The Simulation of a Gyroscope

The Gyroscope is a quite fascinating piece of mechanics, though very simple. I am sure that everybody in their life have at least once seen and played with one.

The part is simple, but the phisics behind it is not so intuitive; I am not going to enter into the equations that gouverne the behaviour of a Gyroscope and nonetheless I wanted to simulate it through an FE model.

The model is a simple one, as you can see from Figure 1; it is basically a steel fly wheel that I have constrained at one end with a connector element that allows only the rotations around the three axes, the translations being constrained to ground.

With these boundary conditions, the object is clearly underconstrained and a static solution would very likely exit with an error. So, because the Gyroscope, by its own name, needs to rotate, we know that a dynamic analysis will be required.

Therefore, as a first step, I am not imposing an angular speed to the object, but just the gravity acceleration along the Y axis and run the analysis. The animation in Figure 2 shows that what happens is what we expected: the Gyroscope tends to drop.

The second step consisted then in imposing an initial angular velocity around the Gyrosope axis, while the gravity acceleration is still applied. The result of this simulation is shown in Figures 3 and 4, where the behaviour is seen from two different points of view.

We notice two facts:

1. The Gyroscope does not drop anymore (Axial Parallelism); the angular speed makes it resisting to the gravity acceleration
2. The main axis of the Gyroscope tends to rotate orthogonally to both the rotation vector and the moment generated by the gravity force (Axial Precession)

Those effects have been widely used to stabilize different systems, from aircrafts to ships to steady state cameras.

As I said, I do not want to enter into the equations (you can check on Wikipedia, for example: https://en.wikipedia.org/wiki/Gyroscope): here I just wanted to show that with a proper simulation (which requires the understanding of the phisycs behind the phenomenon), correct results can be achieved. If we thought that by applying the "centrifugal force" (as it is presented from many pre-processor softwares) we could get the same results we would be wrong: to get the gyroscopic effects we actually need to impose a rotation.

One last simulation aspect concerns the type of solution approach for this problem: Implicit or Explicit.

Because the phenomen is not really a high speed one, because we need few instants to check the behaviour (here I have simulated up to 2 s), because the Explicit is conditionally stable (numerical errors tend sum up), I have decided to use the Implicit approach.