CONTINUUM MECHANICS for ENGINEERS

FIGURE 5.2
Reference frames Ox 1x 2x 3 and O + x 1 + x 2 + x 3 + differing by a superposed rigid body margin.
(5.10-12)
where vectors p i + and pm are defined as shown in Figure 5.2. Here, Q im is
simply the matrix of the direction cosines between Ox 1x 2x 3 and O + x 1 + x2 + x 3 + .
Example 5.10-1
Show that the superposed rigid body motion defined by Eq 5.10-10 is distance
and angle preserving.
Solution
Consider the distance squared between material points X A and Y A in terms
of the superposed motion
Since Q im is orthogonal
+
pi= ci()+ t Qim() t pm
+ + + +
( xi − yi ) ( xi − yi ) = Qim( xm −ym)
Qin( xn − yn)
( ) ( − )
= Q Q x −y
x y
im in m m n n
+ + + +
( xi − yi ) ( xi − yi ) = δmn(
xm −ym)
( xn − yn)
( ) ( − )
= x −y
x y
m m m m
where the delta substitution property has been used. Thus, distance is preserved
in the superposed motion.