1 1

0

-1

TREE

0

0 0

0

1

0

8 0

1 1

0 0

1 0

01 11

COTREE

which has three circuits

3. Creation of the mathematical model In the second section of this article, we attempt to lay the theoretical and conceptual foundation for the study of dynamical systems. Some physical significance can be added to this theory utilizing two general terms: accross-variables { A V} which represent those quantities such as displacements (r), velocities (u) and accelerations (ti) which may be measured uuoss an element, and through-variables ( TV} symbolizing quantities such as forces (F) and torques (T) which act through an element. The vertex postulate states that the algebraic sum of through-variables corresponding to all the terminal graphs incident with any vertex of the graph is, identically, zero. Essentially, this is recognizable as the dynamic force-balance law which requires that force summation upon each body must be equal to zero. The cutset strategy is that all tree elements which contact and exert forces on an inertial tree subsystem but are not part of it are removed and replaced by adjacent cotree elements exerting their internal forces on the isolated system. Clearly, the cutset matrix provides a means by which all through-variables, acting on inertial elements, are expressed and accounted for. Once this critical step is achieved, the (V - 1) node-balance equations can be assembled. Simply, post-multiply the cutset matrix by the through-variables column matrix and equate to zero, to yield:

(5) where { TV} contains c elements. Another governing postulate is the circuit postulate which states that the algebraic sum of across-variables corresponding to all the terminal graphs included in any circuit is, identically, zero. This compatibility concept has been partially applied to mechanical systems for some time. Nevertheless it is mandatory, to some extent, to all dynamic algorithms involved in solving internal kinematic properties. Basically, this postulate represents the geometrical relations guiding the motion of mechanical systems. To be precise, each circuit equation alludes to a closed vector polygon respecting the geometric fit or compatibility law of rigid body dynamics. Similar to equation (5), the (6 - Y + 1) equations are assembled from the circuit matrix post-multiplied by the system across-variables,

(6) where ( A V} encloses

e elements.

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of mechanical systems

571

In addition to satisfying the cutset and circuit interconnection requirements, it is also necessary to satisfy the equilibrium prerequisites of each individual element. The vector-network formulation requires a terminal equation [15] for each element of the system graph. These may be theoretically derived or come from observation and experimentation or simply be the formalization of the designer’s experience. In dynamics, the terminal equations reveal the elements’ empirical characteristics which are based on various laws. For instance, Newton’s second law forms the basis for most of the analysis in dynamics which relates an external force acting on a particle to its acceleration. The same proportionality concept can be observed from Hooke’s law, relating a spring force to its deformation. Infinitely many types of externally-applied forces or torques may be exerted on a dynamic system. These drivers are time or variable dependent and are represented by an appropriate function. Similarly, whenever a specified point or position/velocity driver is fixed or moving as function of time, its corresponding terminal equation must be introduced. Finally, the vector-network can simulate the effect of virtually any type of force/displacement driver given its physical characteristics. In order to maintain a general approach, one may assign subscripts to all common elements encountered in a general 3-D mechanical system. Let a symbolic formulation of each group of elements be represented by (mass elements with significant rotational inertia) N, inertial elements (points on rigid bodies, defined relative to the body’s center of N2 rigid-arm elements mass), (specified points) N3 translational drivers (specified rotations) N4 rotational drivers (springs, dampers and external forces) N5 translational force drivers (dampers and external torques) N6 rotational torque drivers (due to kinematic constraints, pivots, etc) N, dependent force drivers (due to angular constraints on rigid bodies) N, dependent torque drivers (register the torque generated by a translational force). N9 torque coupling elements The actual number of elements in each group can be expressed by n, where i represents the category. This nomenclature encompasses all the different types of elements that can arise in a general three-dimensional mechanical system and will be very useful when deriving the differential equations of motion. Hence according to this symbolic representation, the cutset equations (5) may be written in the form:

TI: TV,

u,

0

0

0

r/;

0

0

0

u,

0

0

0

1

TV,

TV, TV, TV, TV, TV, TV,

= 0

(7)

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M.J. Richard et al. / Modelling of mechanical systems

where sub-matrix [ Aij] is a n, x ni matrix composed of + 1, - 1 and 0 depending whether element N, is incident upon N, and sub-matrix [ Ui] is a unit matrix. The column matrix { TV, } is a nk x 1 representing the through-variables associated with the k th classification. Using the orthogonal relationship, the circuit matrix is obtainable from the cutset matrix. Therefore, the readily derived circuit matrix will yield the circuit equations:

(8)

where sub-matrix [ Bji] is a nj X n, matrix coupled to [ Aij] through the relationship (3) and it specifies which element N, is included in circuit NJ. The column matrix ( A V, } is a nk X 1 matrix symbolizing the across-variables associated with the kth category.

4. Formulation of the equations of motion To illustrate the vector-network technique for obtaining the differential equations of motion, consider a conventional constrained multi-body system as portrayed in Fig. 3 and its vector diagram in Fig. 4. Begin the formulation by deriving the terminal equations for external forces N, [15] as follow, { 4 } and torques { & > as function of across state-variables F, =f(AT/,,

t>

(9)

DRIVER

reference

frame

5.

6

KINEMATICAL JOINT 7, 0

Fig. 3. Common constrained multi-body system.

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H.J. Richard et al. / Modelling of mechanical systems

Fig. 4. Vector-network

diagram

of the common

constrained

multi-body

system.

and

where { A V, } represents the across-variables of inertial elements NI. Once { F5} and { T6} are calculated, the second-order differential equations guiding the motion of a general mechanical system composed of rigid bodies and particles, interconnected with kinematic elements and subjected to some external stimuli, will require the full exploitation of the inertial cutset equations. With the inclusion of internal torques, these are

and U-1) + PMIK~

+

Puma

+ PEJIK~ =o

(14

where is a tiI X ni matrix indicating which forces/ torques are incident upon each inertial element N,, and ( < } or { q.} is a nj X 1 column vector representing the force/ torque variables within the system. The torques { Tg} engendered by cotree forces are functions of external forces { F5} and internal constraining forces { F,} since torsional expressions are developed from an isolated rigid body system. Thus, after dissociating the whole mechanism into n, elements the unknown forces of constraints {F,} will inevitably induce some internal torques. These will automatically be inserted in the algorithm by means of the rigid-arm cutset equations which mathematically state that [A,, I

{I;,)

= -[4&E;)

- [&]{-F7)

where [A*;], for i = 5, 7, is a n2 X n, matrix which is composed the interconnections between elements N2 and N;.

(13) of + 1, - 1 and 0 representing

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The torque-coupling P-9] =

terminal

equations

[_Y,lw2;)

can be recalled

in matrix

notation:

04

where [_rz] is a n, X n2 skew-symmetric matrix performing the matrix equivalent of a cross-product multiplication. If Equation (13) is then inserted into Equation (14) for { E;}, one will obtain:

After multiplying Equation (12) by the transformation matrix [Di] in order to express the general torsional relations along the embedded trihedron of each moving body and after substituting the torque { Tg } and the Euler terminal equations into Equations (11) and (12), the cutset equations become

{&} =

[Gl{h,}~

(174

in which [Mi] is a n, x n, diagonal matrix with mass elements J/r on its diagonal; [II] is the inertial matrix of body elements Nr about the center of gravity; [Ill] is a n, x n, diagonal transformation matrix connecting the moving frames to the inertial frame; [r2] is a n2 x n 2 skew-symmetric matrix performing the matrix equivalent of a cross-product multiplication; [C,] is a n, x n, transformation matrix connecting the Euler systems to the body-fixed systems; { 3, } is a n, x 1 column vector representing Euler angular velocities, and { gr} is a n, x n, skew-symmetric matrix representing the angular velocities of bodies along the instantaneous directions of the body axes; [ Ei] is a constraint unit vector matrix (i = 7, 8) expressed along the inertial reference frame for the kinematic elements. Note that Equations (16) and (17) were derived with respect to the center of gravity of each rigid body and all the terms were measured along their respective moving reference frames. The coefficient matrices [_r2], [C,] and [Or] are all varying as functions of time and only [Ii] along with the cutset sub-matrices remain constant with regard to time. Relations (16) and (17) unfold two sets of equations in four sets of unknown tir, 8i, F, and T,. equations must be added to acquire a unique solution. The Naturally, n, f n8 additional translational and rotational constraints must be expressed from acceleration circuit Equations (8) in order to be compatible with the Newton-Euler equations of motion. Hence, the mathematical model of the physical constraints within rigid body systems required that the acceleration circuit

H.J. Richard et al. / Modelling

equations

be satisfied.

of mechanical systems

575

That is,

[U,] is a ni X n, unit matrix (i = 7, 8), and { yj} represents the absolute angular acceleration with respect to inertial grounds of element j (j= 1, 4, 8). The rigid-arm accelerations { ti, } in Equations (18) must be expressed in terms of inertial across-variables. For a single rigid-arm element, the linear acceleration or terminal equation was developed in previous work [14] and may be recollected as

In order to be a useful asset to the mathematical model, Equation (19) must be expressed in terms of Euler angles. One must exploit the transformation matrices [Q] and [C,] between body axes and the Euler rotational system to yield an expression of the form,

(21) Clearly, the first term of the right hand side of Equation (21) vanishes since it performs a cross-product multiplication between two parallel vectors, { wi }. Hence, after reintroducing Equation (17a) in (21) for { 4, }, the angular acceleration expressions are condensed into

(22) Equations (22) and (20) can be substituted in the circuit relations (18) and (19) and then can be multiplied by [ E,lT and [&IT respectively and can be affixed to the matrix Equations (16) and (17) to give in matrix notation: TREE

COTREE

0 ; -A,,.& MI : . . . . . . . . . . . . . . . . . ..~..........~.~......... 0 . 1, : D,A,,rzA&,: - .S’,TB,, 1 ~;B,~~;B~,D;

0

0 - D,A,&

:

0

. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 0 1 - &B~,D; 1 0 : 0 Across variables

where

1

’

Through variables

(23)

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576

The above matrix could of course be further expanded. However, the relations (23) are the complete equations of motion as generated by the computer program RESTRI and since these are not analytically soluble, one must resort to numerical techniques to secure the dynamical response. The above sets of equations represent, at that instant, the mechanical system in a dynamic multidimensional time-space. Equations (23) contain a set of 2n, + n, + n8 non-homogeneous second-order differential equations in 2n, + n 7 + n8 unknowns. These were exclusively derived from the physical and topological characteristics of the vector-network model. Thus, the simultaneous solution of the cutset, circuit and terminal equations constitutes an efficient algorithm for the integral solution of rigid body systems. The step-by-step vector-network substitution procedure is readily amenable to processing by a digital computer. Consequently, this mathematical model represents a straightforward algorithm by which the behavior of three dimensional dynamic mechanical systems can be simulated. Note that the sparse coefficient matrix of Equations (23) is entirely expressed in terms of system positions or geometrical properties and time. Accordingly, the numerical procedures used for the solution of the variables of Equations (23) must compute these coefficients and forcing vector anew for each evaluation of the accelerations and constraints. Fortunately the anti-symmetric arrangement still prevails and can somewhat ease the calculating time for the regeneration of the coefficient matrix.

5. Numerical simulation Consider the constrained pendulum sketched in Fig. 5. The system consists of two pendulums connected by a third inextensible link (coupler). This mechanism represents a closed loop system in which the masses of each rod have been concentrated at the extremity of each link thereby

(Inertial Reference Fmme)

*x

I

7 Y

(Xz,Y2) WEIGHT

5 WEltliT

Fig. 5. Schematic

of a closed loop constrained

6

pendulum.

H.J. Richard et al. / Modelling

of mechanical systems

577

Y Fig. 6. Vector-network

diagram

of the system in Fig. 5.

idealizing the system into two particles governed by light interconnecting elements. The two masses forming this lumped mechanical system are both under the influence of gravity and are measured at WI, = 2 Kg and m, = 1 Kg. The position of point 3 and 4 in Fig. 5 is taken at 7.0 and 0.0 respectively where distances are in decimeters (dm). The vector-network diagram may be traced from the schematic mechanism (Fig. 5) as shown in Fig. 6. The displacement vectors ri through r4 in solid lines constitute a proper tree and are all emanating from the root of the diagram. Conformably, the vectors r, through r, in dashed lines form the cotree. Hence, the diagram of vectors for this example is composed of nine elements where vectors: represent mass m, and mass m,, r1, r2 represent specified points 3 and 4, r3, r, represent the weight of each mass, and r, 7 rs represent the rigid link elements. r,, rg, 5 From the vector-network diagram, an appropriate input was assembled with the following initial conditions (displacements-dm): xi = 4.0;

J?i = 0.0

Y, = 4.0;

Pi = 0.0

x2=12.0;

J$=o.o

Y,=5.0;

Y2=o.o

The input file was then submitted to RESTRI requesting a 15 second simulation with a maximum relative error of 1O-5 units. From the input description, the computer program will generate the incidence matrix which in turn will be used to create the cutset and circuit equations. The differential equations of motion will automatically be generated and integrated. According to the theory developed in Section 4, the closed loop particle mass system is a trivial example where the dynamical and kinematical properties of a mechanism must be respected. Note that although the simulated system is planar, all three scalar directions will be processed by the computer program but since the direction perpendicular to the plane of motion has no internal/external excitation, the scalar variables perpendicular to the motion will remain in an inactive state retaining their initialized values throughout the computation.

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M.J. Richard et al. / Modelling of mechanical systems

.

A

(RESTRII

-

(DIFF. EQUATION)

21 0

3

6 TIME

9

12

15

(SEC)

Fig. 7. Closed loop particle mass system (mass #l)

vertical displacement

vs time.

For verification purposes, the differential equation of this one degree of freedom system was developed using Lagrange’s equation. Displacements were plotted, in solid lines, in Fig. 7 through 10. RESTRI’s output, in t,riangles, was superposed onto the values projected from the analytical equations to certify the self-formulated results. A perfect correlation was displayed between both predicted values. A typical pendulum response can be observed for the masses m, and m2. The horizontal displacements are permitted to oscillate with full amplitudes as compared to the vertical displacements where the presence of the coupling link imposes a restrained vertical motion at each period as exhibited in Fig. 7 and 9. Although this example is considered to contain one of

6

rs

2

LY ; ,’ * 0

0

-2

2 0 =

-4

N 5

-6

A

(RESTRI)

-

(DIFF.

EQUATION)

-8 0

3

6 TIME

9

12

15

(SEC)

Fig. 8. Closed loop particle mass system (mass # 1) horizontal

displacement

vs time.

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H. J. Richard et al. / Modelling of mechanical systems

k Y

A

( RESTRI

-

( DIFF.

4 0

3

) EQUATION

12

9

6 TIME

)

15

(SEC)

Fig. 9. Closed loop particle mass system (mass #2) vertical displacement

vs time.

5 0

12

!i= J z

9

4 % 0

6

< !ii n s

3

2

0

3

A

(RESTRI

-

(DIFF.

6

) EQUATION)

9 TIME

12

15

(SEC)

Fig. 10. Closed loop particle mass system (mass #2) horizontal

displacement

vs

time.

the simplest kinematic configurations, the dynamics of such a basic system are involved developed as featured in this section by a general-purpose computer program.

and best

6. Conclusion The benefits of the vector-network method lie not only in its use as a computational tool, but rather in the deeper understanding it contributes into the stringent framework of dynamics. The actual formulation of the equations of motion, addressed in Section 4, revealed the paramount importance of the cutset and circuit relationships in the construction of the differential equations

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M.J. Richard et al. / Modelling of mechanical systems

governing the motion of a general system. It was shown that these equations were regulated by a standard anti-symmetric format. The formulation of Equations (23) exhibits most of the vector-network concepts where the TREE and COTREE represents cutset and circuit relations and the across and through coefficients are associated with inertial and internal properties. The vector-network method essentially bisects the dynamical system into two sets of equations; the state and constraint equations, the constraints being non-holonornic expressions in accelerations. The above formulation is equally applicable to closed or open loop systems with geometrical constraints which typify dynamical and linkage machinery. After careful observation of the coefficient matrix, one may deduce that the inertial part on the left hand side of the Equations (23) is solely dependent on the topology of the system and time. Consequently, the forcing vector on the right hand side of Equations (23) contains all the external stimulus which really “drives” the mechanical system. This approach entails the solution of a system of 2n, + n7 + n8 algebraic equations at each step of the integration of 2n, non-linear vector equations to predict the behavior of the mechanism. The most prominent benefit of this methodical model was its suitability for implementation on the digital computer. To demonstrate this application, a self-formulating computer named RESTRI was conceived and tested. From a minimal definition of the system, the computer program is capable of automatically creating and solving the differential equations of motion following a simple substitution procedure.

References [l] P.N. Sheth and J.J. Uicker, IMP (Integrated Mechanisms Program), a computer-aided design analysis system for mechanisms and linkage, J. Engrg. Indust. 94 (1972) 454. [2] N. Orlandea, M.A. Chace and D.A. Calahan, A sparsity-oriented approach to the dynamic analysis and design of mechanical systems - Part 1, J. Engrg. Zndust. 99 (1977) 773. [3] B. Paul and D. Krajcinovic, Computer analysis of machines with planar motion - 1. Kinematics. J. Appl. Mech. 37 (1970) 697. [4] J. Wittenburg, Nonlinear equations of motion for arbitrary systems of interconnected rigid bodies, Dynamics of multibody systems, IUTAM Symposium, Munich, Germany (1977). [5] W.O. Schiehlen and E.J. Kreyzer, Symbolic computerized derivation of equations of motion, in: Dynamics of Multibody Systems, K. Magnus, Ed. (Springer-Verlag, 1978). [6] J. Garcia de Jalon, J. Unda, A. Avello and J.M. Jimenez, Dynamic analysis of three-dimensional mechanisms in natural coordinates, J. Mechan. Transm. Autom. Design 109 (4) (Dec. 1987) 460-465. [7] R.A. Wehage and E.J. Haug, Generalized coordinate partitioning for dimension reduction in analysis of constrained dynamic systems. J. Mech. Design 104 (1982) 247. [8] R.E. Roberson, Generalized correction of numerical error in kinematical differential equations based on Euler-Rodrigues parameters, J. Acta Mechanica (1) (Dec. 1984) 127-132. [9] N.K. Mani, E.J. Haug and K.E. Atkinson, Application of singular value decomposition for analysis of mechanical system dynamics, Design Engineering Technology Conference, Oct. 7- I2 1984 (A.S.M.E., New York, NY) paper 82-DET-89, 6 p. (1984). [lo] P.E. Nikravesh, R.A. Wehage and O.K. Kwon, Euler parameters in computational kinematics and dynamics, Part 1 and Part 2, J. Mechan. Transm. Autom. Design 107 (3) (Sept. 1985) 358-369. [ll] G.C. Andrews, The vector-network model: A topological approach to mechanics, Ph.D. Thesis, University of Waterloo, Waterloo, Ontario (1971). 1121 G.C. Andrews, A general re-statement of the laws of dynamics based on graph theory, in: Problem Analysis in Science and Engineering (Academic Press, 1977).

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[13] G.C. Andrews and H.K. Kesavan, Simulation of multibody systems using the vector-network model, Dynamics of Multibody Systems, IUTAM Symposium, Munich, Germany (1977). [14] M.J. Richard and R.J. Anderson, Dynamic simulation of three-dimensional rigid body systems using vector-network techniques, Internat. J. Math. Comp. Simulation 26 (1984) 289. [15] M.J. Richard, Dynamic simulation of multibody mechanical systems using the vector-network model, Trans. Canad. Sot. Mechan. Enginrg. 12 (1) (1988) 21-30. [16] G.C. Andrews, Dynamics using vector-network techniques, Course Lecture-Notes, Course No. ME-730, Department of Mechanical Engineering, University of Waterloo, Waterloo, Ontario (1975). [17] M. Behzad and G. Chartrand, Introduction to the Theory of Graphs (Allyn and Bacon, 1971). [18] N. Christofides, Graph Theory, An algorithmic Approach (Academic Press, New York, 1975). [19] S. Even, Graph Algorithms (Computer Science Press, 1979). [20] H.E. Koenig and W.S. Blackwell, Linear graph theory - a fundamental engineering discipline, IRE Trans. Ed. E3 (1960) 42.