Dynamics of three-dimensional multi-body systems with elastic components

Pergamon 00457949(94)00607-5

Compur~rs & Slrurrur<,s Vol. 57. No. 2, pp. 309-316, 1995 Copyright 8,: 1995 Elsevier Science Ltd

Prmted in Great Britain. All rights reserved 0045.7949195 $9.50 + 0.00

DYNAMICS OF THREE-DIMENSIONAL MULTI-BODY

SYSTEMS WITH ELASTIC COMPONENTS

Yong Fang and F. W. Liou

Department of Aerospace and Mechanical Engineering, and Engineering Mechanics/Intelligent Systems Center, University of Missouri-Rolla, Rolla, MO 65401, U.S.A.

(Received 4 March 1994)

Abstract-In the mechanical assembling process, sometimes component elastic deformation is involved. To be able to effectively model general behavior of the mechanical assembling process, a more generic formulation of governing equations which includes dynamics of multi-body systems with both changing topologies and elastic deformation is necessary. Presented in this paper is the derivation of the general system governing equations, and a prototyped example to apply these equations to implement an automated computer simulation of a snap fit assembly.

INTRODUCTION

When modeling the assembling process or other manufacturing activities, mechanical components may be subjected to elastic-plastic deformations. For example, referring to the snap fit feature as shown in Fig. 1, to assemble the parts together will require elastic deformation modeling. To be able to automatically model such systems, the following features have to be included in the modeling processes:

(1) Multi-body dynamics-to be general, the assembling process can be modelled as a multi-body system undergoing the sequence of forced/free motion, contact/impact, and then joining process.

(2) Force closed kinematic joints-when modeling the dynamic behavior of mechanical assemblies, the contact condition between assembly components can vary from type to type. For example, a contact point may initially act like a pure rotational joint between two bodies, but in the next instant it may act like a sliding joint. Therefore, the system’s governing equations can be discontinuous, and the formulation would be different from the traditional approach. In the traditional approach, the form-closed kinematic joint only represents a system with fixed configuration. In order to simulate an assembly with changing configurations, the more general constraint, the force-closed point-to-surface constraint, is adopted.

(3) Elastic deformation-for the purpose of simu- lating assembly systems with deformable features, which undergo relatively large overall motion with deformation, nonlinear formulation must be formu- lated to realistically simulate the system.

Some computer simulation algorithms have been presented which enable dynamic analysis of rigid- body systems with changing topologies [l-4]. How- ever, little has been done in this area of mechanical systems with elastic components.

In this paper, the derivation of the equations of motion for general systems has been undertaken from a fundamental point of view by the introduction of an arbitrary representation of the kinematics of the deformation. The formulation is based on the theory of elasto-dynamics. This formulation includes the nonlinear terms, such as the rigid body motion and elastic deformation coupling terms. It can also be used to analyze three-dimensional deformable systems undergoing large overall motion. The formulation is used in a case study to simulate the dynamic behavior of a snap-fit assembly with the model of deformable components and changing topologies.

FORMULATION OF THE EQUATIONS OF MOTION

Generalized coordinates and kinetic energy

Figure 2 depicts a deformable body, where XYZ is a global coordinate system and 5~1 is a local coordinate system fixed with the body. Let q: be the rigid-body coordinates that define the location and orientation of the 5~4 system with respect to the global reference XYZ. The vector q; can be written in the form:

s: = (x0 y. z. e. ei e2 e3 )? = (9; sb )T, (1)

where q; is a set of Cartesian coordinates that define the location of the origin of the 511 system and q$ is a set of rotational coordinates that describe the orientation of the ta[ system.

The global position of an arbitrary point P on body i can be written as

RIP = Rb + Air’, (2)

309

310 Yong Fang and F. W. Liou

EARTHING PLATE

BRASS TERMINALS

EASE

Fig. I. The undeformed mechanical assembly

where r’ is the local position vector of point P and the use of a lower-case letter means with respect to the t;r~c system. A’ is the transformation matrix defined as

ei + ei - f e,e2 - eue3

A’ = 2 e, e2 + e,e, e,2+e$-i (3)

el e3 - eoe2 e2e3 + eOel

Since body i is not rigid, the vector r’ can be written as

r’ = r: + r: (4)

where r: is the position of point P in the undeformed state and ti, is the elastic deformation of the point P. Following the traditional approximation, r: can be represented as

r: = NW, (5)

where N’ = N’(r q {) is a space-dependent shape function, and ui is the vector of time-dependent elastic generalized coordinates (or nodal deformation in finite element approximation) of the deformable body i. Therefore,

Rb = Rb + A’(r: + NW), (6)

where the global position of point P is written in terms of the generalized rigid-body coordinates and the elastic coordinates of body i.

Differentiating eqn (2) with respect to time yields

ki, = lib + AtI + A’i’, (7)

where (‘) denotes differentiation with respect to time. Noting that ri in eqn (6) is constant, i’ can be written as

Substituting eqn (8) into eqn (7) yields

ri> = rib + Air’ + A’N’P’. (9)

The second term on the right hand side of eqn (9) can be written as

kri = A’o’ x r’ = - A’P’L’qb, (10)

where i’ is the skew symmetric matrix defined as

and w = L’q;, is the angular velocity with respect to the local coordinate system. L’ is the transformation matrix defined as

L’=2 -eZ -e, e, e,

i

-PI eu f3 -e2

i

(12)

-e, ez -e, e, ,

Substituting eqn (10) into eqn (9) one obtains

k(z = k; - Af’L’qj, + A ‘N’fi’ = C’@ (13)

where

c’ = (I - A’P’L’A’N’) (14)

q’ = (4: q: ii’)T (15)

and the vector fib is the absolute velocity vector of the origin of the rigid-body in question, while the last term, A’N’I’ is the velocity of P due to the deformation of the body. If the body were rigid, the term A’N’ti’ would be zero. The term, -A’i’L’q;, in eqn (IO) is the result of the differentiation of the transformation matrix with respect to time. It is a function of the rigid body reference rotation as well as the elastic deformation of the body.

i’ = i’ = ,,,‘ti, e (8) Fig. 2. Coordinates of the deformable body

Dynamics of three-dimensional multi-body systems 311

The kinetic energy of the body i is defined as

where p’ and R’ are, respectively, the mass density and volume of body i. Substituting eqn (13) into eqn (16), the kinetic energy can be rewritten as the following:

r=; s

p,qtT~‘T~‘q, dR’. (17)

R,

Since the vector qi is assumed to be time-dependent, and p’ may depend on the location of point P, the equation can be written as

I _ A’f’L’ A’N’

X _ L,T?.‘TA’T LrTfiT&t -tiTf’~’ dR’ (18) N’TA’T _ N’T?‘L NITN’ i

T’ = ;qT~i~,, (19)

where M’ is

M’= p’ s 01

I _ Ai&’ A’N’

X _ LIT~‘TA’T L’T~,T~‘L! _ L”Tf’N’ dR’.

N’TA’T _ ,,,‘Tf’L’ NiTNi

(20)

The mass matrix of eqn (20) can be written in a symbolic form as

(21)

where

(22)

m’ = IP J p'k'N' dR’ (24)

R’

m i,, = - J p’CTi’N’dR’ Cl,

(26)

m :,, = J

P’N’~N’ dR’. (27) fll

It can be seen that the two submatrices, ml, and

rn:,‘,, are defined by rigid-body coordinates and elastic coordinates, respectively, and are constants. Other matrices, however, depend on the system generalized coordinates, and are an implicit function of time. In terms of the submatrices, the kinetic energy can be written as

+ QQ7mbo$, + 2&,‘mb,ti’ + ti”m:,,V). (28)

If the body is rigid, the vector I’ of the elastic coordinates of the body i vanishes, and T’ reduces to

T’= f(tj:Tm:,Qi + 2iljrm:,& + t$~m~,&,). (29)

Further, if the mass center has been selected as the origin of the rigid-body coordinate, the coupling term m:,, will be zero. T’ reduces to

T’ = f($‘m;,tj; + #~m&Q~,). (30)

On the other hand, if the body undergoes only elastic deformation, Q; and & will vanish. Therefore, T’

reduces to T’= ‘$Trnm’ I’

2 cc . (31)

Generalized forces :

In order to derive the equations of motion, the

generalized forces associated with the generalized coordinates of the deformable body i in the multi- body system should be found. The virtual work due to the elastic forces can be written as

(32)

where oi and 6’ are, respectively, the stress and strain vectors and 6 W:, is the virtual work of the elastic forces. Furthermore,

c’ = B’r’ = B’N’u’ r (33)

where B’ is the strain-displacement matrix. For a linear isotropic material, the constitutive equation relating the strain and stress can be written as

(T’ = E’t’ (34)

where E’ is the matrix of elastic coefficients. Substitut- ing eqn (34) into eqn (33) yields

0 ! = E’B’N’“‘. (35)

Therefore, eqn (32) becomes

SW;, = -UiT N’%“E’B’N’dR’6u (36)


6 w:, = -u’%l’, (37)

where the symmetry of the matrix E’ is used. K is the symmetric positive definite stiffness matrix defined as

K= 5

N’%‘rE’B’N’dR’. (38) R’

The virtual work of all external forces acting on the body i in the multi-body system can be written as

sq: sWl,=Q~rsq’=(Q’;‘Q~~Q:‘) sq; ,

ii

(39)

6u’

where Q; and Qi, are, respectively, the associated generalized forces with respect to the translational and rotational coordinates of the rigid-body coordinate reference. QI is the vector of generalized forces associated with the elastic generalized coordinates of body i. The generalized forces depend on both the rigid body motion and the elastic deformation of

body i.

Force closed constraints

The modeling of the three-dimensional multi-body systems with changing topologies is developed based on a force-closed kinematic joint which maintains the kinematic constraint between two bodies by the use of closing force [3, 51. This is in contrast to a form- closed kinematic joint which is constructed such that the kinematic constraint is physically provided by the

Form Closed Constraints

4% F' F

Form Closed Equivalsnfs

NO

sphere Joint

Remlute Joint

Sliier Joint

Fig. 3. Force closed constraints and their equivalents.

+

Fig. 4. A point-to-surface constraint.

bodies. The closing force in the force-closed constraint acts in the same direction as the reaction force would in an equivalent form closed joint. Figure 3 shows the spatial kinematic joints of various degrees of freedom in their form-closed and force-closed versions. As can be seen, any kinematic joint can be represented by one or more force-closed point-to-sur-

face constraints. A general equation for the point-to-surface kin-

ematic constraint, as shown in Fig. 4, is defined as

9(~,t)=n,(x,--x,)+n?.Lv,IY~)+n,(z~-z,) (40)

where

‘x0, y,, + A’WL, ~a, il,)u’ (41)

,z0,

50 + A’ i) 90 + A’ + N’(5,, VII> io)u’, (42)

CO

(43)

where D’ is a transformation matrix which transforms the face from the deformed position to the undeformed position. If the effect of the elastic deformation on the orientation of the outward normal, n, can be neglected, D’ in eqn (43) is an identity matrix. Differentiating eqn (40) yields the velocity constraint

as

4,4 = 4, = 0 (44)

where

&= !!P (

J?!? 2 9 2 !!P aq: aq;, ad aq: aq:, au' >

(45)


Differentiating again, eqn (44) becomes Equation (51) can be written in a more explicit form

4,4 = -(&4),. (46)

This acceleration constraint equation will be used later.

Equations of motion

The equations of motion of the system can be written using Lagrangian formulation with a Lagran- gian multiplier 1, to account for the constraint force in the form

(47)

Using the general expression of the kinetic energy of eqn (19), the first and second terms on the left-hand side of eqn (47) can be expressed as

(48)

where

(49)

Q*i is a quadratic velocity vector which is a function of the generalized coordinates. Therefore, the equations of motion can be written as

M$+K’q’+4:1=Q’+Q*’ i=1,2,...,n, (SO)

where n is the number of the bodies in the multi-body system. Equations (40) and (SO) comprise a mixed system of differential and algebraic equations that govern the dynamics of the system. The combination of eqns (46) and (50) yield a set of matrix equations. They determine the acceleration, Lagrangian multiplier and elastic deformation of the system.

00 0 0 s: Q; + Q:’ 00 0 0 clb Qb + Q,? 0 0 K’ 0

:I: I

= u Q:,+Q:

00 0 0 i, - (Qi3,

(52)

Noting that in a generalized multi-body system with elastic components, the differential equations are coupled together, even though mass center can be selected as the origin of the rigid-body coordinate reference, q,, q. and q, are still coupled due to the elastic deformation. Therefore, some simplification can be made to solve eqn (52).

APPROXIMATION OF THE EQUATIONS OF MOTION

Generally, rigid body motion and elastic deformation are difficult to solve simultaneously, however, in some cases, the effect of the elastic deformation on rigid body motion may be neglected. This assumption may not be valid when high-speed, light-weight mechanical systems are considered, since the coupling terms may be significant. However, the approximation can greatly simplify numerical computation. If it is desirable to increase the computational efficiency, using the approximation, the equations of motion can be decoupled as follows:

mI,,ii’+ K’u’ = Q:, + Qz’ - +fA (54)

where the mass center has been selected as the origin of the rigid-body reference. The terms mi.ii’, mb,,ii’, m:,,ij:, and mb,,@, are neglected. Furthermore, the terms m;,, rnLu, $,,:, $+ Qi, Qb, QT’, and QB’ are assumed not to depend on the elastic deformation of the body. Therefore, the algorithm derived pre- viously [3] can be used to solve for the rigid body reference coordinates, velocities and accelerations, even if the system is discontinuous. The information obtained can then be substituted into eqn (54) to solve for the elastic deformation of the body. In realistic implementation, the finite element method will be used to analyze the deformation of the elastic components.


DETECTION AND REPRESENTATION OF CHANGING

TOPOLOGIES

The topologies of a realistic mechanical assembly are changing during its motion. For example, an old point-to-surface kinematic constraint may be de- stroyed due to the relative sliding or impact response. Conversely, a new point-to-surface kinematic constraint may be formed due to the contact between two bodies during motion. Topology changing not only affects the constraint conditions of the system, but also influences the boundary conditions for the elastic deformation analysis. Therefore, the collision detection and response algorithm is developed to automatically monitor and manage the changing configuration of the system.

Consider motion of an assembly with an initial governing equation, eqn (53). The constraints that are imposed on the system are

&(q, t) = (4, (4, I)> 4*(4, l), ” > 4,,,(Y> r))T = t-4 (55)

where m is the number of point-to-surface constraints. If at a certain time, P, a new contact is detected between two bodies, a corresponding point- to-surface constraint, 4”(q, t) = 0, must be added into the governing equation of the system. The governing equation now becomes

where

$‘(q, 2) = (f$,(q, f)> 42(4> 11, ‘. 1 d&,(43 ?I? d”(q1 t)JT.

(57)

Conversely, an old constraint equation 4,(q, t) = 0

must be deleted from the governing equation due to the destruction of a point-to-surface contact. The governing equation now becomes

where

x 4, , (9, I), / 4,,(4? t))T. (59)

The motion of the system is calculated using eqn (53) at time P. After that instant, the modified eqn (56) or (58) can be used to continue the calculation. The results calculated from eqn (53) at time t’ can be used as the initial condition for the following iteration process. Therefore, the discontinuous governing

equation of the system can be dealt with continuously, taking advantage of the collision detection and response algorithm.

To analyze the elastic deformation, the boundary conditions must also be modified after f’. The relative degrees of freedom between every two bodies is 6 if there are no point-to-surface constraints between them. With formation/destruction of a contact, the relative motion between two bodies is decreased/ increased and a new/old boundary condition (point with known displacement) must be added/deleted into/from the system. When the relative degrees of freedom between two bodies become zero, the relative motion between two bodies is totally governed by the elastic deformation and six point boundary conditions according to six point-to-surface constraints are used to completely filter out the rigid body motion model to facilitate the deformation analysis.

With the help of the collision detection and response analysis, the configuration of the system can be monitored and managed automatically. Not only are the constraint conditions added or deleted, but also the boundary conditions are modified accord- ingly. Therefore, a general mechanical assembly can be simulated realistically without user interface.

CASE STUDY

To demonstrate the above derivation, the approximation method developed is used to simulate a simplified socket assembly model. It is assumed that two bodies will keep contact after impact, when the bodies are under relative low velocity motion. There- fore, the impact response analysis can be greatly

Graphic Simulation

c5 End

No I Conpue Yes

Load Step + 1

Fig. 5. The flow chart of the modeling system.


Fig. 6. Dynamic simulation of the socket assembly (1).




simplified. The modular of dynamic analysis and the modular of collision detection and response are inte- grated to form a modeling system which can simulate the dynamic systems continuously and automatically. Figure 5 shows the flow chart of the implemented modeling system. AutoCAD was used for geometric modeling, and Algor finite element analysis software was used for deformation analysis. A program in C was written to control the overall flow of the data information.

Figure 6 shows the motion of system before two bodies contact each other. Figure 7 depicts the elastic deformation process after contact. The deformation was calculated using the finite element method. Figure 8 shows the final stage of the assembling process.

CONCLUSION

By using the theory of elasto-dynamics, the paper presents a generic formulation of the equations of motion for the three-dimensional multi-body system with elastic components. The topology of the system is also assumed to be changing. The nonlinear terms, such as the coupling between the rigid-body motion and elastic deformation, are included. A simplified linear method has been adopted to solve the equations of motion.

Due to linearization, the solution method presented can only be used in the case of low speed mechanisms or assemblies, such as the example case studied. The components will be assumed to be elastic only when the deformation of the bodies is necessary for the system to be assembled successfully and completely. When this is not the case, the equations of motion must be solved completely.

Acknowledgement-This research was supported by Na- tional Science Foundation grant no. DDM-9210839, and

the Missouri Department of Economic Development through the MRTC grant. Their financial support is greatly appreciated.

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Dynamics of three-dimensional multi-body systems with elastic components