Oscillation of heavy spring

Question

I am trying to derive the solution for oscillation of a mass of a heavy spring. I have classical spring-pendulum (harmonic pendulum) in mind with one side of spring attached to ceiling and suspended mass on the other side. Furthermore I am trying to express force at the ceiling. I want to get some ideas to which direction to go.

As I am understanding I am trying to find a solution of partial differential equation which is $f \left( \frac{\partial \rho}{\partial t}, \frac{\partial \rho}{\partial y}\right)$, where $\rho$ is linear longitudinal-density of a spring and $y$ position on a spring suspended from ceiling.

I am quite sure, it's possible to solve such kind of equation using separation of variables. Is that true? But furthermore I am wandering how to derive it first? I am seeking either for derivation or some ideas, which steps to take to derive it.

My initial guess was to discretize spring as sequence of massless springs and masses, but I wasn't able to integrate over them. The other idea, that I had was from Euler-Lagrange equations: $$ \frac{\partial L}{\partial y}-\frac{d}{dt}\frac{\partial L}{\partial \dot{y}}=0, $$ where $L$ is Lagrangian $L=T-V$. I expressed $dV$ as combination of elastic and gravitational terms, but I am not entirely sure if this makes sense. $$ dV=\frac{k}{2}dy^2+dm^2, $$ which yields integral expression for $V$: $$V= k\int_0^L\left(\frac{\rho(y)}{\rho_0}dx-1\right)+\int_0^L\rho(y)(L_0-y)dy $$ where $L_0$ is the rest length, $y$ is distance from the ceiling, $m$ is spring mass and $\rho$ is the longitudinal density, which is connected to local force. But this expression doesn't seem really useful for future work. What would be the right approach to that kind of derivation?

Answer 1

You won't be able to derive the equations of a massive spring from simple point mechanics, because the deformation of the spring is due to bending of the windings. And bending means that there is a neutral fiber, and everything that is a little away from the neutral fiber is experiencing either longitudinal compression or longitudinal extension. That is the realm of continuum mechanics, that is you need elastic modulus, poisson's ratio, and the equations will possibly include geometric nonlinearity.

Continuum mechanics means that the systen has eigenmodes, and there is a speed of sound propagation. Where do you expect the speed of (transversal) sound to emerge from in your rigid body mechanics approach?

The only low-fi approximation I can think of is if you can somehow relate the bending of the spring windings to an equivalent bending of a straight beam. What you will achieve is likely to be some kind of string equation, with vibration modes being integer multiples of a fundamental frequency. And as you might know, the wave equation is usually solved by separation of variables.

Note that this already implies that the massive spring has infinitely many degrees of freedom because every mass element can be considered a degree of freedom which communicates over its adjacent spring elements.