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Kinematic Equation For Rigid Body Motion 93

3.6 Kinematic Equation For Rigid Body Motion

(a) Rigid body translation: For this motion, the kinematic equation of motion is given by

where c(0) = 0. We note that the displacement vector, u = x-X = c(t), is independent of X.
That is, every material point is displaced in an identical manner, with the same magnitude and
the same direction at time t.
(b) Rigid body rotation about a fixed point: For this motion, the kinematic equation of
motion is given by

where R(t) is a proper orthogonal tensor (i.e., a rotation tensor, see Sect. 2B.10) with R(0) — I,
and b is a constant vector. We note that the material point X = b is always at the spatial point
x = b so that the rotation is about the fixed point x = b.
If the rotation is about the origin, then b = 0 and x = R(f)X.

Example 3.6.1
Show that for motions given by Eq. (3.6.2) there is no change in distance between any pair
of material points.
Solution. Consider two material points x' ' and x' ', we have, from Eq. (3.6.2)

2
That is, the material vector AX=X^-X^ changes to Axsx^-j/ ) where

Now, the square of the length of Ax is given by

The right side of the above equation is, according to the definition of transpose of a tensor
T
T
AX-R(r)R (*)AX. and for an orthogonal tensor, RR = I, so that

In other words, the length of AX does not change.

(c)General rigid body motion: The kinematic equation describing a general rigid body
motion is given by

where R(t) is a rotation tensor with R(0) = I and c(t) is a vector with c(0) = b.