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0593_C11_fm Page 394 Monday, May 6, 2002 2:59 PM

394 Dynamics of Mechanical Systems

In the development of these generalized forces we discovered that the only forces
contributing to these forces were gravity and spring forces. The contact forces between
the smooth tube and the particles did not contribute to the generalized forces; hence, we
should be able to form a potential energy function for the system. To this end, using Eqs.
(11.11.5) and (11.11.10) we obtain the potential energy function:

(
(
P =− (l + )cosθ − mg l + )cosθ − mg l + )cosθ
x
mg
3
x
2
x
1 2 3
(11.11.24)
( ) cosθ 2 2 − ) 2
− Mg L 2 +( )kx1 2 1 +( ) ( k x1 2 2 − ) +( ) ( k x1 2 3 x 2
x
1
where the reference level for the gravitational forces is taken through the support pin O.
From Eq. (11.11.1) we can immediately determine the generalized forces. That is,
∂
F =−∂ P x , F =−∂ P x , F =−∂ P x , F =−∂ ∂θ (11.11.25)
∂
∂
P
x 1 1 x 2 2 x 3 3 θ
By substituting from Eq. (11.11.24) into (11.11.25), we obtain results identical to those of
Eqs. (11.11.20) to (11.11.23).
11.12 Use of Kinetic Energy to Obtain Generalized Inertia Forces

Just as potential energy can be used to obtain generalized applied (active) forces, kinetic
energy can be used to obtain generalized inertia (passive) forces. In each case, the forces
are obtained through differentiation of the energy functions. In this section, we will
establish the procedures for using kinetic energy to obtain the generalized inertia forces.
To begin the analysis, recall Eq. (11.4.7) concerning the projection of the acceleration of
a particle on the partial velocity vectors:

2
2
av = 1 d   ∂ v   − 1   ∂ v   (11.12.1)
⋅
2 dt  q r 2  q r
˙ q r ∂  ∂ 
This expression, which is valid only for holonomic systems (see Section 11.3), provides a
means for relating the generalized inertia forces and kinetic energy. To this end, let P be
a particle with mass m of a holonomic mechanical system S having n degrees of freedom
*
represented by the coordinates q (r = 1,…, n). Then, the inertia force F on P is:
r
*
F =−m a (11.12.2)
where a is the acceleration of P in an inertial reference frame R. Recall further that the
kinetic energy K of P is:

K = 1 v 2 (11.12.3)
2