3 Experimental Analysis of a Softening-Hardening Nonlinear Oscillator using Control-Based Continuation 23 Remarks: • Every modification of the applied excitation changes the system response and requires to solve Eq. (3.9) again. • All the harmonics in Eqs. (3.12)–(3.13) do not need to be considered and the size of the Jacobian J can often be tailored to the first few most important harmonics. • The perturbation used to computeJusing finite differences might need to be adapted for each harmonic depending on the amplification or attenuation introduced by the excitation system transfer function. • Depending on the characteristics of the excitation system, the Jacobian J might not need to be updated at each periodic solution. 3.3 Description of the Experimental Set-Up The methods presented in Sect. 3.2 are demonstrated on the system shown in Fig. 3.1a. This system is made of a thin steel plate clamped at one end on an aluminum armature. At the other end of the plate, two sets of neodymium magnets are attached. The system acts as a SDOF oscillator and is fixed vertically to avoid gravity-induced deformations transverse to the plate thickness. Under base excitation, the moving magnets interact with a laminated iron stator and a coil. The magnetic interactions introduce a complex nonlinear restoring force with hardening, softening-hardening, or bi-stable characteristics depending on the distance between the magnets and the iron stator. In this study, the distance is such that the system presents a softening-hardening restoring force. The frequency range of interest is 20.5–21.35 Hz and corresponds to oscillation amplitudes of maximum 2.5 mm. The damping in the system can be adjusted with the load connected to the coil. Here, the circuit is left open producing the smallest possible damping. Base and plate tip absolute displacements are measured using two Omron lasers, ZX2-LD50 and ZX2-LD100, respectively. Their sampling period is set to 60 s. A strain gauge also measures the plate deformation at the clamping (see Fig. 3.1b). The nonlinear oscillator is excited at the base by a long-stroke electrodynamic shaker, model APS 113, equipped with linear bearings and operated in current control mode using a Maxon ADS-50/10-4QDC motor controller. Typical base displacements are sinusoidal with a frequency ranging from 20.5 to 21.35 Hz and an amplitude ranging from 0 to 0.3 mm. A PID feedback control system is used to center the position of the shaker’s arm. Proportional, derivative and integral gains are 0.09, 0.0085, and 0.008, respectively. The fine tuning of the control gains was not necessary for CBC to work. A second-order IIR Butterworth filter with a cutoff frequency at 500 Hz was applied to the error signal. The real-time control of the oscillator is achieved through a PD control system implemented in parallel with the PID base displacement controller. Proportional and derivative gains are 0.05 and 0.003, respectively. The error signal is based on the strain gauge signal. The latter presents a very low noise level such that filtering the error signal was not required. Fig. 3.1 (a) Picture of the nonlinear oscillator. (b) Picture of the experimental set-up
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