Page 255 - Dynamics of Mechanical Systems

0593_C07_fm Page 236 Monday, May 6, 2002 2:42 PM

236 Dynamics of Mechanical Systems

Determine the second moment vectors I SO I , SO , and I SO .
a b c
P7.5.3: See Problem P7.5.1. Determine the moments and products of inertia I S O (i, j = 1, 2, 3).
ij
P7.5.4: See Problem P7.5.2. Let the transformation matrix between n , n , n and n , n , n 3
1
2
c
a
b
have elements S (j = 1, 2, 3; α = a, b, c) defined as:
jα
⋅
S = nn α
jα
j
Find the S .
jα
P7.5.5: See Problems P7.5.1 to P7.5.4. Find the moments and products of inertia I SO
αβ
(α, β = a, b, c). Also verify that:
I SO = SS β j I SO
αβ
ij
α i
and
I SO = SS I SO
ij iα jβαβ
P7.5.6: See Problem P7.5.3. Find the inertia dyadic I S/O . Express the results in terms of the
unit vectors n , n , and n of Figure P7.5.1.
3
1
2
P7.5.7: See Problems P7.5.5 and P7.5.6. Verify that I SO (α, β = a, b, c) is given by:
αβ
⋅
I SO = n I SO n ⋅ β
αβ
α
P7.5.8: See Problems P7.5.3 and P7.5.5. Verify that:

I SO + I SO + I SO = I SO + I SO + I SO
11 22 33 aa bb cc
P7.5.9: A 3-ft bar B weighs 18 pounds. Let the bar be homogeneous and uniform so that
its mass center G is at the geometric center. Let the bar be placed on an X–Y plane so that
it is inclined at 30° to the X-axis as shown in Figure P7.5.9. It is known that the moment
of inertia of a homogeneous, uniform bar relative to its center is zero for directions parallel
2
to the bar and m /12 for directions perpendicular to the bar where m is the bar mass
and is its length (see Appendix II). It is also known that the products of inertia for a bar
for directions parallel and perpendicular to the bar are zero. Determine the moments and
products of inertia: I BG I , BG I , BG I , BG I , BG , and I BG .
xx yy zz xy xz yz

Y
= 3 ft
weight = 18 lb
B
G 30°
X
O
FIGURE P7.5.9
A homogeneous bar in the X–Y plane
with center at the origin.