s-2m-1s 1-d rel

A working note for system of two masses and one spring

$$\tag{1} m_1 \frac{d^2 x_1}{dt^2} = -k(x_1 - x_2) - k l_0$$

and

$$\tag{2} m_2 \frac{d^2 x_2}{dt^2} = -k(x_2 - x_1) + k l_0,$$

where $x_3 > x_2 > x_1$.

Multiply Eqn (1) with $m_2$ and Eqn (2) with $m_1$ will turn previous equations into

$$\tag{3} m_2 m_1 \frac{d^2 x_1}{dt^2} = -k m_2 (x_1 - x_2) - k m_2 l_0$$

and

$$\tag{4} m_1 m_2 \frac{d^2 x_2}{dt^2} = -k m_1 (x_2 - x_1) + k m_1 l_0.$$

Substract Eqn (4) with Eqn (3) will produce

$$\tag{5} \begin{array}{rcl} \displaystyle m_2 m_1 \frac{d^2}{dt^2} (x_2 - x_1) & = & -k (m_1 + m_2) (x_2 - x_1) + k (m_1 + m_2) l_0 \newline \newline \displaystyle \left( \frac{m_2 m_1}{m_1 + m_2} \right) \frac{d^2}{dt^2} (x_2 - x_1) & = & -k (x_2 - x_1) + k l_0 \newline \newline \displaystyle \mu \frac{d^2 x_{21}}{dt^2} & = & k x_{21} + k l_0 \newline \newline \displaystyle \mu \frac{d^2 (x_{21} + l_0)}{dt^2} & = & -k(x_{21} - l_0) \newline \newline \displaystyle \frac{d^2 (x_{21} - l_0)}{dt^2} & = & \displaystyle -\left( \frac{k}{\mu} \right) (x_{21} - l_0) \newline \newline \displaystyle \frac{d^2 y}{dt^2} & = & \displaystyle -\omega^2 y. \end{array}$$

Solution of previous final equation is

$$\begin{array}{rcl}\tag{6} y & = & A \sin (\omega t + \phi) \newline \newline x_{21} - l_0 & = & A \sin (\omega t + \phi) \newline \newline x_{21} & = & l_0 + A \sin (\omega t + \phi), \end{array}$$

with

$$\tag{7} \omega = \sqrt{\frac{k}{\mu}}$$

and

$$\tag{8} \mu = \frac{m_1 m_2}{m_1 + m_2},$$

which is known as effective mass ¹.