Page 378 - Dynamics of Mechanical Systems

0593_C11_fm Page 359 Monday, May 6, 2002 2:59 PM

Generalized Dynamics: Kinematics and Kinetics 359

Observe that Eqs. (11.3.8), (11.3.9), and (11.3.10) constitute three constraint equations.
Hence, the disk D has three degrees of freedom, as expected. Observe next that the third
equation, Eq. (11.3.10), is integrable in terms of elementary functions. That is,

z = cosθ (11.3.11)
r
From Figure 11.3.1, we see that this expression is simply a requirement that D remains in
contact with the surface S — a geometric, or holonomic, constraint. Finally, observe that
Eqs. (11.3.8) and (11.3.9) are not integrable in terms of elementary functions. Instead, they
are coupled nonlinear differential equations requiring numerical solution. These expres-
sions arise from the requirement that C must have zero velocity in S — a kinematic, or
nonholonomic, constraint.
In the following and subsequent sections we will see that simple analyses using energy
functions (that is, the use of kinetic energy and Lagrange’s equations) are precluded with
nonholonomic systems. With Kane’s equations, however, the analysis is essentially the
same for both holonomic and nonholonomic systems. Nevertheless, our focus will be on
holonomic systems because, as noted earlier, the vast majority of mechanical systems of
interest are holonomic systems.

11.4 Vector Functions, Partial Velocity, and Partial Angular Velocity

Consider a holonomic mechanical system S having n degrees of freedom represented by
the coordinates q (r = 1,…, n). Let P be a typical point of S. Then, a position vector p
r
locating P relative to a fixed point O in an inertia frame R will, in general, be a function
of the q and time t. That is,
r

p = p(qt ) (11.4.1)
,
r
Then, from the chain rule for differentiation, the velocity of P in R is:

n
v = d p = ∂ p + ∑ ∂ p q ˙ (11.4.2)
dt ∂t ∂q r
r =1 r

The terms ∂p/∂t and ∂p/∂q (r = 1,…, n) are called partial velocity vectors. These vectors
r
are fundamental vectors in the development of generalized dynamics theories and pro-
cedures. Indeed, they may be viewed as base vectors in the generalized space of motion
of a mechanical system. Because of their fundamental nature and widespread use, they
are given a separate notation defined as:

D
v = ∂ p t and v = ∂ p q r (11.4.3)
∂
∂
D
˙
t
q r
Then, from Eq. (11.4.2), we have:
v =∂ v q ˙ (11.4.4)
∂
˙ q r r