Coupled Oscillator: Coupled Mass-Spring System with Input

Setup State Space Equation

Problem Statement

Find an input function such that the lower mass, $m_{1}$, is stationary in the steady state. Find the equation of motion for the upper mass, $m_{2}$.

The use of one spring between the masses is just a simplification of a multi-spring system, so the possibility of being off-kilter is neglected and just the vertical forces are considered.

Initial Conditions and Values

$m_{1}=140kg{\frac {}{}}$

$k_{1}=108000{\frac {N}{m}}$

$m_{2}=80kg{\frac {}{}}$

$k_{2}=80000{\frac {N}{m}}$

Let the initial conditions be zero for the time being.

Force Equations

Sum of the forces in the x direction yields

For $m_{1}$

$+\uparrow \sum F_{x_{2}}=m_{2}{\ddot {x}}_{2}$

$\Rightarrow \ m_{2}{\ddot {x}}_{2}=F_{s_{2}}-m_{2}g$

Since $F_{s}=-k\,x$

$\Rightarrow \ {\ddot {x}}_{2}={\frac {-k_{2}}{m_{2}}}\,x_{2}-g$

And for $m_{1}$

$+\uparrow \sum F_{x_{1}}=m_{1}{\ddot {x}}_{1}$

$\Rightarrow \ m_{1}{\ddot {x}}_{1}=F_{s_{1}}+F-m_{1}g-F_{s_{2}}$

Where $F{\frac {}{}}=F(t)$ is the input force

$\Rightarrow \ {\ddot {x}}_{1}={\frac {-k_{1}}{m_{1}}}\,x_{1}+{\frac {k_{2}}{m_{1}}}\,x_{2}-g+{\frac {F}{m_{1}}}$