Steady-state responses of axially accelerating viscoelastic beams: Approximate analysis and numerical confirmation

Nonlinear parametric vibration of axially accelerating viscoelastic beams is investigated via an approximate analytical method with numerical confirmations. Based on nonlinear models of a finite-small-stretching slender beam moving at a speed with a periodic fluctuation, a solvability condition is established via the method of multiple scales for subharmonic resonance. Therefore, the amplitudes of steady-state periodic responses and their existence conditions are derived. The amplitudes of stable steady-state responses increase with the amplitude of the axial speed fluctuation, and decrease with the viscosity coefficient and the nonlinear coefficient. The minimum of the detuning parameter which causes the existence of a stable steady-state periodic response decreases with the amplitude of the axial speed fluctuation, and increases with the viscosity coefficient. Numerical solutions are sought via the finite difference scheme for a nonlinear partial-differential equation and a nonlinear integro-partial-differential equation. The calculation results qualitatively confirm the effects of the related parameters predicted by the approximate analysis on the amplitude and the existence condition of the stable steady-state periodic responses. Quantitative comparisons demonstrate that the approximate analysis results have rather high precision. Supported by the National Outstanding Young Scientists Foundation of China (Grant No. 10725209), the National Natural Science Foundation of China (Grant No. 10672092), Scientific Research Project of Shanghai Municipal Education Commission (Grant No. 07ZZ07), and Shanghai Leading Academic Discipline Project (Grant No. Y0103)     

Duffing系统的主-超谐联合共振

非线性动力系统极易发生共振,在多频激励下可能发生联合共振或组合共振,目前关于非线性系统的主-超谐联合共振的研究少见报道.本文以Duffing系统为对象,研究系统在主-超谐联合共振时的周期运动和通往混沌的道路.应用多尺度法得到系统的近似解析解,并利用数值方法对解析解进行验证,结果吻合良好.基于Lyapunov第一方法得到稳态周期解的稳定性条件,并分析了非线性刚度对稳态周期解的幅值和稳定性的影响.此外,由于近似解只能描述周期运动,不足以描述系统的全局特性,因而应用Melnikov方法对系统进行全局分析,得到系统进入Smale马蹄意义下混沌的条件,依据该条件以及主-超谐联合共振的条件选取一组参数进行数值仿真.分岔图和最大Lyapunov指数显示出两个临界值:当激励幅值通过第一个临界值时,异宿轨道破裂,混沌吸引子突然出现,系统以激变方式进入混沌;激励幅值通过第二个临界值时,系统在混沌态下再次发生激变,进入另一种混沌态.利用Melnikov方法考察了第一个临界值在多种频率组合下的变化趋势,并用数值仿真验证了解析结果的正确性.     

Non-linear structural vibrations under combined parametric and external excitations

A set of second order equations with weak quadratic and cubic non-linearities is considered. Simultaneous parametric and external (forcing) excitations are included. The frequency of the parametric excitation is near a natural frequency of the system, and three cases are analyzed: (i) the external excitation is absent; (ii) the external excitation is present but is not involved in a resonance; and (iii) the external frequency is the same as the parametric frequency. Results are obtained by the method of multiple scales. Frequency-response curves are presented for various combinations of excitation amplitudes, damping coefficients, and phase shift between the excitations. It is found that stable multi-modal responses may exist in the first-order asymptotic solution, even though only one mode is involved in the resonance and no internal resonance condition is present.     

Response of self-excited three-degree-of-freedom systems to multifrequency excitations

The response to multifrequency excitation of a three-degree-of-freedom self-excited system is analyzed by using multiple scales. Five cases of resonance are considered: Harmonic, subharmonic, superharmonic, simultaneous sub/superharmonic, and combination resonances. The steady-state amplitudes for each case are plotted, showing the influences of the several parameters. Approximate solutions are found and stability analyses are carried out for each case.     

Galerkin method for steady-state response of nonlinear forced vibration of axially moving beams at supercritical speeds

The present paper investigates the steady-state periodic response of an axially moving viscoelastic beam in the supercritical speed range. The straight equilibrium configuration bifurcates in multiple equilibrium positions in the supercritical regime. It is assumed that the excitation of the forced vibration is spatially uniform and temporally harmonic. Under the quasi-static stretch assumption, a nonlinear integro-partial-differential equation governs the transverse motion of the axially moving beam. The equation is cast in the standard form of continuous gyroscopic systems via introducing a coordinate transform for non-trivial equilibrium configuration. For a beam constituted by the Kelvin model, the primary resonance is analyzed via the Galerkin method under the simply supported boundary conditions. Based on the Galerkin truncation, the finite difference schemes are developed to verify the results via the method of multiple scales. Numerical simulations demonstrate that the steady-state periodic responses exist in the transverse vibration and a resonance with a softening-type behavior occurs if the external load frequency approaches the linear natural frequency in the supercritical regime. The effects of the viscoelastic damping, external excitation amplitude, and nonlinearity on the steady-state response amplitude for the first mode are illustrated.     

Principal resonance response of a stochastic elastic impact oscillator under nonlinear delayed state feedback

In this paper, the principal resonance response of a stochastically driven elastic impact(EI) system with time-delayed cubic velocity feedback is investigated. Firstly, based on the method of multiple scales, the steady-state response and its dynamic stability are analyzed in deterministic and stochastic cases, respectively. It is shown that for the case of the multivalued response with the frequency island phenomenon, only the smallest amplitude of the steady-state response is stable under a certain time delay, which is different from the case of the traditional frequency response. Then, a design criterion is proposed to suppress the jump phenomenon, which is induced by the saddle-node bifurcation. The effects of the feedback parameters on the steady-state responses, as well as the size, shape, and location of stability regions are studied. Results show that the system responses and the stability boundaries are highly dependent on these parameters. Furthermore, with the purpose of suppressing the amplitude peak and governing the resonance stability, appropriate feedback gain and time delay are derived.     

Experimental evidence of simultaneous multi-resonance noise reduction using an absorber with essential nonlinearity under two excitation frequencies

The addition of an essentially nonlinear membrane absorber to a linear vibroacoustic system with multiple resonances is studied experimentally, using quasiperiodic excitation. An extended experimental dataset of the system response is analyzed under steady-state excitation at two frequencies. Thresholds between low and high damping states within the system and associated noise reduction are observed and quantified thanks to frequency conversion and RMS efficiency indicators. Following previous numerical results, it is shown that the membrane NES (Nonlinear Energy Sink) acts simultaneously and efficiently on two acoustic resonances. In all cases, the introduction of energy at a second excitation frequency appears favorable to lower the frequency conversion threshold and to lower the noise within the system. In particular, a simultaneous control of two one-to-one resonances by the NES is observed. Exploration of energy conversion in the two excitation amplitudes plane advocates for a linear dependence of the frequency conversion thresholds on the two excitation amplitudes.     

Combinational parametric resonance under imperfectly periodic excitation

Conditions for mean square stability of a two-degree-of-freedom system have been obtained for the case of sum combinational resonance due to sinusoidal parametric excitation with constant amplitude and white-noise random temporal variations in the instantaneous frequency. Analysis is based on the asymptotic Krylov–Bogoliubov averaging method. The resulting set of ten deterministic differential equations for second-order moments of the four new state variables (inphase and quadrature modal responses) was solved analytically for its neutral stability boundary. The imperfections in periodicity may lead to either stabilization or destabilization of the system and the corresponding conditions have been clearly established from the solution obtained. Furthermore, conditions for almost sure stability have been obtained for a special symmetric case of identical modal damping factors and identical modal excitation amplitudes.     

Nonlinear dynamics of oscillators with bilinear hysteresis and sinusoidal excitation

The transient and steady-state response of an oscillator with hysteretic restoring force and sinusoidal excitation are investigated. Hysteresis is modeled by using the bilinear model of Caughey with a hybrid system formulation. A novel method for obtaining the exact transient and steady-state response of the system is discussed. Stability and bifurcations of periodic orbits are studied using Poincaré maps. Results are compared with asymptotic expansions obtained by Caughey. The bilinear hysteretic element is found to act like a ‘soft spring’. Several sub-harmonic resonances are found in the system, however, no chaotic behavior is observed. Away from the sub-harmonic resonance the asymptotic expansions and the exact steady-state response of the system are seen to match with good accuracy.     

Chaotic rocking behavior of freestanding objects with sliding motion

This paper examines the influence of effects of sliding on the non-linear rocking response behavior of freestanding rigid objects (blocks) subjected to harmonic horizontal and vertical excitations. It is well known that the rocking responses depend strongly on the impact effect between object and the base, which takes place with abrupt reduction in kinetic energy. In this study, it is shown that the rocking behavior is significantly affected by the presence of the sliding motion. A parametric response analysis is carried out over a range of excitation amplitudes and frequencies. Chaotic responses are observed over a wide response region, particularly for the case of large vertical amplitude excitation with significant sliding motions. The chaotic characteristics are demonstrated using time histories, Poincaré sections, power spectral density and Lyapunov exponents of the rocking responses. The complex chaotic response behavior is illustrated by Poincaré section in the phase space. The distribution of various types of rocking responses and the effects of sliding motion are examined via bifurcation diagrams and examples of typical rocking responses.     

Nonlinear vibration of a curved beam under uniform base harmonic excitation with quadratic and cubic nonlinearities

This paper presents nonlinear vibration analysis of a curved beam subject to uniform base harmonic excitation with both quadratic and cubic nonlinearities. The Galerkin method is employed to discretize the governing equations. A high-dimensional model that can take nonlinear model coupling into account is derived, and the incremental harmonic balance (IHB) method is employed to obtain the steady-state response of the curved beam. The cases investigated include softening stiffness, hardening stiffness and modal energy transfer. The stability of the periodic solutions for given parameters is determined by the multi-variable Floquet theory using Hsu's method. Particular attention is paid to the anti-symmetric response with and without excitation, as the excitation frequency is close to the first and third natural frequencies of the system. The results obtained with the IHB method compare very well with those obtained via numerical integration.     

Stability analysis of ultrasound thick-shell contrast agents

The stability of thick shell encapsulated bubbles is studied analytically. 3-D small perturbations are introduced to the spherical oscillations of a contrast agent bubble in response to a sinusoidal acoustic field with different amplitudes of excitation. The equations of the perturbation amplitudes are derived using asymptotic expansions and linear stability analysis is then applied to the resulting differential equations. The stability of the encapsulated microbubbles to nonspherical small perturbations is examined by solving an eigenvalue problem. The approach then identifies the fastest growing perturbations which could lead to the breakup of the encapsulated microbubble or contrast agent.     

Coupling-induced excitation of a forbidden surface plasmon mode of a gold nanorod

Using the finite-difference time-domain (FDTD) method, we simulate the coupling between a gold nanorod and gold nanoparticles with different plasmonic resonant frequencies/volumes as well as that between the nanorod and a dielectric nanosphere. The influences of coupling with different nanoparticles on the excitation of a forbidden longitudinal surface plasmon mode of the nanorod under normal incidence are investigated. It is found that the cause of this excitation is the broken symmetry of the local electric field experienced by the nanorod resulting from the charge pileup on the other nanoparticle. This result is valuable for understanding the near-field optical characterization of plasmonic metal nanoparticles. Supported by the National Natural Science Foundation of China (Grant Nos. 10821062 and 10804004), the National Basic Research Program of China (Grant No. 2007CB307001), and the Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 200800011023) Contributed by GONG QiHuang     

Experimental analysis of the linear and nonlinear behaviour of composites with delaminations

An analysis of the linear and nonlinear vibration responses of composites with delaminations is presented. The effect of delamination size on the linear and nonlinear vibration response is studied. The composite material used in this paper is a glass fibre reinforced plastic (GFRP) having delaminations at the plies interfaces. The experimental procedure consists in inducing the specimen on its resonance flexural modes with different excitation levels (amplitudes) for six bending modes and for each delamination length. The presence of the nonlinearity introduced by the delamination was clearly identified by the variation of natural frequencies for increasing excitation levels. Then, nonlinear elastic parameters for progressive delamination length were determined and discussed for the first six bending modes. The linear and the nonlinear elastic parameters were compared in their sensitive modes.     

Dynamic responses of axially moving viscoelastic beam under a randomly disordered periodic excitation

We investigate dynamic responses of axially moving viscoelastic beam subject to a randomly disordered periodic excitation. The method of multiple scales is used to derive the analytical expression of first-order uniform expansion of the solution. Based on the largest Lyapunov exponent, the almost sure stability of the trivial steady-state solution is examined. Meanwhile, we obtain the first-order and the second-order steady-state moments for the non-trivial steady-state solutions. Specially, we discuss the first mode theoretically and numerically. Results show that under the same conditions of the parameters, as the intensity of the random excitation increases, non-trivial steady-state solution fluctuation will become strenuous, which will result in the non-trivial steady-state solution lose stability and the trivial steady-state solution can be a possible. In the case of parametric principal resonance, the stochastic jump is observed for the first mode, which indicates that the stationary joint probability density concentrates at the non-trivial solution branch when the random excitation is small, but with the increase of intensity of the random excitation, the probability of the trivial steady-state solution will become larger. This phenomenon of stochastic jump can be defined as a stochastic bifurcation.     

Parametric excitation of two internally resonant oscillators

The response of two-degree-of-freedom systems with quadratic non-linearities to a principal parametric resonance in the presence of two-to-one internal resonances is investigated. The method of multiple scales is used to construct a first-order uniform expansion yielding four first-order non-linear ordinary differential (averaged) equations governing the modulation of the amplitudes and the phases of the two modes. These equations are used to determine steady state responses and their stability. When the higher mode is excited by a principal parametric resonance, the non-trivial steady state value of its amplitude is a constant that is independent of the excitation amplitude, whereas the amplitude of the lower mode, which is indirectly excited through the internal resonance, increases with the amplitude of the excitation. However, in addition to Poincaré-type bifurcations, this response exhibits a Hopf bifurcation leading to amplitude- and phase-modulated motions. When the lower mode is excited by a principal parametric resonance, the averaged equations exhibit both Poincaré and Hopf bifurcations. In some intervals of the parameters, the periodic solutions of the averaged equations, in the latter case, experience period-doubling bifurcations, leading to chaos.     

弹性微管内气泡的非线性受迫振动

将弹性管壁视为膜弹性结构, 探索在外部声场作用下弹性微管内液柱-气泡-管壁构成耦合振动系统的非线性特征. 利用逐级近似法对系统非线性共振频率、基频和三倍频振动幅值响应、 分频激励共振机理等进行了理论分析. 基频和三倍频振动的幅-频响应数值结果表明: 气泡的轴向共振和管壁共振不能同时出现; 两垂直方向的振动均表现出幅值响应多值性, 进而可能引起系统的不稳定声响应; 三倍频振动在低频区的声响应强于高频区. 关键词： 弹性微管 受迫振动 非线性振动 气泡声响应     

The high frequency, transient response of a panel with attached masses

The general, high-frequency response of a panel with attached masses is approximated using a transient form of asymptotic modal analysis (AMA). This method is derived by applying asymptotic simplifications to classical solutions in both the time and frequency domains. These relations are applied to a panel with one or more attached masses that is excited by impulsive loads. Predictions are made of the mean-squared, transverse displacement histories as well as localized responses near the added masses. It is shown that the latter compare well with experimental data when the masses are separated by more than the mean wavelength of the frequency band. The approximate solutions are shown to require relatively little computational time and memory and are applicable to general forms of excitation.     

Flow-induced oscillations of a cantilevered pipe conveying fluid with base excitation

It is known that a plain cantilevered pipe conveying fluid loses its stability by a Hopf bifurcation, leading to either planar or non-planar flutter for flow velocities beyond the critical flow velocity for Hopf bifurcation. If an external mass is attached to the end of the pipe (an end-mass), the resulting dynamics become much richer, showing 2D and 3D quasiperiodic and chaotic oscillations at high flow velocities. In this paper, a cantilevered pipe, with and without an end-mass, subjected to a small-amplitude periodic base excitation is considered. A set of three-dimensional nonlinear equations is used to analyze the pipe?s response at various flow velocities and with different amplitudes and frequencies of base excitation. The nonlinear equations are discretized using the Galerkin technique and the resulting set of equations is solved using Houbolt?s finite difference method. It is shown that for a plain pipe (with no end-mass), non-planar post-instability oscillations can be reduced to planar periodic oscillations for a range of base excitation frequencies and amplitudes. For a pipe with an end-mass, similarly to a plain pipe, three-dimensional period oscillations can be reduced to planar ones. At flow velocities beyond the critical flow velocity for torus instability, the three-dimensional quasiperiodic oscillations can be reduced to two-dimensional quasiperiodic or periodic oscillations, depending on the frequency of base excitation. In all these cases, a low-amplitude base excitation results in reducing the three-dimensional oscillations of the pipe to purely two-dimensional oscillations, over a range of excitation frequencies. These numerical results are in agreement with the previous experimental work.