Models and simulations are employed to analyze the motion of a spring-supported piston in a vibrated liquid-filled cylinder. The piston motion is damped by forcing liquid through a narrow gap between a hole through the piston and a post fixed to the housing. As the piston moves, the length of this gap changes, so the piston damping coefficient depends on the piston position. This produces a nonlinear damper, even for highly viscous flow.

When gas is absent, the vibration response is overdamped. However, adding a little gas changes the response of this spring-mass-damper system to vibration. During vibration, Bjerknes forces cause some of the gas to migrate below the piston. The resulting pneumatic spring enables the liquid to move with the piston so as to force very little liquid through the gap. Thus, this “Couette mode” has low damping and a strong resonance near the frequency given by the pneumatic spring constant and the total mass of the piston and the liquid. Near this frequency, the amplitude of the piston motion is large, so the nonlinear damper produces a large net force on the piston.

To analyze the effect of this nonlinear damper in detail, a surrogate system is developed by modifying the original system in two ways. First, the gas regions are replaced by upper and lower bellows with similar compressibility to give a well-defined “pneumatic” spring. Second, the upper stop against which the piston is pushed by its lower supporting spring is replaced with an upper spring, thereby removing the nonlinearity from the stop.

An ordinary-differential-equation (ODE) drift model based on quasi-steady Stokes flow is used to produce a regime map of the vibration amplitudes and frequencies for which the piston is up or down for conditions of experimental interest. These results agree fairly well with Arbitrary Lagrangian Eulerian (ALE) simulations of the incompressible Navier-Stokes (NS) equations for the liquid and Newton’s 2^{nd} Law for the piston and bellows.

A quantitative understanding of this nonlinear behavior may enable the development of novel tunable dampers for sensing vibrations of specified amplitudes and frequencies.