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
Periodic isolator is well known for its wave filtering characteristic.While in middle and high frequencies,the internal resonances of the periodic isolator are evident especially when damping is small.This study propo...Periodic isolator is well known for its wave filtering characteristic.While in middle and high frequencies,the internal resonances of the periodic isolator are evident especially when damping is small.This study proposes a novel aperiodic vibration isolation for improving the internal resonances control of the periodic isolator.The mechanism of the internal resonances control by the aperiodic isolator is firstly explained.For comparing the internal resonances suppression effect of the aperiodic isolator with the periodic isolator,a dynamic model combing the rigid machine,the isolator,and the flexible plate is derived through multi subsystem modeling method and transfer matrix method,whose accuracy is verified through the finite element method.The influences of the aperiodicity and damping of the isolator on the vibration isolation performance and internal resonances suppression effect are investigated by numerical analysis.The numerical results demonstrate that vibration attenuation performances of the periodic isolator and aperiodic isolator are greatly over than that of the continuous isolator in middle and high frequencies.The aperiodic isolator opens the stop bandgaps comparing with the periodic isolator where the pass bandgaps are periodically existed.The damping of the isolator has the stop bandgap widening effect on both the periodic isolator and the aperiodic isolator.In addition,a parameter optimization algorithm of the aperiodic isolator is presented for improving the internal resonances control effect.It is shown that the vibration peaks within the target frequency band of the aperiodic isolator are effectively reduced after the optimization.Finally,the experiments of the three different vibration isolation systems are conducted for verifying the analysis work.展开更多
This study investigates the nonlinear dynamics of geometrically imperfect graphene platelet-reinforced metal foam(GPLRMF)fluid-conveying pipes under the 1:1 internal resonance condition.With simply supported boundary ...This study investigates the nonlinear dynamics of geometrically imperfect graphene platelet-reinforced metal foam(GPLRMF)fluid-conveying pipes under the 1:1 internal resonance condition.With simply supported boundary conditions,the system is subject to the combined external lateral loads and internal pulsating fluid excitations.The nonlinear dynamic model is established with the Euler-Lagrange equations and then systematically discretized via the Galerkin method.The multi-scale analysis reveals how material properties and geometric imperfections influence the internal resonance.Particular emphasis is placed on elucidating,through the modal energy analysis,the energy exchange mechanisms between the first two vibration modes.展开更多
Under the 3：1 internal resonance condition, the steady-state periodic response of the forced vibration of a traveling viscoelastic beam is studied. The viscoelastic behaviors of the traveling beam are described by th...Under the 3：1 internal resonance condition, the steady-state periodic response of the forced vibration of a traveling viscoelastic beam is studied. The viscoelastic behaviors of the traveling beam are described by the standard linear solid model, and the material time derivative is adopted in the viscoelastic constitutive relation. The direct multi-scale method is used to derive the relationships between the excitation frequency and the response amplitudes. For the first time, the real modal functions are employed to analytically investigate the periodic response of the axially traveling beam. The unde- termined coefficient method is used to approximately establish the real modal functions. The approximate analytical results are confirmed by the Galerkin truncation. Numerical examples are presented to highlight the effects of the viscoelastic behaviors on the steady-state periodic responses. To illustrate the effect of the internal resonance, the energy transfer between the internal resonance modes and the saturation-like phenomena in the steady-state responses is presented.展开更多
Applying the multidimensional Lindstedt-Poincare （MDLP） method, we study the forced vibrations with internal resonance of a clamped-clamped pipe conveying fluid under ex- ternal periodic excitation. The frequency-am...Applying the multidimensional Lindstedt-Poincare （MDLP） method, we study the forced vibrations with internal resonance of a clamped-clamped pipe conveying fluid under ex- ternal periodic excitation. The frequency-amplitude response curves of the first-mode resonance with internal resonance are obtained and its characteristics are discussed; moreover, the motions of the first two modes are also analyzed in detail. The present results reveal rich and complex dynamic behaviors caused by internal resonance and that some of the internal resonances are de- cided by the excitation amplitude. The MDLP method is also proved to be a simple and efficient technique to deal with nonlinear dynamics.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
Through the Galerkin method the nonlinear ordinary differential equations （ODEs） in time are obtained from the nonlinear partial differential equations （PDEs） to describe the mo- tion of the coupled structure of a...Through the Galerkin method the nonlinear ordinary differential equations （ODEs） in time are obtained from the nonlinear partial differential equations （PDEs） to describe the mo- tion of the coupled structure of a suspended-cable-stayed beam. In the PDEs, the curvature of main cables and the deformation of cable stays are taken into account. The dynamics of the struc- ture is investigated based on the ODEs when the structure is subjected to a harmonic excitation in the presence of both high-frequency principle resonance and 1：2 internal resonance. It is found that there are typical jumps and saturation phenomena of the vibration amplitude in the struc- ture. And the structure may present quasi-periodic vibration or chaos, if the stiffness of the cable stays membrane and frequency of external excitation are disturbed.展开更多
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
In this study, the vibrations of multiple stepped beams with cubic nonlinearities are considered. A three-to-one internal resonance case is investigated for the system. A general approximate solution to the problem is...In this study, the vibrations of multiple stepped beams with cubic nonlinearities are considered. A three-to-one internal resonance case is investigated for the system. A general approximate solution to the problem is found using the method of multiple scales （a perturbation technique）. The modulation equations of the amplitudes and the phases are derived for two modes. These equations are utilized to determine steady state solutions and their stabilities. It is assumed that the external forcing frequency is close to the lower frequency. For the numeric part of the study, the three-to-one ratio in natural frequencies is investigated. These values are observed to be between the first and second natural frequencies in the cases of the clamped-clamped and clamped-pinned supports, and between the second and third natural frequencies in the case of the pinned-pinned support. Finally, a numeric algorithm is used to solve the three-to-one internal resonance. The first mode is externally excited for the clamped-clamped and clamped-pinned supports, and the second mode is externally excited for the pinned-pinned support. Then, the amplitudes of the first and second modes are investigated when the first mode is externally excited. The amplitudes of the second and third modes are investigated when the second mode is externally excited. The force-response, damping-response, and .frequency- response curves are plotted for the internal resonance modes of vibrations. The stability analysis is carried out for these plots.展开更多
The Newtonian method is employed to obtain nonlinear mathematical model of motion of a horizontally cantilevered and inflexible pipe conveying fluid. The order magnitudes of relevant physical parameters are analyzed q...The Newtonian method is employed to obtain nonlinear mathematical model of motion of a horizontally cantilevered and inflexible pipe conveying fluid. The order magnitudes of relevant physical parameters are analyzed qualitatively to establish a foundation on the further study of the model. The method of multiple scales is used to obtain eigenfunctions of the linear free-vibration modes of the pipe. The boundary conditions yield the characteristic equations from which eigenvalues can be derived. It is found that flow velocity in the pipe may induced the 3：1, 2：1 and 1：1 internal resonances between the first and second modes such that the mechanism of flow-induced internal resonances in the pipe under consideration is explained theoretically. The 3：1 internal resonance first occurs in the system and is, thus, the most important since it corresponds to the minimum critical velocity.展开更多
A vibration-based energy harvester is essentially a resonator working in a limited frequency range.To increase the working frequency range is a challenging problem.This paper reveals a novel possibility for enhancing ...A vibration-based energy harvester is essentially a resonator working in a limited frequency range.To increase the working frequency range is a challenging problem.This paper reveals a novel possibility for enhancing energy harvesting via internal resonance.An internal resonance energy harvester is proposed.The excitation is successively assumed as the Gaussian white noise,the colored noise defined by a second-order filter,the narrow-band noise,and exponentially correlated noise.The corresponding averaged root-meansquare output voltages are computed.Numerical results demonstrate that the internal resonance increases the operating bandwidth and the output voltage.展开更多
The 1:2 internal resonance of coupled dynamic system with quadratic and cubic nonlinearities is studied. The normal forms of this system in 1 :2 internal resonance were derived by using the direct method of normal for...The 1:2 internal resonance of coupled dynamic system with quadratic and cubic nonlinearities is studied. The normal forms of this system in 1 :2 internal resonance were derived by using the direct method of normal form. In the normal,forms, quadratic and cubic nonlinearities were remained. Based on a new convenient transformation technique, the 4-dimension bifurcation equations were reduced to 3-dimension. A bifurcation equation with one-dimension was obtained. Then the bifurcation behaviors of a universal unfolding were studied by using the singularity theory. The method of this paper can be applied to analyze the bifurcation behavior in strong internal resonance on 4-dimension center manifolds.展开更多
The nonlinear behavior of a cantilevered fluid conveying pipe subjected to principal parametric and internal resonances is investigated in this paper.The flow velocity is divided into constant and sinusoidal parts.The...The nonlinear behavior of a cantilevered fluid conveying pipe subjected to principal parametric and internal resonances is investigated in this paper.The flow velocity is divided into constant and sinusoidal parts.The velocity value of the constant part is so adjusted such that the system exhibits 3:1 internal resonances for the first two modes.The method of multiple scales is employed to obtain the response of the system and a set of four first-order nonlinear ordinary- differential equations for governing the amplitude of the response.The eigenvalues of the Jacobian matrix are used to assess the stability of the equilibrium solutions with varying parameters.The co- dimension 2 derived from the double-zero eigenvalues is analyzed in detail.The results show that the response amplitude may undergo saddle-node,pitchfork,Hopf,homoclinic loop and period- doubling bifurcations depending on the frequency and amplitude of the sinusoidal flow.When the frequency of the sinusoidal flow equals exactly half of the first-mode frequency,the system has a route to chaos by period-doubling bifurcation and then returns to a periodic motion as the amplitude of the sinusoidal flow increases.展开更多
The nonlinear normal modes (NNMs) associated with integrnal resonance can be classified into two kinds: uncoupled and coupled. The bifurcation problem of the coupled NNM of system with 1 : 2 : 5 dual internal resonanc...The nonlinear normal modes (NNMs) associated with integrnal resonance can be classified into two kinds: uncoupled and coupled. The bifurcation problem of the coupled NNM of system with 1 : 2 : 5 dual internal resonance is in two variables. The singular analysis of it is presented after separating the two variables by taking advantage of Maple algebra, and some new bifurcation patterns are found. Different from the NNMs of systems with single internal resonance, the number of the NNMs of systems with dual internal resonance may be more or less than the number of the degrees of freedom. At last, it is pointed out that bifurcation problems in two variables can be conveniently solved by separating variables as well as using coupling equations.展开更多
In previous research on the nonlinear dynamics of cable-stayed bridges,boundary conditions were not properly modeled in the modeling.In order to obtain the nonlinear dynamics of cable-stayed bridges more accurately,a ...In previous research on the nonlinear dynamics of cable-stayed bridges,boundary conditions were not properly modeled in the modeling.In order to obtain the nonlinear dynamics of cable-stayed bridges more accurately,a double-cable-stayed shallow-arch model with elastic supports at both ends and the initial configuration of bridge deck included in the modeling is developed in this study.The in-plane eigenvalue problems of the model are solved by dividing the shallow arch(SA)into three partitions according to the number of cables and the piecewise functions are taken as trial functions of the SA.Then,the in-plane one-toone-to-one internal resonance among the global mode and the local modes(two cables’modes)is investigated when external primary resonance occurs.The ordinary differential equations(ODEs)are obtained by Galerkin’s method and solved by the method of multiple time scales.The stable equilibrium solutions of modulation equations are obtained by using the NewtonRaphson method.In addition,the frequency-/force-response curves under different vertical stiffness are provided to study the nonlinear dynamic behaviors of the elastically supported model.To validate the theoretical analyses,the Runge-Kutta method is applied to obtain the numerical solutions.Finally,some interesting conclusions are drawn.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
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 bifurcation dynamics of shallow arch which possesses initial deflection under periodic excitation for the case of 1:2 internal resonance is studied in this paper. The whole parametric plane is divided into several...The bifurcation dynamics of shallow arch which possesses initial deflection under periodic excitation for the case of 1:2 internal resonance is studied in this paper. The whole parametric plane is divided into several different regions according to lire types of motions; then the distribution of steady state motions of shallow arch on the plane of physical parameters is obtained. Combining with numerical method, the dynamics of the system in different regions, especially in the Hopf bifurcation region, is studied in detail. The rule of the mode interaction and the route to chaos of the system is also analysed at the end.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
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 Key Research and Development Plan of China (Grant No.2023YFB3406302)Guangdong Basic and Applied Basic Research Foundation (Grant No.2024A1515011126)the Key Research and Development Plan of Shanxi (Grant No.2024GH-ZDXM-29)。
文摘Periodic isolator is well known for its wave filtering characteristic.While in middle and high frequencies,the internal resonances of the periodic isolator are evident especially when damping is small.This study proposes a novel aperiodic vibration isolation for improving the internal resonances control of the periodic isolator.The mechanism of the internal resonances control by the aperiodic isolator is firstly explained.For comparing the internal resonances suppression effect of the aperiodic isolator with the periodic isolator,a dynamic model combing the rigid machine,the isolator,and the flexible plate is derived through multi subsystem modeling method and transfer matrix method,whose accuracy is verified through the finite element method.The influences of the aperiodicity and damping of the isolator on the vibration isolation performance and internal resonances suppression effect are investigated by numerical analysis.The numerical results demonstrate that vibration attenuation performances of the periodic isolator and aperiodic isolator are greatly over than that of the continuous isolator in middle and high frequencies.The aperiodic isolator opens the stop bandgaps comparing with the periodic isolator where the pass bandgaps are periodically existed.The damping of the isolator has the stop bandgap widening effect on both the periodic isolator and the aperiodic isolator.In addition,a parameter optimization algorithm of the aperiodic isolator is presented for improving the internal resonances control effect.It is shown that the vibration peaks within the target frequency band of the aperiodic isolator are effectively reduced after the optimization.Finally,the experiments of the three different vibration isolation systems are conducted for verifying the analysis work.
基金supported by the Shanghai Shuguang Program(No.18SG36)。
文摘This study investigates the nonlinear dynamics of geometrically imperfect graphene platelet-reinforced metal foam(GPLRMF)fluid-conveying pipes under the 1:1 internal resonance condition.With simply supported boundary conditions,the system is subject to the combined external lateral loads and internal pulsating fluid excitations.The nonlinear dynamic model is established with the Euler-Lagrange equations and then systematically discretized via the Galerkin method.The multi-scale analysis reveals how material properties and geometric imperfections influence the internal resonance.Particular emphasis is placed on elucidating,through the modal energy analysis,the energy exchange mechanisms between the first two vibration modes.
基金Project supported by the State Key Program of the National Natural Science Foundation of China(No.11232009)the National Natural Science Foundation of China(Nos.11372171 and 11422214)
文摘Under the 3：1 internal resonance condition, the steady-state periodic response of the forced vibration of a traveling viscoelastic beam is studied. The viscoelastic behaviors of the traveling beam are described by the standard linear solid model, and the material time derivative is adopted in the viscoelastic constitutive relation. The direct multi-scale method is used to derive the relationships between the excitation frequency and the response amplitudes. For the first time, the real modal functions are employed to analytically investigate the periodic response of the axially traveling beam. The unde- termined coefficient method is used to approximately establish the real modal functions. The approximate analytical results are confirmed by the Galerkin truncation. Numerical examples are presented to highlight the effects of the viscoelastic behaviors on the steady-state periodic responses. To illustrate the effect of the internal resonance, the energy transfer between the internal resonance modes and the saturation-like phenomena in the steady-state responses is presented.
基金the National Natural Science Foundation of China (Nos. 10702045 and 10872135)the Aerospace Foundation of China (No. 2009ZA018)the Natural Science Foundation of Liaoning Province (No. 2009A572)
文摘Applying the multidimensional Lindstedt-Poincare （MDLP） method, we study the forced vibrations with internal resonance of a clamped-clamped pipe conveying fluid under ex- ternal periodic excitation. The frequency-amplitude response curves of the first-mode resonance with internal resonance are obtained and its characteristics are discussed; moreover, the motions of the first two modes are also analyzed in detail. The present results reveal rich and complex dynamic behaviors caused by internal resonance and that some of the internal resonances are de- cided by the excitation amplitude. The MDLP method is also proved to be a simple and efficient technique to deal with nonlinear dynamics.
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supported by the National Natural Science Foundation of China(Nos.10672121 and 11072125)
文摘Through the Galerkin method the nonlinear ordinary differential equations （ODEs） in time are obtained from the nonlinear partial differential equations （PDEs） to describe the mo- tion of the coupled structure of a suspended-cable-stayed beam. In the PDEs, the curvature of main cables and the deformation of cable stays are taken into account. The dynamics of the struc- ture is investigated based on the ODEs when the structure is subjected to a harmonic excitation in the presence of both high-frequency principle resonance and 1：2 internal resonance. It is found that there are typical jumps and saturation phenomena of the vibration amplitude in the struc- ture. And the structure may present quasi-periodic vibration or chaos, if the stiffness of the cable stays membrane and frequency of external excitation are disturbed.
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 Scientific and Technical Research Council of Turkey (TUBITAK) under project No. 104M427
文摘In this study, the vibrations of multiple stepped beams with cubic nonlinearities are considered. A three-to-one internal resonance case is investigated for the system. A general approximate solution to the problem is found using the method of multiple scales （a perturbation technique）. The modulation equations of the amplitudes and the phases are derived for two modes. These equations are utilized to determine steady state solutions and their stabilities. It is assumed that the external forcing frequency is close to the lower frequency. For the numeric part of the study, the three-to-one ratio in natural frequencies is investigated. These values are observed to be between the first and second natural frequencies in the cases of the clamped-clamped and clamped-pinned supports, and between the second and third natural frequencies in the case of the pinned-pinned support. Finally, a numeric algorithm is used to solve the three-to-one internal resonance. The first mode is externally excited for the clamped-clamped and clamped-pinned supports, and the second mode is externally excited for the pinned-pinned support. Then, the amplitudes of the first and second modes are investigated when the first mode is externally excited. The amplitudes of the second and third modes are investigated when the second mode is externally excited. The force-response, damping-response, and .frequency- response curves are plotted for the internal resonance modes of vibrations. The stability analysis is carried out for these plots.
基金Project supported by the National Natural Science Foundation of China (No.10472083) and the National Natural Science Key Foundation of China (No.10532050)
文摘The Newtonian method is employed to obtain nonlinear mathematical model of motion of a horizontally cantilevered and inflexible pipe conveying fluid. The order magnitudes of relevant physical parameters are analyzed qualitatively to establish a foundation on the further study of the model. The method of multiple scales is used to obtain eigenfunctions of the linear free-vibration modes of the pipe. The boundary conditions yield the characteristic equations from which eigenvalues can be derived. It is found that flow velocity in the pipe may induced the 3：1, 2：1 and 1：1 internal resonances between the first and second modes such that the mechanism of flow-induced internal resonances in the pipe under consideration is explained theoretically. The 3：1 internal resonance first occurs in the system and is, thus, the most important since it corresponds to the minimum critical velocity.
基金supported by the State Key Program of National Natural Science of China(Grant No.11232009)Shanghai Leading Academic Discipline Project(Grant No.S30106)
文摘A vibration-based energy harvester is essentially a resonator working in a limited frequency range.To increase the working frequency range is a challenging problem.This paper reveals a novel possibility for enhancing energy harvesting via internal resonance.An internal resonance energy harvester is proposed.The excitation is successively assumed as the Gaussian white noise,the colored noise defined by a second-order filter,the narrow-band noise,and exponentially correlated noise.The corresponding averaged root-meansquare output voltages are computed.Numerical results demonstrate that the internal resonance increases the operating bandwidth and the output voltage.
文摘The 1:2 internal resonance of coupled dynamic system with quadratic and cubic nonlinearities is studied. The normal forms of this system in 1 :2 internal resonance were derived by using the direct method of normal form. In the normal,forms, quadratic and cubic nonlinearities were remained. Based on a new convenient transformation technique, the 4-dimension bifurcation equations were reduced to 3-dimension. A bifurcation equation with one-dimension was obtained. Then the bifurcation behaviors of a universal unfolding were studied by using the singularity theory. The method of this paper can be applied to analyze the bifurcation behavior in strong internal resonance on 4-dimension center manifolds.
基金Project supported by the National Natural Science Foundation of China(No.10072039)RGC in City University of Hong Kong(No.7001206 and No.7001338).
文摘The nonlinear behavior of a cantilevered fluid conveying pipe subjected to principal parametric and internal resonances is investigated in this paper.The flow velocity is divided into constant and sinusoidal parts.The velocity value of the constant part is so adjusted such that the system exhibits 3:1 internal resonances for the first two modes.The method of multiple scales is employed to obtain the response of the system and a set of four first-order nonlinear ordinary- differential equations for governing the amplitude of the response.The eigenvalues of the Jacobian matrix are used to assess the stability of the equilibrium solutions with varying parameters.The co- dimension 2 derived from the double-zero eigenvalues is analyzed in detail.The results show that the response amplitude may undergo saddle-node,pitchfork,Hopf,homoclinic loop and period- doubling bifurcations depending on the frequency and amplitude of the sinusoidal flow.When the frequency of the sinusoidal flow equals exactly half of the first-mode frequency,the system has a route to chaos by period-doubling bifurcation and then returns to a periodic motion as the amplitude of the sinusoidal flow increases.
文摘The nonlinear normal modes (NNMs) associated with integrnal resonance can be classified into two kinds: uncoupled and coupled. The bifurcation problem of the coupled NNM of system with 1 : 2 : 5 dual internal resonance is in two variables. The singular analysis of it is presented after separating the two variables by taking advantage of Maple algebra, and some new bifurcation patterns are found. Different from the NNMs of systems with single internal resonance, the number of the NNMs of systems with dual internal resonance may be more or less than the number of the degrees of freedom. At last, it is pointed out that bifurcation problems in two variables can be conveniently solved by separating variables as well as using coupling equations.
基金This work was supported by the National Natural Science Foundation of China(Grant Nos.11972151 and 11872176)。
文摘In previous research on the nonlinear dynamics of cable-stayed bridges,boundary conditions were not properly modeled in the modeling.In order to obtain the nonlinear dynamics of cable-stayed bridges more accurately,a double-cable-stayed shallow-arch model with elastic supports at both ends and the initial configuration of bridge deck included in the modeling is developed in this study.The in-plane eigenvalue problems of the model are solved by dividing the shallow arch(SA)into three partitions according to the number of cables and the piecewise functions are taken as trial functions of the SA.Then,the in-plane one-toone-to-one internal resonance among the global mode and the local modes(two cables’modes)is investigated when external primary resonance occurs.The ordinary differential equations(ODEs)are obtained by Galerkin’s method and solved by the method of multiple time scales.The stable equilibrium solutions of modulation equations are obtained by using the NewtonRaphson method.In addition,the frequency-/force-response curves under different vertical stiffness are provided to study the nonlinear dynamic behaviors of the elastically supported model.To validate the theoretical analyses,the Runge-Kutta method is applied to obtain the numerical solutions.Finally,some interesting conclusions are drawn.
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(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 bifurcation dynamics of shallow arch which possesses initial deflection under periodic excitation for the case of 1:2 internal resonance is studied in this paper. The whole parametric plane is divided into several different regions according to lire types of motions; then the distribution of steady state motions of shallow arch on the plane of physical parameters is obtained. Combining with numerical method, the dynamics of the system in different regions, especially in the Hopf bifurcation region, is studied in detail. The rule of the mode interaction and the route to chaos of the system is also analysed at the end.
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.
