Download Chapter 5 Forces in Multibody Dynamics

Chapter 5 Forces in Multibody Dynamics
5.1 Introduction
The science of dynamics deals with the effects of forces on material bodies. Therefore,
a clear concept of force is necessary for a study of dynamics. The modeling of forces
in the study of multibody dynamics is undoubtedly important. The objective of this
chapter is to review the concepts of forces and their classification, then an extension
of these concepts into a general form applicable to multibody systems is developed.
5.2 Forces
A force can be presented by a vector, then it can be defined as a quantity having
magnitude, direction, and sense. Equation (5.2.1) represents a force vector.




F  Fx i  Fy j  Fz k
(5.2.1)
Forces are classified as free, sliding, and bound forces. A force is not associated with a
specific location called a free force, for example a force acting on a rigid body when
translational motion of the body is concerned. A bound force is one with a specified
point of application, for example a force acting on deformable or rigid body. The
application point of a force vector determines the effect it has on a material system.


For example, the forces, FA and FB , in Figure 5.2.1 have a different effect upon the
multibody system. A sliding force is one which can be moved along a given line
collinear with the force itself, as in Figure 5.2.2.
1
Figure 5.2.1 Forces applied at different points of a multibody system
2
Figure 5.2.2 A Sliding Force
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5.3 Moment of a Force About a Point and About a Line
A moment can be defined as the rotational effect of a force about a point or about a


line. The moment, M o of a force F about the O may be defined as a vector with

magnitude – magnitude F of the force times the perpendicular distance from O to F ,
direction – perpendicular to the plane at ), and sense – sense of advance according to

the right-hand screw rule. Let a be a position vector from O to any point A on line L.


Then the moment of F about O, M o , is defined as

 
Mo  a  F
(5.3.1)
Example 5.3.1. Moment of a force about a point
 


Given: F  4 i  3 j  2k
N,
which passes through the origin. Find the moment of this force about a point O
(2,1,7)m.
Solution: We have
 

 


 
M o  a  F  (2 i  j  7k)  (4 i  3 j  2k)






 (6k  4 j)  (4k  2 i )  (28 j  21i )



 19 i  24 j  2k
NM
4
Figure 5.3.1 Force acting along line L, and point O
5

The moment of a force about a line is defined similarly. Let us consider a force F

with line of action L. Let Z be a line about which the moment of F is to be taken.

Let r be the position vector from a point on line Z, the line of application of the

force. Let u be the unit vector along Z. Figure 5.3.2 shows such a situation.


Then the moment of F about Z, M Z , is defined as

   
M Z  [( r  F)  u ] u
(5.3.2)

Figure 5.3.2 Force F , line of action L, and line Z.
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5.4 Moment of a System of Forces
consider a rigid body B as shown in Figure 5.4.1, where it is subjected to an arbitrary


forces system consisting of N forces. The resultant R of all the forces Fi ( i  1,..., N )
applied to the rigid body B is then
N 

R   Fi
(5.4.1)
i 1
Then resultant moment of the forces about point O can be determined by vector
addition resulting from successive applications of Equation 5.3.1. This resultant can
be written as
N

 
M O   a i  Fi
(5.4.2)
i 1


where a ( i  1,..., N ) locates a point on the line of action of Fi , as shown in Figure
5.4.1.
7
Figure 5.4.1 A set of forces acting on a rigid body
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5.5 Generalized Applied (Active) Forces
Applied (active) forces are composed of externally applied forces, gravitational forces,
spring and damper forces, and contact forces.
5.5.1 Externally Applied Forces
Consider a multibody system shown in Figure 5.5.1. Suppose the system has n
degrees of freedom described by generalized coordinates, x r ( r = 1, …, n). The
generalized coordinates describe the relative orientation and translation between
bodies. Let one body of the system be subject to an externally applied force at point P.
Then the generalized active forces are defined as

 v p
Fr  F 
x r
(5.5.1)


where v p is the velocity of point P in the inertial reference frame R and v p x r

is the partial velocity of P with respect to x r in R. The velocity v p can be written
in the form


v p  v prm x r n om
( sum on r and m)
(5.5.2)

where the n om are unit vectors fixed in the inertial frame R. Then the partial velocity
of point P can be written in the form


v p x r  v prm n om
(5.5.3)

Unlike F , the generalized force Fr is a scalar. From Eq. (5.5.1), we know that the
number of generalized forces and the system have same degrees of freedom. If the


force F is perpendicular to the partial velocity, v p x r , then the generalized force


Fr is zero. Also, Fr could be zero if either v p x r is zero, F is zero.
9
Figure 5.5.1 Force acting at point P on one body of multibody system
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
For a multibody system is subjected to a set of forces Fi (i  1,..., n ) applied at point
Pi of the system as shown in Figure 5.5.2. Then the generalized forces for this set of
forces with respect to the generalized coordinate x r , are obtained by adding the
generalized forces from the individual forces. That is,
N 

Fr   Fi  v Pi x r
(5.5.4)
i 1
Consider a typical body of a multibody system is subjected to an externally applied
force field and the force field can be replaced by a system of forces consisting of


single force Fk passing through G k together with a couple with torque Tk
applied to B k (see Figure 5.5.3). Then the generalized active forces are defined as

 v k  
Fr  F 
T k
x r
x r
(5.5.5)


where v k and k are the mass center velocity and angular velocity of body k, B k ,
in the inertial reference frame R. The generalized active forces can also given by the
expression
Fr  v kmr Fkm   krp Tkm
(5.5.6)



where Fkm and Tkm are the n om components of Fk and Tk . There is a sum over the
repeated indices.
11
Figure 5.5.2 A system of forces acting in a multibody system.
12
Figure 5.5.3 Typical body B k of a multibody system and equivalent applied force
system.
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5.5.2 Gravitational Forces
For bodies near the surface of the earth, the gravitational (or weight) forces are nearly
constant. The gravitational forces acted on two points not too apart are almost parallel.
We may replace the forces on bodies by a single force passing through the center of
gravity of the body. In the vicinity of the earth’s system, for a body having dimensions
small composed with the radius of the earth, the center of gravity can be identified
with the mass center of the body. Gravitational force will give a freely body a

downward acceleration. We can denote this acceleration by a vector g and which is
a constant if air resistance is neglected.
The gravity forces exerted by the earth on a multibody system are given by


FkG  m k g (k  1,..., N)
(n sum on k)
(5.5.7)

where N is the number of bodies, m k is the mass of body k, g is the acceleration
vector of the gravity.
The contribution of the gravitational forces to the generalized active forces are
 G v k
Fr  Fk 
x r
(5.5.8)

where v k is the velocity of the mass center in inertial reference frame R of body k.
5.5.3 Spring and Dampers Forces
In addition to the externally applied forces and torques on the system, the active
forces exerted multibody systems by translational and torsional spring and damper
between each body of the system are of great interest. Consider two typical adjoining
bodies B j and B k shown in Figure 5.5.4. Let b measure the displacement between

the two bodies along S. S is the line connecting the two typical body. Let the n be
the unit vector parallel to line S. From the law of action and reaction, the spring and
damper could be represented by forces equal in magnitude and opposite in direction

acting B j and B k . Let the force acting on B j be Fj and the force acting on B k
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
be Fk .


The Fj and Fk can be expressed as



Fj  Fk  f (b, b ) n
(5.5.9)
where f (b, b ) is a function of the properties of the spring and damper. For a linear
spring, f (b, b ) has the form
f (b, b )  k 0  k1b
(5.5.10)
where k 0 and k 1 are constants. For a linear damper f (b, b ) has the form
f (b, b )  c 0  c1b
(5.5.11)
where c 0 and c1 are constants.


The contributions of Fj and Fk to the generalized active force Fr could be obtain
as
 v Pj  v Pk
Fr  Fj 
 Fk 
x r
x r
(5.5.12)


Since Fj  Fk
 v Pj v Pk
Fr  Fj  [

]
x r
x r
(5.5.13)
15
Next, consider two typical bodies, B j and B k , are hinged with torsional spring and
damper shown in Figure 5.5.5. Let the two typical bodies, B j and B k , be connected

with a hinge where axis is parallel to a unit vector m as shown in Figure 5.5.5. If we
denote the relative hinge rotation by  , the torques exerted on bodies B j and
B k by the torsional spring and damper might be expressed in the form



Tj  Tk  g(,  ) m
(5.5.14)
where g(,  ) is a function of the properties of the torsional spring and damper. For
a linear torsional spring, g(,  ) has the form
g(,  )  r0  r1
(5.5.15)
where r0 and r1 are constants. For a torsional damper g(,  ) has the form
g(,  )  v 0  v1 b
(5.5.16)
where v 0 and v1 are constants.


The contributions of Tj and Tk to the generalized active force Fr could be obtain
as
 Pj 

 
 Pk
Fr  Tj 
 Tk 
x r
x r
(5.5.17)


Since Tj  Tk
16
Figure 5.5.4 Two typical adjoining bodies B j and B k with translational springs and
dampers
Figure 5.5.5 Two contact bodies B j and B k
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 Pj

 
 Pk
Fr  Tj  [

]
x r
x r
(5.5.18)
5.5.4 Contact Forces
Contact forces are very difficult to model. It contribute nothing to the generalized
forces, Fr . For example, in a hinge joint, the contribution of the interaction forces is
equal to zero. To illustrate this, consider two contact bodies B j and B k , let the
point of contact be express by point c of B j and c of B k , as in Figure 5.5.5. In
contact condition, the velocities of point c and c in an inertial frame should be
equal. The contact forces exerted on B j and B k , should have same magnitude and
opposite direction. That is,


Fj  Fk
(5.5.19)
Then we will have
 v c  v c  v c
 v c
Fr  Fj 
 Fk 
 Fj 
 (Fj ) 
0
x r
x r
x r
x r
(5.5.20)
5.6 Generalized Inertia (Passive) Forces
As with applied forces, we can define generalized inertia forces in the same way. The
generalized inertia (passive) force is defined as the projection of an inertia force along
a partial velocity vector. Consider a particle P having mass m and is subjected to a

force F as in Figure 5.6.1. From Newton’s Law, the particle acceleration is
proportional to the force. That is


F  ma
(5.6.1)
According to the d’Alembert principle, Eq. (5.6.1) could be written as
18


F  ma  0
(5.6.2)
or
 
F  F*  0
(5.6.3)


where F*  ma is the inertia force.
For a multibody system, let the inertia force system on a typical body B k be

represented by the single force Fk* passing through the mass center together with a



couple with torque Tk* (see Figure 5.6.2). Then Fk* and Tk* can be written as:


Fk*  m k a k
(no sum on k)
(5.6.4)
and



Tk*  I k   k  k  (I k  k )
(no sum on k)
(5.6.5)
where m k is the mass of B k and I k is the inertia dyadic of B k relative to its
mass center.
The contributions of inertia forces to the generalized inertia force F * r could be
obtain as

 * v  * 
F r  Fk 
 Tk 
x r
x r
*
(5.5.12)



where v is the mass center velocity of the rigid body,  its angular velocity, Fk*

the inertia force, Tk* the inertia torque, and x r the time derivative of the
generalized coordinates.
19
Figure 5.6.1 Inertia force on particle P.
20
Figure 5.6.2 Typical body B k of a multibody system and equivalent inertia force
system.
Reference
[1] Kane,T.R. and Levinson,D.A., Dynamics: Theory and Application, McGraw-Hill
Book Company, New York, 1985.
[2] Huston, R.L., and C.E. Passerello, and M.W. Harlow, “Dynamics of
Multirigid-Body Systems,” Journal of Applied Mechanics, Vol. 45, 1978, pp. 889-894.
[3] Huston,R.L., Multibody Dynamics, Butterworth-Heinemann, 1990.
[4] Josephs, H., R.L., Huston, Dynamics of Mechanical Systems, CRC Press, 2002.
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