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0593_C04*_fm Page 107 Monday, May 6, 2002 2:06 PM

Kinematics of a Rigid Body 107

Rolling motion is governed by the magnitude and direction of the angular velocity of
B in S. Let n be a unit vector normal to S at C. Then B has pure rolling in S if the angular
velocity of B in S is perpendicular to n. That is,

S
B
PureRolling: ωω⋅n = 0 (4.11.2)
S
B
B is pivoting in S if ω is parallel to n. That is,
ω
Pivoting: ωω = S ωω n = n (4.11.3)
B
B
S
B is at rest relative to S if ω is zero:
B
S
B
S
Rest: ωω= 0 (4.11.4)
Finally, B has general rolling in S if B is neither at rest nor pivoting or has pure rolling
in S. (Pure rolling is desired in machine elements to reduce the wear of the rolling surfaces.)
Consider again Figure 4.11.1. Let P be an arbitrary point of B. Because C is fixed in B
S
C
and because V is zero, Eqs. (3.4.6), (4.5.2), and (4.9.4) show that the velocity of P in S is
simply:
S V = ωω B × p (4.11.5)
S
P
where p is the position vector locating P relative to C.
The acceleration of P in S may be obtained by differentiating in Eq. (4.11.5).

4.12 The Rolling Disk and Rolling Wheel

As an illustration of these ideas, consider a uniform circular disk D rolling on a horizontal
flat surface S as depicted in Figure 4.12.1 (see References 4.1 to 4.4). Let C be the contact
point between D and S, and let G be the center of D. Let axes X, Y, and Z form a Cartesian
reference frame fixed relative to S with the Z-axis being normal to S. Let N , N , and N 3
1
2
be unit vectors parallel to X, Y, and Z. Let T be a line in the X–Y plane (the plane of S)
which is at all times tangent to D at C. Then, the angle φ between T and the X-axis defines
the turning of D. Let L be a radial line fixed in D. Let L be a diametral line passing
R
D
through G and C. Then, the angle ψ between L and L measures the roll of D. Finally, the
D
R
angle θ between L and a vertical line measures the lean of D.
D
Because D rolls on S, the kinematics of D in S can be expressed in terms of θ, φ, and ψ.
To see this, consider the angular velocity of D in S. Let n , n , and n be unit vectors parallel
3
2
1
to T, to the axis of D, and to L as shown. Let ˆ n 1 , ˆ n 2 , and n be mutually perpendicular
3
D
unit vectors with ˆ n 1 parallel to n , ˆ n 2 in the plane of S and perpendicular to n (as shown),
1
1
and ˆ n 3 parallel to N . Finally, let d , d , and d be mutually perpendicular unit vectors
2
3
3
1
fixed in D with d parallel to n and d parallel to L .
R
2
3
2
Let S (N , N , N ), R ( ˆ n 1 , ˆ n 2 , ˆ n 3 ), R (n , n , n ), and D (d , d , d ) be reference frames
1
3
2
1
3
2
2
1
2
1
3
containing the unit vector sets as indicated. Then, the configuration graph defining the
orientation of D in S is shown in Figure 4.12.2.