Page 114 - Robot Builders Source Book - Gordon McComb

3.7 Drive with a Variable Moment of Inertia 103

Substituting «! into Equation (3.160), we obtain

Hence,

For the initial condition t= 0, CD = CO Q, we find the value of a:

Finally, the solution of Equation (3.160) has the form

Integrating this expression we will find the function 0(£). Indeed, from Equation (3.163
it follows that

or

Equations (3.150), (3.153), (3.158), and (3.164) answer the question formulated at the
beginning of this section by showing the dependencies <j)(f) for all four cases under
consideration. Equations (3.149), (3.152), (3.159), and (3.163) express the changes in
speed in terms of time, thus describing the form of the function (o(t).

3.7 Drive with a Variable Moment of Inertia

We can imagine a number of systems in which the moment of inertia changes
during rotation—for instance, the situation represented in Figure 3.28a). In this
example a vessel rotating around a vertical axis during its rotation is filled with some
granular material (say, sand). The rotation is achieved by a specific drive means. Obvi-
ously, the rotating mass changes in a certain manner and, thus, its moment of inertia
changes. An analogous example is given in Figure 3.28b). Here the vessels are driven
translationally and filled by some material while being accelerated.
Other examples are shown in Figure 3.29a) and b). In Figure 3.29a) the column
rotates around the vertical axis carrying a horizontal beam on which a sliding body
with an inertial mass m is mounted. The distance r between the mass center of the
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