When a simple mass-spring system is excited by a piston that pushes and pulls on the spring in a sinusoidal manner, the oscillatory response of the of the mass-spring system can be described by the equation:
where $x(t)$ is the position of the mass, $b$ is the damping coefficient, $\omega_o$ is the natural frequency of oscillation and $X(t)$ is the position of the piston that drives the system. Let the motion of the piston be of the form:
The response of the mass-spring system to the driving force of the piston depends strongly on the frequency of oscillation, $\omega$, of the piston motion, and is also strongly affected by the value of the damping coefficient $b$.
The response can be characterized by a dimensionless parameter $\xi$ equal to the ratio of the amplitude of oscillation, $x_o$ to the amplitude of the piston motion, $X_o$:
Relative to the natural frequency of oscillation, $\omega_o$, at what frequency, $\omega$, should the piston oscillate to maximize $\xi$?