Interval type-2 fuzzy fractional harmonic-balance optimization of a quasi-zero-stiffness isolator for robust low-frequency vibration suppression

Abstract

Quasi-zero-stiffness (QZS) isolators possess a combination of high load capacity and low dynamic stiffness; however, their characteristics are sensitive to geometric nonlinearity, frequency-dependent dissipation, excitation amplitude and epistemic uncertainty. In this study, a new design scheme is proposed to produce the constant geometric restoring force of a three-spring QZS isolator subject to base excitation. Caputo fractional damping + exact first-harmonic stiffness (in terms of complete elliptic integrals) + multi-harmonic alternating frequency–time harmonic balance (to check the harmonic content and residual error). Interval type-2 fuzzy numbers account for uncertainty in the following quantities: geometric ratio, residual tangent stiffness, fractional damping intensity (or order), and base-motion amplitude. Differential evolution of upper-tail resonance risk and lower-tail isolation losses. A limiting-case validation against a literature benchmark verifies that the fractional-order models recover the classic viscously damped three-spring QZS equations for the case of approaching finite response amplitude and unity fractional order. For the 50 kg realization, the optimized design has a geometric ratio of 1.1824, residual stiffness of 0.02110, damping ratio of 0.02581 and fractional order of 0.9063. Its 95th-percentile peak transmissibility is 3.358 across 120 outer-footprint scenarios, which is 28.9% below the optimized viscous-limit QZS reference. Adverse-tail isolation onset is 23.3% earlier than deterministic fractional tuning completion, and adverse-tail mean attenuation improvements are as high as 1.08 dB. The five-harmonic verification ensures that the ratio of the third harmonic stays below 0.31% in all ranges mentioned in operating range.