Motion of a spring that has mass

[jaumzaum]

Hello!

I was trying to find the equations of motion for a spring with uniform distribution of mass (uniform just in t=0, because after a while the distribution will be non-uniform).
I tried to attack this problem first in the discrete (non-continuous) way:

"Consider N springs with elastic constant k joining N masses m. Find the acceleration of the i-th mass over time)".

Then I found the following equation for the motion:

$$k(x_{i+1}-2x_{i}+x_{i-1})=ma_{i}$$
I know the first term seems like a second derivative, however I was not able to either solve this system nor extrapolate that in the continuous way.
Can you guys help me with this problem (for example, trying to help me to find the equations of motion or showing me any paper or website that explains how to find them)?

 

[Dr.D]

The mass distribution remains uniform provided us specify location using a material (body) coordinate.

 

[Chestermiller]

Following up on what Dr. D said, let L be the unstretched length of the spring, and let s be a material coordinate that runs from s = 0 at one end of the spring to s = L at the other end of the spring. Also, let x(s,t) be the location at time t of the material element situated at material location s along unstretched configuration of the spring. Then based on this, the local tension T in the spring at material location s and time t is given by $$T(s,t)=kL\left(\frac{\partial x}{\partial s}-1\right)$$ Also, the mass between material locations s and $s+\Delta s$ is given by: $$\rho \Delta s$$ where $\rho$ is the linear density of the unstretched spring. So a force balance on a short section of the spring between material coordinates s and $s+\Delta s$ becomes: $$T(s+\Delta s,t)-T(s,t)=\rho \Delta s\frac{\partial ^ 2x}{\partial t^2}$$

 

[vanhees71]

Here's a nice EJP paper on the subject (open access):

https://doi.org/10.1088/1361-6404/abcddf