Page 171 - Dynamics of Mechanical Systems

0593_C05_fm Page 152 Monday, May 6, 2002 2:15 PM

152 Dynamics of Mechanical Systems

Alternatively, Eqs. (5.8.11) and (5.8.12) may be used to obtain the angular speed ω and
the angular acceleration α of B if the acceleration of typical points P and Q are known.
To see this, observe first that for point Q expressions analogous to Eqs. (5.8.11) and (5.8.12)
are:
2
ω ˙˙ y
x − α˙˙
x = x + Q Q (5.8.14)
C Q α 2 + ω 4
and

x + ω ˙˙
α˙˙ 2 y
y = y + α Q 2 + ω 4 Q (5.8.15)
Q
C
Next, by subtracting these expressions from Eqs. (5.8.11) and (5.8.12) we have:

2
α ˙˙ P ( y Q) + ( x − ˙˙ P)
ω ˙˙
y − ˙˙
x
x − x = Q (5.8.16)
P Q α 2 + ω 4
x − ) + (
2
α ˙˙ Q ( x ˙˙ ω ˙˙ y P)
y − ˙˙
y − y = P Q (5.8.17)
P Q α 2 + ω 4
The expressions may be solved for ω and α as follows: Let ξ and η be defined as:
α ω 2
D
ξ = and η = (5.8.18)
D
α + ω 4 α + ω 4
2
2
Then, Eqs. (5.8.16) and (5.8.17) become:
P ( Q) +(
˙˙ y − ˙˙ y ξ ˙˙ x − ˙˙ x P) =η x − x Q (5.8.19)
Q
P
and
Q ( P) +(
˙˙ x − ˙˙ x ξ ˙˙ y − ˙˙ y P) =η y − y Q (5.8.20)
Q
P
Solving for ξ and η we obtain:
ξ= ( [ − )( y ˙˙ y ˙˙ − )( x ˙˙ − )]
1
x ˙˙
∆ x P x Q Q − ) −( y P y Q Q P (5.8.21)
P
η= ( [ − )( y ˙˙ y ˙˙ − )( x ˙˙ − )]
1
x ˙˙
∆ y P y Q P − ) −( x P x Q Q P (5.8.22)
Q
where ∆, the determinant of the coefficients, is:

 2 − ) 2 
∆= − ( y ˙˙ − ) +( x ˙˙ ˙˙ x (5.8.23)
˙˙ y
  P Q Q P  