Page 241 - Dynamics of Mechanical Systems

0593_C07_fm Page 222 Monday, May 6, 2002 2:42 PM

222 Dynamics of Mechanical Systems

and
aa [
aa ]
a = ( I − ) I +( I − ) 2 12
2
2 11 I 12 11 I (7.9.46)
When a plane of symmetry is identified or when a system or body is planar, the analysis
is two dimensional, as above. In this case, the transformation between different unit vector
sets is also simplified. Recall from Eqs. (7.5.5) and (7.5.7) that the moments and products
of inertia referred to unit vectors ˆ n and n are related by the expression:
k i
ˆ I =
ik jl
kl S S I ij (7.9.47)
where the transformation matrix components are defined by Eq. (7.5.6) as:

⋅
S = nn ˆ (7.9.48)
ik i k
Suppose the unit vectors n , n , and n are principal unit vectors, with n being normal
c
c
b
a
to a plane of symmetry. Let n , n , and n be mutually perpendicular unit vectors, with
2
3
1
n being parallel to n . Let θ be the angle between n and n as in Figure 7.9.2. Then, the
3
a
c
1
transformation matrix components between the n (i = 1, 2, 3) and the n (α = a, b, c) are:
α
i
cos θ sin θ  0

⋅
nn = −sin θ cos θ 0  (7.9.49)
i α  
0 0  1
Using Eq. (7.9.47), the moments and products of inertia I , relative to n and n , then become:
j
i
ij
I = cos θ I + sin θ I (7.9.50)
2
2
11 aa bb
aa (
θ
I = I = −sin cosθ I − ) (7.9.51)
I
12 21 bb
I = sin θ I + cos θ I (7.9.52)
2
2
22 aa bb
I = I = I = I = 0 (7.9.53)
13 31 23 32
I = I (7.9.54)
33 cc
Equations (7.9.50), (7.9.51), and (7.9.52) may be expressed in an even simpler form by
using the trigonometric identities:
sin θ ≡ 1 − 1 cos2θ
2
2 2
cos ≡ 1 + 1 cos2θ (7.9.55)
2
2 2
θ
sin cosθ ≡ 1 sin2θ
2