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62 Multibody Systems Approach to Vehicle Dynamics
I
2zz
I
2zz
I
2zz
I
2zz
I
2zz
3 2 3 2
d / 2 ⎛ b y b⎞
⎛ b y b⎞
t
∫ dy
d
/ 2 ⎜
⎝ 24 2
⎟ ⎜
⎠ ⎝ 24 2
⎟
⎠
t
∫
d / 2
d
/ 2
3
⎛ b
⎜
⎝ 12
3 3
⎡by
yb⎤
t
⎢ ⎥
⎣ 12 3 ⎦
2 ⎞
yb⎟
dy
⎠
d / 2
d
/ 2
3 3 3 3
⎛ bd db⎞
⎛ bd db⎞
t
⎜
⎝ 24 24
⎟ ⎜
⎠ ⎝ 24 24
⎟
⎠
3 3
⎛ bd
db⎞
t
⎜
(2.186)
⎝ 12
⎟
⎠
Since the mass of the rectangle body m 2 is given by m 2 tbd we can write
I m b 2
⎛ d
2zz
2 ⎜
⎝ 12
2
⎞
⎟
⎠
(2.187)
Another example of a standard shape is the ring shown in Figure 2.30. This
again considers the two-dimensional case for plane motion where the rectangular
body, Body 2, is constrained to move only in the X 1 Y 1 plane of frame O 1 .
Taking this body to have a thickness of t and a density of we can say that
the mass m of the small elemental ring of mass, at a radius R |R| 2 , with
radial width r, is given by
m t2Rr (2.188)
The moment of inertia I 2zz is again found by summing the second moments
of the elements of mass over the volume of the body:
I 2zz Σ R 2 m
2t Σ R R 2 r (2.189)
Body 2
Ro
Y 1
Ri
Y 2
{R } 2
O 2
X 2
δr
GRF
O 1
Fig. 2.30
X 1
Moment of inertia for a ring