Effect of Feedback Delays on Nonlinear Vibrations of Cantilever Beams

The authors present a comprehensive investigation of the effect of feedback delays on the nonlinear vibrations of piezoelectrically-actuated cantilever beams. More specifically, in the first part of this work, we examine the free response of a cantilever beam subjected to delayed-acceleration feedback. We characterize the stability of the trivial solutions and determine the normal form of the bifurcation at the stability boundary. We show that the trivial solutions lose stability via a Hopf bifurcation leading to limit-cycle oscillations (LCO). We assess the stability of the resulting LCO close to the stability boundary by determining the nature of the Hopf bifurcation (sub-or supercritical). We show that the bifurcation type depends only on the frequency of the delayed-response at the bifurcation point and the coefficients of the beam geometric and inertia nonlinearities. To analyze the stability of the LCO in the postbifurcation region, we utilize the Method of Harmonic Balance and the Floquet Theory. We observe that, increasing the gain magnitude for certain feedback delays may culminate in a chaotic response. In the second part of this study, we analyze the effect of feedback delays on a cantilever beam subjected to primary base excitations. We find that the nature of the forced response is largely determined by the stability of the trivial solutions of the unforced response. For stable trivial solutions (i.e., inside the stability boundaries of the linear system), the free response emanating from delayed feedback diminishes leaving only the particular solution resulting from the external excitation. In that case, delayed feedback acts as a vibration absorber. On the other hand, for unstable trivial solutions, the response contains two coexisting frequencies. Therefore, depending on the excitation amplitude and the closeness of the frequency of the delayed response to the excitation frequency, the response is either periodic or quasiperiodic. Finally, we study the effect of higher vibration modes on the beam response. We show that the validity of a single-mode analysis is dependent on the gain-delay combination utilized for feedback as well as the position and size of the piezoelectric patch.