Page 164 - Dynamics of Mechanical Systems

0593_C05_fm Page 145 Monday, May 6, 2002 2:15 PM

Planar Motion of Rigid Bodies — Methods of Analysis 145

and

× ωω
2 /
a GO 2 = αα ×(l 2) n + ωω 2 [ ×(l 2) n 23]
2 23 2
(5.6.10)
= (l 2)θ ˙˙ n −(l 2)θ 2 ˙ n
2 21 2 23
(G may be viewed as moving on a circle about O .) Hence, by substituting into Eq. (5.6.8),
2
2
we have:

v G 2 = lθ ˙ n +(l 2)θ ˙ n (5.6.11)
111 2 21
and

˙˙
˙˙
a G 2 = lθ n − lθ 2 ˙ n +(l 2)θ n −(l 1)θ 2 ˙ n (5.6.12)
111 1 13 2 21 2 23
In terms of N and N , these expressions become:
X
Z
1 [
˙
θ
v G 2 = l cosθ 1 +(l 2)θ ˙ 2 cosθ 2] N X
[
˙
−
+−l sinθ 1 (l 2)θ ˙ 2 sinθ 2] N Z (5.6.13)
θ
1
and
1 [
θ
˙˙
˙˙
a G 2 = l cosθ 1 − lθ 2 ˙ 1 sinθ 1 +(l 2)θ 2 cosθ 2 −(l 2)θ 2 ˙ 2 sin θ 2] N X
[
˙˙
˙˙
θ
+−l sinθ 1 − lθ 2 ˙ 1 cosθ 1 (l 2)θ 2 sinθ 2 (l 2)θ 2 ˙ 2 cosθ 2] N Z (5.6.14)
−
−
1
Similarly, the velocity of acceleration of O is:
3
˙˙
v O 3 = lθ ˙ n + lθ n (5.6.15)
111 2 21
and
˙˙
˙˙
a O 3 = lθ n − lθ 2 ˙ n + lθ n − lθ 2 ˙ n (5.6.16)
111 1 13 2 21 2 23
In terms of N and N , these expressions become:
X
Z
1 [
θ
˙
v O 3 = l cosθ 1 + lθ ˙ 2 cosθ 2] N X
[
˙
+−l sinθ 1 − lθ ˙ 2 sinθ 2] N Z (5.6.17)
θ
1