Steady state responses of oscillators having nonlinear fractional derivatives

Abstract

The application of viscoelastic materials to structures and mechanical systems has been intensively investigated in recent decades. To accurately describe the viscoelastic behavior of the materials, the fractional calculus concept has been introduced into the stress-strain relations. The main purpose of this thesis is to study the dynamic behaviors of the fractional oscillators, which include fractional autonomous and non-autonomous systems, and nonlinear fractional oscillators associated with viscoelastic structures, e.g. viscoelastic arch system, viscoelastic plane truss system and viscoelastic column system with time delay. Owing to the unavailability of the closed-form solutions, many approximately analytical methods are always used to investigate nonlinear dynamic systems containing fractional derivatives. The residue harmonic balance method, proposed recently for higher-order approximate solutions to nonlinear dynamic systems, is extended to analyze the steady state responses of nonlinear fractional oscillators, including a generalized van der Pol oscillator having nonlinear fractional derivative and a fractional Duffing-van der Pol oscillator with time-delayed state feedback. Comparisons between the obtained approximations and numerical results indicate the effectiveness of this method. By combining the harmonic balance method with the polynomial homotopy continuation technique, the steady state responses of harmonically forced fractional Rayleigh oscillator and a generalized van der Pol oscillator involving nonlinear fractional derivatives are considered. Parametric studies are carried out to analyze the effects of fractional orders and the imposed excitations on the system by using response curves. The Neimark bifurcations are captured to delineate regions of instability. The stabilities of steady state solutions are detected by the numerical integration method and are verified with the help of the linear averaging procedure. Many investigators have demonstrated that the nonlinear influences both of the constitutive relationship of materials and the mechanical structures are very important for the dynamic behavior of the system. Therefore, in this thesis, the author focuses on the dynamic analysis of viscoelastic structures whose viscoelastic damping is described by the fractional operators involving linear and nonlinear terms of displacement of response. As the applications in structural engineering, steady state response analysis of viscoelastic arch and viscoelastic plane truss systems is performed. The constitutive behavior of the viscoelastic material of these structures is characterized by using fractional Kelvin-Voigt model based on the Caputo fractional derivative. The equations of motion governing dynamic behaviors of structures are first constructed and simplified by the Galerkin method to obtain fractional oscillators. Then, steady state responses are studied by the harmonic balance method along with the homotopy continuation technique. Multiple solutions, saddle node bifurcations, jump phenomena and even chaos are found and illustrated for combinations of system parameters. It has been shown that the structural stabilization is achieved by increasing the fractional order and material modulus ratio by eliminating the saddle nodes or shrinking the hysteresis area. The dynamic snap-through phenomenon is observed in truss system, when the forcing amplitude increases beyond a critical value. Finally, the feedback control strategy is employed to suppress undesirable vibrations and bifurcations of the system. As well known, time delays are ubiquitous and always exist in the controlled systems. The combined effects of fractional derivatives and time-delayed feedback on the system behavior, however, have rarely been considered before, especially by an analytical method. As illustrative examples, the Duffing oscillator under linear-plus-nonlinear feedback control and a viscoelastic column with timedelayed feedback are investigated by the first order averaging method. Various bifurcation structures are obtained analytically and verified numerically. It is shown that the proposed control law is effective in preventing the undesirable vibrations and bifurcations.

Date of Award	2 Oct 2013
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	Yee Tak Andrew LEUNG (Supervisor)