Multi-mass Spring System

If two or more masses are connected by spring elements, the basic principles remain the same.

We have:

$$\begin{array}{rl}{m}_1{\ddot{x}}_1& =f-k_1(x_1-x_2)\\ m_2{\ddot{x}}_2& =k_1(x_1-x_2)-k_2{x}_2\end{array}$$

If we move all terms to the left side of the equal sign except for the external force $f$, we have:

$$\begin{array}{rl}{m}_1\ddot{x_1}+k_1(x_1-x_2)& =f\\ m_2\ddot{x_2}-k_1(x_1-x_2)+k_2{x}_2& =0\end{array}$$

We have to make some assumptions about the relative motions of each mass.

• For example, we assumed that the $x_1>x_2$, which results in part (b) of the diagram.
• If we instead assumed that $x_2>x_1$, we would have part (c), where the directions of the forces associated with $k_1$ are the opposite. However, this assumption still generates the same equations of motion.

An easy way to visualize the relative motions of the masses would be to imagine that $x_1=0$ and only $x_2$ has been displaced, or vice versa.