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

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!

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. $$m_1x''+c_1x'+k_1x +k_3(x-y)=f_1(t)\\ m_2y''+c_2y'+k_2y-k_3(x-y)=f_2(t)$$
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.