Page 305 - Dynamics of Mechanical Systems

0593_C09_fm Page 286 Monday, May 6, 2002 2:50 PM

286 Dynamics of Mechanical Systems

FIGURE 9.5.2
A set of particles moving in an iner-
tial reference frame R.

frame R as in Figure 9.5.2. Let the particles be acted upon by forces F (i = 1,…, N) as
i
shown. Then, from Newton’s law, we have for each particle:
F = m a ( i = 1 ,K , N) (nosum ) (9.5.4)
i i i

where a is the acceleration of P in R.
i
i
Let G be the mass center of S. Then,
a = a + d 2 r dt 2 (9.5.5)
i G i
where r locates P relative to G as in Figure 9.5.2.
i
i
Let the system of forces F be represented by an equivalent force system (see Section 6.5)
i
consisting of a single force F passing through G together with a couple with torque T.
Then, F and T are:
N N
F = F and T = r × F (9.5.6)
∑ i ∑ i i
= i 1 = i 1

Hence, from Eqs. (9.5.4) and (9.5.5), we have:

N
N
a =
F = ∑ m ii ∑ m ( i a + d 2 r dt 2 )
i
G
= i 1 = i 1
(9.5.7)
 N  2  N 
= ∑ m i  a + d 2 ∑ m i i  a G
r = M
G
 i =1  dt  i =1 
where M is the total mass of S and where the operations of summation and integration
may be interchanged because N is finite. (The sum
N
∑ m r
ii
i=1

is zero because G is the mass center.)