Based on the piston theory of supersonic flow and the energy method, a two dimensional wing with a control surface in supersonic flow is theoretically modeled, in which the cubic stiffness in the torsional direction o...Based on the piston theory of supersonic flow and the energy method, a two dimensional wing with a control surface in supersonic flow is theoretically modeled, in which the cubic stiffness in the torsional direction of the control surface is considered. An approximate method of the cha- otic response analysis of the nonlinear aeroelastic system is studied, the main idea of which is that under the condi- tion of stable limit cycle flutter of the aeroelastic system, the vibrations in the plunging and pitching of the wing can approximately be considered to be simple harmonic excita- tion to the control surface. The motion of the control surface can approximately be modeled by a nonlinear oscillation of one-degree-of-freedom. The range of the chaotic response of the aeroelastic system is approximately determined by means of the chaotic response of the nonlinear oscillator. The rich dynamic behaviors of the control surface are represented as bifurcation diagrams, phase-plane portraits and PS diagrams. The theoretical analysis is verified by the numerical results.展开更多
In this work, we investigate the amplitude death in coupled system with small number of nonlinear oscillators. We show how the transitions to the partial and the complete amplitude deathes happen. We also show that th...In this work, we investigate the amplitude death in coupled system with small number of nonlinear oscillators. We show how the transitions to the partial and the complete amplitude deathes happen. We also show that the partial amplitude death can be found in globally coupled oscillators either.展开更多
Periodic solutions occur commonly in linear or nonlinear dynamical systems. In some cases, the stability of a periodic solution holds if the rightmost characteristic root has negative real part. Based on the Lambert W...Periodic solutions occur commonly in linear or nonlinear dynamical systems. In some cases, the stability of a periodic solution holds if the rightmost characteristic root has negative real part. Based on the Lambert W function, this paper presents a simple algorithm for locating the rightmost characteristic root of periodic solutions of some nonlinear oscillators with large time delay. As application, the proposed algorithm is used to study the primary resonance and 1/3 subharmonic resonance of the Duffing oscillator under harmonic excitation and delayed feedback, as well as the control problem of the van der Pol oscillator under harmonic excitation by using delayed feedback, with a number of case studies. The main advantage of this algorithm is that though very simple in implementation, it works effectively with high accuracy even if the delay is large.展开更多
The global bifurcation of strongly nonlinear oscillator induced by parametric and external excitation is researched. It is known that the parametric and external excitation may induce additional saddle states, and res...The global bifurcation of strongly nonlinear oscillator induced by parametric and external excitation is researched. It is known that the parametric and external excitation may induce additional saddle states, and result in chaos in the phase space, which cannot be detected by applying the Melnikov method directly. A feasible solution for this problem is the combination of the averaged equations and Melnikov method. Therefore, we consider the averaged equations of the system subject to Duffing-Van der Pol strong nonlinearity by introducing the undetermined fundamental frequency. Then the bifurcation values of homoclinic structure formation are detected through the combined application of the new averaged equations with Melnikov integration. Finally, the explicit application shows the analytical conditions coincide with the results of numerical simulation even disturbing parameter is of arbitrary magnitude.展开更多
First-passage failure of multiple-degree-of-freedom nonlinear oscillators with lightly nonlinear dampings and strongly nonlinear stiffness subject to additive and/or parametric Gaussian white noise excitations is stud...First-passage failure of multiple-degree-of-freedom nonlinear oscillators with lightly nonlinear dampings and strongly nonlinear stiffness subject to additive and/or parametric Gaussian white noise excitations is studied. First, by using the stochastic averaging method based on the generalized harmonic functions, the averaged It stochastic differential equation for the amplitudes of the nonlinear oscillators can be derived. Then the associated backward Kolmogorov equation of the conditional reliability function is established, and the conditional reliability is approximately expressed as a series expansion in terms of Kummer functions with time-dependent coefficients. By using the Galerkin method, the time dependent coefficients of the associated conditional reliability function can be solved by a set of differential equations. Finally, the proposed procedure is applied to Duffing-Van der Pol systems under external and/or parametric excitations of Gaussian white noises. The results are also verified by those obtained from Monte Carlo simulation of the original system. The effects of system parameters on first-passage failure are discussed briefly.展开更多
文摘Based on the piston theory of supersonic flow and the energy method, a two dimensional wing with a control surface in supersonic flow is theoretically modeled, in which the cubic stiffness in the torsional direction of the control surface is considered. An approximate method of the cha- otic response analysis of the nonlinear aeroelastic system is studied, the main idea of which is that under the condi- tion of stable limit cycle flutter of the aeroelastic system, the vibrations in the plunging and pitching of the wing can approximately be considered to be simple harmonic excita- tion to the control surface. The motion of the control surface can approximately be modeled by a nonlinear oscillation of one-degree-of-freedom. The range of the chaotic response of the aeroelastic system is approximately determined by means of the chaotic response of the nonlinear oscillator. The rich dynamic behaviors of the control surface are represented as bifurcation diagrams, phase-plane portraits and PS diagrams. The theoretical analysis is verified by the numerical results.
基金supported by National Natural Science Foundation of China under Grant No.10775022the New Century Excellent Talent Project of the Ministry of Education of China under Grant No.07-0112
文摘In this work, we investigate the amplitude death in coupled system with small number of nonlinear oscillators. We show how the transitions to the partial and the complete amplitude deathes happen. We also show that the partial amplitude death can be found in globally coupled oscillators either.
基金supported by the National Natural Science Foundation of China (Grant Nos. 10825207, 11032009)by Program for Changjiang Scholars and Innovative Research Team in University (Grant No. IRT0968)
文摘Periodic solutions occur commonly in linear or nonlinear dynamical systems. In some cases, the stability of a periodic solution holds if the rightmost characteristic root has negative real part. Based on the Lambert W function, this paper presents a simple algorithm for locating the rightmost characteristic root of periodic solutions of some nonlinear oscillators with large time delay. As application, the proposed algorithm is used to study the primary resonance and 1/3 subharmonic resonance of the Duffing oscillator under harmonic excitation and delayed feedback, as well as the control problem of the van der Pol oscillator under harmonic excitation by using delayed feedback, with a number of case studies. The main advantage of this algorithm is that though very simple in implementation, it works effectively with high accuracy even if the delay is large.
基金supported by the National Natural Science Foundation of China (Grant Nos. 10872141, 11072168)the National Hi-Tech Research and Development Program of China ("863" Project) (Grant No. 2008AA042406)
文摘The global bifurcation of strongly nonlinear oscillator induced by parametric and external excitation is researched. It is known that the parametric and external excitation may induce additional saddle states, and result in chaos in the phase space, which cannot be detected by applying the Melnikov method directly. A feasible solution for this problem is the combination of the averaged equations and Melnikov method. Therefore, we consider the averaged equations of the system subject to Duffing-Van der Pol strong nonlinearity by introducing the undetermined fundamental frequency. Then the bifurcation values of homoclinic structure formation are detected through the combined application of the new averaged equations with Melnikov integration. Finally, the explicit application shows the analytical conditions coincide with the results of numerical simulation even disturbing parameter is of arbitrary magnitude.
基金supported by the National Natural Science Foundation of China (Grant No. 11025211)the Natural Science Foundation of Zhejiang Province (Grant No. 26090125)the Special Fund for National Excellent PhD Dissertation
文摘First-passage failure of multiple-degree-of-freedom nonlinear oscillators with lightly nonlinear dampings and strongly nonlinear stiffness subject to additive and/or parametric Gaussian white noise excitations is studied. First, by using the stochastic averaging method based on the generalized harmonic functions, the averaged It stochastic differential equation for the amplitudes of the nonlinear oscillators can be derived. Then the associated backward Kolmogorov equation of the conditional reliability function is established, and the conditional reliability is approximately expressed as a series expansion in terms of Kummer functions with time-dependent coefficients. By using the Galerkin method, the time dependent coefficients of the associated conditional reliability function can be solved by a set of differential equations. Finally, the proposed procedure is applied to Duffing-Van der Pol systems under external and/or parametric excitations of Gaussian white noises. The results are also verified by those obtained from Monte Carlo simulation of the original system. The effects of system parameters on first-passage failure are discussed briefly.
