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1.
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 sinusoidai 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 codimension 2 derived from the double-zero eigenvaiues 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.  相似文献   

2.
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.  相似文献   

3.
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.  相似文献   

4.
Bifurcations of an airfoil with nonlinear pitching stiffness in incompressible flow are investigated. The pitching spring is regarded as a spring with cubic stiffness. The motion equations of the airfoil are written as the four dimensional one order differential equations. Taking air speed and the linear part of pitching stiffness as the parameters, the analytic solutions of the critical boundaries of pitchfork bifurcations and Hopf bifurcations are obtained in 2 dimensional parameter plane. The stabilities of the equilibrium points and the limit cycles in different regions of 2 dimensional parameter plane are analyzed. By means of harmonic balance method, the approximate critical boundaries of 2-multiple semi-stable limit cycle bifurcations are obtained, and the bifurcation points of supercritical or subcritical Hopf bifurcation are found. Some numerical simulation results are given.  相似文献   

5.
Linear and weakly nonlinear analyses are made for the Rayleigh-Bénard convection in two-component couple-stress liquids with the Soret effect. Conditions for pitchfork, Hopf, Takens-Bogdanov, and codimension-two bifurcations are presented. The Lorenz model is used to study the inverted bifurcation. Positive values of the Soret coefficient favor a pitchfork bifurcation, whereas negative values favor a Hopf bifurcation.Takens-Bogdanov and codimension-two bifurcations are not as much influenced by the Soret coefficient as pitchfork and Hopf bifurcations. The influence of the Soret coefficient on the inverted bifurcation is similar to the influence on the pitchfork bifurcation. The influence of other parameters on the aforementioned bifurcations is also similar as reported earlier in the literature. Using the Newell-Whitehead-Segel equation, the condition for occurrence of Eckhaus and zigzag secondary instabilities is obtained. The domain of appearance of Eckhaus and zigzag instabilities expands due to the presence of the Soret coefficient for positive values. The Soret coefficient with negative values enhances heat transport, while positive values diminish it in comparison with heat transport for the case without the Soret effect. The dual nature of other parameters in influencing heat and mass transport is shown by considering positive and negative values of the Soret coefficient.  相似文献   

6.
The Hopf bifurcations of an airfoil flutter system with a cubic nonlinearity are investigated, with the flow speed as the bifurcation parameter. The center manifold theory and complex normal form method are Used to obtain the bifurcation equation. Interestingly, for a certain linear pitching stiffness the Hopf bifurcation is both supercritical and subcritical. It is found, mathematically, this is caused by the fact that one coefficient in the bifurcation equation does not contain the first power of the bifurcation parameter. The solutions of the bifurcation equation are validated by the equivalent linearization method and incremental harmonic balance method.  相似文献   

7.
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.  相似文献   

8.
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.  相似文献   

9.
The dynamical behaviour of a parametrically excited Duffing-van der Pol oscillator under linear-plus-nonlinear state feedback control with a time delay is concerned. By means of the method of averaging together with truncation of Taylor expansions, two slow-flow equations on the amplitude and phase of response were derived for the case of principal parametric resonance. It is shown that the stability condition for the trivial solution is only associated with the linear terms in the original systems besides the amplitude and frequency of parametric excitation. And the trivial solution can be stabilized by appreciate choice of gains and time delay in feedback control. Different from the case of the trivial solution, the stability condition for nontrivial solutions is also associated with nonlinear terms besides linear terms in the original system. It is demonstrated that nontrivial steady state responses may lose their stability by saddle-node (SN) or Hopf bifurcation (HB) as parameters vary. The simulations, obtained by numerically integrating the original system, are in good agreement with the analytical results.  相似文献   

10.
Based on the Timoshenko beam theory, the finite-deflection and the axial inertia are taken into account, and the nonlinear partial differential equations for flexural waves in a beam are derived. Using the traveling wave method and integration skills, the nonlinear partial differential equations can be converted into an ordinary differential equation. The qualitative analysis indicates that the corresponding dynamic system has a heteroclinic orbit under a certain condition. An exact periodic solution of the nonlinear wave equation is obtained using the Jacobi elliptic function expansion. When the modulus of the Jacobi elliptic function tends to one in the degenerate case, a shock wave solution is given. The small perturbations are further introduced, arising from the damping and the external load to an original Hamilton system, and the threshold condition of the existence of the transverse heteroclinic point is obtained using Melnikov's method. It is shown that the perturbed system has a chaotic property under the Smale horseshoe transform.  相似文献   

11.
The subharmonic resonance and bifurcations of a clamped-clamped buckled beam under base harmonic excitations are investigated. The nonlinear partial integrodifferential equation of the motion of the buckled beam with both quadratic and cubic nonlinearities is given by using Hamilton's principle. A set of second-order nonlinear ordinary differential equations are obtained by spatial discretization with the Galerkin method. A high-dimensional model of the buckled beam is derived, concerning nonlinear coupling. The incremental harmonic balance (IHB) method is used to achieve the periodic solutions of the high-dimensional model of the buckled beam to observe the nonlinear frequency response curve and the nonlinear amplitude response curve, and the Floquet theory is used to analyze the stability of the periodic solutions. Attention is focused on the subharmonic resonance caused by the internal resonance as the excitation frequency near twice of the first natural frequency of the buckled beam with/without the antisymmetric modes being excited. Bifurcations including the saddle-node, Hopf, perioddoubling, and symmetry-breaking bifurcations are observed. Furthermore, quasi-periodic motion is observed by using the fourth-order Runge-Kutta method, which results from the Hopf bifurcation of the response of the buckled beam with the anti-symmetric modes being excited.  相似文献   

12.
Three-to-One Internal Resonances in Hinged-Clamped Beams   总被引:7,自引:0,他引:7  
Chin  Char-Ming  Nayfeh  Ali H. 《Nonlinear dynamics》1997,12(2):129-154
The nonlinear planar response of a hinged-clamped beam to a primary excitation of either its first mode or its second mode is investigated. The analysis accounts for mid-plane stretching, a static axial load and a restraining spring at one end, and modal damping. For a range of axial loads, the second natural frequency is approximately three times the first natural frequency and hence the first and second modes may interact due to a three-to-one internal resonance. The method of multiple scales is used to attack directly the governing nonlinear partial-differential equation and derive two sets of four first-order nonlinear ordinary-differential equations describing the modulation of the amplitudes and phases of the first two modes in the case of primary resonance of either the first or the second mode. Periodic motions and periodically and chaotically modulated motions of the beam are determined by investigating the equilibrium and dynamic solutions of the modulation equations. For the case of primary resonance of the first mode, only two-mode solutions are possible, whereas for the case of primary resonance of the second mode, single- and two-mode solutions are possible. The two-mode equilibrium solutions of the modulation equations may undergo a supercritical or a subcritical Hopf bifurcation, depending on the magnitude of the axial load. A shooting technique is used to calculate limit cycles of the modulation equations and Floquet theory is used to ascertain their stability. The limit cycles correspond to periodically modulated motions of the beam. The limit cycles are found to undergo cyclic-fold bifurcations and period-doubling bifurcations, leading to chaos. The chaotic attractors may undergo boundary crises, resulting in the destruction of the chaotic attractors and their basins of attraction.  相似文献   

13.
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 steadystate 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. Project supported by the National Natural Science Foundation of China (19472046)  相似文献   

14.
内共振条件下直线运动梁的动力稳定性   总被引:35,自引:4,他引:31  
冯志华  胡海岩 《力学学报》2002,34(3):389-400
基于Kane方程,建立起了包含有耦合的三次几何及惯性非线性项大范围直线运动梁动力学控制方程.利用多尺度法并结合笛卡尔坐标变换,对所得方程进行一次近似展开,着重对满足一、二阶模态间3:1内共振现象的两端铰支梁参激振动平凡解稳定性进行了详尽的分析,得出了稳定性边界的解析表达式.采用中心流形定理对调制微分方程组进行降维处理,分析了相应Hopf分岔类型并通过数值计算发现了稳定的极限环存在.  相似文献   

15.
Chin  Char-Ming  Nayfeh  Ali H. 《Nonlinear dynamics》1999,20(2):131-158
The nonlinear planar response of a hinged-clamped beam to a principal parametric resonance of either its first or second mode or a combination parametric resonance of the additive type of its first two modes is investigated. The analysis accounts for mid-plane stretching, a static axial load, a restraining spring at one end, and modal damping. The natural frequency of the second mode is approximately three times the natural frequency of the first mode for a range of static axial loads, resulting in a three-to-one internal resonance. The method of multiple scales is used to attack directly the governing nonlinear integral-partial-differential equation and associated boundary conditions and derive three sets of four first-order nonlinear ordinary-differential equations describing the modulation of the amplitudes and phases of the first two modes in the cases of (a) principal parametric resonance of either the first or the second mode, and (b) a combination parametric resonance of the additive type of these modes. Periodic motions and periodically and chaotically modulated motions of the beam are determined by investigating the equilibrium and dynamic solutions of the modulation equations. For the case of principal parametric resonance of the first mode or combination parametric resonance of the additive type, trivial and two-mode solutions are possible, whereas for the case of parametric resonance of the second mode, trivial, single, and two-mode solutions are possible. The trivial and two-mode equilibrium solutions of the modulation equations may undergo either a supercritical or a subcritical Hopf bifurcation, depending on the magnitude of the axial load. For some excitation parameters, we found complex responses including period-doubling bifurcations and blue-sky catastrophes.  相似文献   

16.
J. C. Ji 《Nonlinear dynamics》2014,78(3):2161-2184
Stable bifurcating solutions may appear in an autonomous time-delayed nonlinear oscillator having quadratic nonlinearity after the trivial equilibrium loses its stability via two-to-one resonant Hopf bifurcations. For the corresponding non-autonomous time-delayed nonlinear oscillator, the dynamic interactions between the periodic excitation and the stable bifurcating solutions can induce resonant behaviour in the forced response when the forcing frequency and the frequencies of Hopf bifurcations satisfy certain relationships. Under hard excitations, the forced response of the time-delayed nonlinear oscillator can exhibit three types of secondary resonances, which are super-harmonic resonance at half the lower Hopf bifurcation frequency, sub-harmonic resonance at two times the higher Hopf bifurcation frequency and additive resonance at the sum of two Hopf bifurcation frequencies. With the help of centre manifold theorem and the method of multiple scales, the secondary resonance response of the time-delayed nonlinear oscillator following two-to-one resonant Hopf bifurcations is studied based on a set of four averaged equations for the amplitudes and phases of the free-oscillation terms, which are obtained from the reduced four-dimensional ordinary differential equations for the flow on the centre manifold. The first-order approximate solutions and the nonlinear algebraic equations for the amplitudes and phases of the free-oscillation terms in the steady state solutions are derived for three secondary resonances. Frequency-response curves, time trajectories, phase portraits and Poincare sections are numerically obtained to show the secondary resonance response. Analytical results are found to be in good agreement with those of direct numerical integrations.  相似文献   

17.
Nonlinear Responses of Buckled Beams to Subharmonic-Resonance Excitations   总被引:4,自引:0,他引:4  
Emam  Samir A.  Nayfeh  Ali H. 《Nonlinear dynamics》2004,35(2):105-122
We investigated theoretically and experimentally the nonlinear responseof a clamped-clamped buckled beam to a subharmonic resonance of orderone-half of its first vibration mode. We used a multi-mode Galerkindiscretization to reduce the governing nonlinear partial-differentialequation in space and time into a set of nonlinearly coupledordinary-differential equations in time only. We solved the discretizedequations using the method of multiple scales to obtain a second-orderapproximate solution, including the modulation equations governing itsamplitude and phase, the effective nonlinearity, and the effectiveforcing. To investigate the large-amplitude dynamics, we numericallyintegrated the discretized equations using a shooting method to computeperiodic orbits and used Floquet theory to investigate their stabilityand bifurcations. We obtained interesting dynamics, such as phase-lockedand quasiperiodic motions, resulting from a Hopf bifurcation,snapthrough motions, and a sequence of period-doubling bifurcationsleading to chaos. Some of these nonlinear phenomena, such as Hopfbifurcation, cannot be predicted using a single-mode Galerkindiscretization. We carried out an experiment and obtained results ingood qualitative agreement with the theoretical results.  相似文献   

18.
The nonlinear harmonic response of an autoparametric system comprised of a linear oscillator with a vertically attached flexural beam is investigated and the capability of the beam as a vibration absorber is assessed. A weak torsional spring is used for constraining the rotation of the beam giving rise to an almost non-flexural rotational mode with a low frequency. The system parameters are also tuned to enforce the zero-to-one-to-one internal resonance condition. The Lagrange’s formulation accompanied by the assumed-mode method is used to derive the discretized equations based on the order three nonlinear Euler–Bernoulli beam theory. An analytical solution is developed based on the method of multiple scales where the generalized coordinate corresponding to the non-flexural rotational mode is approximated by higher order perturbation expansion than the other coordinates, due to much larger contribution of the non-flexural rotation to the response. Comprehensive response and bifurcation analysis are performed using analytical and direct numerical solutions. The results are obtained for vertically-aligned and also initially inclined beams and various interesting behaviors are recognized for different non-dimensional system parameters. Different types of bifurcations such as the Pitch-fork, Hopf, Period-doubling and symmetry breaking bifurcations are observed in the solution of slow-flow equations and some of them are found to be beneficial for vibration absorption in a limited range of excitation amplitudes and frequencies.  相似文献   

19.
Garg  Anshul  Dwivedy  Santosha K. 《Nonlinear dynamics》2020,101(4):2107-2129

In this work, theoretical and experimental analysis of a piezoelectric energy harvester with parametric base excitation is presented under combination parametric resonance condition. The harvester consists of a cantilever beam with a piezoelectric patch and an attached mass, which is positioned in such a way that the system exhibits 1:3 internal resonance. The generalized Galerkin’s method up to two modes is used to obtain the temporal form of the nonlinear electromechanical governing equation of motion. The method of multiple scales is used to reduce the equations of motion into a set of first-order differential equations. The fixed-point response and the stability of the system under combination parametric resonance are studied. The multi-branched non-trivial response exhibits bifurcations such as turning point and Hopf bifurcations. Experiments are performed under various resonance conditions. This study on the parametric excitation along with combination and internal resonances will help to harvest energy for a wider frequency range from ambient vibrations.

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20.
Three nonlinear integro-differential equations of motion derived in Part I are used to investigate the forced nonlinear vibration of a symmetrically laminated graphite-epoxy composite beam. The analysis focuses on the case of primary resonance of the first in-plane flexural (chordwise) mode when its frequency is approximately twice the frequency of the first out-of-plane flexural-torsional (flapwise-torsional) mode. A combination of the fundamental-matrix method and the method of multiple scales is used to derive four first-order ordinary-differential equations describing the modulation of the amplitudes and phases of the interacting modes with damping, nonlinearity, and resonances. The eigenvalues of the Jacobian matrix of the modulation equations are used to determine the stability of their constant solutions, and Floquet theory is used to determine the stability and bifurcations of their limit-cycle solutions. Hopf bifurcations, symmetry-breaking bifurcations, period-multiplying sequences, and chaotic motions of the modulation equations are studied. The results show that the motion can be nonplanar although the input force is planar. Nonplanar responses may be periodic, periodically modulated, or chaotically modulated motions.  相似文献   

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