Identification of Secondary Resonances of Nonlinear Systems using Phase-Locked Loop Testing

TL;DR

The paper tackles the challenge of identifying secondary nonlinear resonances, such as superharmonics and subharmonics, which complicate conventional experimental modal analysis. It proposes phase-locked loop testing (PLLT) augmented with adaptive LMS-based Fourier decomposition to enforce a resonant phase lag and to extract NFRCs and backbone curves without prior system knowledge. The key contributions include a practical PLLT framework that can uncover unstable branches and isolated resonance curves, demonstrated on both a numerical Duffing oscillator and a physical clamped beam, with automatic state transfer to subharmonic isolas and backbone tracking. This approach enables robust, automatic mapping of nonlinear resonance landscapes, with potential implications for improved design, monitoring, and control of nonlinear mechanical systems; the method is validated against harmonic balance and SST benchmarks and shows good agreement despite experimental variability.

Abstract

One unique feature of nonlinear dynamical systems is the existence of superharmonic and subharmonic resonances in addition to primary resonances. In this study, an effective vibration testing methodology is introduced for the experimental identification of these secondary resonances. The proposed method relies on phase-locked loop control combined with adaptive filters for online Fourier decomposition. To this end, the concept of a resonant phase lag is exploited to define the target phase lag to be followed during the experimental continuation process. The method is demonstrated using two systems featuring cubic nonlinearities, namely a numerical Duffing oscillator and a physical experiment comprising a clamped-clamped thin beam. The obtained results highlight that the control scheme can accurately characterize secondary resonances as well as track their backbone curves. A particularly salient feature of the developed algorithm is that, starting from the rest position, it facilitates an automatic and smooth dynamic state transfer toward one point of a subharmonic isolated branch, hence, inducing branch switching.

Figures (24)

• Figure 1: NFRCs of a hardening Duffing oscillator computed at different forcing amplitudes by the harmonic balance method. The unstable responses are indicated by dashed lines. (a) Amplitude and (b) phase lag.
• Figure 2: PLL scheme with adaptive filters.
• Figure 3: NFRCs () and the backbone curve () of the primary resonance identified by PLLT. The reference solution is provided by HBM in gray; light gray corresponds to unstable responses. (a) Amplitude and (b) phase lag.
• Figure 4: NFRCs () at $F=0.3$N and backbone curves () of the 3:1 and 2:1 resonances identified by PLLT. The reference solution is provided by HBM in gray; light gray corresponds to unstable responses. The fold and branch-point bifurcations are marked with circles and crosses, respectively. (a) Amplitude and (b) phase lag.
• Figure 5: NFRCs () at $F=0.5$N and backbone curves () of the 3:1 and 2:1 resonances identified by PLLT. The reference solution is provided by HBM in gray; light gray corresponds to unstable responses. The fold and branch-point bifurcations are marked with circles and crosses, respectively. (a) Amplitude and (b) phase lag.