Page 286 - Dynamics of Mechanical Systems

0593_C08_fm Page 267 Monday, May 6, 2002 2:45 PM

Principles of Dynamics: Newton’s Laws and d’Alembert’s Principle 267

FIGURE 8.12.4
An incorrect free-body diagram of the
rotating rod.

FIGURE 8.13.1
A circular disk rolling on a horizontal
surface S.

8.13 The Rolling Circular Disk

For another illustration of a body with nonplanar motion, consider again the rolling
circular disk (or rolling coin) as discussed in Section 4.12. That is, let D be a circular disk
with mass m and radius r rolling on a “perfectly rough” flat horizontal surface S as in
Figure 8.13.1. Let the orientation of D be defined by the roll, lean, and turning angles ψ,
θ, and φ, as shown. Let G be the mass center of D, and let C be the contact point of D with S.
When we studied the kinematics of D in Section 4.12, we discovered that the requirement
that D rolls on S led to the following expressions for the velocity and acceleration of G
and the angular velocity and angular acceleration of D in an inertia frame R (in which S
is fixed) (see Eqs. (4.12.2), (4.12.3), (4.12.7), and (4.12.8)):

v = ( r ˙ ψφθ n ) − rθ ˙ n (8.13.1)
˙
+ sin
1 2
+ sinθ
θ
a = ( r ˙˙ ψ φ + φθ2 ˙ ˙ cosθ n ) 1 ( θ ψφ cosθ + φ ˙ 2 sin cosθ n ) 2
+− + ˙ ˙
˙˙
˙˙
r
−
˙ ˙
+− ( r ψφθ φ ˙ 2 sin θ − θ ˙ 2 n ) (8.13.2)
2
sin
3
˙
˙
ωω= θn +( ψ + sinφ ˙ ) θ n + cosφ ˙ θn (8.13.3)
1 2 3