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Chapter 6: Problem 25

Three identical rigid disks, each of mass \(m\) and radius \(R\), are attached at their centers to an elastic shaft of area polar moment of inertia \(J\) and shear modulus \(G\). The ends of the rod are embedded in rigid supports as shown. The spans between the disks and between the disks and the supports are each of length \(L\). Derive the equations of angular motion for the system if the disks are subjected to the twisting moments \(M_{1}, M_{2}\) and \(M_{3}\), respectively.

Short Answer

Expert verified
Based on the given step by step solution, derive the equations of angular motion for the three disks subjected to twisting moments M1, M2, and M3. The equations are: 1) For Disk 1: $$M_1 - GJ\frac{\theta_2-\theta_1}{L} = I_{1}\frac{d^2\theta_1}{dt^2}$$ 2) For Disk 2: $$M_2 + GJ\frac{\theta_2-\theta_1}{L} - GJ\frac{\theta_3-\theta_2}{L} = I_{2}\frac{d^2\theta_2}{dt^2}$$ 3) For Disk 3: $$M_3 - GJ\frac{\theta_3-\theta_2}{L} = I_{3}\frac{d^2\theta_3}{dt^2}$$

Step by step solution

01

Define Angular Displacements of the Disks

Let the angular displacements of the three disks be denoted as \(\theta_1\), \(\theta_2\), and \(\theta_3\). Since the shaft is elastic, the angular displacements of the disks are not equal.
02

Determine the Elastic Torque in the Shaft

The shaft experiences elastic torsion due to the relative angular displacements of the disks. The elastic torque in the shaft between Disk 1 and Disk 2 can be found using Hooke's law for torsion: $$T_{12} = GJ\frac{\theta_2-\theta_1}{L}$$ Similarly, the elastic torque in the shaft between Disk 2 and Disk 3 can also be calculated using Hooke's law for torsion: $$T_{23} = GJ\frac{\theta_3-\theta_2}{L}$$
03

Apply Newton's Second Law for Disk 1

For Disk 1, the net torque acting on it is: $$M_1 - T_{12} = I_{1}\frac{d^2\theta_1}{dt^2}$$ Where \(I_1\) is the moment of inertia of Disk 1, and \(\frac{d^2\theta_1}{dt^2}\) represents its angular acceleration. Substitute the value of \(T_{12}\) from Step 2: $$M_1 - GJ\frac{\theta_2-\theta_1}{L} = I_{1}\frac{d^2\theta_1}{dt^2}$$
04

Apply Newton's Second Law for Disk 2

For Disk 2, the net torque acting on it is: $$M_2 + T_{12} - T_{23} = I_{2}\frac{d^2\theta_2}{dt^2}$$ Where \(I_2\) is the moment of inertia of Disk 2, and \(\frac{d^2\theta_2}{dt^2}\) represents its angular acceleration. Substitute the values of \(T_{12}\) and \(T_{23}\) from Step 2: $$M_2 + GJ\frac{\theta_2-\theta_1}{L}- GJ\frac{\theta_3-\theta_2}{L} = I_{2}\frac{d^2\theta_2}{dt^2}$$
05

Apply Newton's Second Law for Disk 3

For Disk 3, the net torque acting on it is: $$M_3 - T_{23} = I_{3}\frac{d^2\theta_3}{dt^2}$$ Where \(I_3\) is the moment of inertia of Disk 3, and \(\frac{d^2\theta_3}{dt^2}\) represents its angular acceleration. Substitute the value of \(T_{23}\) from Step 2: $$M_3 - GJ\frac{\theta_3-\theta_2}{L} = I_{3}\frac{d^2\theta_3}{dt^2}$$ So we have now derived the equations of angular motion for the three disks subjected to the twisting moments \(M_1, M_2,\) and \(M_3\).

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