Page 397 - Dynamics of Mechanical Systems

0593_C11_fm Page 378 Monday, May 6, 2002 2:59 PM

378 Dynamics of Mechanical Systems

P (m )
2
2
P(m)
P (m )
1
1
S
P (m )
i
i
P (m )
N
N
R
R
FIGURE 11.9.1 FIGURE 11.9.2
A particle P moving in an inertial reference A set S of N particles moving in an inertial reference
frame R. frame R.
where a is the acceleration of P in R. If P is part of a mechanical system S having n degrees
of freedom represented by coordinates q (r = 1,…, n), then the generalized inertia forces
r
F * associated with these coordinates are defined as:
r
⋅
* D
*
a
F = Fv q r ˙ = − m ⋅v ˙ ( r = 1 ,… n , ) (11.9.2)
r
q r
Consider next a set S of N particles P (i = 1,…, N) having masses m and moving in an
i
i
inertial reference frame R as depicted in Figure 11.9.2. Let S have n degrees of freedom
*
F
with coordinates q (r = 1,…, n). Then, the generalized inertia forces on S are defined as:
r
r
N
r ∑
⋅
˙ = −
⋅
F = N Fv q r ∑ m a v P i ˙ ( r = 1 ,… n , ) (11.9.3)
* D
*
P i
i i
i
q r
i=1 i=1
where, as before, F * is the inertia force on P and a is the acceleration of P in R.
i i i i
Finally, consider a rigid body B that is part of a mechanical system S. As before, let S
have n degrees of freedom represented by coordinates q (r = 1,…, n). Also, as we have
r
done before, let B be considered to be made up of particles P (i = 1,…, N) having masses
i
m as in Figure 11.9.3 where G is the mass center of B and R is an inertial reference frame.
i
Then, based upon the definitions of Eqs. (11.9.2) and (11.9.3), the generalized inertia forces
F * on B are:
r
N
N
r ∑ q r ∑ m a v ( n , )
⋅
*
F = Fv ˙ = − i i ⋅ q r ˙ r = 1 ,… (11.9.4)
P i
P i
i *
i=1 i=1
where, as before, F * is the inertia force on P and a is the acceleration of P in R.
i i i i
We can simplify Eq. (11.9.4) by taking advantage of the rigidity of B. Specifically, we
*
can represent the system of inertia forces on B by a single force F passing through the
mass center G together with a couple with torque T , where from Eqs. (8.6.5) and (8.6.6)
*
*
F and T are:
*
α
F =− Ma and T =− ⋅ −αωω × I⋅ ( ωω ) (11.9.5)
*
G
*
I