80 Alexandre Spits et. al. Non-invasiveness is not guaranteed throughout the arclength sweep. It is only achieved when the controlled non-fundamental component of the reference sx∗,l matches that of the response sx,l, i.e., su,l = 0. An FRC solution is found when the arclength sweep intersects with the desired FRC and the non-fundamental control ceases to be invasive. Control of the resonating non-fundamental harmonic is performed simultaneously with the fundamental sweep, which is assumed to be faster than the non-fundamental sweep. This allows the non-fundamental reference to remain approximately constant while the fundamental component is adjusted. By changing the sign of the controller gain ki,nf and the initial value of the controlled Fourier coefficient sx∗,l, distinct solutions can be identified at a specific excitation frequency, including isolated branches of response. This approach is valid as long as the arc is sufficiently large in the fundamental amplitude versus the excitation frequency plane. Analysis The x-ACBC algorithm was tested experimentally on an electronic Duffing oscillator subjected to harmonic forcing. The setup is depicted in Figure 4. The electronic Duffing oscillator [7] is an electronic circuit designed to implement a weakly dissipative oscillator with very strong nonlinearity and relatively small resonance frequencies. This setup offers a significant advantage, allowing x-ACBC to face traditional experimental challenges without involving shaker-structure interactions. The electronic system can be associated with equivalent coefficients similar to those in the Duffing equation, which is expressed as follows m¨x+c ˙x+kx+k3 x 3 =psinωt (1) The equivalent coefficients are m=10−4 [s2], c =4.9e −4 [s], k =1.68 [-], and k3 =0.99 [V−2]. The forcing level pis 2 [V]. FRCs are identified with the ACBC and x-ACBC. Forcing signal Velocity signal Damping knob Damping switch Test points Displacement signal Power supply Frequency knob Nonlinearity switch Nonlinearity knob Frequency switch Single/double-well switch Fig. 4 Setup of the electronic Duffing system [7]. Figure 5 represents the identified bifurcation diagram of the electronic Duffing. Assessing stability in CBC presents a significant challenge in experimental research. The objective of this study is to identify both stable and unstable open-loop responses. The subsequent figures will not distinguish between stable and unstable solutions. The fundamental resonance was identified using ACBC. Similar to a hardening Duffing oscillator, the fundamental resonance of the electronic oscillator shifts to higher frequencies with an increase in the forcing level. Behaviors such as multistability and amplitude-frequency dependence are observed. The complete identification of secondary resonances is only possible using the x-ACBC algorithm. Three secondary resonances are identified: two superharmonic resonances (2 : 1 and3 : 1), and one subharmonic resonance (1 : 3). Secondary resonances occur at values greater than one-third, one-half, or three times the natural frequency (130 rad/s). The system exhibits strong nonlinear behavior due to its high forcing level and cubic stiffness. The identification of other secondary resonances was not pursued in this study. Unlike in the numerical Duffing oscillator, the pair-secondary resonances (here the 2 : 1 resonance) appear attached to the main branch due to the asymmetry present in the electronic system. These asymmetries come from unavoidable offsets. Isolated branches of solution require specific identification methods as they do not lie in the direct continuation of the frequency response branch. Using the x-ACBC algorithm, the 1 : 3subharmonic resonance detection begins from a point on
RkJQdWJsaXNoZXIy MTMzNzEzMQ==