Mitigating quasiperiodic vibrations in forced Rayleigh oscillator via shunt damping: application to harmonically excited galloping oscillator

Abstract A realizable system governed by a forced Rayleigh equation is a mechanical oscillator subject to simultaneous galloping and harmonic excitations, where quasiperiodic vibrations may emerge in post-galloping regions. The complexity and larger amplitude of quasiperiodic regimes relative to adjacent periodic regimes necessitate vibration annihilation in harmonically forced galloping oscillators. However, environmental changes, such as temperature, can vary the periodic and quasiperiodic ranges, calling for adaptive vibration reduction techniques. This research investigates piezoelectric (PZT) shunt damping to reduce oscillation amplitude and eliminate quasiperiodic vibrations from a forced galloping oscillator. The system is composed of a cantilever beam carrying a bluff body and exposed to simultaneous galloping and harmonic forcing. Two PZT layers are used to mitigate the oscillations through the shunt damping effect. First, the nonlinear dimensionless equations are derived using the extended Hamilton's principle, a quasi-steady aerodynamic model, and Gauss's law. Then, using the multiple-scales method, an approximate analytical solution governing the oscillator deflection is extracted. Next, the oscillator dynamics and vibration isolation are investigated by studying the fixed points of the modulation equations, their stability, margins of stability, bifurcations, and frequency response curves. The results show that, in post-galloping regions, the oscillator loses its synchronized or phase-locked periodic solution via cyclic-fold or Neimark–Sacker bifurcations, resulting in large-amplitude quasiperiodic oscillations. However, by leveraging the shunt damping effect, the quasiperiodic regimes are eliminated, culminating in a periodic solution with reduced amplitude. Examining the effects of temperature on the response shows that decreasing the temperature increases both the periodic response amplitude and the quasiperiodic frequency range. The results are validated by numerical integration, and excellent agreement is observed. Additionally, there exists an optimum time constant ratio that maximizes vibration suppression. Therefore, this research shows that shunt damping can effectively eliminate quasiperiodic oscillations and suppress vibration amplitude.