Page 170 - Dynamics of Mechanical Systems

0593_C05_fm Page 151 Monday, May 6, 2002 2:15 PM

Planar Motion of Rigid Bodies — Methods of Analysis 151

Hence, Eq. (5.8.2) becomes:

a = ˙˙ n + ˙˙ n = ( r sinα θ + rω 2 cosθ n )
P
y
x
P x P y x
(5.8.7)
r (
+− α cosθ + rω 2 sin n ) θ
y
Then, ˙˙ x and ˙˙ y are:
P P
˙˙ x = rα sin +θ rω 2 cosθ
P
and

˙˙ y =− rα cos +θ rω 2 sinθ (5.8.8)
P
Solving for r sinθ and r cosθ we obtain:

α x ˙˙ + ω 2 y ˙˙
rsinθ = P P
α + ω 4
2
and
ω 2 x ˙˙ − α y ˙˙
rcosθ = P P (5.8.9)
α + ω 4
2
From Figure 5.8.1, we see that C may be located relative to O by the equation:

p = x n + y n = p + r
C C x C y P
(5.8.10)
= x n + y n + cosθ n + sinθ n
r
r
P x P y x y
This leads to the component and coordinate expressions:
α
ω 2 x − ˙˙
˙˙
y
x = x + cosθ = x + P P (5.8.11)
r
C P P α 2 + ω 4
and
α x + ω 2 y ˙˙
˙˙
y = y + sinθ = y + P P (5.8.12)
r
C P P α 2 + ω 4
Equations (5.8.11) and (5.8.12) can be used to locate C if we know the position and
acceleration of a typical point P of B and the angular speed and angular acceleration of
B. Then, once C is located, the acceleration of any other point Q may be obtained from
the expression:

a = αα × q + ωω ×(ωω × q) (5.8.13)
Q
where q is a vector locating Q relative to C.