Stability and performance benefits of system tautochrones for vibration control

• Published in: Journal of mathematical analysis and applications 2024-03, Vol.531 (1), p.127777, Article 127777
• Main Authors: Monroe, Ryan J., Geist, Bruce K., Shaw, Steven W.
• Format: Article
• Language: English
• Subjects: Brachistochrone; Calculus of variations; Centrifugal pendulum vibration absorber (CPVA); Geometric analysis; Nonlinear analysis; Tautochrone
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/j.jmaa.2023.127777

Centrifugal pendulum vibration absorbers have become a prominent means of controlling torsional oscillations that are typically generated by the combustion pulses within an internal combustion engine. The dynamic stability and performance of devices designed to smooth torsional oscillation are highly dependent upon the motion path defined for their pendulous masses. Generally speaking, dynamic responses are more difficult to manage when pendulum and system resonances shift as a function of the excitation levels due to nonlinear effects. As a result, paths for pendulums that are so-called tautochronic, meaning their resonance does not change as the pendulum amplitude varies, facilitate designing vibration absorbing systems that efficiently reduce vibration even as excitation levels increase. Approximate tautochronic paths for centrifugal pendulum vibration absorbers (CPVAs) are typically derived by making simplifying assumptions, which uncouple the pendulum dynamics from that of the rotor (base) system in which it acts upon. The resulting family of motion paths enable absorber designs that generate desirable and reliable vibration absorption, when slightly over-tuned with respect to the excitation frequency. In this paper, we begin the process of relaxing the assumptions that isolate a pendulum subsystem from its operating context by considering the complete system, which consists of a pendulum with inertial-coupling to the base system. For a simplified vibration system related to a CPVA, wherein a base mass rolls on a horizontal plane and a pendulous mass slides within a cut-out on the sliding base mass, we identify shapes of cut-outs that produce system brachistochrone curves. As in the case of a classical brachistochrone for a fixed base mass, we show the system brachistochrone is a system tautochrone, meaning for a range of input energy, system response that includes pendulum and base mass motions together have a constant period. We show that system tautochrones have distinct vibration control advantages, including superior dynamic stability and vibration absorbing performance. Our results provide a guide to how one can find solutions for the analogous CPVA system. •New family of brachistochrones for a mass sliding in a base mass in a gravity field.•Brachistochrone is a tautochrone, where system period is independent of amplitude.•System tautochrone produces superior vibration absorption compared to cycloid.•System tautochrone path design can benefit ce