In this article, the nonlinear dynamic responses of sandwich functionally graded(FG) porous cylindrical shell embedded in elastic media are investigated. The shell studied here consists of three layers, of which the o...In this article, the nonlinear dynamic responses of sandwich functionally graded(FG) porous cylindrical shell embedded in elastic media are investigated. The shell studied here consists of three layers, of which the outer and inner skins are made of solid metal, while the core is FG porous metal foam. Partial differential equations are derived by utilizing the improved Donnell's nonlinear shell theory and Hamilton's principle. Afterwards, the Galerkin method is used to transform the governing equations into nonlinear ordinary differential equations, and an approximate analytical solution is obtained by using the multiple scales method. The effects of various system parameters,specifically, the radial load, core thickness, foam type, foam coefficient, structure damping,and Winkler-Pasternak foundation parameters on nonlinear internal resonance of the sandwich FG porous thin shells are evaluated.展开更多
Composite cylindrical shells,as key components,are widely employed in large rotating machines.However,due to the frequency bifurcations and dense frequency spectra caused by rotation,the nonlinear vibration usually ha...Composite cylindrical shells,as key components,are widely employed in large rotating machines.However,due to the frequency bifurcations and dense frequency spectra caused by rotation,the nonlinear vibration usually has the behavior of complex multiple internal resonances.In addition,the varying temperature fields make the responses of the system further difficult to obtain.Therefore,the multiple internal resonances of composite cylindrical shells with porosities induced by rotation with varying temperature fields are studied in this paper.Three different types of the temperature fields,the Coriolis forces,and the centrifugal force are considered here.The Hamilton principle and the modified Donnell nonlinear shell theory are used to obtain the equilibrium equations of the system,which are transformed into the ordinary differential equations(ODEs)by the multi-mode Galerkin technique.Thereafter,the pseudo-arclength continuation method,which can identify the regions of instability,is introduced to obtain the numerical results.The detailed parametric analysis of the rotating composite shells is performed.Multiple internal resonances caused by the interaction between backward and forward wave modes and the energy transfer phenomenon are detected.Besides,the nonlinear amplitude-frequency response curves are different under different temperature fields.展开更多
This paper investigates the transverse 3:1 internal resonance of an axially transporting nonlinear viscoelastic Euler-Bernoulli beam with a two-frequency parametric excitation caused by a speed perturbation.The Kelvin...This paper investigates the transverse 3:1 internal resonance of an axially transporting nonlinear viscoelastic Euler-Bernoulli beam with a two-frequency parametric excitation caused by a speed perturbation.The Kelvin-Voigt model is introduced to describe the viscoelastic characteristics of the axially transporting beam.The governing equation and the associated boundary conditions are obtained by Newton’s second law.The method of multiple scales is utilized to obtain the steady-state responses.The RouthHurwitz criterion is used to determine the stabilities and bifurcations of the steady-state responses.The effects of the material viscoelastic coefficient on the dynamics of the transporting beam are studied in detail by a series of numerical demonstrations.Interesting phenomena of the steady-state responses are revealed in the 3:1 internal resonance and two-frequency parametric excitation.The approximate analytical method is validated via a differential quadrature method.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
The bifurcation analysis of a simple electric power system involving two synchronous generators connected by a transmission network to an infinite-bus is carried out in this paper. In this system, the infinite-bus vol...The bifurcation analysis of a simple electric power system involving two synchronous generators connected by a transmission network to an infinite-bus is carried out in this paper. In this system, the infinite-bus voltage are considered to maintain two fluctuations in the amplitude and phase angle. The case of 1:3 internal resonance between the two modes in the presence of parametric principal resonance is considered and examined. The method of multiple scales is used to obtain the bifurcation equations of this system. Then, by employing the singularity method, the transition sets determining different bifurcation patterns of the system are obtained and analyzed, which reveal the effects of the infinite-bus voltage amplitude and phase fluctuations on bifurcation patterns of this system. Finally, the bifurcation patterns are all examined by bifurcation diagrams.The results obtained in this paper will contribute to a better understanding of the complex nonlinear dynamic behaviors in a two-machine infinite-bus(TMIB) power system.展开更多
We design an electromechanical transducer harvesting system with one-to-one internal resonance that can emerge a broader spectrum vibrations. The novel harvester is composed of a Duffing electrical circuit coupled to ...We design an electromechanical transducer harvesting system with one-to-one internal resonance that can emerge a broader spectrum vibrations. The novel harvester is composed of a Duffing electrical circuit coupled to a mobile rod, and the coupling between both components is realized via the electromagnetic force. Approximate analytical solutions of the electromechanical system are carried out by introducing the multiple scales analysis, also the nonlinear modulation equation for one-to-one internal resonance is obtained. The character of broadband harvesting performance are analyzed, the two peaks and one jump phenomenon bending to the right for variation of control parameters are observed. It is shown that an advanced bandwidth over a corresponding linear model that does not possess a modal energy interchange.展开更多
This paper proposes a two-to-one internal resonance to widen the bandwidth of vibratory energy harvesters.To describe the improved characteristic,an electromagnetic spring-pendulum harvester is designed.Approximate an...This paper proposes a two-to-one internal resonance to widen the bandwidth of vibratory energy harvesters.To describe the improved characteristic,an electromagnetic spring-pendulum harvester is designed.Approximate analytical solutions of the electromechanical coupled system are carried out by introducing the method of multiple scales,and the frequency response relationships of the displacement and the current are obtained.The character of broadband harvesting performance is examined,the two peaks and double jump phenomena for variation of design parameters were observed.The effect of key control parameters on the harvesters bandwidth is considered,and the nonlinear behaviors of the harvester are validated via numerical results.展开更多
A dynamic model for an inclined carbon ?ber reinforced polymer(CFRP)cable is established, and the linear and nonlinear dynamic behaviors are investigated in detail. The partial differential equations for both the in-p...A dynamic model for an inclined carbon ?ber reinforced polymer(CFRP)cable is established, and the linear and nonlinear dynamic behaviors are investigated in detail. The partial differential equations for both the in-plane and out-of-plane dynamics of the inclined CFRP cable are obtained by Hamilton's principle. The linear eigenvalues are explored theoretically. Then, the ordinary differential equations for analyzing the dynamic behaviors are obtained by the Galerkin integral and dimensionless treatments.The steady-state solutions of the nonlinear equations are obtained by the multiple scale method(MSM) and the Newton-Raphson method. The frequency-and force-response curves are used to investigate the dynamic behaviors of the inclined CFRP cable under simultaneous internal(between the lowest in-plane and out-of-plane modes) and external resonances, i.e., the primary resonances induced by the excitations of the in-plane mode,the out-of-plane mode, and both the in-plane mode and the out-of-plane mode, respectively. The effects of the key parameters, e.g., Young's modulus, the excitation amplitude,and the frequency on the dynamic behaviors, are discussed in detail. Some interesting phenomena and results are observed and concluded.展开更多
The present paper deals with the investigation of dynamic responses of a rotating pre-deformed blade in four cases of resonance,including two subharmonic resonances and two combination resonances.The dimensionless gas...The present paper deals with the investigation of dynamic responses of a rotating pre-deformed blade in four cases of resonance,including two subharmonic resonances and two combination resonances.The dimensionless gas excitation amplitude is assumed to share the same order with the dimensionless vibration displacement.Four cases of resonance are confirmed by examining the secular terms.The theoretical analysis framework is established for each resonance case based on the method of multiple scales.The original dynamic system is integrated numerically by the Runge–Kutta method.The frequency components and phases obtained from fast Fourier transform of the numerical response are used to verify the theoretical results.For the purpose of contrast,modulation equations are also integrated numerically.In all four resonance cases,the theoretical results agree well with the numerical simulation.Parameter studies are conducted to clarify the effects of system parameters on the perturbation curves.Various results are obtained for the rotating blade.A quasi-saturation phenomenon occurs in both combination resonances of summed type and difference type,and the corresponding limit value of the second-mode response can be reduced by decreasing the external detuning parameter.The quasi-saturation phenomenon of rotating blade only appears with high gas pressure.The subharmonic resonance of second mode and the combination resonance of summed type are hard to excite in practice compared with the other two cases.展开更多
A stochastic averaging method of quasi integrable and resonant Hamiltonian systems under excitation of fractional Gaussian noise(fGn) with the Hurst index 1/2 < H < 1 is proposed. First, the definition and the b...A stochastic averaging method of quasi integrable and resonant Hamiltonian systems under excitation of fractional Gaussian noise(fGn) with the Hurst index 1/2 < H < 1 is proposed. First, the definition and the basic property of f Gn and related fractional Brownian motion(fBm) are briefly introduced. Then, the averaged fractional stochastic differential equations(SDEs) for the first integrals and combinations of angle variables of the associated Hamiltonian systems are derived. The stationary probability density and statistics of the original systems are then obtained approximately by simulating the averaged SDEs numerically. An example is worked out to illustrate the proposed stochastic averaging method. It is shown that the results obtained by using the proposed stochastic averaging method and those from digital simulation of original system agree well.展开更多
基金Project supported by the National Natural Science Foundation of China (No. 11972204)。
文摘In this article, the nonlinear dynamic responses of sandwich functionally graded(FG) porous cylindrical shell embedded in elastic media are investigated. The shell studied here consists of three layers, of which the outer and inner skins are made of solid metal, while the core is FG porous metal foam. Partial differential equations are derived by utilizing the improved Donnell's nonlinear shell theory and Hamilton's principle. Afterwards, the Galerkin method is used to transform the governing equations into nonlinear ordinary differential equations, and an approximate analytical solution is obtained by using the multiple scales method. The effects of various system parameters,specifically, the radial load, core thickness, foam type, foam coefficient, structure damping,and Winkler-Pasternak foundation parameters on nonlinear internal resonance of the sandwich FG porous thin shells are evaluated.
基金supported by the National Natural Science Foundation of China(No.11972204)。
文摘Composite cylindrical shells,as key components,are widely employed in large rotating machines.However,due to the frequency bifurcations and dense frequency spectra caused by rotation,the nonlinear vibration usually has the behavior of complex multiple internal resonances.In addition,the varying temperature fields make the responses of the system further difficult to obtain.Therefore,the multiple internal resonances of composite cylindrical shells with porosities induced by rotation with varying temperature fields are studied in this paper.Three different types of the temperature fields,the Coriolis forces,and the centrifugal force are considered here.The Hamilton principle and the modified Donnell nonlinear shell theory are used to obtain the equilibrium equations of the system,which are transformed into the ordinary differential equations(ODEs)by the multi-mode Galerkin technique.Thereafter,the pseudo-arclength continuation method,which can identify the regions of instability,is introduced to obtain the numerical results.The detailed parametric analysis of the rotating composite shells is performed.Multiple internal resonances caused by the interaction between backward and forward wave modes and the energy transfer phenomenon are detected.Besides,the nonlinear amplitude-frequency response curves are different under different temperature fields.
基金Project supported by the National Natural Science Foundation of China (Nos.12002142,1187215951976087)+1 种基金the National Natural Science Foundation of Shanghai of China (No.21ZR1462500)the Natural Science Foundation of Shandong Province of China (No.ZR2021QB137)。
文摘This paper investigates the transverse 3:1 internal resonance of an axially transporting nonlinear viscoelastic Euler-Bernoulli beam with a two-frequency parametric excitation caused by a speed perturbation.The Kelvin-Voigt model is introduced to describe the viscoelastic characteristics of the axially transporting beam.The governing equation and the associated boundary conditions are obtained by Newton’s second law.The method of multiple scales is utilized to obtain the steady-state responses.The RouthHurwitz criterion is used to determine the stabilities and bifurcations of the steady-state responses.The effects of the material viscoelastic coefficient on the dynamics of the transporting beam are studied in detail by a series of numerical demonstrations.Interesting phenomena of the steady-state responses are revealed in the 3:1 internal resonance and two-frequency parametric excitation.The approximate analytical method is validated via a differential quadrature method.
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Project supported by the National Natural Science Foundation of China(No.10632040)
文摘The bifurcation analysis of a simple electric power system involving two synchronous generators connected by a transmission network to an infinite-bus is carried out in this paper. In this system, the infinite-bus voltage are considered to maintain two fluctuations in the amplitude and phase angle. The case of 1:3 internal resonance between the two modes in the presence of parametric principal resonance is considered and examined. The method of multiple scales is used to obtain the bifurcation equations of this system. Then, by employing the singularity method, the transition sets determining different bifurcation patterns of the system are obtained and analyzed, which reveal the effects of the infinite-bus voltage amplitude and phase fluctuations on bifurcation patterns of this system. Finally, the bifurcation patterns are all examined by bifurcation diagrams.The results obtained in this paper will contribute to a better understanding of the complex nonlinear dynamic behaviors in a two-machine infinite-bus(TMIB) power system.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 11632008 and 11702119)the Natural Science Foundation of Jiangsu Province, China (Grant No. BK20170565)+1 种基金China Postdoctoral Science Foundation (Grant No. 2020M671353)Jiangsu Planned Projects for Postdoctoral Research Funds, China (Grant No. 2020Z376)。
文摘We design an electromechanical transducer harvesting system with one-to-one internal resonance that can emerge a broader spectrum vibrations. The novel harvester is composed of a Duffing electrical circuit coupled to a mobile rod, and the coupling between both components is realized via the electromagnetic force. Approximate analytical solutions of the electromechanical system are carried out by introducing the multiple scales analysis, also the nonlinear modulation equation for one-to-one internal resonance is obtained. The character of broadband harvesting performance are analyzed, the two peaks and one jump phenomenon bending to the right for variation of control parameters are observed. It is shown that an advanced bandwidth over a corresponding linear model that does not possess a modal energy interchange.
基金supported by the National Natural Science Foundation of China(Grants 11632008,11702119,and 11972173)the Natural Science Foundation of Jiangsu Province(Grant BK20170565)+1 种基金the Qing Lan Project of Jiangsu Provincethe Training program for Young Talents of Jiangsu University.
文摘This paper proposes a two-to-one internal resonance to widen the bandwidth of vibratory energy harvesters.To describe the improved characteristic,an electromagnetic spring-pendulum harvester is designed.Approximate analytical solutions of the electromechanical coupled system are carried out by introducing the method of multiple scales,and the frequency response relationships of the displacement and the current are obtained.The character of broadband harvesting performance is examined,the two peaks and double jump phenomena for variation of design parameters were observed.The effect of key control parameters on the harvesters bandwidth is considered,and the nonlinear behaviors of the harvester are validated via numerical results.
基金Project supported by the National Natural Science Foundation of China(Nos.11572117 and 11502076)
文摘A dynamic model for an inclined carbon ?ber reinforced polymer(CFRP)cable is established, and the linear and nonlinear dynamic behaviors are investigated in detail. The partial differential equations for both the in-plane and out-of-plane dynamics of the inclined CFRP cable are obtained by Hamilton's principle. The linear eigenvalues are explored theoretically. Then, the ordinary differential equations for analyzing the dynamic behaviors are obtained by the Galerkin integral and dimensionless treatments.The steady-state solutions of the nonlinear equations are obtained by the multiple scale method(MSM) and the Newton-Raphson method. The frequency-and force-response curves are used to investigate the dynamic behaviors of the inclined CFRP cable under simultaneous internal(between the lowest in-plane and out-of-plane modes) and external resonances, i.e., the primary resonances induced by the excitations of the in-plane mode,the out-of-plane mode, and both the in-plane mode and the out-of-plane mode, respectively. The effects of the key parameters, e.g., Young's modulus, the excitation amplitude,and the frequency on the dynamic behaviors, are discussed in detail. Some interesting phenomena and results are observed and concluded.
基金This project is supported by the National Natural Science Foundation of China(Grant Nos.11702033 and 11872159)the Fundamental Research Funds for the Central Universities,CHD(Grant Nos.300102120106,300102128107)the Innovation Program of Shanghai Municipal Education Commission(No.2017-01-07-00-09-E00019).
文摘The present paper deals with the investigation of dynamic responses of a rotating pre-deformed blade in four cases of resonance,including two subharmonic resonances and two combination resonances.The dimensionless gas excitation amplitude is assumed to share the same order with the dimensionless vibration displacement.Four cases of resonance are confirmed by examining the secular terms.The theoretical analysis framework is established for each resonance case based on the method of multiple scales.The original dynamic system is integrated numerically by the Runge–Kutta method.The frequency components and phases obtained from fast Fourier transform of the numerical response are used to verify the theoretical results.For the purpose of contrast,modulation equations are also integrated numerically.In all four resonance cases,the theoretical results agree well with the numerical simulation.Parameter studies are conducted to clarify the effects of system parameters on the perturbation curves.Various results are obtained for the rotating blade.A quasi-saturation phenomenon occurs in both combination resonances of summed type and difference type,and the corresponding limit value of the second-mode response can be reduced by decreasing the external detuning parameter.The quasi-saturation phenomenon of rotating blade only appears with high gas pressure.The subharmonic resonance of second mode and the combination resonance of summed type are hard to excite in practice compared with the other two cases.
基金supported by the National Natural Science Foundation of China under grants nos.:11272279,11321202 and 11432012
文摘A stochastic averaging method of quasi integrable and resonant Hamiltonian systems under excitation of fractional Gaussian noise(fGn) with the Hurst index 1/2 < H < 1 is proposed. First, the definition and the basic property of f Gn and related fractional Brownian motion(fBm) are briefly introduced. Then, the averaged fractional stochastic differential equations(SDEs) for the first integrals and combinations of angle variables of the associated Hamiltonian systems are derived. The stationary probability density and statistics of the original systems are then obtained approximately by simulating the averaged SDEs numerically. An example is worked out to illustrate the proposed stochastic averaging method. It is shown that the results obtained by using the proposed stochastic averaging method and those from digital simulation of original system agree well.