Page 208 - Intro to Tensor Calculus

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Consider the velocity of a point which is rotating with the rigid body. Denote by v i = v i (S), for
i =1, 2, 3, the velocity components relative to the S reference frame and by v i = v i (S),i =1, 2, 3the
velocity components of the same point relative to the body-fixed axes. In terms of the basis vectors we can
write
~
V = v 1 (S) ˆ e 1 + v 2 (S) ˆ e 2 + v 3 (S) ˆ e 3 = dx i ˆ e i (2.2.52)
dt
as the velocity in the S reference frame. Similarly, we write

~ dx i
V = v 1 (S)e 1 + v 2 (S)e 2 + v 3 (S)e 3 = b (2.2.53)
b
b
b
e i
dt
as the velocity components relative to the body-fixed reference frame. There are occasions when it is desirable
~ ~
to represent V in the S frame of reference and V in the S frame of reference. In these instances we can write
~
b
b
b
V = v 1 (S)e 1 + v 2 (S)e 2 + v 3 (S)e 3 (2.2.54)
and
~
V = v 1 (S) ˆ e 1 + v 2 (S) ˆ e 2 + v 3 (S) ˆ e 3 . (2.2.55)
Here we have adopted the notation that v i (S) are the velocity components relative to the S reference frame
and v i (S) are the same velocity components relative to the S reference frame. Similarly, v i (S) denotes the
velocity components relative to the S reference frame, while v i (S) denotes the same velocity components
relative to the S reference frame.
~
~
Here both V and V are vectors and so their components are first order tensors and satisfy the transfor-
mation laws
˙
and v i (S)= ` ij v j (S)= ` ij x j . (2.2.56)
v i (S)= ` ji v j (S)= ` ji ˙x j
The equations (2.2.56) define the relative velocity components as functions of time t. By differentiating the
equations (2.2.50) we obtain
dx i ˙
= v i (S)= ` ji ˙x j + ` ji x j (2.2.57)
dt
and
dx i ˙
˙
= v i (S)= ` ij x j + ` ij x j . (2.2.58)
dt
Multiply the equation (2.2.57) by ` mi and multiply the equation (2.2.58) by ` im and derive the relations
˙
v m (S)= v m (S)+ ` mi ` ji x j (2.2.59)

and
˙
v m (S)= v m (S)+ ` im ` ij x j . (2.2.60)
The equations (2.2.59) and (2.2.60) describe the transformation laws of the velocity components upon chang-
ing from the S to the S reference frame. These equations can be expressed in terms of the angular velocity
by making certain substitutions which are now defined.
The first order angular velocity vector ω i is related to the second order skew-symmetric angular velocity
tensor ω ij by the defining equation
ω mn = e imn ω i . (2.2.61)