Kinematics of General Spatial Mechanical Systems. M. Kemal Ozgoren
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Equation (2.76) is the expression of the rotation about rotated axis formula.
1 Shifting Formulas for the Rotation Matrices
The following two formulas, which are called shifting formulas, can be obtained as two consequences of Eq. (2.76).
(2.77)
(2.78)
2.7.2 Mathematical Properties of the Basic Rotation Matrices
In the following formulas, σijk is defined as in Chapter 1. That is, for distinct indices only,
(2.79)
1 Expansion Formulas
If j ≠ i,
(2.80)
(2.81)
If j = i,
(2.82)
1 Shifting Formulas with Quarter and Half Rotations
If j ≠ i,
(2.84)
If j = i,
(2.85)
1 Three Successive Half Rotations About Mutually Orthogonal Axes
Provided that i ≠ j ≠ k,
2.8 Examples Involving Rotation Matrices
2.8.1 Example 2.1
The first basis vector of a reference frame
(2.88)
(2.89)
In the first sequence,
Noting that
(2.92)
(2.93)
In the second sequence,
(2.94)
(2.95)
The corresponding vector equations can be written as follows, like those written for
(2.96)
(2.97)