Page 399 - Dynamics of Mechanical Systems

0593_C11_fm Page 380 Monday, May 6, 2002 2:59 PM

380 Dynamics of Mechanical Systems

and

˙˙
a =−lθ ˙ 2 n + lθ n (11.10.2)
P
r θ
where is the pendulum length, θ is the orientation angle, and n and n are the radial
θ
r
and transverse unit vectors, respectively, shown in Figure 11.10.1. The inertia force F on
*
P is, then,
˙˙
*
P
F =−m a =−mlθ ˙ 2 n − mlθ n (11.10.3)
r θ
where m is the mass of P.
θ
˙
From Eqs. (11.4.4) and (11.10.1), the partial velocity v of P with respect to is:
θ
v = l n θ (11.10.4)
˙ θ
From Eq. (11.9.2), the generalized inertia force F θ * is:

˙˙
⋅
*
*
2
l
F = Fv θ ˙ = − m θ (11.10.5)
θ
Example 11.10.2: Rod Pendulum
Consider next the rod pendulum consisting of a uniform rod B having mass m, length ,
and mass center G. Let B be pinned at one end so that it is free to move in a vertical plane
as described by the angle θ as in Figure 11.10.2. The velocity and acceleration of G and
the angular velocity and angular acceleration of B itself in an inertial reference frame R are:
˙˙
v = ( ) 2 θ ˙ n and a = ( ) 2 θ n −( ) 2 θ ˙ n r (11.10.6)
G
G
l
l
l
θ
θ
and
˙˙
˙
ωω = θn and αα = θn (11.10.7)
z z
where n , n , and n are mutually perpendicular unit vectors as shown in Figure 11.10.2.
θ
z
r
Let the inertia force system on B be represented by an equivalent force system consisting
*
*
of a single force F passing through G together with a couple with torque T. Then, F and
T are:
*
˙˙
*
G
m l
m l
F =−m a =− ( ) 2 θ n + ( ) 2 θ ˙ n r (11.10.8)
θ
and
T =− (l 12 θ n ) (11.10.9)
˙˙
*
2
m
z
The partial velocity of G and the partial angular velocity of B with respect to θ are:
G
l
v = ( ) 2 n and ωω ˙ θ = n z (11.10.10)
θ
˙ θ