ematical dampings, respectively, were estimated from energy- balance relations and .... These equations may be assembled in two different forms depending on ... equations of motion, an appropriate set of initial conditions must be defined.
H. M. Lankarani Assistant Professor. Mechanical Engineering Department, Wichita State University, Wichita, KS 67208
P. E. Nikravesh Professor. Department of Aerospace and Mechanical Engineering, The University of Arizona, Tucson, AZ 85721
Canonical Impulse-Momentum Equations for Impact Analysis of iultibody Systems For mechanical systems that undergo intermittent motion, the usual formulation of the equations of motion is not valid over the periods of discontinuity, and a procedure for balancing the momenta of these systems is often performed. A canonical form of the equations of motion is used here as the differential equations of motion. A set of momentum balance-impulse equations is derived in terms of a system total momenta by explicitly integrating the canonical equations. The method is stable when the canonical equations are numerically integrated and it is efficient when the derived momentum balance-impulse equations are solved. The method shows that the constraint violation phenomenon, which is usually caused by the numerical integration error, can be substantially reduced as compared to the numerical integration of the standard Newtonian form of equations of motion. Examples are provided to illustrate the validity of the method.
1
Introduction On the subject of impulsive motion, analytical dynamics books [1-3] mostly deal with systems of particles or systems of rigid bodies with no kinematic joints. Impulsive motion can also occur in constrained multibody systems due to impulsive forces, or impulsive constraints, or a combination of the two [4]. The effort here is concentrated on the dynamic analysis of those constrained systems for which the impulsive motion is caused by a collision or an impact. One method of solving such problems has been the use of a continuous contact force model. In reference [5], a logical spring-damper represents the force element between a pair of colliding rigid bodies in a system. Both the stiffness and the damping coefficients, representing the elasticity of the contact surfaces and the mathematical dampings, respectively, were estimated from energybalance relations and impulse-momentum equations. The logical spring-damper is activated for the short duration of contact. This model shows good results only for the collision between the bodies of an unconstrained system of rigid bodies. For kinematically constrained systems, the impulsive effects are not only due to the impulse of the collision forces, but also due to the impulse of the constraint forces. This model does not account for the existence of and change in the constraint forces while estimating the stiffness and the damping coefficients. A compensation for the joint forces was made using an effective mass concept with the linear contact force model [6]. However, as Hunt and Grossley showed, the linear springdamping (Kelvin-Voigt) model does not represent the physical nature of the energy transfer process [7]. Instead, they used a Hertzian law with a damping function to model the contact force variations. Their analysis was, however, confined to two free bodies impacting at fairly low velocities. In general, the Contributed by the Design Automation Committee for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received March 1989; revised December 1990.
inclusion of the contact force in the system's differential equations of motion results in computational inefficiency of failure of numerical integration routines. The most popular method of solving the intermittent motion problems has been the piecewise analysis method. The analysis starts by integrating the system's equations of motion until the discontinuous event occurs at the time of impact. At this time, integration is halted, and a momentum balance is performed to calculate velocities after impact. Integration of the system's equations of motion is resumed with new velocities until the next discontinuity. For mathematical treatment of such systems, some simplifying assumptions must be made. One must assume that the impact is frictionless and that the coefficient of restitution is a known quantity. Furthermore, it must be assumed that the impact occurs instantaneously. This is valid if the duration of contact is so small that the change in the configuration of the system is negligible. The classical development of the momentum balance-impulse equations has been only for systems of particles or unconstrained rigid bodies (with no kinematic joint) [1-3]. Wittenburg discussed the impulsive motion of multibody constrained systems [8]. Wehage provided a numerical solution for such problems [9]. The obtained momentum balance-impulse equations were a set of linear equations, and the solution to these equations gave rise to the velocities after impact. The procedure for the method of this paper follows steps similar to Wehage's except that a more suitable form of the equations of motion are employed. A canonical form of the equations of motion in terms of the time derivative of the system's modified momenta is used to represent the transient response of the system. The numerical integration of the canonical equations provides a solution that is more stable than the solution obtained from the standard form of the equations of motions in terms of accelerations. Furthermore, by implicitly integrating the canonical equations for the duration of impact, a Transactions of the ASME
180/Vol. 114, MARCH 1992
Copyright © 1992 by ASME
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depending on whether the system velocities or momenta are used. The two sets of equations are discussed next. 2.1 Equations of Motion in Terms of Acclerations. The equations of motion may be derived in terms of accelerations v as a set of n second-order ordinary differential equations of the form M v = g + D r X,
Fig. 1
(5)
where M is the system mass matrix, vector g refers to the applied forces plus the gyroscopic forces in the Euler rotational equations of motion [10], and X is a vector of m Lagrange multipliers. The term DrX represents the constraint forces, for which the symbol Tperforms the transpose operation. For this standard set of second-order differential equations, an appropriate set of initial conditions is
Body-fixed and global coordinate systems
v(0) = vu
q(0) = q°, set of momentum balance-impulse equations is obtained which is a natural continuation of the canonical equations since both sets are expressed in terms of the components of momenta. 2
Equations of Motion The equations of motion can be described in terms of different sets of coordinates. If the number of coordinates is greater than the number of system's degrees of freedom, then algebraic equations are required to show the dependency of the coordinates. One such set of coordinates which leads to defining algebraic constraints for the kinematic joints is the so-called absolute coordinates. In order to specify the position of a rigid body in a global nonmoving XYZ coordinate system, it is sufficient to specify the spatial location of the origin (center of mass) and the angular orientation of a body-fixed £ijf coordinate system (refer to Fig. 1). For the ith body in a multibody system, vector q, denotes a vector of coordinates which contains a vector of Cartesian translational coordinates r, and a set of rotational coordinates. Matrix A, represents the rotational transformation of the £,-ij/f,- axes relative to the XYZ axes. A vector of velocities for body / is defined as v,-, which contains a 3-vector of translational velocities r, and a 3-vector of angular velocities co/. The components of the angular velocity vector co/ are defined in the £,-T),-|",- coordinate system rather than in the XYZ coordinate system. A vector of accelerations for this body is denoted by v,-, which contains r,- and cb/. For a multibody system containing b bodies, the vectors of coordinates, velocities and accelerations are q, v, and v, which contain the elements of q„ v„ and v„ respectively, i = 1, .... b. In constrained systems, assume that the algebraic equations are presented by m independent holonomic constraint equations: *(q) = 0. (1) The kinematic velocity and acceleration equations are obtained by taking the first and second time derivatives of Eq. (1) as
(6)
such that the coordinates q° and velocities v° satisfy the constraint and velocity equations, respectively. 2.2 Canonical Equations of Motion. The equations of motion for a system can also be derived in terms of the total momenta of the system. Baumgarte [11] expressed the canonical equations of motion for a constrained system by rearranging the Euler-Lagrange equations. The canoncial equations are, however, a direct consequence of the system's Hamiltonian function. The derivation of the canoncial equations through the Hamiltonian is given in many classical mechanics books for unconstrained systems [12, 13]. The derivation may, however, be extended to a constrained system. The Lagrangian function L for a dynamical system is expressed as L(q,v,t) - T — V, where Tand F a r e the kinetic and potential energies of the system, respectively. By the method of Lagrange [3], a new Lagrangian L* is introduced for the system by enforcing the kinematic velocity Eq. (2) as L*(q,v,t)=L-oTi,
(7)
where a represents a vector of m Lagrange multipliers associated with the momenta of the constraint's *. The Hamiltonian fucntion H associated with the Lagrangian L* is expressed as H(q,y,t)=-L*+L*y.
(8)
The components of the vector of modified momenta p are defined from the Hamiltonian by the quantity p = (L*)T, or p = (7' v -ff 7 b) 7 '.
(9)
The canonical equations are, in general, a set of In firstorder differential equations in terms of v and p , and are derived from the Hamiltonian function as y = Hl
(10)
P=-Hl
* = Dv = 0
(2)
However, since v is not directly integrable, q, which is evaluated from Eq. (4), is integrated. For the particular Hamiltonian associated with a dynamical system, the second set of the canonical Eqs. (10) yields
* = D v + b v = 0,
(3)
i>=-(Vq +
*M _ 1 (Ap + D 7 ' A ( r ) = - ( e + l ) * M " 1 [ p ( - ) + D V - ) ] ,
fdt=¥Tir. J,(->
Substitution of this impulse into Eq. (19) yields lis impulse into Eq. (19) yi f'< + )
(20)
r
Ap = ^ 7 T HTodt. The second term in Eq. (20) can be integrated by parts to obtain /(+)
DTadt=
•>/
BTa
_,< + )
-
\ J (
L J/(->
DTodt = DTA
y
/
\
-
, 1
\
'\ \ \
M
\ \ x, (rn/s) i
\
i
-
\ \
1
^
!
1
(m/fc2) / 1 1 1 1
1 1
0.5
i
1.0
i
i
i
1.5
1 1 1 1
2.0
' ' ' '
Time (s)
Fig. 5 Position, velocity, and acceleration of pendulum 1 in X-direction (e = 0.5)
Fig. 6
Impact of a slider-crank mechanism with a free-block
The system consists of two moving bodies, each connected each by a pin joint to the ground. The initial configuration of the system is shown as it is released from rest. The dynamic analysis of the system was performed by the numerical integration of the system canonical equations of motion while a geometric condition was checked for the possibility of penetration of the two spheres. The normal vector c has the following components at the time of contact:
c= -AXX-X2, a
Yi-Y*
Of,
where d = {(X\ - X2f + (F, - Y2f\A. The composite normal vector is then obtained as Journal of Mechanical Design
where ciX), C(Y)> and c (Z) are the components of c in the XYZ reference frame. Different coefficients of restitution were considered to represent the energy loss in the collision of the two spheres as the system experienced multiple impacts. The motion of pendulum 1 is shown in Fig. 5 for a restitution coefficient of 0.5. As a second example, the impact between the slider of a slider-crank mechanism and another sliding block is considered [6]. A schematic representation of the system is shown in Fig. 6. Physical data for this example are provided in the Appendix. If a set of Cartesian coordinates is employed to describe the configuration of the system, then the system is kinematically constrained. The constraint equations correspond to the prismatic and the pin joints connecting different components. The slider-crank is driven by a restoring torque such that the crank maintains almost a constant angular velocity. At some instant, the slider impacts a free block which is inertially driven to the left at a constant speed. If the coefficient of restitution between the blocks is known, the piecewise analysis of the previous section may be performed. However since this parameter is unknown, a procedure must be followed in order to obtain an estimate for it, and also to validate the results obtained with this estimate. For this reason, two different methods of analysis were used for simulation. In the first method, a continuous motion analysis was performed. Although the main assumption for an impact is its discontinuous nature, in reality the impulsive force acts in a continuous manner. If the value of the force acting on two bodies in contact is known, a continuous motion analysis can be performed by adding the contact force directly to the vector of forces associated with the two bodies. Thus, the equations of motion are integrated over the period of contact. In this example, a variation of the Hertz contact force model was assumed between the two blocks [18]. The model assumed that the local plasticity effect was the sole factor responsible for the energy loss in impact. All the parameters in the forceindentation law were taken from an earlier study [19]. With this model, the numerical integration of the equations of motion for the system was carried over the period of contact. A variable step-size algorithm [17] was used in this process so that the smaller step sizes during the period of contact were automatically adjusted for. An integration restart was necessary at the initial time of contact. A plot of the slider velocity versus time is shown in Fig. 7. Other results from this continuous motion analysis are provided in Table 3. MARCH 1992, Vol. 114/185
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Table 3 Summary of the results for the slider-crank impact example Variable Velocity before impact (m/s) Velocity after impact (m/s)
Piecewise Analysis
*
