s-3m-2s 1-d

A working note for system of three masses and two spring

Suppose there are three masses $m_1$, $m_2$, $m_3$ and two springs with constant $k$, that are put in series. Then using Newton’s second law of motion we can have

$$\tag{1} m_1 \ddot{x}_1 = -k(x_1 - x_2) - k l_0, $$

$$\tag{2} m_2 \ddot{x}_2 = -k(x_2 - x_1) - k(x_2 - x_3), $$

$$\tag{3} m_3 \ddot{x}_2 = -k(x_3 - x_2) + k l_0, $$

where $x_3 > x_2 > x_1$. Previous equations can be presented in matrix form as follow

$$\tag{4} \left[ \begin{matrix} \ddot{x}_1 \newline \ddot{x}_2 \newline \ddot{x}_3 \newline \end{matrix} \right] = \left[ \begin{matrix} -k & k & 0 \newline k & -2k & k \newline 0 & k & -k \newline \end{matrix} \right] \left[ \begin{matrix} x_1 \newline x_2 \newline x_3 \newline \end{matrix} \right] + \left[ \begin{matrix} -k l_0 \newline 0 \newline k l_0 \newline \end{matrix} \right] $$

and generalized as

$$\tag{5} \left[ \begin{matrix} \ddot{x}_1 \newline \ddot{x}_2 \newline \ddot{x}_3 \newline \vdots \newline \ddot{x} _{n-2} \newline \ddot{x} _{n-1} \newline \ddot{x}_n \newline \end{matrix} \right] = \left[ \begin{matrix} -k & k & 0 & \cdots & 0 & 0 & 0 \newline k & -2k & k & \cdots & 0 & 0 & 0 \newline 0 & k & -2k & \cdots & 0 & 0 & 0 \newline \vdots & \vdots & \vdots & \ddots &\vdots & \vdots & \vdots \newline 0 & 0 & 0 & \cdots & -2k & k & 0 \newline 0 & 0 & 0 & \cdots & k & -2k & k \newline 0 & 0 & 0 & \cdots & 0 & k & -k \newline \end{matrix} \right] \left[ \begin{matrix} x_1 \newline x_2 \newline x_3 \newline \vdots \newline x _{n-2} \newline x _{n-1} \newline x_n \newline \end{matrix} \right] + \left[ \begin{matrix} -k l_0 \newline 0 \newline 0 \newline \vdots \newline 0 \newline 0 \newline k l_0 \newline \end{matrix} \right] $$

The most right term in Eqn (5) can not be neglected if the displacement is large ¹.