The Practice of Engineering Dynamics. Ronald J. Anderson

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is desired.

Schematic illustration oaf rigid rod rotating about a fixed point.

      By definition, the velocity of images relative to images is the time derivative of the position of images relative to images. This position vector is designated images and is shown in the figure.

      An expression for the position of images relative to images in this system is,

      (1.7)equation

      (1.8)equation

      In this coordinate system, it is clear that there is a rate of change of magnitude of the vector only and the velocity of point images relative to images after performing the simple differentiation is,

      Another derivation of the velocity of images relative to images might use the system of unit vectors images that are fixed in the rod. The advantage of using this system is that the position vector is easily expressed as,

      (1.10)equation

      Note that the length of this vector is a constant so that the total derivative must come from its rate of change of direction. The angular velocity of the coordinate system is equal to the angular velocity of the rod since the coordinate system is fixed in the rod. That is,

      (1.11)equation

      (1.12)equation

      Since images is constant, images, and the final result is,

      Keep in mind that sequential sets of unit vectors are related to each other by simple plane rotations. Also note that the unit vectors are not related to any point in the system – they simply express directions. Given these two facts, we can relate the two sets of unit vectors we have been using by noting that images (i.e. the plane rotation relating the two sets is a rotation about the images or images axis). The relationships between the two sets of unit vectors can be expressed as follows.

      or

      (1.16)equation

      This is the same result as that shown in Equation 1.9.

      (1.17)equation

      Expanding this

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