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Figure 1.6 Relative position vectors.

      We must, in fact, work with relative position vectors that go from the fixed point to the point of interest by passing from joint to joint. In this case, we define first a vector that goes from to () and then add to it a vector that goes from to (). That is,

(1.23)

By definition, the absolute velocity of is the time rate of change of its position with respect to a fixed point. That is,

      (1.24)

which, upon substitution of Equation 1.23, becomes,

      (1.25)

      which in turn becomes,

(1.26)

      where we see that the absolute velocity of can be expressed as the sum of the velocities of points in the vector chain relative to previous points in the chain so long as the first point is stationary.

Simply differentiating Equation 1.26 with respect to time gives the corresponding expression for accelerations.

      (1.27)

      With respect to the particular system being considered here, we can write,

      (1.28)

      and,

      (1.29)

      Considering first the position of with respect to we see that the length is constant so there will be no rate of change of magnitude of the vector but there will be a rate of change of direction since the coordinate system is rotating. We find,

      (1.30)

      We differentiate again, noting that is not constant so that there will be a rate of change of magnitude this time, and find,

      (1.31)

The rate of change of is a little more complicated for two reasons. First, the vector is not in the body to which the coordinate system being used is fixed so there will be a rate of change of magnitude arising from the time derivatives of the trigonometric functions. Second, the vector itself is not of fixed length so terms involving the rate of change of will appear. Differentiating yields,

      (1.32)

      which, noting that (see Figure 1.5), expands to,

      (1.33) Скачать книгу