Math 6410 • ODE

The displacement \(x\) of a spring--mounted mass under the action of dry friction
is assumed to satisfy
\[
  \ddot{x}+x=F_0\operatorname{sgn}(v_0-\dot{x})
\]
\begin{enumerate}
\item[(a)] Calculate the phase paths in the \((x,y)\) plane and draw the phase diagram.
Deduce that the system ultimately converges into a limit cycle oscillation. What happens
when \(v_0=0\)?

\item[(b)] Suppose \(v_0=0\) and the initial conditons 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}