Page 419 - Dynamics of Mechanical Systems

0593_C11_fm Page 400 Monday, May 6, 2002 2:59 PM

400 Dynamics of Mechanical Systems

The first two of these equations are nonintegrable in terms of elementary functions;
therefore, the system is nonholonomic.
With three constraint equations, the disk has three degrees of freedom. These may
conveniently be represented by the angles θ, φ, and ψ, and in terms of these angles the
mass center velocity and the disk angular velocity are (see Eqs. (11.10.41) and (11.10.42)):

v = ( ψφ ˙ n ) θ − rθ ˙ n (11.12.25)
r ˙
+ sin
G 1 2
and

)
φ
˙
˙
˙
φ
ωω= θn +( ψ ˙ + sinθ n + cosθn (11.12.26)
1 2 3
Hence, the kinetic energy of the disk is:
K = ( ) m( ) + ωω ⋅ ⋅ωω
1
2
12
v
G I
2
2
= 1 mr ( ψφ ˙ ) + r θ 2  (11.12.27)
2 ˙
+ sinθ
 2
˙
2    
2
+ 1  I θ 2 ˙ + ( ψI ˙ + sinφ ˙ θ ) + ( cosφI ˙ ) θ 2 
2   11 22 33  
where the moments of inertia I , I , and I are:
33
22
11
I = I = mr 2 4 , I = mr 2 2 (11.12.28)
11 33 22
Assuming (erroneously) that Eq. (11.12.5) can be used to determine the generalized
inertia forces, we have:
∂ 
F =− d  K  + ∂K

*
˙
θ
dt  ∂ θ  ∂ θ
(11.12.29)
˙ ˙
θ
θ
˙ 2
=−mr 2 ( [ 54 ˙˙ 3 ψ φcos −( ) φ sin cos θ ]
θ ) −( ) 2
54
θ
˙˙
*
F =−mr 2 [ ( ) ψ ˙˙ sin +( ) ˙˙ 2 θ +( ) 4 φcos 2 θ
32
32
1
φsin
θ
˙ ˙
θ
θ
˙ ˙
3
+( ) 2 ψθ cos +(11 4 ) θφsin cos θ ] (11.12.30)
˙ ˙
˙˙ +
+
˙˙
*
32
F =−mr 2 [ ( )( ψφsin θ φθcos θ) (11.12.31)
ψ
*
F
In Section 11.10, in Eqs. (11.10.53), (11.10.54), and (11.10.55), we found , F φ * , and F ψ * to be:
θ
θ
˙˙
˙ ˙
θ
*
˙ 2
3
F =−mr 2 ( [ 54 θ ) −( ) 2 ψ φ cos −( ) φ sin cos θ ] (11.12.32)
54
θ