Stochastic Hopf Bifurcation of MDOF Quasi-Integrable Hamiltonian Systems with Delayed Feedback Fractional-Order PD Controller

Hopf bifurcation, as the most representative dynamic bifurcation, is closely related to the stability of many engineering structures. In this work, the stochastic Hopf bifurcation (SHB) of a controlled quasi-integrable Hamiltonian system (H.S.) of multi-degree-of-freedom (MDOF) is investigated, where the system is subjected to wide-band noise and controlled by a Fractional-order Proportional-Derivative (FOPD) controller with time delay. By decoupling FOPD control force and simplifying it without time delay, the averaged Itô differential equations of the approximated system are derived with the technique of stochastic averaging. Then, the average bifurcation parameter expression of system is obtained, which can determine the criterion of the SHB deduced by the FOPD control force. Last, an illustration of coupled Rayleigh oscillators is given to demonstrate the validity of the procedure. The influences of time delay, noise intensities and fractional order on the system SHB are discussed.

Hopf Bifurcation of Compound Stochastic van der Pol System

Hopf bifurcation analysis for compound stochastic van der Pol system with a bound random parameter and Gaussian white noise is investigated in this paper. By the Karhunen-Loeve (K-L) expansion and the orthogonal polynomial approximation, the equivalent deterministic van der Pol system can be deduced. Based on the bifurcation theory of nonlinear deterministic system, the critical value of bifurcation parameter is obtained and the influence of random strengthδand noise intensityσon stochastic Hopf bifurcation in compound stochastic system is discussed. At last we found that increasedδcan relocate the critical value of bifurcation parameter forward while increasedσmakes it backward and the influence ofδis more sensitive thanσ. The results are verified by numerical simulations.

Nonlinear Dynamic Characteristics and Optimal Control of SMA Composite Wings Subjected to Stochastic Excitation

A kind of high-aspect-ratio shape memory alloy (SMA) composite wing is proposed to reduce the wing’s fluttering. The nonlinear dynamic characteristics and optimal control of the SMA composite wings subjected to in-plane stochastic excitation are investigated where the great bending under the flight loads is considered. The stochastic stability of the system is analyzed, and the system’s response is obtained. The conditions of stochastic Hopf bifurcation are determined, and the probability density of the first-passage time is obtained. Finally, the optimal control strategy is proposed. Numerical simulation shows that the stability of the system varies with bifurcation parameters, and stochastic Hopf bifurcation appears in the process; the reliability of the system is improved through optimal control, and the first-passage time is delayed. Finally, the effects of the control strategy are proved by experiments. The results of this paper are helpful for engineering applications of SMA.

Hysteretic Nonlinear Characteristics and Stochastic Bifurcation of Cantilevered Piezoelectric Energy Harvester

Hysteretic nonlinear characteristics and stochastic bifurcation of cantilevered piezoelectric energy harvester was studied in this paper. Piezoelectric ceramics was adhesively bonded on the substrate of cantilever beam to make piezoelectric cantilever beam. Von de Pol difference item was introduced to interpret the hysteretic phenomena of piezoelectric ceramics, and then the nonlinear dynamic model of piezoelectric cantilever beam subjected to axial stochastic excitation was developed. The stochastic stability of the system was analyzed, and the steady-state probability density function and the joint probability density function of the dynamic response of the system were obtained. Finally, the conditions of stochastic Hopf bifurcation were determined. Numerical simulation shows that stochastic Hopf bifurcation appears when bifurcation parameter varies, which can increase vibration amplitude of cantilever beam system and improve the efficiency of piezoelectric energy harvester. The results of this paper are helpful to application of cantilevered piezoelectric energy harvester in engineering fields.

STOCHASTIC POINCARÉ–BENDIXSON THEOREM AND ITS APPLICATION ON STOCHASTIC HOPF BIFURCATION

A stochastic Poincaré–Bendixson theorem and its generalized theorem are given in this paper. The aim of these theorems is to show the existence of a crater-like stationary distribution for a stochastic planar autonomous system. The main theorems are applied on a stochastic predator-prey system as an instruction. For example, stochastic Hopf bifurcation phenomenon is observed when the qualitative change of the shape of the stationary distribution is taken as an indicator of bifurcation.