- Email: [email protected]
Computational methods for studying impact in multibody systems
Pergamon
0045-7949(94)00631-B
COMPUTATIONAL
Compurers & Slrucrurrs Vol. 57. No. 3. pp. 421-%25. 1995 Copyright 0 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0045.7949195 $9.50 + O.&l
METHODS FOR STUDYING MULTIBODY SYSTEMS
IMPACT IN
Chau-Chin Changt and R. L. Huston$$ TDepartment of Mechanical Engineering, Nan-Tai College of Technology, Tainan 71008, Taiwan, Republic of China SDepartment of Mechanical, Industrial and Nuclear Engineering, College of Engineering, University of Cincinnati, Cincinnati, OH 45221-0072, U.S.A. (Received 18 March 1994)
Abstract-This paper presents a computational procedure for deriving and solving the governing dynamical equations of multibody systems subjected to impact. The procedure is developed by assuming that the duration of the impact is very short and that during the impact there is little change in the configuration of the system-although the velocities of the system may undergo significant changes. These assumptions lead to a set of linear algebraic equations for the velocity increments. These equations may then be solved to obtain initial conditions for the analysis of the subsequent motion of the system. Two examples are presented to illustrate and validate the procedure.
INTRODUCTION
The analysis of multibody system dynamics has received considerable attention from dynamicists in recent years. References [l-7] and their bibliographies summarize a vast number of research efforts and writings. An area which has received little attention in all this research, however, is the effect of impact upon multibody systems. In this paper we consider this phenomena and we formulate a procedure for making efficient computational analyses of it. Our fundamental assumption is that during impact there occur large forces (“impulsive” forces) occurring over a short time interval. We assume that the system configuration remains essentially unchanged during this short impact time interval, but that system velocities may change significantly. In our analysis we use Kane’s method [8] to obtain the governing equations. Kane’s equations have been shown to be ideally suited for the analysis of multibody system dynamics [7]. They lead to equations which are readily converted into algorithms for numerical analyses. By integrating the dynamical equations through the impact time interval we obtain a balance of generalized impulses and generalized momenta. These equations in turn lead to a set of linear algebraic equations for velocity increments. The solution of these equations then provides the basis for studying the movement of the system during the post-impact period. The balance of the paper is divided into four parts with the first of these providing a summary of multibody dynamics analysis useful in the sequel. The §To whom correspondence
next part presents the impact analysis itself. The final two parts contain the examples and concluding remarks. PRELIMINARY
Multibody
CONSIDERATIONS
systems
A multibody system is a collection of bodies with a given connection configuration. The system may contain closed loops and there may be separation between adjoining bodies. The system may be subjected to both geometric and kinematic constraints. Forces may be applied to the bodies of the system both externally (such as through gravity and contact forces) and internally between the bodies (such as through spring, damping, and actuator forces). Kinematics
Suppose a multibody system has N bodies with n degrees of freedom. Let the movement of such a system be described by n generalized coordinates x, (/ = I,. . , n). (These coordinates will, in general, include both translation and rotation variables.) Let n variables y, (I = I, . , n), called “generalized speeds” [8], be introduced such that the y, are independent linear combinations of the derivatives of the generalized coordinates. For example, the individual y, could be set equal to the individual 1,, respectively. It is often convenient to let the y, associated with the rotational degrees of freedom be components of the relative angular velocities of the bodies. In Ref. [7] it is shown that the angular velocity ok and the mass center velocity vk of a typical body & (k = 1, . . . , N), of the system may be expressed as:
should be addressed.
0, = o+n~lnn,n 421
(1)
Chau-Chin Chang and R. L. Huston
422
ated with generalized may be written as
and
speed y, of the entire
system
(2)
where the no,n(m = 1,2,3) are mutually perpendicular unit vectors fixed in an inertia frame R. The coefficients w~,,~and vkbn(1 = 1, . , n), called “partial angular velocity components” and “partial velocity components” [7], form arrays which depend only upon the geometrical parameters of the system. [Regarding notation, repeated indices, such as 1 and m in eqns (1) and (2), indicate a sum over the range of the index.] By differentiating in eqns (1) and (2), the angular acceleration and the mass center acceleration of BL in R may be expressed as
= $ (v;.F, !.=I
ak
=
(vkidj,
+
(4)
tiU,,,.hhtM.
Observe that the We,,,,and the Q,,~ are coefficients the generalized speeds y,. That is
F, = ~‘h/,n FL,,,+ Wk,,nMk,n
Similarly, let the inertia forces on B, be represented by a single force Fr passing through G, together with a couple with torque M,*. Then F: and Mf may be expressed as F: = -m,a, (11) M;=
-I,.
(12)
r,-u,.(L,-u,)
where m, is the mass of Bk and I, is the central inertia dyadic. The contribution of F: and MC to the generalized inertia force F: is then
(13)
ay, 4
or F? = L.~,,,,F?,,,+ QJ~,,,,M?,,, (no sum on k)
and no,n=
dVh= -aahn = aY, dYi v’ h
-
(10)
of
&J, da, D w~,,~n,,,, = = = dY/
VI&
(9)
or simply as
(3) and
+w:.M,)
(6)
where of and v: are the “partial angular velocity” and the “partial velocity” of BL and its mass center in R.
(14)
where F&, and ML,, are the n,,,, components of F: and M; The generalized inertia force F: associated with generalized speed y, of the entire system may then be written as h F;= c vt.F;+w;,M: (15) i=, or
Kinetics Let the multibody system be subjected to a force field which may be represented on a typical body B, by a single force F, passing through mass center G,! together with a couple with torque M,. Then the contribution of FI, and M, to the generalized active force F, associated with y, is
F: = L’i,?,,F”i,r, = u~,,,M:,,. I
(16)
Finally. from Kane’s equations [8], the dynamical equations of motion may be written as F,+F:=O
(/=I
,...,
n).
(17)
IMPACT ANALYSIS
F, = 3
M, = v; FA+ 0;
Fk + 3
a.Y,
M,
ay,
By using eqns (5) and (6) eqn (7) may be written
F, =
vkbn Fk,n+
(7)
Wk,,n Mh,, b’
Fkm and Mk,n are the no,” components
sumon k)
as
(8)
of F, and MA. Regarding the index notation, if there is a sum over k, eqn (8) represents the contribution from all the bodies. Hence, the generalized active force F, associ-
An “impulsive force” is defined as a force which is large but acts only over a short time interval. The term “large” means that the impulsive force has a significantly greater magnitude than other forces exerted on the system during the impact time interval. Suppose a multibody system has N bodies and n degrees of freedom which can be characterized by n generalized coordinates X, (I = 1, , n). Let n generalized speeds y, be defined as relative angular velocity components and relative displacement derivatives between adjoining bodies and between a reference
Impact in muitibody systems body of the system and the inertia frame. Let the impulsive forces on a typical body Bk be equivalent to a force Fk passing through mass center G, together with a couple with torque Mk. Then the generalized impulse ZI acting on the system and associated with the generalized speed y, is defined as:
423
By using eqns (1 I), (12) and (15), the inertia force term of eqn (20) may be written as
.(/=l,...,n).
..,n)
x(I=l,.
(23)
(18)
where [t,, t2] is the impact time interval. The generalized momentum P, of the system for the generalized speed y, is defined as
By using the foregoing assumption of essentially unchanged configurations during the impact, the first term may be expressed as:
s 11 N
N
P,’
1 (v:~mkvk+of[~Ik.mk)
(I=1 ,...,
n). (19)
k=l
,
(I = 1, . . , n).
k=l
where vk(tZ) and v,(t,) beginning and the end Similarly, the second (23) may be expressed
=-
=z,
(Z=l,...,
’
bk@d - vkh>l
k=l
(24)
are the velocities of G, at the of the impact. term on the right side of eqn as:
(21)
In general, the partial velocities v: and the partial angular velocities CD:are functions of time and the configuration (orientation and position) of the bodies of the system. Our assumption is that, during the impact time interval, the velocity of points of the system and the angular velocity of bodies of the system may change incrementally, but the configuration of the system remains essentially the same. Since the partial velocities and the partial angular velocities do not depend upon the velocity changes, but only upon the configuration, they may be treated as constant vectors. Hence, eqn (21) may be written as:
j-:‘F,df =;,(I:;+Fkdt
11
5 mkti
= (20)
By using eqn (9), the first term eqn (20) may be written as
(Z=l,...,n).
5 rnvf.s”ak dt
=-
By integrating Kane’s equation [eqn (17)] over the impact time interval we have:
j-:‘W+I:’F: dt = 0
kg,V;’ (-ma/c)dt
,,
+~&M,dt)
n).
(22)
of
’ Ik.
bkb)
-
wk(t,)l
(25)
k=l
where ok(t2) and mk(t,) are angular velocities of Bk at the beginning and end of the impact. Finally, by using eqns (1) and (5), the third term on the right side of eqn (24) may be written as
O:. t-d
x Ik
d>
where the Z,, are the nOs and non components of central inertia dyadic I, and where ermris the standard permutation symbol [7]. Although the generalized speeds yp and y, in eqn (26) are not necessarily constant during the impact time interval, they are nevertheless bounded. The value of this term is equal to the mean value of the integrand multiplied by the impact time interval.
Chau-Chin Chang and R. L. Huston
424
Since the duration of impact is very short, this term can be neglected, compared with the first two, &-f s 1,
mkv:’
bk(b)
-Vk(fl)l
k=l
-k&&r,
=
bktb)
mkVk(tZ)
-kg,P:'
+
f
-
+
wk(t,)l
d'
Ik.
Wk(fZ)l
[V:'mkvk(t,)+w:.Ik'wk(t,)]
1=,
=
-P,(t2)+
P,(l,)
(I =
,n)
1,.
(27)
where P,(t,) and P,(t2) are generalized momentum of the system with respect to the generalized speeds y, at beginning and end of the impact. By substituting from eqns (22) and (27) into eqn (20) we obtain P,(t*) - P,(t,) = I,
(I = 1,. . , n).
(28)
That is, during the impact time interval, the changes in generalized momenta are equal to the generalized impulses for each generalized speed. Substituting eqns (1) (2), (5) and (6) into eqn (19) the generalized P, may be expressed as p, = (mk vhl!n vkptn
+
zk!n~!WkpmWkp~r)yp
where the ag are generalized alP =
mkUhlmUkp#n
+
=
al,,yn
(29)
masses given by iktn!t~kb~~Whpt?.
(30)
Substituting eqn (24) into eqn (30) the governing dynamical equation may be written as a,/&&)
- a,&&,)
= 1, (I = 1, . 1n)
(31)
or a,,,Ay,,=I,
(/=l,...,
n),
(32)
where y,(t,) and y,,(tZ) are generalized speeds y,, at and where Ay,, is the time t, and t,, respectively, change in the generalized speed during the impact time interval. Equations (32) form a set of linear algebraic equations, which in turn form the basis for the analysis of the effects of impulsive forces upon multibody systems. By solving these equations for the Ay,, we obtain Ay,,=a~‘l,
@ = 1,. . .,n).
quantities in turn produce initial conditions for the subsequent motion of the system. A feature and benefit of eqn (32) is that there are exactly the same number of equations as there are degrees-of-freedom of the system. Hence, the unknown generalized speeds are uniquely determined. Sometimes, however, it is of interest to determine internal impulsive forces and moments at joints of the system where the movement is constrained. These are called “constraint impulses”. They may be determined as follows: let the constrained joint be temporarily unconstrained. Then let the constraint impulses be exerted as though they were externally applied. The system will then have additional degrees of freedom corresponding to the removed constraints, leading to additional dynamical equations containing the desired constraint impulsive force and moment components. Moreover, if relative angular velocity components and relative displacement speeds between the bodies are used as the generalized speeds, then these constraint impulsive force and moment components will appear singly (that is, uncoupled) in these additional equations. Finally, by specifying the incremental values of the generalized speeds corresponding to the removed constraints, we can determine the values of the constraint impulsive force and moment components. (The incremental values of the generalized speeds are frequently zero.) EXAMPLES
To illustrate these procedures, consider the classical problem of a straight chain of pin-connected links being struck at its end, as depicted in Fig. 1. (This problem is considered by Whittaker [9].) Specifically, consider a system of 10 identical links each having a mass of I .O kg and a length of 1 .O m. When the end is struck, as in Fig. 1, the links will begin to rotate relative to each other. The rate of rotation may be determined from eqns (32): let the applied impulse be 5.0 Ns. Then the solution of eqn (32) for this configuration leads to incremental joint velocities as listed in Table 1. (This solution for joint velocities was determined from link rotations using DYNOCOMBS [lo]-an automated multibody dynamics simulation computer program.) These results are identical to those listed by Whittaker [9]. The movement of the system after the impulse is depicted in Fig. 2, where Ai and B, (i = 1,2, 3,4) represent the position of ends A and B of the system for various times after impact.
J,AJr
(33)
The Ay,, can be used to determine mass center velocity changes and body angular velocity changes. These
I Fig. I. An impulsive
force applied chain.
at the end of a IO-link
Impact in multibody systems Table I. Incremental joint velocities for the system in Fig. 2 Joint
Velocity (m s-l)
Joint
17.321 -4.641 1.244 -0.333 0.089 - 0.024
1
2 3 4 5 6
Velocity (m s-‘)
7 8 9
0.006 - 0.002 0.0005
10
-0.0001
II
0.00007
15
-
14
-
13
-
12
-
11
-
10
-
9
-
b =
8
*
7:
As a second example consider a IO-link hanging pendulum. Let the pendulum be hanging vertically in its equilibrium configuration and let the end of the lower line be struck by a horizontal impulsive force. Let each link have a mass of 1.0 slug and a length of 1.0 ft and let the impulse have magnitude: 30 lb. The movement of this system after impact is depicted in Fig. 3.
6
-
5
-
425 It = 0.8
12
4 7t
0
1
2
3
4
5
6
7
8
9
10 II 12 13 14 15
x (f0)
Fig. 3. Motion DISCUSSION
9
5. 6.
t = 2.0
A.
6
4
4.
9.
7
10.
11.
12.
1 0
I,,
I
I,
2
3
4
I
5
6
I
I
7
8
I
9
I
10
I
11
X (m)
Fig. 2. Movement
of lo-link
chain
12
13.
14. after impact.
was partially under Grant
REFERENCES
8.
8
*
Acknowledgemenr-Research for this paper supported by the National Science Foundation MSS8912521 to the University of Cincinnati.
7.
10
5
after impact.
AND CONCLUSIONS
The paragraphs at the end of the impact section describe the impact analysis procedures including the procedures for determining constraint impulse force and moment components. The entire set of procedures are readily converted to algorithms for numerical evaluation. The examples illustrate the efficacy of the procedures. The elementary nature of the examples provides the means for verifying the numerical results. By using a multibody dynamics simulation computer program, such as DYNOCOMBS [IO], it is possible to use the impact analysis to study impact in a wide variety of systems, including those with grippers, with hammers, with systems having combustion and with accidentally colliding systems.
g
of IO-link pendulum
J. Wittenburg, Dynamics of Systems of Rigid Bodies. Stuttgart, Germany (1977). T. R. Kane and D. A. Levinson, Multibody dynamics. J. appl. Mech. 50, 1071-1078 (1983). P. E. Nikravesh, Computer-Aided Analysis of Mechanical Systems. Prentice Hall, Englewood Cliffs, NJ (1988). R. E. Roberson and R. Schwertassek, Dynamics of Multibody Systems. Springer, Berlin (1988). A. A. Shabana, Dynamics of Multibody Systems. Wiley, New York (1989). W. Schielen, Multibody Systems Handbook. Springer, Berlin (1990). R. L. Huston, Mulribody Dynamics. ButterworthHeinemann, Stoneham, MA (1990). T. R. Kane and D. A. Levinson, Dynamics: Theory and Applications. McGraw Hill, New York (1985). E. T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, p. 175. Cambridge University Press, London (1937). J. E. Kamman and R. L. Huston, User’s Manual for UCIN DYNOCOMBS, National Technical Information Service, Springfield, VA, Report no. PB85240075/AS (1985). R. L. Huston, C. E. Passerello and M. W. Harlow, Dynamics of multi-rigid-body systems. J. uppl. Mech. 45, 889-894 (1978). T. R. Kane, Impulsive motions. J. uppl. Mech. 29, 715-718 (1961). T. R. Kane and D. A. Levinson, Formulation of equations for motion for complex spacecraft. J. Guidance Con@. 3, 99-l 12 (1980). J. B. Keller, Impact with friction. J. appl. Mech. 53, l-4 (1986).
0045-7949(94)00631-B
COMPUTATIONAL
Compurers & Slrucrurrs Vol. 57. No. 3. pp. 421-%25. 1995 Copyright 0 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0045.7949195 $9.50 + O.&l
METHODS FOR STUDYING MULTIBODY SYSTEMS
IMPACT IN
Chau-Chin Changt and R. L. Huston$$ TDepartment of Mechanical Engineering, Nan-Tai College of Technology, Tainan 71008, Taiwan, Republic of China SDepartment of Mechanical, Industrial and Nuclear Engineering, College of Engineering, University of Cincinnati, Cincinnati, OH 45221-0072, U.S.A. (Received 18 March 1994)
Abstract-This paper presents a computational procedure for deriving and solving the governing dynamical equations of multibody systems subjected to impact. The procedure is developed by assuming that the duration of the impact is very short and that during the impact there is little change in the configuration of the system-although the velocities of the system may undergo significant changes. These assumptions lead to a set of linear algebraic equations for the velocity increments. These equations may then be solved to obtain initial conditions for the analysis of the subsequent motion of the system. Two examples are presented to illustrate and validate the procedure.
INTRODUCTION
The analysis of multibody system dynamics has received considerable attention from dynamicists in recent years. References [l-7] and their bibliographies summarize a vast number of research efforts and writings. An area which has received little attention in all this research, however, is the effect of impact upon multibody systems. In this paper we consider this phenomena and we formulate a procedure for making efficient computational analyses of it. Our fundamental assumption is that during impact there occur large forces (“impulsive” forces) occurring over a short time interval. We assume that the system configuration remains essentially unchanged during this short impact time interval, but that system velocities may change significantly. In our analysis we use Kane’s method [8] to obtain the governing equations. Kane’s equations have been shown to be ideally suited for the analysis of multibody system dynamics [7]. They lead to equations which are readily converted into algorithms for numerical analyses. By integrating the dynamical equations through the impact time interval we obtain a balance of generalized impulses and generalized momenta. These equations in turn lead to a set of linear algebraic equations for velocity increments. The solution of these equations then provides the basis for studying the movement of the system during the post-impact period. The balance of the paper is divided into four parts with the first of these providing a summary of multibody dynamics analysis useful in the sequel. The §To whom correspondence
next part presents the impact analysis itself. The final two parts contain the examples and concluding remarks. PRELIMINARY
Multibody
CONSIDERATIONS
systems
A multibody system is a collection of bodies with a given connection configuration. The system may contain closed loops and there may be separation between adjoining bodies. The system may be subjected to both geometric and kinematic constraints. Forces may be applied to the bodies of the system both externally (such as through gravity and contact forces) and internally between the bodies (such as through spring, damping, and actuator forces). Kinematics
Suppose a multibody system has N bodies with n degrees of freedom. Let the movement of such a system be described by n generalized coordinates x, (/ = I,. . , n). (These coordinates will, in general, include both translation and rotation variables.) Let n variables y, (I = I, . , n), called “generalized speeds” [8], be introduced such that the y, are independent linear combinations of the derivatives of the generalized coordinates. For example, the individual y, could be set equal to the individual 1,, respectively. It is often convenient to let the y, associated with the rotational degrees of freedom be components of the relative angular velocities of the bodies. In Ref. [7] it is shown that the angular velocity ok and the mass center velocity vk of a typical body & (k = 1, . . . , N), of the system may be expressed as:
should be addressed.
0, = o+n~lnn,n 421
(1)
Chau-Chin Chang and R. L. Huston
422
ated with generalized may be written as
and
speed y, of the entire
system
(2)
where the no,n(m = 1,2,3) are mutually perpendicular unit vectors fixed in an inertia frame R. The coefficients w~,,~and vkbn(1 = 1, . , n), called “partial angular velocity components” and “partial velocity components” [7], form arrays which depend only upon the geometrical parameters of the system. [Regarding notation, repeated indices, such as 1 and m in eqns (1) and (2), indicate a sum over the range of the index.] By differentiating in eqns (1) and (2), the angular acceleration and the mass center acceleration of BL in R may be expressed as
= $ (v;.F, !.=I
ak
=
(vkidj,
+
(4)
tiU,,,.hhtM.
Observe that the We,,,,and the Q,,~ are coefficients the generalized speeds y,. That is
F, = ~‘h/,n FL,,,+ Wk,,nMk,n
Similarly, let the inertia forces on B, be represented by a single force Fr passing through G, together with a couple with torque M,*. Then F: and Mf may be expressed as F: = -m,a, (11) M;=
-I,.
(12)
r,-u,.(L,-u,)
where m, is the mass of Bk and I, is the central inertia dyadic. The contribution of F: and MC to the generalized inertia force F: is then
(13)
ay, 4
or F? = L.~,,,,F?,,,+ QJ~,,,,M?,,, (no sum on k)
and no,n=
dVh= -aahn = aY, dYi v’ h
-
(10)
of
&J, da, D w~,,~n,,,, = = = dY/
VI&
(9)
or simply as
(3) and
+w:.M,)
(6)
where of and v: are the “partial angular velocity” and the “partial velocity” of BL and its mass center in R.
(14)
where F&, and ML,, are the n,,,, components of F: and M; The generalized inertia force F: associated with generalized speed y, of the entire system may then be written as h F;= c vt.F;+w;,M: (15) i=, or
Kinetics Let the multibody system be subjected to a force field which may be represented on a typical body B, by a single force F, passing through mass center G,! together with a couple with torque M,. Then the contribution of FI, and M, to the generalized active force F, associated with y, is
F: = L’i,?,,F”i,r, = u~,,,M:,,. I
(16)
Finally. from Kane’s equations [8], the dynamical equations of motion may be written as F,+F:=O
(/=I
,...,
n).
(17)
IMPACT ANALYSIS
F, = 3
M, = v; FA+ 0;
Fk + 3
a.Y,
M,
ay,
By using eqns (5) and (6) eqn (7) may be written
F, =
vkbn Fk,n+
(7)
Wk,,n Mh,, b’
Fkm and Mk,n are the no,” components
sumon k)
as
(8)
of F, and MA. Regarding the index notation, if there is a sum over k, eqn (8) represents the contribution from all the bodies. Hence, the generalized active force F, associ-
An “impulsive force” is defined as a force which is large but acts only over a short time interval. The term “large” means that the impulsive force has a significantly greater magnitude than other forces exerted on the system during the impact time interval. Suppose a multibody system has N bodies and n degrees of freedom which can be characterized by n generalized coordinates X, (I = 1, , n). Let n generalized speeds y, be defined as relative angular velocity components and relative displacement derivatives between adjoining bodies and between a reference
Impact in muitibody systems body of the system and the inertia frame. Let the impulsive forces on a typical body Bk be equivalent to a force Fk passing through mass center G, together with a couple with torque Mk. Then the generalized impulse ZI acting on the system and associated with the generalized speed y, is defined as:
423
By using eqns (1 I), (12) and (15), the inertia force term of eqn (20) may be written as
.(/=l,...,n).
..,n)
x(I=l,.
(23)
(18)
where [t,, t2] is the impact time interval. The generalized momentum P, of the system for the generalized speed y, is defined as
By using the foregoing assumption of essentially unchanged configurations during the impact, the first term may be expressed as:
s 11 N
N
P,’
1 (v:~mkvk+of[~Ik.mk)
(I=1 ,...,
n). (19)
k=l
,
(I = 1, . . , n).
k=l
where vk(tZ) and v,(t,) beginning and the end Similarly, the second (23) may be expressed
=-
=z,
(Z=l,...,
’
bk@d - vkh>l
k=l
(24)
are the velocities of G, at the of the impact. term on the right side of eqn as:
(21)
In general, the partial velocities v: and the partial angular velocities CD:are functions of time and the configuration (orientation and position) of the bodies of the system. Our assumption is that, during the impact time interval, the velocity of points of the system and the angular velocity of bodies of the system may change incrementally, but the configuration of the system remains essentially the same. Since the partial velocities and the partial angular velocities do not depend upon the velocity changes, but only upon the configuration, they may be treated as constant vectors. Hence, eqn (21) may be written as:
j-:‘F,df =;,(I:;+Fkdt
11
5 mkti
= (20)
By using eqn (9), the first term eqn (20) may be written as
(Z=l,...,n).
5 rnvf.s”ak dt
=-
By integrating Kane’s equation [eqn (17)] over the impact time interval we have:
j-:‘W+I:’F: dt = 0
kg,V;’ (-ma/c)dt
,,
+~&M,dt)
n).
(22)
of
’ Ik.
bkb)
-
wk(t,)l
(25)
k=l
where ok(t2) and mk(t,) are angular velocities of Bk at the beginning and end of the impact. Finally, by using eqns (1) and (5), the third term on the right side of eqn (24) may be written as
O:. t-d
x Ik
d>
where the Z,, are the nOs and non components of central inertia dyadic I, and where ermris the standard permutation symbol [7]. Although the generalized speeds yp and y, in eqn (26) are not necessarily constant during the impact time interval, they are nevertheless bounded. The value of this term is equal to the mean value of the integrand multiplied by the impact time interval.
Chau-Chin Chang and R. L. Huston
424
Since the duration of impact is very short, this term can be neglected, compared with the first two, &-f s 1,
mkv:’
bk(b)
-Vk(fl)l
k=l
-k&&r,
=
bktb)
mkVk(tZ)
-kg,P:'
+
f
-
+
wk(t,)l
d'
Ik.
Wk(fZ)l
[V:'mkvk(t,)+w:.Ik'wk(t,)]
1=,
=
-P,(t2)+
P,(l,)
(I =
,n)
1,.
(27)
where P,(t,) and P,(t2) are generalized momentum of the system with respect to the generalized speeds y, at beginning and end of the impact. By substituting from eqns (22) and (27) into eqn (20) we obtain P,(t*) - P,(t,) = I,
(I = 1,. . , n).
(28)
That is, during the impact time interval, the changes in generalized momenta are equal to the generalized impulses for each generalized speed. Substituting eqns (1) (2), (5) and (6) into eqn (19) the generalized P, may be expressed as p, = (mk vhl!n vkptn
+
zk!n~!WkpmWkp~r)yp
where the ag are generalized alP =
mkUhlmUkp#n
+
=
al,,yn
(29)
masses given by iktn!t~kb~~Whpt?.
(30)
Substituting eqn (24) into eqn (30) the governing dynamical equation may be written as a,/&&)
- a,&&,)
= 1, (I = 1, . 1n)
(31)
or a,,,Ay,,=I,
(/=l,...,
n),
(32)
where y,(t,) and y,,(tZ) are generalized speeds y,, at and where Ay,, is the time t, and t,, respectively, change in the generalized speed during the impact time interval. Equations (32) form a set of linear algebraic equations, which in turn form the basis for the analysis of the effects of impulsive forces upon multibody systems. By solving these equations for the Ay,, we obtain Ay,,=a~‘l,
@ = 1,. . .,n).
quantities in turn produce initial conditions for the subsequent motion of the system. A feature and benefit of eqn (32) is that there are exactly the same number of equations as there are degrees-of-freedom of the system. Hence, the unknown generalized speeds are uniquely determined. Sometimes, however, it is of interest to determine internal impulsive forces and moments at joints of the system where the movement is constrained. These are called “constraint impulses”. They may be determined as follows: let the constrained joint be temporarily unconstrained. Then let the constraint impulses be exerted as though they were externally applied. The system will then have additional degrees of freedom corresponding to the removed constraints, leading to additional dynamical equations containing the desired constraint impulsive force and moment components. Moreover, if relative angular velocity components and relative displacement speeds between the bodies are used as the generalized speeds, then these constraint impulsive force and moment components will appear singly (that is, uncoupled) in these additional equations. Finally, by specifying the incremental values of the generalized speeds corresponding to the removed constraints, we can determine the values of the constraint impulsive force and moment components. (The incremental values of the generalized speeds are frequently zero.) EXAMPLES
To illustrate these procedures, consider the classical problem of a straight chain of pin-connected links being struck at its end, as depicted in Fig. 1. (This problem is considered by Whittaker [9].) Specifically, consider a system of 10 identical links each having a mass of I .O kg and a length of 1 .O m. When the end is struck, as in Fig. 1, the links will begin to rotate relative to each other. The rate of rotation may be determined from eqns (32): let the applied impulse be 5.0 Ns. Then the solution of eqn (32) for this configuration leads to incremental joint velocities as listed in Table 1. (This solution for joint velocities was determined from link rotations using DYNOCOMBS [lo]-an automated multibody dynamics simulation computer program.) These results are identical to those listed by Whittaker [9]. The movement of the system after the impulse is depicted in Fig. 2, where Ai and B, (i = 1,2, 3,4) represent the position of ends A and B of the system for various times after impact.
J,AJr
(33)
The Ay,, can be used to determine mass center velocity changes and body angular velocity changes. These
I Fig. I. An impulsive
force applied chain.
at the end of a IO-link
Impact in multibody systems Table I. Incremental joint velocities for the system in Fig. 2 Joint
Velocity (m s-l)
Joint
17.321 -4.641 1.244 -0.333 0.089 - 0.024
1
2 3 4 5 6
Velocity (m s-‘)
7 8 9
0.006 - 0.002 0.0005
10
-0.0001
II
0.00007
15
-
14
-
13
-
12
-
11
-
10
-
9
-
b =
8
*
7:
As a second example consider a IO-link hanging pendulum. Let the pendulum be hanging vertically in its equilibrium configuration and let the end of the lower line be struck by a horizontal impulsive force. Let each link have a mass of 1.0 slug and a length of 1.0 ft and let the impulse have magnitude: 30 lb. The movement of this system after impact is depicted in Fig. 3.
6
-
5
-
425 It = 0.8
12
4 7t
0
1
2
3
4
5
6
7
8
9
10 II 12 13 14 15
x (f0)
Fig. 3. Motion DISCUSSION
9
5. 6.
t = 2.0
A.
6
4
4.
9.
7
10.
11.
12.
1 0
I,,
I
I,
2
3
4
I
5
6
I
I
7
8
I
9
I
10
I
11
X (m)
Fig. 2. Movement
of lo-link
chain
12
13.
14. after impact.
was partially under Grant
REFERENCES
8.
8
*
Acknowledgemenr-Research for this paper supported by the National Science Foundation MSS8912521 to the University of Cincinnati.
7.
10
5
after impact.
AND CONCLUSIONS
The paragraphs at the end of the impact section describe the impact analysis procedures including the procedures for determining constraint impulse force and moment components. The entire set of procedures are readily converted to algorithms for numerical evaluation. The examples illustrate the efficacy of the procedures. The elementary nature of the examples provides the means for verifying the numerical results. By using a multibody dynamics simulation computer program, such as DYNOCOMBS [IO], it is possible to use the impact analysis to study impact in a wide variety of systems, including those with grippers, with hammers, with systems having combustion and with accidentally colliding systems.
g
of IO-link pendulum
J. Wittenburg, Dynamics of Systems of Rigid Bodies. Stuttgart, Germany (1977). T. R. Kane and D. A. Levinson, Multibody dynamics. J. appl. Mech. 50, 1071-1078 (1983). P. E. Nikravesh, Computer-Aided Analysis of Mechanical Systems. Prentice Hall, Englewood Cliffs, NJ (1988). R. E. Roberson and R. Schwertassek, Dynamics of Multibody Systems. Springer, Berlin (1988). A. A. Shabana, Dynamics of Multibody Systems. Wiley, New York (1989). W. Schielen, Multibody Systems Handbook. Springer, Berlin (1990). R. L. Huston, Mulribody Dynamics. ButterworthHeinemann, Stoneham, MA (1990). T. R. Kane and D. A. Levinson, Dynamics: Theory and Applications. McGraw Hill, New York (1985). E. T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, p. 175. Cambridge University Press, London (1937). J. E. Kamman and R. L. Huston, User’s Manual for UCIN DYNOCOMBS, National Technical Information Service, Springfield, VA, Report no. PB85240075/AS (1985). R. L. Huston, C. E. Passerello and M. W. Harlow, Dynamics of multi-rigid-body systems. J. uppl. Mech. 45, 889-894 (1978). T. R. Kane, Impulsive motions. J. uppl. Mech. 29, 715-718 (1961). T. R. Kane and D. A. Levinson, Formulation of equations for motion for complex spacecraft. J. Guidance Con@. 3, 99-l 12 (1980). J. B. Keller, Impact with friction. J. appl. Mech. 53, l-4 (1986).