Parametric resonances in a base-excited double pendulum

Two parametrically-induced phenomena are addressed in the context of a double pendulum subject to a vertical base excitation. First, the parametric resonances that cause the stable downward vertical equilibrium to bifurcate into large-amplitude periodic solutions are investigated extensively. Then the stabilization of the unstable upward equilibrium states through the parametric action of the high-frequency base motion is documented in the experiments and in the simulations. It is shown that there is a region in the plane of the excitation frequency and amplitude where all four unstable equilibrium states can be stabilized simultaneously in the double pendulum. The parametric resonances of the two modes of the base-excited double pendulum are studied both theoretically and experimentally. The transition curves (i.e., boundaries of the dynamic instability regions) are constructed asymptotically via the method of multiple scales including higher-order effects. The bifurcations characterizing the transitions from the trivial equilibrium to the periodic solutions are computed by either continuation methods and or by time integration and compared with the theoretical and experimental results.     

Road to chaos for a soft spring system under weak periodic disturbance

The Mel’nikov-Holmes method is used to discuss the road to chaos for an elastic system with soft spring constant and under weak periodic disturbance. It is found that according to how the disturbance is applied, forced vibration or parametric excitation, the response may pass to chaos through different sequences of subharmonic bifurcations, and that the orders in the sequences may be different for different disturbance frequencies.     

随机干扰与随机参数激励联合作用下的Hopf分叉

本文研究van der Pol-Duffing型的非线性振子在随机干扰和随机参数联合作用下的Hopf分叉现象。本文所得结果证实了当系统处在于Hopf分叉点附近时,对系统的参数的变化具有敏感性。在研究过程中,我们利用Markov扩散过程逼近系统的随机响应,得到了沿稳定矩的概率1稳定和矩稳定的条件。对于非线性振子,我们得到了振幅过程的稳态概论密度函数。研究发现,确定性系统的Hopf分叉点在随机参数作用下具有漂移现象,这种漂移是由系统的性质所决定的,当分叉点为超临界的,分叉点向前漂移;而当分叉点为亚临界时,这种漂移是向后的。当系统处在外部随机干扰作用下时,系统出现非零响应。另外我们发现,稳态矩的分叉与其阶数无关。     

Nonlinear dynamic analysis of dielectric elastomer membrane with electrostriction

The dielectric elastomer (DE) is an important intelligent soft material widely used in soft actuators, and the dynamic response of the DE is highly nonlinear due to the material properties. In the DE, electrostriction denotes the deformation-dependent permittivity. In the present study, we formulate the nonlinear dynamic governing equations of the DE membrane considering the electrostriction effect. The free vibration and parametric excitation of the DE membrane with different geometric sizes are calculated. The free vibration bifurcations induced by the initial location and the voltage are both discussed according to an energy-based approach. The amplitude-frequency characteristics and bifurcation diagrams of parametric excitation are also given. The results show that electrostriction decreases the free vibration amplitude and increases the frequency, but it has less influence on the parametric excitation oscillation frequency and decreases the parametric excitation amplitude except when the membrane resonates. The initial location and the applied voltage can induce the snap-through instability of the free vibration. A large geometric size will lead to a much lower resonance frequency. The resonance amplitudes increase while the resonance frequencies decrease with the increase in the applied voltage. The critical voltage of snap-through instability for the parametric excitation is larger than that for the free vibration one.

     

Global Bifurcations and Chaotic Dynamics in Nonlinear Nonplanar Oscillations of a Parametrically Excited Cantilever Beam

This paper presents the analysis of the global bifurcations and chaotic dynamics for the nonlinear nonplanar oscillations of a cantilever beam subjected to a harmonic axial excitation and transverse excitations at the free end. The governing nonlinear equations of nonplanar motion with parametric and external excitations are obtained. The Galerkin procedure is applied to the partial differential governing equation to obtain a two-degree-of-freedom nonlinear system with parametric and forcing excitations. The resonant case considered here is 2:1 internal resonance, principal parametric resonance-1/2 subharmonic resonance for the in-plane mode and fundamental parametric resonance–primary resonance for the out-of-plane mode. The parametrically and externally excited system is transformed to the averaged equations by using the method of multiple scales. From the averaged equation obtained here, the theory of normal form is applied to find the explicit formulas of normal forms associated with a double zero and a pair of pure imaginary eigenvalues. Based on the normal form obtained above, a global perturbation method is utilized to analyze the global bifurcations and chaotic dynamics in the nonlinear nonplanar oscillations of the cantilever beam. The global bifurcation analysis indicates that there exist the heteroclinic bifurcations and the Silnikov type single-pulse homoclinic orbit in the averaged equation for the nonlinear nonplanar oscillations of the cantilever beam. These results show that the chaotic motions can occur in the nonlinear nonplanar oscillations of the cantilever beam. Numerical simulations verify the analytical predictions.     

Local and global bifurcations of valve mechanism

In this paper we study in detail problems of nonlinear oscillations of valve mechanism at internal combustion engine. The practical measurement indicates that stiffness of valve mechanism is not constant but is a function of the rotational angle of the cam. For simplicity of analysis we replace valve mechanism of internal combustion engine with a nonlinear oscillator of single degree of freedom under combined parametric and forcing excitation. We use the method of multiple scales and normal form theory to study local and global bifurcations of valve mechanism at internal combustion engine.     

Dynamic analysis of piezoelectric energy harvester under combination parametric and internal resonance: a theoretical and experimental study

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.

     

Three-to-One Internal Resonances in Parametrically Excited Hinged-Clamped Beams

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.     

参外联合激励下的簇发振荡及其机理

当周期激励频率远小于系统固有频率时,会存在快慢耦合效应,与单项激励不同,参外联合激励不仅会导致快子系统平衡曲线和分岔行为的复杂化,也会产生一些特殊的非线性现象,为此,本文以两耦合Hodgkin-Huxley细胞模型为例,引入周期参外联合激励,探讨在频域不同尺度耦合时该系统的簇发振荡的特点及其分岔机制．通过建立相应的快慢子系统,得到慢变参数变化下的快子系统的各种分岔模式以及相应的分岔行为,结合转换相图,揭示耦合系统随激励幅值变化时的动力学行为及其机理．研究表明,在激励幅值较小时,系统表现为概周期振荡,两频率分别近似于快子系统平衡曲线由Hopf分岔引起的两稳定极限环的振荡频率．概周期解随激励幅值的增加进入簇发振荡,导致这些簇发振荡的主要原因是在慢变参数变化的部分区间内,存在唯一稳定的平衡曲线,使得系统的轨迹逐渐趋向该平衡曲线,产生沉寂态,并随着慢变参数的变化,由分岔进入激发态．同时,快子系统中参与簇发振荡的稳定吸引子随激励幅值的变化也会不同,导致不同形式的簇发振荡．另外,与单项激励下的情形不同,联合激励时快子系统的部分稳定吸引子掩埋在其它稳定吸引子内,从而失去对簇发振荡的影响．     

双频激励下Filippov系统的非光滑簇发振荡机理

旨在揭示含双频周期激励的不同尺度Filippov系统的非光滑簇发振荡模式及分岔机制. 以Duffing和Van der Pol耦合振子作为动力系统模型,引入周期变化的双频激励项,当两激励频率与固有频率存在量级差时,将两周期激励项表示为可以作为一慢变参数的单一周期激励项的代数表达式,给出了当保持外部激励频率不变,改变参数激励频率的情况下,快子系统随慢变参数变化的平衡曲线及因系统出现的fold分岔或Hopf分岔导致的系统分岔行为的演化机制.结合转换相图和由Hopf分岔产生稳定极限环的演化过程,得到了由慢变参数确定的同宿分岔、多滑分岔的临界情形及因慢变参数改变而出现的混合振荡模式,并详细阐述了系统的簇发振荡机制和非光滑动力学行为特性.通过对比两种不同情形下的平衡曲线及分岔图,指出虽然系统有相似的平衡曲线结构, 却因参数激励频率取值的不同,致使平衡曲线发生了更多的曲折,对应的极值点的个数也有所改变,并通过数值模拟, 对结果进行了验证.      

Development of a steady relief at the interface of fluids in a vibrational field

The vibrations of a vessel strongly influence the behavior of the interface of the fluids in it. Thus, vertical vibrations can lead both to the parametric excitation of waves (Faraday ripples) and to the suppression of the Rayleigh-Taylor instability [1–2]. At the present time, the influence of vertical vibrations on the behavior of a fluid surface have been studied in sufficient detail (see, for example, review [3]). The behavior of an interface of fluids in the case of horizontal vibrations has been studied less. An interesting phenomenon has been revealed in the experimental papers [4, 5]: in the case of fairly strong horizontal vibrations of a vessel containing a fluid with a free surface, the fluid collects near one of the vertical vessel walls, the free surface being practically plane and stationary with respect to the vessel, while its angle of inclination to the horizon depends on the vibration rate. But if there is a system of immiscible fluids with comparable but different densities in the vessel, horizontal vibrations lead to the formation of a steady wave relief at the interface. An explanation of the behavior of a fluid with a free boundary was given in [6] on the basis of averaged equations of fluid motion in a vibrational field. The present paper is devoted to an analysis of the behavior of the interface of fluids with comparable densities in a high-frequency vibrational field. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 8–13, November–December, 1986.     

1／2亚谐共振──主参数共振情况下非线性振动系统的余维2退化分叉

本文应用Normal Form理论和退化向量场的普适开折理论研究了参数激励与强迫激励联合作用下非线性振动系统的余维2退化分叉,用Melnikov方法讨论了全局分叉的存在性.     

Theoretical and Experimental Study on the Parametrically Excited Vibration of Mass-Loaded String

The parametrically excited lateral vibration of a mass-loaded string is investigated in this paper. Supposing that the mass at the lower end of the string is subjected to a vertical harmonic excitation and neglecting the higher-order vibration modes, the equation of motion for the mass-loaded string can be represented by a Mathieu's equation with cubic nonlinearity. Based on the stability criterion for Mathieu's equation, the critical conditions inducing parametric resonance are clarified. Theoretical analysis shows that when the natural frequency f _s of the string lateral vibration and the vertical excitation frequency f satisfy f _s= (n/2)f, n= 1, 2, 3, ..., parametric resonance occurs in the case of no damping. For a damped system, parametric resonance most likely occurs when f is close to 2f _s, and depends on the damping of the system and the vertical excitation. The critical excitation has been derived at different frequencies. If the natural frequency of the mass vertical vibration happens to be twice that of the string lateral vibration, the parametric resonance may occur due to a small disturbance. Numerical simulations show that the lateral vibration of the string does not increase infinitely at parametric resonance because the parametric excitation is self-tuned due to the coupling between the vertical and lateral vibrations. Finally, the theoretical results are supported by some experimental work.     

Periodic and Chaotic Motions of a Rotor-Active Magnetic Bearing with Quadratic and Cubic Terms and Time-Varying Stiffness

In this paper, we use the asymptotic perturbation method to investigate nonlinear oscillations and chaotic dynamics in a rotor-active magnetic bearings (AMB) system with 8-pole legs and the time-varying stiffness. The stiffness in the AMB is considered as the time varying in a periodic form. Because of considering the weight of the rotor, the formulation on the electromagnetic force resultants includes the quadratic and cubic nonlinearities. The resulting dimensionless equations of motion for the rotor-AMB system with the time-varying stiffness in the horizontal and vertical directions are a two-degree-of-freedom nonlinear system with quadratic and cubic nonlinearities and parametric excitation. The asymptotic perturbation method is used to obtain the averaged equations in the case of primary parametric resonance and 1/2 subharmonic resonance. It is found that there exist period-3, period-4, period-6, period-7, period-8, quasiperiodic and chaotic modulated amplitude oscillations in the rotor-AMB system with the time-varying stiffness. It is seen from the numerical results that there are the phenomena of the multiple solutions and the soft-spring type and the hardening-spring type in nonlinear frequency-response curves for the rotor-AMB system. The parametric excitation, or the time-varying stiffness produced by the PD controller is considered to be a controlling force which can control the chaotic response in the rotor-AMB system to a period n motion.     

Slepian Approach to a Gaussian Excited Elasto-Plastic Frame Including Geometric Nonlinearity

Taking geometric non-linearity into account, an oscillator inthe form of a portal frame with a rigid traverse and withideal-elastic-ideal-plastic clamped-in columns behaves under horizontalexcitation as an ideal-elastic-hardening/softening-plastic oscillatorgiven that the columns carry a tension/compression axial force. Assumingthat the horizontal excitation of the traverse is Gaussian white noise,statistics related to the plastic displacement response are determinedby use of simulation based on the Slepian model process method combinedwith envelope excursion properties. Besides giving a physical insight,the method gives good approximations to results obtained by slow directsimulation of the total response. Moreover, the influence of a randomlyvarying axial column force is investigated by direct responsesimulation. This case corresponds to parametric excitation as generatedby the vertical acceleration component in an earthquake.     

Heteroclinic orbit and subharmonic bifurcations and chaos of nonlinear oscillator

Dynamical behavior of nonlinear oscillator under combined parametric and forcing excitation, which includes van der Pol damping, is very complex. In this paper, Melnikov'smethod is used to study the heteroclinic orbit bifurcations, subharmonic bifurcations and chaos in this system. Smale horseshoes and chaotic motions can occur from odd subharmonic bifurcation of infinite order in this system for various resonant cases.Finally the numerical computing method is used to study chaotic motions of this system. The results achieved reveal some new phenomena.     

Bifurcation Analyses in the Parametrically Excited Vibrations of Cylindrical Panels

We consider parametrically excited vibrations of shallow cylindrical panels. The governing system of two coupled nonlinear partial differential equations is discretized by using the Bubnov–Galerkin method. The computations are simplified significantly by the application of computer algebra, and as a result low dimensional models of shell vibrations are readily obtained. After applying numerical continuation techniques and ideas from dynamical systems theory, complete bifurcation diagrams are constructed. Our principal aim is to investigate the interaction between different modes of shell vibrations under parametric excitation. Results for system models with four of the lowest modes are reported. We essentially investigate periodic solutions, their stability and bifurcations within the range of excitation frequency that corresponds to the parametric resonances at the lowest mode of vibration.     

Influence of Horizontal Excitations on Dynamic Stability of a Slender Beam Under Vertical Excitation

Experimental studies have been conducted to clarify the influence of horizontal harmonic excitations on the dynamic stability of a slender cantilever beam under vertical harmonic excitation. Three kinds of aluminum test beams with rectangular cross section have been used. The test beam being clamped at one end and free at the other end, was vertically stood, and was harmonically excited to both vertical and horizontal directions simultaneously. The direction of the horizontal excitation was taken parallel to one of the beam side faces, i.e. two directions were considered as X and Y directions which have the largest and smallest flexural rigidity, respectively. By varying the horizontal excitation amplitude, keeping the amplitude of excitation in the vertical direction, the influence of the horizontal excitation has been investigated on the principal instability regions in which unstable vibration of the fundamental vibration mode occurs. The excitation frequency in the vertical excitation was taken around twice the fundamental natural frequency 2f _{Y 1} in smallest rigidity direction, while that in the horizontal direction was taken around both the fundamental natural frequency f _{Y 1} and twice of it 2f _{Y 1}. Obtained experimental results present useful fundamental data for aseismatic design of structures under earthquake containing both vertical and horizontal excitation components.     

Internal resonance of an axially transporting beam with a two-frequency parametric excitation

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 Routh-Hurwitz 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.     

Growth predictions of two interacting cracks

The elastic fracture behavior of a plate subjected to uniform stress surrounding two equal cracks inclined at an angle is investigated. The orientation of the crack plane with applied stress can be varied. Among the cases are: (1) two cracks inclined symmetrically with respect to the vertical and horizontal applied stress, and (2) one crack is horizontal while the other is inclined to the vertical applied stress. The strain energy density criterion is used for determining the combined crack and load arrangement that correspond to the lowest critical load at global instability. The direction of crack initiation is also determined. Quantitative results pertaining to the fracture characteristics are given in graphical forms.