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INTERNAL RESONANT INTERACTIONS OF THREE FREE SURFACE-WAVES IN A CIRCULAR CYLINDRICAL BASIN
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作者 马晨明 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 2003年第12期1411-1420,共10页
The basic equations of free capillary_gravity surface_waves in a circular cylindrical basin were derived from Luke's principle. Taking Galerkin's expansion of the velocity potential and the free surface elevat... The basic equations of free capillary_gravity surface_waves in a circular cylindrical basin were derived from Luke's principle. Taking Galerkin's expansion of the velocity potential and the free surface elevation, the second_order perturbation equations were derived by use of expansion of multiple scale. The nonlinear interactions with the second order internal resonance of three free surface_waves were discussed based on the above. The results include:derivation of the couple equations of resonant interactions among three waves and the conservation laws; analysis of the positions of equilibrium points in phase plane; study of the resonant parameters and the non_resonant parameters respectively in all kinds of circumstances; derivation of the stationary solutions of the second_order interaction equations corresponding to different parameters and analysis of the stability property of the solutions; discussion of the effective solutions only in the limited time range. The analysis makes it clear that the energy transformation mode among three waves differs because of the different initial conditions under nontrivial circumstance. The energy may either exchange among three waves periodically or damp or increase in single waves. 展开更多
关键词 free surface-wave internal resonant interaction stationary solution
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Inter-well internal resonance analysis of rectangular asymmetric cross-ply bistable composite laminated cantilever shell under transverse foundation excitation
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作者 Lele REN Wei ZHANG Yufei ZHANG 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI CSCD 2024年第8期1353-1370,共18页
The chaotic dynamic snap-through and complex nonlinear vibrations are investigated in a rectangular asymmetric cross-ply bistable composite laminated cantilever shell,in cases of 1:2 inter-well internal resonance and ... The chaotic dynamic snap-through and complex nonlinear vibrations are investigated in a rectangular asymmetric cross-ply bistable composite laminated cantilever shell,in cases of 1:2 inter-well internal resonance and primary resonance.The transverse foundation excitation is applied to the fixed end of the structure,and the other end is in a free state.The first-order approximate multiple scales method is employed to perform the perturbation analysis on the dimensionless two-degree-of-freedom ordinary differential motion control equation.The four-dimensional averaged equations are derived in both polar and rectangular coordinate forms.Deriving from the obtained frequency-amplitude and force-amplitude response curves,a detailed analysis is conducted to examine the impacts of excitation amplitude,damping coefficient,and tuning parameter on the nonlinear internal resonance characteristics of the system.The nonlinear softening characteristic is exhibited in the upper stable-state,while the lower stable-state demonstrates the softening and linearity characteristics.Numerical simulation is carried out using the fourth-order Runge-Kutta method,and a series of nonlinear response curves are plotted.Increasing the excitation amplitude further elucidates the global bifurcation and chaotic dynamic snap-through characteristics of the bistable cantilever shell. 展开更多
关键词 bistable composite laminated cantilever shell inter-well internal resonance primary resonance chaotic dynamic snap-through complex nonlinear vibration
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Primary resonance of traveling viscoelastic beam under internal resonance 被引量:10
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作者 Hu DING Linglu HUANG +1 位作者 Xiaoye MAO Liqun CHEN 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI CSCD 2017年第1期1-14,共14页
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
关键词 traveling beam nonlinear vibration VISCOELASTICITY primary resonance internal resonance
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Nonlinear dynamic responses of sandwich functionally graded porous cylindrical shells embedded in elastic media under 1:1 internal resonance 被引量:5
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作者 Yunfei LIU Zhaoye QINT Fulei CHU 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI CSCD 2021年第6期805-818,共14页
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
关键词 nonlinear internal resonance sandwich functionally graded(FG)porous shell improved Donnell's nonlinear shell theory multiple scales method Galerkin method
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FLOW-INDUCED INTERNAL RESONANCES AND MODE EXCHANGE IN HORIZONTAL CANTILEVERED PIPE CONVEYING FLUID (Ⅱ) 被引量:4
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作者 徐鉴 杨前彪 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 2006年第7期935-941,共7页
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
关键词 pipe conveying fluid internal resonance STABILITY BIFURCATION
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Three-to-one internal resonance in multiple stepped beam systems 被引量:3
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作者 A. Tekin E. zkaya S. M. Bagdatlh 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 2009年第9期1131-1142,共12页
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
关键词 stepped beam three-to-one internal resonance stability analysis nonlinear vibration perturbation method
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A piezoelectric energy harvester based on internal resonance 被引量:13
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作者 Liqun Chen Wenan Jiang 《Acta Mechanica Sinica》 SCIE EI CAS CSCD 2015年第2期223-228,共6页
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
关键词 Vibration energy harvesting internal resonance Stochastic excitations
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1∶2 INTERNAL RESONANCE OF COUPLED DYNAMIC SYSTEM WITH QUADRATIC AND CUBIC NONLINEARITIES 被引量:3
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作者 陈予恕 杨彩霞 +1 位作者 吴志强 陈芳启 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 2001年第8期917-924,共8页
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
关键词 quadratic and cubic nonlinearities Normal Form 1 : 2 internal resonance
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CO-DIMENSION 2 BIFURCATIONS AND CHAOS IN CANTILEVERED PIPE CONVEYING TIME VARYING FLUID WITH THREE-TO-ONE INTERNAL RESONANCES 被引量:2
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作者 XuJian ChungKwokWai ChanHenryShuiYing 《Acta Mechanica Solida Sinica》 SCIE EI 2003年第3期245-255,共11页
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
关键词 nonlinear dynamics BIFURCATION stability fluid-solid interaction internal resonance
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Size-dependent modal interactions of a piezoelectrically laminated microarch resonator with 3:1 internal resonance 被引量:1
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作者 A.NIKPOURIAN M.R.GHAZAVI S.AZIZI 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI CSCD 2020年第10期1517-1538,共22页
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
关键词 microarch resonator internal resonance multiple time scales method micro-electro-mechanical system(MEMS) piezoelectric actuation electrostatic excitation
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SINGULAR ANALYSIS OF BIFURCATION OF NONLINEAR NORMAL MODES FOR A CLASS OF SYSTEMS WITH DUAL INTERNAL RESONANCES 被引量:1
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作者 李欣业 陈予恕 吴志强 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 2002年第10期1122-1133,共12页
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
关键词 dual internal resonance nonlinear normal mode mode coupling mode bifurcation the singularity theory
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Broadband energy harvesting based on one-to-one internal resonance 被引量:1
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作者 Wen-An Jiang Xin-Dong Ma +2 位作者 Xiu-Jing Han Li-Qun Chen Qin-Sheng Bi 《Chinese Physics B》 SCIE EI CAS CSCD 2020年第10期195-201,共7页
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
关键词 energy harvesting internal resonance BROADBAND nonlinear modal interactions
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Multiple internal resonances of rotating composite cylindrical shells under varying temperature fields
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作者 Yunfei LIU Jun WANG +2 位作者 Jiaxin HU Zhaoye QIN Fulei CHU 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI CSCD 2022年第10期1543-1554,共12页
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
关键词 multiple internal resonances rotation effect nonlinear temperature variation multi-mode Galerkin technique pseudo-arclength continuation method coupled effect
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Internal resonance of an axially transporting beam with a two-frequency parametric excitation
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作者 Dengbo ZHANG Youqi TANG +2 位作者 Ruquan LIANG Yuanmei SONG Liqun CHEN 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI CSCD 2022年第12期1805-1820,共16页
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
关键词 axially transporting viscoelastic beam internal resonance two-frequency excitation nonlinear vibration perturbation technique
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STEADY STATE MOTIONS OF SHALLOW ARCH UNDER PERIODIC FORCE WITH 1:2 INTERNAL RESONANCEON THE PLANE OF PHYSICAL PARAMETERS
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作者 毕勤胜 陈予恕 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 1998年第7期625-635,共11页
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
关键词 shallow arch internal resonance steady state motion BIFURCATION CHAOS
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Dynamical modeling and non-planar coupled behavior of inclined CFRP cables under simultaneous internal and external resonances
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作者 Houjun KANG Tieding GUO +2 位作者 Weidong ZHU Junyi SU Bingyu ZHAO 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI CSCD 2019年第5期649-678,共30页
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
关键词 inclined carbon fiber reinforced polymer(CFRP)cable BIFURCATION nonlinear dynamics internal resonance external resonance
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BIFURCATION ANALYSIS OF A DOUBLE PENDULUM WITH INTERNAL RESONANCE
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作者 毕勤胜 陈予恕 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 2000年第3期255-264,共10页
By employing the normal form theory, the Hopf bifurcation and the transition boundary of an autonomous double pendulum with 1:1 internal resonance at the critical point is studied. The results are compared with numeri... By employing the normal form theory, the Hopf bifurcation and the transition boundary of an autonomous double pendulum with 1:1 internal resonance at the critical point is studied. The results are compared with numerical solutions. Further, by numerical methods, the road to chaos of a non-autonomous system is presented in the end. 展开更多
关键词 internal resonance Hopf bifurcation transition boundary CHAOS
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Nonlinear dynamic singularity analysis of two interconnected synchronous generator system with 1:3 internal resonance and parametric principal resonance
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作者 Xiaodong WANG Yushu CHEN Lei HOU 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI CSCD 2015年第8期985-1004,共20页
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
关键词 parametric principal resonance internal resonance singularity method bifurcation
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BIFURCATION IN A PARAMETRICALLY EXCITED TWO-DEGREE-OF-FREEDOM NONLINEAR OSCILLATING SYSTEM WITH 1∶2 INTERNAL RESONANCE
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作者 季进臣 陈予恕 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 1999年第4期11-20,共10页
The nonlinear response of a two_degree_of_freedom nonlinear oscillating system to parametric excitation is examined for the case of 1∶2 internal resonance and, principal parametric resonance with respect to the lower... The nonlinear response of a two_degree_of_freedom nonlinear oscillating system to parametric excitation is examined for the case of 1∶2 internal resonance and, principal parametric resonance with respect to the lower mode. The method of multiple scales is used to derive four first_order autonomous ordinary differential equations for the modulation of the amplitudes and phases. The steady_state solutions of the modulated equations and their stability are investigated. The trivial solutions lose their stability through pitchfork bifurcation giving rise to coupled mode solutions. The Melnikov method is used to study the global bifurcation behavior, the critical parameter is determined at which the dynamical system possesses a Smale horseshoe type of chaos. 展开更多
关键词 parametric excitation internal resonance Melnikov method
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Nonlinear study of the dynamic behavior of a string-beam coupled system under combined excitations 被引量:3
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作者 Y.S. Hamed M. Sayed +1 位作者 D.-X. Cao W. Zhang 《Acta Mechanica Sinica》 SCIE EI CAS CSCD 2011年第6期1034-1051,共18页
In this paper,the nonlinear dynamic behavior of a string-beam coupled system subjected to external,parametric and tuned excitations is presented.The governing equations of motion are obtained for the nonlinear transve... In this paper,the nonlinear dynamic behavior of a string-beam coupled system subjected to external,parametric and tuned excitations is presented.The governing equations of motion are obtained for the nonlinear transverse vibrations of the string-beam coupled system which are described by a set of ordinary differential equations with two degrees of freedom.The case of 1:1 internal resonance between the modes of the beam and string,and the primary and combined resonance for the beam is considered.The method of multiple scales is utilized to analyze the nonlinear responses of the string-beam coupled system and obtain approximate solutions up to and including the second-order approximations.All resonance cases are extracted and investigated.Stability of the system is studied using frequency response equations and the phase-plane method.Numerical solutions are carried out and the results are presented graphically and discussed.The effects of the different parameters on both response and stability of the system are investigated.The reported results are compared to the available published work. 展开更多
关键词 1:1 internal resonance - String-beam - Multiple scales Chaotic response Stability
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