Coupled Oscillator: Coupled Mass-Spring System with Damping

Problem Statement

For the below system set up a set of state variable equations, and then solve using Laplace transformations. Assume all motion takes place in the vertical directions.

Fig. 1

Solution

Initial Values

For the upper mass:

$m_1=80kg\frac{}{}$

$k_1=60000\frac{N}{m}$

$b_1=0.1\frac{}{}$

And for the lower mass:

$m_2=400kg\frac{}{}$

$k_2=120000\frac{N}{m}$

$b_2=0.2\frac{}{}$

Find the Force Equations

First we need to sum forces in the y-direction for each block.

For mass 1:

$$\begin{gathered} +\uparrow \sum F_{y_1}=m_1{\ddot{x}}_1\Rightarrow \\ {m}_1{\ddot{x}}_1=-2b_1{\dot{x}}_1-k_1{x}_1+k_2(x_2-x_1)+m_{1}g \end{gathered}$$

For mass 2:

$$\begin{gathered} +\uparrow \sum F_{y_2}=m_2{\ddot{x}}_2\Rightarrow \\ {m}_2{\ddot{x}}_2=-2b_2{\dot{x}}_2-k_2(x_2-x_1)+m_{2}g \end{gathered}$$

So if we put the equations above into the correct form we have:

${\ddot{x}}_1=-\frac{2b_1}{m_1}{\dot{x}}_1-(\frac{k_1}{m_1}+\frac{k_2}{m_2})x_1-\frac{k_2}{m_2}{x}_2+g$

and

${\ddot{x}}_2=-\frac{2b_2}{m_2}{\dot{x}}_2-\frac{k_2}{m_2}{x}_2+\frac{k_2}{m_2}{x}_1+g$

State Space Equation

The general form for the state equation is as shown below:

$\underset{\_}{\dot{x}}(t)=\hat{A}\underset{\_}{x}(t)+\hat{B}\underset{\_}{u}(t)$

Where $\hat{M}$ denotes a matrix and $\underset{\_}{v}$ denotes a vector.

If we let $x_1\frac{}{}$, ${\dot{x}}_1\frac{}{}$, $x_2\frac{}{}$, and $\dot{x_2}\frac{}{}$ be the state variables, then

$\left[\begin{array}{c}{\dot{x}}_1\\ {\ddot{x}}_1\\ {\dot{x}}_2\\ {\ddot{x}}_2\end{array}\right]=$ $\left[\begin{array}{cccc}0& 1& 0& 0\\ -(\frac{k_1}{m_1}+\frac{k_2}{m_2})& -\frac{2b_1}{m_1}& \frac{k_1}{m_1}& 0\\ 0& 0& 0& 1\\ \frac{k_2}{m_2}& 0& -\frac{k_2}{m_2}& -\frac{2b_2}{m_2}\\ \end{array}\right]\left[\begin{array}{c}{x}_1\\ {\dot{x}}_1\\ x_2\\ {\dot{x}}_2\end{array}\right]+\left[\begin{array}{cc}0& 0\\ 0& 0\\ 0& 0\\ 0& 0\\ \end{array}\right]\left[\begin{array}{c}0\\ \end{array}\right]$

We don't need to include gravity here if we allow are initial conditions for the spring to be zero with gravity accounted for.