A homogeneous disk of weight \(W\) and radius \(r\) is supported by two identical cylindrical steel shafts of length \(l\), as shown in Figure 2.70. From solid mechanics, for each shaft the relation between the moment on the disk \(M\) and the angle of rotation of this disk \(\theta\) is

\[ M=\frac{G J}{l} \theta \]

where \(G\) is the shear modulus of the material, \(J\) is the polar moment of inertia of the cross section of each shaft, and \(G J\) is known as the torsional rigidity. Suppose this system is being designed for a particular application that requires the frequency of vibration to be \(f\). Find the value of \(r\) in terms of the parameters to satisfy this requirement.

Transcribed Image Text:

Figure 2.70: A homogeneous disk of weight W and radius r is supported by two identical cylindrical steel shafts of length between support and face of disk.