Page 347 - Dynamics of Mechanical Systems

0593_C10_fm Page 328 Monday, May 6, 2002 2:57 PM

328 Dynamics of Mechanical Systems

If P is a particle with mass m, the kinetic energy K of P is defined as:

D 1
K = mv 2 (10.5.1)
2
where v is the velocity of P in an inertial reference frame R. The factor 1/2 is introduced
for convenience in relating kinetic energy to work and to other energy functions.
If S is a set of particles, the kinetic energy of S is defined as the sum of the kinetic
energies of the individual particles. Specifically, if S contains N particles P with masses
i
m and velocities v (i = 1,…, N) in inertial frame R, the kinetic energy of S is defined as:
i
i
N
K = ∑ 1 m v 2 i (10.5.2)
D
i
i=1 2
For a rigid body B we can define the kinetic energy of B as the sum of the kinetic energies
of the particles making up B. To see this, consider a depiction of B as in Figure 10.5.1. Let
G be the mass center of B (see Section 6.8). Then, from Eq. (4.9.4), the velocity v of typical
i
particle P of B in an inertial reference frame R is:
i
v = v + ωω × r (10.5.3)
i G i

where v is the velocity of G in R, ωω ωω is the angular velocity of B in R, and r locates P i
G
i
relative to G. Then, from Eq. (10.5.2), the kinetic energy of B is:
N
i ∑
K = ∑ 1 m v 2 i = N 1 m (v G + ωω × i) r 2 (10.5.4)
i
i=1 2 i=1 2
By expanding the terms in Eq. (10.5.4), the kinetic energy becomes:
N
N
i ∑
+
ωω
K = 1 ∑ m v G ∑ m v ⋅ × + 1 N m (ωω × i) r 2
2
r
2 i i G 2 i
i=1 i=1 i=1

N
N
i i ∑
= 1  ∑ m v 2 + v ⋅ ×  ∑ m r  + 1 N m (ωω × i) r 2 (10.5.5)
ωω
i
2 i  =1  G G i  =1  2 i=1 i
N
= 1 Mv 2 + + 1 ∑ m (ωω r i) 2
×
0
2 G 2 i
i i=1
P (m ) P (m )
2 2
1 1 B
G P (m )
r i i i
P (m )
N N
R
FIGURE 10.5.1
A rigid body composed.