An iteration-free approach to excitation harmonization

TL;DR

The paper addresses the problem of excitation harmonics generated by exciter-structure interaction in nonlinear vibration tests. It introduces an iteration-free harmonization approach that injects higher harmonics into the shaker command and uses a PI feedback loop, together with a steady-flow adaptive filter, to cancel all harmonics beyond the fundamental, achieving nearly purely sinusoidal excitation. Analytical stability insights guide drive-point selection and parameter ranges, while a heuristic tuning procedure enables robust, fixed gains across harmonics. Virtual and real experiments demonstrate superior speed and lower harmonic distortion compared with state-of-the-art iterative methods, and robustness across force and base-excitation configurations, making the approach practically attractive for nonlinear vibration testing.

Abstract

Sinusoidal excitation is particularly popular for testing structures in the nonlinear regime. Due to the nonlinear behavior and the inevitable feedback of the structure on the exciter, higher harmonics in the applied excitation are generated. This is undesired, because the acquired response may deviate substantially from that of the structure under purely sinusoidal excitation, in particular if one of the higher harmonics engages into resonance. We present a new approach to suppress those higher excitation harmonics and thus the unwanted exciter-structure interaction: Higher harmonics are added to the voltage input to the shaker whose Fourier coefficients are adjusted via feedback control until the excitation is purely sinusoidal. The stability of this method is analyzed for a simplified model; the resulting closed-form expressions are useful, among others, to select an appropriate exciter configuration, including the drive point. A practical procedure for the control design is suggested. The proposed method is validated in virtual and real experiments of internally resonant structures, in the two common configurations of force excitation via a stinger and base excitation. Excellent performance is achieved already when using the same control gains for all harmonics, throughout the tested range of amplitudes and frequencies, even in the strongly nonlinear regime. Compared to the iterative state of the art, it is found that the proposed method is simpler to implement, enables faster testing and it is easy to achieve a lower harmonic distortion.

Figures (9)

• Figure 1: Conventional shaker-based sinusoidal vibration test (force excitation) extended by proposed harmonization module. SUT: structure under test.
• Figure 2: Virtual experiment: Schematic illustration Shaw.2016.
• Figure 3: Real part of complex transfer function in Eq. (\ref{['eq:stability']}) for (a) drive point $x_1$, and (b) drive point $x_2$.
• Figure 4: Tuning of the controller gains in the virtual experiment. Left column: tuning of the proportional gain ($k_{\mathrm{i}} = 0$), (a) $\bar{k}_{\mathrm{p}} = 2$, (c) $\bar{k}_{\mathrm{p}} = 4$, (e) $\bar{k}_{\mathrm{p}} = 6$, and (g) selected value $\bar{k}_{\mathrm{p}} = 3$. Right column: tuning of the integral gain ($\bar{k}_{\mathrm{p}} = 3$), (b) $\bar{k}_{\mathrm{i}} = 1$, (d) $\bar{k}_{\mathrm{i}} = 2.5$, (f) $\bar{k}_{\mathrm{i}} = 4$, and (h) selected value $\bar{k}_{\mathrm{i}} = 2$. $\bar{k}_{\mathrm{p}} = k_{\mathrm{p}} G/R$, $\bar{k}_{\mathrm{i}} = k_{\mathrm{i}} G/R/\omega_{\mathrm{LP}}$
• Figure 5: Virtual experiment: Stepped sine results of main branch: Fourier coefficients of applied force (a) without ($\mathcal{H}=\lbrace\rbrace$) and (b) with active harmonization ($\mathcal{H}=\lbrace 2,\ldots,7\rbrace$); magnitude of (a) fundamental and (b) third Fourier coefficient of response displacement; phase projection at $\Omega/\omega_1 = 1.2$ on (e) lower and (f) upper part of main branch. Asymptotically stable response regimes are indicated by solid, and unstable by dashed parts of the reference curve.