Turning points of an underdamped release from rest ($t_n=n\pi /{\omega }_d,x(t_n)=(-1{)}^n{x}_0{e}^{-\beta n\pi /{\omega }_d}$)

For $\ddot{x}+2\beta \dot{x}+{\omega }_0^{2}x=0$ with $0<\beta <{\omega }_0$, put ${\omega }_d=\sqrt{{\omega }_0^2-{\beta }^2}$. The initial conditions $x(0)=x_0$ and $\dot{x}(0)=0$ giveThus the turning points have the times and displacements in the title. For release from $x_0<0$, the first positive maximum is $|x_0|e^{-\beta \pi /{\omega }_d}$, sufficient to test whether the body reaches a fixed positive stop before that first maximum. Every later positive maximum is smaller.