Equation of Motion for a Structure Vibrating near the Speed of Light
For a standard $N$ DOF system we can find the eigenfrequencies and eigenmodes of the system by setting up an eigenvalue problem in the form of $$ ([M]-\omega^{2}[K])\phi=0. $$
For a continuous structure, we can set up equations in some coordinate system and solve.
Consider a structure, such as a beam, vibrating near the speed of light. How can we factor this into our equations of motion?
Note 1: These equations of motion are obviously simplified because there's nothing in them stopping a velocity greater than the speed of light.
Note 2: For reference, the following is the Euler-Bernoulli Equations for a beam. I would consider the one of the simpler continuous cases and may be a good starting point.
$$ EI\frac{\partial^{4}w}{\partial x^{4}}+\mu\frac{\partial^{2}w}{\partial t^{2}}=f(x) $$
$w$: deflection of beam
$E$: Modulus of Elasticity
$I$: Bending Moment of Inertia
$\mu$: Mass per Unit Length
Edit 1:
For a simple example consider the classical single degree of freedom rotating imbalance system.
Let $\beta$, the mass ratio be defined as, $$ \beta = \frac{m_{2}}{m_{1}+m_{2}}, $$
where $m_{1}$ is the larger non-rotating mass, and $m_{2}$ is the rotating mass.
Let $$ \omega_{n}^{2}=\frac{k}{m_{1}+m_{2}}. 453
$$
Let $$ 2\zeta\omega_{n}=\frac{c}{m_{1}+m_{2}}. $$
Let $$ \Omega = \frac{\omega}{\omega_{n}}, $$ where $\omega$ is the frequency that $m_{2}$ is rotating at.
And finally, let $\epsilon$ be the radius that $m_{2}$ is rotating about.
We can derive the response of the system, and it's magnitude is given by $$ \frac{|X|}{\epsilon \beta}=\frac{\Omega^{2}}{\sqrt{(1-\Omega^{2})^{2}+(2\zeta \Omega)^{2}}} $$
Note that $m_{1}$ responds in the form of $$ X(t)=|X|\sin \omega t, $$ with $$ \dot{X}(t)=\omega |X|\cos \omega t. $$
The reason why I chose this system is because the limit of $\frac{|X|}{\epsilon \beta}$ as $\Omega$ tends toward infinity is 1. For this system there's nothing stopping us from arbitrarily increasing $\omega$ until $\omega |X|>c$ or until $\epsilon \omega>c$.
I would imagine that people who study the oscillations of neutron stars might know something about this, since I know that they rotate very quickly. 37