A model of vibrating device coupling two pendulums (VDP) which is highly nonlinear was put forward to conduct vibration analysis. Based on energy analysis, dynamic equations with cubic nonlinearities were established ...A model of vibrating device coupling two pendulums (VDP) which is highly nonlinear was put forward to conduct vibration analysis. Based on energy analysis, dynamic equations with cubic nonlinearities were established using Lagrange's equation. In order to obtain approximate solution, multiple time scales method, one of perturbation technique, was applied. Cases of non-resonant and 1:1:2:2 internal resonant were discussed. In the non-resonant case, the validity of multiple time scales method is confirmed, comparing numerical results derived from fourth order Runge-Kutta method with analytical results derived from first order approximate expression. In the 1:1:2:2 internal resonant case, modal amplitudes of Aa1 and Ab2 increase, respectively, from 0.38 to 0.63 and from 0.19 to 0.32, while the corresponding frequencies have an increase of almost 1.6 times with changes of initial conditions, indicating the existence of typical nonlinear phenomenon. In addition, the chaotic motion is found under this condition.展开更多
An externally excited Duffing oscillator under feedback control is discussed and analyzed under the worst resonance case.Multiple time scales method is applied for this system to find analytic solution with the existe...An externally excited Duffing oscillator under feedback control is discussed and analyzed under the worst resonance case.Multiple time scales method is applied for this system to find analytic solution with the existence and nonexistence of the time delay on control loop.An appropriate stability analysis is also performed and appropriate choices for the feedback gains and the time delay are found in order to reduce the amplitude peak.Different response curves are involved to show and compare controller effects.In addition,analytic solutions are compared with numerical approximation solutions using Rung-Kutta method of fourth order.展开更多
The nonlinear interactions of a microarch resonator with 3:1 internal resonance are studied.The microarch is subjected to a combination of direct current(DC)and alternating current(AC)electric voltages.Thin piezoelect...The nonlinear interactions of a microarch resonator with 3:1 internal resonance are studied.The microarch is subjected to a combination of direct current(DC)and alternating current(AC)electric voltages.Thin piezoelectric layers are thoroughly bonded on the top and bottom surfaces of the microarch.The piezoelectric actuation is not only used to modulate the stiffness and resonance frequency of the resonator but also to provide the suitable linear frequency ratio for the activation of the internal resonance.The size effect is incorporated by using the so-called modified strain gradient theory.The system is highly nonlinear due to the co-existence of the initial curvature,the mid-plane stretching resulting from clamped anchors,and the electrostatic excitation.The eigenvalue problem is solved to conduct a frequency analysis and identify the possible regions for activating the internal resonance.The effects of the piezoelectric actuation,the electric excitation,and the small-scale effect are investigated on the internal resonance.Exclusive nonlinear phenomena such as Hopf bifurcation and hysteresis are identified in the microarch response.It is shown that by applying appropriate piezoelectric actuation,one is able to activate microarch internal resonance regardless of the initial rise level of the microarch.It is also disclosed that among all the parameters,AC electric voltage has the greatest effect on the energy exchange between the interacting modes.The results can be used to design resonators and internal resonance based micro-electro-mechanical system(MEMS)energy harvesters.展开更多
Steady-state periodical response is investigated for an axially moving viscoelastic beam with hybrid supports via approximate analysis with numerical confirmation. It is assumed that the excitation is spatially unifor...Steady-state periodical response is investigated for an axially moving viscoelastic beam with hybrid supports via approximate analysis with numerical confirmation. It is assumed that the excitation is spatially uniform and temporally harmonic. The transverse motion of axially moving beams is governed by a nonlinear partial-differential equation and a nonlinear integro-partial-differential equation. The material time derivative is used in the viscoelastic constitutive relation. The method of multiple scales is applied to the governing equations to investigate primary resonances under general boundary conditions. It is demonstrated that the mode uninvolved in the resonance has no effect on the steady-state response. Numerical examples are presented to demonstrate the effects of the boundary constraint stiffness on the amplitude and the stability of the steady-state response. The results derived for two governing equations are qualitatively the same,but quantitatively different. The differential quadrature schemes are developed to verify those results via the method of multiple scales.展开更多
基金Projects(50574091, 50774084) supported by the National Natural Science Foundation of ChinaProject supported by the Priority Academic Program Development of Jiangsu Higher Education Institutions+1 种基金Project(CXLX12_0949) supported by Research and Innovation Project for College Graduates of Jiangsu Province, ChinaProject(2013DXS03) supported by the Fundamental Research Funds for the Central Universities, China
文摘A model of vibrating device coupling two pendulums (VDP) which is highly nonlinear was put forward to conduct vibration analysis. Based on energy analysis, dynamic equations with cubic nonlinearities were established using Lagrange's equation. In order to obtain approximate solution, multiple time scales method, one of perturbation technique, was applied. Cases of non-resonant and 1:1:2:2 internal resonant were discussed. In the non-resonant case, the validity of multiple time scales method is confirmed, comparing numerical results derived from fourth order Runge-Kutta method with analytical results derived from first order approximate expression. In the 1:1:2:2 internal resonant case, modal amplitudes of Aa1 and Ab2 increase, respectively, from 0.38 to 0.63 and from 0.19 to 0.32, while the corresponding frequencies have an increase of almost 1.6 times with changes of initial conditions, indicating the existence of typical nonlinear phenomenon. In addition, the chaotic motion is found under this condition.
文摘An externally excited Duffing oscillator under feedback control is discussed and analyzed under the worst resonance case.Multiple time scales method is applied for this system to find analytic solution with the existence and nonexistence of the time delay on control loop.An appropriate stability analysis is also performed and appropriate choices for the feedback gains and the time delay are found in order to reduce the amplitude peak.Different response curves are involved to show and compare controller effects.In addition,analytic solutions are compared with numerical approximation solutions using Rung-Kutta method of fourth order.
文摘The nonlinear interactions of a microarch resonator with 3:1 internal resonance are studied.The microarch is subjected to a combination of direct current(DC)and alternating current(AC)electric voltages.Thin piezoelectric layers are thoroughly bonded on the top and bottom surfaces of the microarch.The piezoelectric actuation is not only used to modulate the stiffness and resonance frequency of the resonator but also to provide the suitable linear frequency ratio for the activation of the internal resonance.The size effect is incorporated by using the so-called modified strain gradient theory.The system is highly nonlinear due to the co-existence of the initial curvature,the mid-plane stretching resulting from clamped anchors,and the electrostatic excitation.The eigenvalue problem is solved to conduct a frequency analysis and identify the possible regions for activating the internal resonance.The effects of the piezoelectric actuation,the electric excitation,and the small-scale effect are investigated on the internal resonance.Exclusive nonlinear phenomena such as Hopf bifurcation and hysteresis are identified in the microarch response.It is shown that by applying appropriate piezoelectric actuation,one is able to activate microarch internal resonance regardless of the initial rise level of the microarch.It is also disclosed that among all the parameters,AC electric voltage has the greatest effect on the energy exchange between the interacting modes.The results can be used to design resonators and internal resonance based micro-electro-mechanical system(MEMS)energy harvesters.
基金supported by the National Natural Science Foundation of China (10902064 and 10932006)China National Funds for Distinguished Young Scientists (10725209)+2 种基金the Program of Shanghai Subject Chief Scientist (09XD1401700)Shanghai Leading Talent Program,Shanghai Leading Academic Discipline Project (S30106)the program for Cheung Kong Scholars Programme and Innovative Research Team in University (IRT0844)
文摘Steady-state periodical response is investigated for an axially moving viscoelastic beam with hybrid supports via approximate analysis with numerical confirmation. It is assumed that the excitation is spatially uniform and temporally harmonic. The transverse motion of axially moving beams is governed by a nonlinear partial-differential equation and a nonlinear integro-partial-differential equation. The material time derivative is used in the viscoelastic constitutive relation. The method of multiple scales is applied to the governing equations to investigate primary resonances under general boundary conditions. It is demonstrated that the mode uninvolved in the resonance has no effect on the steady-state response. Numerical examples are presented to demonstrate the effects of the boundary constraint stiffness on the amplitude and the stability of the steady-state response. The results derived for two governing equations are qualitatively the same,but quantitatively different. The differential quadrature schemes are developed to verify those results via the method of multiple scales.
