Inertia of body that is oriented at some angle from axis of rotation

[xeltius]

1. Problem I am attempting to model the dynamics of the system shown in the attached image. I have a DC Motor with two disks attached to it via springs and something similar to a servo horn. I have two questions. The first is how to determine the inertia of the disk and the spring about the servo axis (we can call it z-z). The next is I do not know if the body will naturally want to precess or not. Intuitively, I want to think it will behave like an amusement park ride such that the disk rises, become parallel with the bottom face of the motor. However, just as all of our intuition about gyroscopes was wrong before we were shown equations and demonstrations, I admit that my own intuition about the dynamic behavior of the spring-disk assembly could be flawed.
2. Assumptions that we can make are that the weight of the disk large enough to cause elongation of the spring at a sufficient velocity (In other words, the spring is NOT too stiff)3. I have drawn a diagram as shown in the bottom third of my attached image. I have attempted to break the shape down into simple shapes and sum the inertias. I was unsuccessful with that method. The last dynamics course I took was an undergraduate class which followed a semester-long Statics course. We used the Hibbeler text in both cases. The issue with this text is that all of the examples have some arbitrary curve bounded by the x- and y-axes. In my case, my object is created by four lines, none of which are aligned with an axis that is orthogonal or parallel with the axis of rotation. I have looked into an inertia tensor. Essentially, my mindset is that instead of my mass being at an arbitrary orientation to a standard axis, I could consider my axis being arbitrary. Then, theoretically, the inertia calculation would not depend on any ``complex" geometry. However, I do not want to waste hours trying to teach myself about tensors and how to use them if that is not the correct way to solve the problem.

EDIT: All masses and springs are identical to one another, respectively.

 

[Nate810]

The best way to solve this problem would be to use the Parallel Axis Theorem. This theorem states that the moment of inertia of an arbitrary object about any axis can be found by using the moment of inertia about its center of mass and the distance between the two axes. In this case, the moment of inertia of the disk and spring about the servo axis (z-z) can be determined by first finding the moment of inertia about the center of mass and then adding the product of the mass of the disk and spring and the square of the distance between the z-z axis and the center of mass. You can also use the parallel axis theorem to calculate the moment of inertia of the disk and spring about any other axis. As for your second question, it is difficult to tell how the body will naturally want to precess without knowing more information about the system. Generally speaking, a gyroscope will naturally precess in the direction of the applied torque, but it is difficult to predict the behavior of your system without more information.