Page 405 - Dynamics of Mechanical Systems

0593_C11_fm Page 386 Monday, May 6, 2002 2:59 PM

386 Dynamics of Mechanical Systems

degrees of freedom. Recall that if we introduce six parameters (say, x, y, z, θ, φ, and ψ) to
define the position and orientation of D we find that the conditions of rolling lead to the
constraint equations (see Eq. (11.3.8), (11.3.9), and (11.3.10)):

˙ x = ( [ ˙ + sin ) θ cos +φ θ ˙ cos sinθ ] (11.10.34)
θ
r ψφ

˙ y = ( [ ˙ + sin ) θ sin −φ θ ˙ cos cosφ ] (11.10.35)
r ψφ
θ

˙ z =− θ ˙ sinθ (11.10.36)
r

where x, y, and z are the Cartesian coordinates of G and θ, φ, and ψ are the orientation
angles of D as in Figure 11.10.7.
The last of these equations is integrable leading to the expression:

z = cosθ (11.10.37)
r

This expression simply means that D must remain in contact with the surface S. It is
therefore a geometric (or holonomic) constraint. Equations (11.10.34) and (11.10.35), how-
ever, are not integrable in terms of elementary functions. These equations are kinematic
(or nonholonomic) constraints. They ensure that the instantaneous velocity of the contact
point C, relative to S, is zero.
To obtain the generalized inertia forces for this nonholonomic system we may simply
select three of the six parameters as our independent variables. Then the partial velocities
and partial angular velocities may be determined from the coefficients of the derivatives
of these three variables in the expressions for the velocities and angular velocity. That is,
observe that the coordinate derivatives are linearly related in Eqs. (11.10.34), (11.10.35),
and (11.10.36). This means that we can readily solve for the nonselected coordinate deriv-
atives in terms of the selected coordinate derivatives. To illustrate this, suppose we want
to describe the movement of D in terms of the orientation angles θ, φ, and ψ. We may
express the velocity of the mass center G as:

G
v = ˙ x N + ˙ y N + ˙ z N (11.10.38)
1 2 3
where N , N , and N are unit vectors parallel to the X-, Y-, and Z-axes as in Figure 11.10.7.
2
3
1
Then, from Eqs. (11.10.34), (11.10.35), and (11.10.36), we may express v as:
G
v = ( [ ˙ + sinθ + cos sinθ ˙ θ φ ]
˙
r ψφ
G ) cosφ N 1
r + ( [ ˙ ψφ ˙ ) sinφ − cos cosθ ˙ θ φ ] N 2 (11.10.39)
+ sinθ
r − θ ˙ sinθ N
3