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. 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:

$+\uparrow \sum F_{y_{1}}=m_{1}{\ddot {x}}_{1}\Rightarrow \ m_{1}{\ddot {x}}_{1}=-2b_{1}{\dot {x}}_{1}-k_{1}x_{1}+k_{2}(x_{2}-x_{1})+m_{1}g$

For mass 2:

$+\uparrow \sum F_{y_{2}}=m_{2}{\ddot {x}}_{2}\Rightarrow \ m_{2}{\ddot {x}}_{2}=-2b_{2}{\dot {x}}_{2}-k_{2}(x_{2}-x_{1})+m_{2}g$

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:

${\underline {\dot {x}}}(t)={\widehat {A}}\,{\underline {x}}(t)+{\widehat {B}}\,{\underline {u}}(t)$

Where ${\widehat {M}}$ denotes a matrix and ${\underline {v}}$ 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

${\begin{bmatrix}{\dot {x}}_{1}\\{\ddot {x}}_{1}\\{\dot {x}}_{2}\\{\ddot {x}}_{2}\end{bmatrix}}=$ ${\begin{bmatrix}0&1&0&0\\-({\frac {k_{1}}{m_{1}}}+{\frac {k_{2}}{m_{2}}})&-{\frac {2b_{1}}{m_{1}}}&{\frac {k_{1}}{m_{1}}}&0\\0&0&0&1\\{\frac {k_{2}}{m_{2}}}&0&-{\frac {k_{2}}{m_{2}}}&-{\frac {2b_{2}}{m_{2}}}\\\end{bmatrix}}{\begin{bmatrix}x_{1}\\{\dot {x}}_{1}\\x_{2}\\{\dot {x}}_{2}\end{bmatrix}}+{\begin{bmatrix}0&0\\0&0\\0&0\\0&0\\\end{bmatrix}}{\begin{bmatrix}0\\\end{bmatrix}}$

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