Page 435 - Dynamics of Mechanical Systems

0593_C12_fm Page 416 Monday, May 6, 2002 3:11 PM

416 Dynamics of Mechanical Systems

S having n degrees of freedom, represented by generalized coordinates q , Kane’s equations
r
state that:

F + F = 0 (12.2.1)
*
r r
Professor Thomas R. Kane first published Eq. (12.2.1) in 1961 [12.1]. The equations were
intended primarily to be a means for studying nonholonomic systems. The objective was
to obtain a “Lagrangian” formulation that would automatically eliminate nonworking
constraint forces from the analysis. As such, Kane’s equations were originally called
Lagrange’s form of d’Alembert’s principle [12.2, 12.3].
Some writers suggest that Kane’s equations are simply a reformulation of principles
developed earlier by Appell and Jourdain [12.9]. It appears, however, that those principles
and their resulting equations are not as simple nor as intuitive as the expressions of Eq.
(12.2.1).
With the advent of high-speed digital computers and the corresponding development
of procedures in computational mechanics, Kane’s equations have found application in
areas far beyond those envisioned in 1961. Indeed, Kane’s equations are currently the
equations of choice for automated (numerical) formulation of the governing equations of
motion for large mechanical systems [12.5, 12.6].
Intuitively, Kane’s equations may be interpreted as follows: if the partial velocity vectors
define the directions of motion of a mechanical system, then Kane’s equations represent
a projection of the applied and inertia forces along those directions.
From this perspective, Kane’s equations may be developed from d’Alembert’s principle
(see Section 8.3) by simply projecting the forces along the partial velocity vectors v .
˙ q r
Specifically, let a mechanical system S be regarded as a set of N particles P (i = 1,…, N)
i
having masses m . Let F represent the resultant of the applied forces on S, and let F *
i
represent the resultant of the inertia forces on S. That is,
N N
F = F and F = ∑ − ( m a ) (12.2.2)
*
P i
∑ i i
= i 1 = i 1
where F represents the applied forces on P , and a P i is the acceleration of P in an inertial
i
i
i
frame R. Then, d’Alembert’s principle states that [see Eq. (8.3.3)]:
+
*
FF = 0 (12.2.3)
Then, by taking the scalar product (projection) of Eq. (12.2.3) with the partial velocity
vectors v (r = 1,…, n), we have:
˙ q r
⋅
Fv + F v = 0 or F + F * = 0 (r = … ) n (12.2.4)
⋅
*
,
1
,
˙ q r ˙ q r r r
(See Eqs. (11.5.1) and (11.9.2).)
Because d’Alembert’s principle may be obtained from Newton’s laws, as discussed in
Section 8.3, we may consider Kane’s equations to be derived from Newton’s laws. But,
unlike Newton’s laws or d’Alembert’s principle, Kane’s equations provide for the auto-
matic elimination of nonworking constraint forces. Also, Kane’s equations produce exactly
the same number of equations as the degrees of freedom. That is, analysts need not be