CONTINUUM MECHANICS for ENGINEERS

Recall the definition t + = t + a from which it is obvious that dt/dt + = 1. As a
result of this, an expression for the velocity under a superposed rigid body
motion is obtained by taking the time derivatives of the bracketed term of
the preceding equation:
Define
or, written an alternative way by post-multiplying by Q jk
(5.10-14)
(5.10-15)
(5.10-16)
Substituting Eq 5.10-16 into the second of Eq 5.10-14 yields an expression for
the velocity field of a continuum undergoing a superposed rigid body rotation
(5.10-17)
Note that Ω ij is skew-symmetric by taking the time derivative of the orthogonality
condition for Q ij. That is,
When the derivatives are taken
v a˙t Q˙ t x Q t x˙
+ = ()+ () + ()
i i im m im m
= a˙ ()+ t Q˙ () t x + Q () t v
i im m im m
()= t Q˙ () t Q () t
Ω ij im jm
() tQ ()= t Q˙ () tQ () tQ ()= t Q˙ () t
Ω ij jk im jm jk ik
˙Q im
+
v = a˙+ Ω Q x + Q v
i i ij jk k im m
d d
[ QimQin]= []= 1 0
dt dt
Q˙ Q + Q Q˙
=0
im in im in
and the definition of used from Eq 5.10-16 substituted into this expression,
it yields
[ ] = 0
Q Ω + Ω Q
im ji ij jm
This is true only if the bracketed term is zero, thus
Ωij =−Ωji
(5.10-18)