Question

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

Answer: The final equations of angular motion for the system are: \[ \frac{d^{2}\theta_{1}}{dt^{2}} = \frac{2k_{t}}{mR^{2}}(\theta_{1} - \theta_{2}) \] \[ \frac{d^{2}\theta_{2}}{dt^{2}} = \frac{2k_{t}}{mR^{2}}(\theta_{2} - \theta_{1}) + \frac{2k_{t}}{mR^{2}}(\theta_{2} - \theta_{3}) \] \[ \frac{d^{2}\theta_{3}}{dt^{2}} = \frac{2k_{t}}{mR^{2}}(\theta_{3} - \theta_{2}) \]

Step-by-step solution

Define the Variables

Let \(\theta_{1}, \theta_{2}\), and \(\theta_{3}\) represent the angular displacements of the three disks respectively. We will also represent the torques (moment) acting on the disks as \(M_{1}, M_{2}\), and \(M_{3}\).

Modeling the Torque

According to the torsional relationship, the torque, \(M_{i}\), and the angular displacement, \(\theta_{i}\), are related through the torsional stiffness, \(k_{t}\). Torsional stiffness \(k_t\) can be calculated by \(k_{t} = \frac{GJ}{L}\), where \(G\) is the shear modulus, \(J\) is the area polar moment of inertia, and \(L\) is the length of the span. The torque acting on each disk is: \[ M_{1} = k_{t}(\theta_{1} - \theta_{2}) \] \[ M_{2} = k_{t}(\theta_{2} - \theta_{1}) + k_{t}(\theta_{2} - \theta_{3}) \] \[ M_{3} = k_{t}(\theta_{3} - \theta_{2}) \]

Equations of Motion

The equations of motion for each disk are derived from the relationship between the torque and angular acceleration. The angular acceleration is also related to the mass moment of inertia \(I\) of the rigid disk, which is given by \(I = \frac{1}{2}mR^{2}\), where \(m\) is the mass of the disk and \(R\) is its radius. The equations of motion for the disks are: \[ I\frac{d^{2}\theta_{1}}{dt^{2}} = M_{1} = k_{t}(\theta_{1} - \theta_{2}) \] \[ I\frac{d^{2}\theta_{2}}{dt^{2}} = M_{2} = k_{t}(\theta_{2} - \theta_{1}) + k_{t}(\theta_{2} - \theta_{3}) \] \[ I\frac{d^{2}\theta_{3}}{dt^{2}} = M_{3} = k_{t}(\theta_{3} - \theta_{2}) \]

Final Equations of Angular Motion

Now, we have the final equations of angular motion for the system: \[ \frac{d^{2}\theta_{1}}{dt^{2}} = \frac{2k_{t}}{mR^{2}}(\theta_{1} - \theta_{2}) \] \[ \frac{d^{2}\theta_{2}}{dt^{2}} = \frac{2k_{t}}{mR^{2}}(\theta_{2} - \theta_{1}) + \frac{2k_{t}}{mR^{2}}(\theta_{2} - \theta_{3}) \] \[ \frac{d^{2}\theta_{3}}{dt^{2}} = \frac{2k_{t}}{mR^{2}}(\theta_{3} - \theta_{2}) \] These are the equations of angular motion for the given system when the disks are subjected to twisting moments \(M_{1}, M_{2}\) and \(M_{3}\).

Key concepts

Torsional Stiffness

Torsional stiffness is a parameter that characterizes how resistant a shaft or rod is to twisting when subjected to a torque. It's a measure of the rigidity of an object—how much it resists being twisted. In simple terms, think of it like the stiffness of a spring, but for rotational movement.

The equation to calculate torsional stiffness is given by the formula: \[\begin{equation}k_{t} = \frac{GJ}{L},\[\begin{equation} where \(G\) is the shear modulus, representing the material's rigidity to shear stress, \(J\) is the polar moment of inertia that quantifies how the object's mass is distributed about the axis of rotation, and \(L\) is the length over which the torque is applied. The concept is key when analyzing objects like the disks on the elastic shaft in the given exercise. If torsional stiffness is high, it means the system will show less angular displacement under a particular torque, which directly feeds into how we predict the system's behavior under stress.

Shear Modulus

The shear modulus, often denoted as \(G\), is a property that reflects a material's ability to resist deformation from shear stress. Imagine pushing the top of a pack of cards held at the bottom: the way the cards slide over each other is akin to shear deformation.

The shear modulus is part of the torsional stiffness calculation and plays a critical role in determining how the material will respond when it experiences a force trying to twist it. To put it in perspective, materials with a high shear modulus are typically stiffer and less likely to deform, much like how a thick steel rod is harder to twist than a rubber tube.

Mass Moment of Inertia

The mass moment of inertia, symbolized as \(I\), represents the rotational inertia of an object—basically, how difficult it is to change the object's rotational speed. The higher the mass moment of inertia, the harder it is to start or stop the rotation.

In the context of the given exercise, using disks, the mass moment of inertia is crucial for determining the angular acceleration when a torque (moment) is applied. For a disk, this is calculated with the formula \(I = \frac{1}{2}mR^{2}\), where \(m\) is the mass, and \(R\) is the radius. This relationship shows that the further out the mass is distributed from the center, the larger the moment of inertia, akin to how a figure skater's spin slows when they extend their arms.

Angular Displacement

Angular displacement refers to the angle through which a point or line has been rotated in a specified sense about a specified axis. It's basically the rotational equivalent of linear distance. In our exercise, angular displacement is denoted by \(\theta_{i}\) for each disk, and it's a measure of how much the disk has turned due to the applied torques.

The whole setup with the disks and the elastic shaft essentially acts like a complex spring system that twists around with different amounts of angular displacement for each disk. By applying the equations of angular motion, we can see how the different forces and moments interact with the material properties to result in rotation, dictating the angular responses of the system when subjected to external torques.