System Tautochronic Path for Pendulum Vibration Absorber Based on Simple Harmonic Oscillator Transformation

Abstract Centrifugal pendulum vibration absorbers (CPVAs) are essentially collections of pendulums attached to a rotor or rotating component or components within a mechanical system for the purpose of mitigating the typical torsional surging that is inherent to internal combustion engines and electric motors. The dynamic stability and performance of CPVAs are highly dependent on the motion path defined for their pendulous masses. Assemblies of absorbers are tuned by adjusting these paths such that the pendulums respond to problematic orders (multiples of average rotation speed) in a way that smooths the rotational accelerations arising from combustion or other non-uniform rotational acceleration events. For most motion paths, pendulum tuning indeed shifts as a function of the pendulum response amplitude. For a given motion path, the tuning shift that occurs as pendulum amplitude varies produces potentially undesirable dynamic instabilities. Large amplitude pendulum motion that mitigates a high percentage of torsional oscillation while avoiding instabilities brought on by tuning shift introduces complexity and hazards into CPVA design processes. Therefore, identifying pendulum paths whose tuning order does not shift as the pendulum amplitude varies, so-called tautochronic paths, may greatly simplify engineering design processes for generating high-performing CPVAs. This paper expands on the work of Sabatini [1], in which a mathematical condition for tautochronicity is identified for a class of differential equations that includes those that arise in the modeling of the motion of a pendulum in a centrifugal field. The approach is based on a transformation from the physical coordinate to a standard Hamiltonian system. We show that transforming a nonlinear oscillator made tautochronic through path modification actually transforms the nonlinear oscillator into a simple harmonic oscillator. To illustrate the new approach and results, the technique is applied to the simplified problem of determining the cut-out shape that produces tautochronic motion for a mass sliding in the cut-out of a larger mass that is free to translate horizontally without friction. In the simplified problem, centrifugal acceleration is replaced by constant gravitational acceleration and rotation of the rotor inertia is replaced by the translation of the large base mass.