Solving a specific system of differential equations for the Wilberforce Pendulum

Question

I am investigating the motion of the Wilberforce pendulum and need to find a equation for position and angle over time.

I have found equations for the acceleration of the pendulum from the forces, and now have equations of the form:

$$m\frac{d^2z}{dt^2}=kz + Ɛθ$$ $$I\frac{d^2θ}{dt^2}=Jθ + Ɛz$$

Here, M, I, k, J, and Ɛ are constant.

Now, I need to solve these, but don't quite know how to. I have dealt with solving DEs, but not simultaneously. I have done some reading, and I have heard of using the Laplace transform, but I sadly don't know how to use that method, and the online resources that I have looked at have been not very helpful.

If anyone knows of any methods of solving these types of problems to find position and angle over time then I would really appreciate either dropping the name of the method and a place that I could learn about how to use it, or even some working for this particular problem.

Some specifics: I don't want to assume that the frequencies match, and want prefer to have most, if not all initial conditions as variables I can set, such as mass, inertia, initial displacement/angle, etc.

Answer 1

I suppose $m,k,\varepsilon,I,J$ are constant, i.e. do not depend on the time $t$? If so you can do the following:

1. Rewrite your system into the form $$\frac{d x}{dt} = A\cdot x$$ with $x(t)\in \mathbb{R}^4$ and $A\in\mathbb{R}^{4\times 4}$ a Matrix. Hence you rewrite your system into one, in which only first and zero order differential terms appear. An Example how this can be done is found here How to express a 2nd order ODE as 1st order ODE's?
2. Now you can employ the Matrix Exponential to solve the system for any given intial data. The wikipedia page actually calculates some examples and gives a general guideline on how to proceed, see https://en.wikipedia.org/wiki/Matrix_differential_equation