Math 6410 • ODE

The displacement \(x\) of a spring-mounted mass under the action of dry friction is
assumed to satisfy
\[
  m\ddot{x}+kx=-F_0\sgn(\dot{x})
\]
An example would be a mass \(m\) connected to a fixed support by a spring of stiffness
\(k\) and resting on a surface with frictional force \(F_0\), \(F_0>0\). Set \(m=k=1\) for
convenience and let \(y=\dot{x}\).
\begin{enumerate}
\item[(a)] Consider the initial conditions \(x=x_0>0\), \(\dot{x}=0\) at \(t=0\). Show that
the phase path will spiral exactly \(n\) times before reaching equilibrium if
\[
  (4n-1)F_0<x_0<(4n+1)F_0.
\]
\item[(b)] Suppose the initial conditions at \(t=0\) are \(x=x_0>3F_0\) and \(\dot{x}=0\).
Subsequently, whenever \(x=-\alpha\), where \(2F_0-x_0<-\alpha<0\) and \(\dot{x}>0\),
a trigger operates to increase suddenly the forward velocity so that the kinetic
energy increases by a constant amount \(E\). Show that if \(E>8F_0^2\) then a periodic
motion is approached, and show that the largest value of \(x\) in the periodic motion
is equal to \(F_0+E/(4F_0)\).
\end{enumerate}