Coupled mass spring system with damping, I need help with the equation

Question

I know that the equation $mx''+cx'+kx=f(t)$ is used for a normal mass spring system, but I don't know how to express the differential equation for a coupled mass spring system with damping. These are the values:
First spring: $c=1$ $k=3$
Second spring: $c=3$ $k=1$
(no mass are given, so $m=1$)

The system is without external force, but placed in vertical position so $f(t)=mg$

Do I sum the constants of both springs so I can use the equation $mx''+cx'+kx=f(t)$? Or do I solve the equations for separate and then sum the final result? Thanks for your help!

Answer 1

Normally for coupled systems you have two position variables, one for each mass. So you have $$m_1x''+c_1x'+k_1x=f_1(t)\\ m_2y''+c_2y'+k_2y=f_2(t)$$ These are still not coupled. You need a term that is usually $k_3(x-y)$ added to the first and subtracted from the second to represent the coupling. You can write this as a single matrix equation where the $m$ and $c$ matrices are diagonal, but the $k$ matrix has off-diagonal terms to represent the coupling. $$m_1x''+c_1x'+k_1x +k_3(x-y)=f_1(t)\\ m_2y''+c_2y'+k_2y-k_3(x-y)=f_2(t)$$

Answer 2

This is more of a physics question, but if you have two masses connected by a spring then you need to write express Newton's 2nd law for each mass in some appropriate reference frame.

In the end it would be a system of two equations. One gives you the motion of the first mass, while the other gives you the motion of the second mass.

Answer 3

Assuming that the first mass-spring system is pinned with respect to the world frame, and the second mass-spring system is pinned to the first, you have $$ f_1(t) = m_1 x_1'' + c_1 x_1' + k_1 x_1 $$ and $$ f_2(t) = m_2 x_2'' + c_2 x_2' + k_2 x_2 $$ separately.