Measurement of nonlinear normal modes using multi-harmonic stepped force appropriation and free decay

• Published in: Mechanical systems and signal processing 2016-08, Vol.76-77, p.612-633
• Main Authors: Ehrhardt, David A., Allen, Matthew S.
• Format: Article
• Language: English
• Subjects: Backbone; Computer simulation; Continuous-scan laser Doppler vibrometry; Decay; Dynamical systems; Excitation; Experimental force appropriation; Mathematical models; Nonlinear dynamics; Nonlinear normal mode; Nonlinearity
• ISSN: 0888-3270; 1096-1216
• DOI: 10.1016/j.ymssp.2016.02.063

Nonlinear Normal Modes (NNMs) offer tremendous insight into the dynamic behavior of a nonlinear system, extending many concepts that are familiar in linear modal analysis. Hence there is interest in developing methods to experimentally and numerically determine a system׳s NNMs for model updating or simply to characterize its dynamic response. Previous experimental work has shown that a mono-harmonic excitation can be used to isolate a system׳s dynamic response in the neighborhood of a NNM along the main backbones of a system. This work shows that a multi-harmonic excitation is needed to isolate a NNM when well separated linear modes of a structure couple to produce an internal resonance. It is shown that one can tune the multiple harmonics of the input excitation using a plot of the input force versus the response velocity until the area enclosed by the force–velocity curve is minimized. Once an appropriated NNM is measured, one can increase the force level and retune the frequency to obtain a NNM at a higher amplitude or remove the excitation and measure the structure׳s decay down a NNM backbone. This work explores both methods using simulations and measurements of a nominally-flat clamped–clamped beam excited at a single point with a magnetic force. Numerical simulations are used to validate the method in a well defined environment and to provide comparison with the experimentally measured NNMs. The experimental results seem to produce a good estimate of two NNMs along their backbone and part of an internal resonance branch. Full-field measurements are then used to further explore the couplings between the underlying linear modes along the identified NNMs. •This work explores a means whereby multi-harmonic excitation can be used to isolate a NNM.•The theory shows that a plot of the input force versus response velocity can be used to monitor the phase condition of the structural response in real time, as well as understand which harmonics should be added to the input force to minimize the phase condition.•This is numerically demonstrated using a representative nonlinear reduced order model with modal damping.•The numerical results show that this metric is more useful than the phase of the response and its harmonics, the phase being found to be highly sensitive to perturbations.•The methods are examined experimentally by measuring two NNMs of a flat clamped–clamped beam.•Modal interactions from internal resonances are detected experimentally and isolated t