Page 165 - Dynamics of Mechanical Systems

0593_C05_fm Page 146 Monday, May 6, 2002 2:15 PM

146 Dynamics of Mechanical Systems

and
[
θ
˙˙
˙˙
a O 3 = l cosθ 1 − lθ 2 ˙ 1 sinθ 1 + lθ 2 cosθ 2 − lθ 2 ˙ 2 sinθ 2] N X
[
˙˙
+−lθ 1 sinθ 1 − lθ 2 ˙ 1 cosθ 1 − lθ ˙˙ 2 sinθ 2 − lθ 2 ˙ 2 cosθ 2] N Z (5.6.18)
Observe that using the local unit vectors again leads to simpler expressions (compare
Eqs. (5.6.11) and (5.6.12) with Eqs. (5.6.13) and (5.6.14)). Nevertheless, with the use of the
local unit vectors we have mixed sets in the individual equations. For example, in Eq.
(5.6.11), the unit vectors are neither parallel nor perpendicular; hence, the components are
not readily added. Therefore, for computational purposes, the use of the global unit vectors
is preferred.
The velocities and accelerations of the remaining points of the system may be obtained
similarly. Indeed, we can inductively determine the velocity and acceleration of the center
of a typical bar B as:
k

l
v G k = lθ ˙ n + lθ ˙ n +…+ lθ ˙ n +( )θ ˙ n (5.6.19)
2
1 11 2 21 j j1 k k1
and
˙˙
˙˙
l
a G k = lθ n + lθ n +…+ lθ ˙˙ n +( )θ ˙˙ n
2
1 11 2 21 j j1 k k1
(5.6.20)
− lθ ˙ 2 n − lθ ˙ 2 n −…− lθ ˙ 2 n −( )θ ˙ 2 n
l
2
1 13 2 23 j j3 k k3
where n , n , and θ are associated with the bar B , immediately preceding B . In terms of
j1
j3
j
j
k
N and N , these expressions become:
X
Z
[
θ
˙
˙
θ
+ l
v G K = l cosθ 1 + lθ ˙ 2 cosθ 2 +…+l cosθ j ( )θ2 ˙ k cosθ k] N X
j
1
[
˙
˙
θ
+−l sinθ 1 − lθ ˙ 2 sinθ 2 −…− l sinθ j ( )θ2 ˙ k sinθ k] N Z (5.6.21)
θ
− l
j
1
and
[
˙˙
˙˙
˙˙
l 2
a G k = lθ 1 cosθ 1 + lθ ˙˙ 2 cosθ 2 +…+ lθ j cosθ j +( )θ k cosθ k
− lθ ˙ 2 sinθ − lθ ˙ 2 sinθ −…− lθ ˙ 2 sinθ −( )θ ˙ 2 sinθ N ]
l 2
1 1 2 2 j j k k X
(5.6.22)
˙˙
+− [ lθ ˙˙ sinθ − lθ sinθ −…− lθ sinθ −( ) 2 θ sinθ
˙˙
˙˙
l
1 1 2 2 j j j k k
− θ l ˙ 2 cosθ − l θ ˙ 2 cosθ −…− l θ ˙ 2 cosθ −( ) 2 θ ˙ 2 cosθ
l
1 1 2 2 j j k k] N Z
The velocity and acceleration of O may be obtained from these latter expressions by
3
simply replacing the fraction ( /2) by .