双介质耦合刚性基弹性层平面应变型导波模式及界面散射能量分配

过渡段动力稳定性问题已成为制约400 km/h及以上高铁路基设计的关键难题,亟需从波动和能量的角度探究由基础非均匀引发的线路系统动力响应放大机理.文章将轨下基础简化为上表面自由、底端固定的刚性基弹性层,将高铁过渡段车致弹性波传播问题提炼为非均匀介质刚性基弹性层中波的散射问题,建立双介质耦合刚性基弹性层平面应变模型,优化该类波导结构频散方程在复平面求根方法,并结合岩土类介质特征展开刚性基弹性层频散分析,以明确其多模式导波特性及散射能量分配,最后,围绕弹性层厚度、刚度比等影响因素开展对比分析.结果表明:刚性基弹性层各模式导波均具有截止频率,弹性层厚度越小,杨氏模量越大,各阶导波模式的截止频率越高;入射波在双介质刚性基弹性层发生散射后,透射场基阶模式导波会占据主体能量,随着高阶导波模式被逐一激发,反射场及透射场高阶模式能量占比会在全频率范围呈现“此消彼长”状态;交换两侧弹性层材料,改变弹性层厚度及两弹性层刚度比不会显著改变能量分布规律,但总体来看,能量更易集中在较软侧弹性层中,各模式导波在激发初始频段会更为活跃,可分配到更多能量.     

Random Vibrations with Impacts: A Review

Analytical methods and specific results are presented for random vibrations of systems with lumped parameters and “classical” impacts whereby finite relations between impact/rebound velocities are imposed at the impact instants that are not known in advance but rather governed by the equations of motion. Emphasis is placed on the procedures using special piecewise-linear transformation of state variables that exclude the velocity jumps at impacts or makes them small if impact losses are present. In the former case, exact analyses for stationary probability densities of the response to white-noise excitation are possible, whereas the stochastic averaging method is applied in the latter case. Furthermore, the special case of an isochronous system permits a more profound response analysis, such as predicting the spectral density of the response or subharmonic response to narrow-band excitation. The method of direct energy balance is also illustrated, based on direct application of the stochastic differential equation calculus between impacts. Certain two-degree-of-freedom impacting systems are considered, with application to moored systems, as used in ocean engineering.     

Rotary motion of the parametric and planar pendulum under stochastic wave excitation

In this paper a rotary motion of a pendulum subjected to a parametric and planar excitation of its pivot mimicking random nature of sea waves has been studied. The vertical motion of the sea surface has been modelled and simulated as a stochastic process, based on the Shinozuka approach and using the spectral representation of the sea state proposed by Pierson–Moskowitz model. It has been investigated how the number of wave frequency components used in the simulation can be reduced without the loss of accuracy and how the model relates to the real data. The generated stochastic wave has been used as an excitation to the pendulum system in numerical and experimental studies. For the first time, the rotary response of a pendulum under stochastic wave excitation has been studied. The rotational number has been used for statistical analysis of the results in the numerical and experimental studies. It has been demonstrated how the forcing arrangement affects the probability of rotation of the parametric pendulum.     

Seismic behaviour of isolated multi-span continuous bridge to nonstationary random seismic excitation

This paper concentrates on the results of responses of a multi-span continuous bridge isolated with double concave friction pendulum bearings subjected to non-stationary random seismic excitation characterized by the incoherence, the wave-passage, and the site-response effects. The earthquake excitation is modelled as a non-stationary random process as uniformly modulated broad-band excitation. To perform the seismic isolation procedure, the double concave friction pendulum bearings which are sliding devices that utilize two spherical concave surfaces are placed at each of the six support points of the deck. The non-stationary response of the isolated bridge is compared with the corresponding stationary response in order to study the effects of non-stationary characteristics of the earthquake input motion. Solutions obtained from the stationary and non-stationary stochastic analyses for the isolated bridge to spatially varying earthquake ground motions are compared for the special cases of the earthquake ground motion model. The spatially varying earthquake ground motions are described stochastically based on an empirical coherency loss function and a filtered power spectral density function. The site effect is considered by a transfer function derived from one dimensional wave propagation theory. It is observed that the stationary assumption is reasonable for the considered ground motion duration.     

Parametric random excitation of nonlinear coupled oscillators

The stochastic bifurcation and response statistics of nonlinear modal interaction under parametric random excitation are studied analytically, numerically and experimentally. Two basic definitions of stochastic bifurcation are first introduced. These are bifurcation in distribution and bifurcation in moments. bifurcation in moments is examined for the case of a coupled oscillator subjected to parametric filtered white noise. The center frequency of the excitation is selected to be close to either twice the first mode or second mode natural frequencies or the sum of the two. The stochastic bifurcation in moments is predicted using the Fokker-Planck equation together with gaussian and non-Gaussian closures and numerically using the Monte Carlo simulation. When one mode is parametrically excited it transfers energy to the other mode due to nonlinear modal interaction. The Gaussian closure solution gives close results to those predicted numerically only in regions well remote from bifurcation points. However, bifurcation points predicted by the non-Gaussian closure are in good agreement with those estimated by numerical simulation. Depending on the excitation level, the probability density of the excited mode is strongly non-Gaussian and exhibits multi-maxima as predicted by Monte Carlo simulation. Experimental tests are carried out at relatively low excitation levels. In the neighborhood of stochastic bifurcation in mean square the measured results exhibit different regimes of response characteristics including zero motion and occasional small random motion regimes. These two regimes are characterized by the phenomenon of on-off intermittency. Both regimes overlap and thus it is difficult to locate experimentally the bifurcation point.     

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.

     

Random Oscillations of an Antiphase-Excited Aeroelastic System with Synchronization

We examine synchronization of the oscillatory motion of thin elastic cylindrical plates forming the walls of a channel filled by a gas. The gas motion in the channel is described by a system of Navier–Stokes equations solved by the MacCormack method of second-order accuracy. The motion of the channel walls is described by a system of dynamic, geometrically nonlinear equations of the thin-shell theory; this system is solved by the finite difference method. Kinematic and dynamic contact conditions are set at the interface between the media. By means of a numerical experiment, possible scenarios of the transition of the aeroelastic system to in-phase oscillations were identified, and the regime of random oscillations in the system with synchronization under antiphase external excitation was found.     

Nonlinear Random Response of Ocean Structures Using First- and Second-Order Stochastic Averaging

This paper deals with the dynamic response of nonlinear elastic structure subjected to random hydrodynamic forces and parametric excitation using a first- and second-order stochastic averaging method. The governing equation of motion is derived by using Hamilton's principle, taking into account inertia and curvature nonlinearities and work done due to hydrodynamic forces. Within the framework of first-order stochastic averaging, the system response statistics and stability boundaries are obtained. Unfortunately, the effects of nonlinear inertia and curvature are not reflected in the final results, since the contribution of these nonlinearities is lost during the averaging process. In the absence of hydrodynamic forces, the method fails to give bounded response statistics, and the analysis yields stability conditions. It is the second-order stochastic averaging which can capture the influence of stiffness and inertia nonlinearities that were lost in the first-order averaging process. The results of the second-order averaging are compared with those predicted by Gaussian and non-Gaussian closures and by Monte Carlo simulation. In the absence of parametric excitation, the non-Gaussian closure solutions are in good agreement with Monte Carlo simulation. On the other hand, in the absence of hydrodynamic forces, second-order averaging gives more reliable results in the neighborhood of stochastic bifurcation. However, under pure parametric random excitation, the stochastic averaging and Monte Carlo simulation predict the on-off intermittency phenomenon near bifurcation point, in addition to stochastic bifurcation in probability.     

Dynamic response of a deepwater riser subjected to combined axial and transverse excitation by the nonlinear coupled model

In offshore engineering long slender risers are simultaneously subjected to both axial and transverse excitations. The axial load is the fluctuating top tension which is induced by the floater’s heave motion, while the transverse excitation comes from environmental loads such as waves. As the time-varying axial load may trigger classical parametric resonance, dynamic analysis of a deepwater riser with combined axial and transverse excitations becomes more complex. In this study, to fully capture the coupling effect between the planar axial and transverse vibrations, the nonlinear coupled equations of a riser’s dynamic motion are formulated and then solved by the central difference method in the time domain. For comparison, numerical simulations are carried out for both linear and nonlinear models. The results show that the transverse displacements predicted by both models are similar to each other when only the random transverse excitation is applied. However, when the combined axial dynamic tension and transverse wave forces are both considered, the linear model underestimates the response because it ignores the coupling effect. Thus the coupled model is more appropriate for deep water. It is also found that the axial excitation can significantly increase the riser’s transverse response and hence the bending stress, especially for cases when the time-varying tension is located at the classical parametric resonance region. Such time-varying effects should be taken into account in fatigue safety assessment.     

Stochastic jump and bifurcation of a slender cantilever beam carrying a lumped mass under narrow-band principal parametric excitation

A detailed theoretical investigation into the single-mode approximate response of a slender cantilever beam carrying a lumped mass subjected to base narrow-band random excitation is presented for the first time. The method of multiple scales is used and the stochastic jump and bifurcation have been investigated for the principal parametric resonance of the system using the stationary joint probability. Results show that stochastic jump occurs mainly in the region of triple-valued solution. For the frequency-response domain, if the excitation central frequency is a variable and others keep constant, the basic phenomena imply that the higher the frequency, the more probable the jump from the stationary non-trivial branch to the stationary trivial one once the frequency exceeds a certain value. If the bandwidth is a variable and others keep constant, the basic phenomena indicate that the most probable motion is around the non-trivial branch when the bandwidth is smaller, whereas the most probable motion gradually approaches the trivial one when the bandwidth becomes higher. For the force-response domain, there is a region of excitation acceleration within which the joint probability density has two peaks: an outer flabellate peak and a central volcano peak. Results show that the outer flabellate peak decreases while the central volcano peak increases as the value of the excitation acceleration decreases.     

Footing Settlement on a Consolidating Soil Layer with Stochastic Properties

Geotechnical engineering applications are characterized by various sources of uncertainties, most of them attributed to the stochastic nature of soil parameters and their properties. In particular, soil’s inherent random heterogeneity, inexact measurements and insufficient data necessitate numerical methods that incorporate the stochastic soil properties for a realistic representation of the soil behavior. In this paper, the process of consolidation of saturated soils is examined on the basis of the coupled u–p finite element formulation. A generalized Newmark implicit time integration scheme is implemented to treat the time integration of the coupled consolidation equations. A benchmark geotechnical engineering problem of a strip footing resting on a saturated soil layer is analyzed. The soil permeability coefficient k, as well as the elastic modulus E, are treated as lognormal random fields in two dimensions. The investigation of the effect of the spatial variability of the soil properties on the response of a footing–soil system is examined by means of the direct Monte Carlo simulation. The influence of the coefficient of variation and correlation length of the stochastic fields is quantified in terms of footing settlements, as well as excess soil water pore pressure. The effects of spatial variability of the permeability coefficient k and the elastic modulus E on the system response are demonstrated. It is shown that the footing differential settlement, along with generated excess pore pressures, is highly affected by the variation of the soil properties considered, as well as the correlation length of the underlying random fields.     

On the stability of nonlinear vibrations of coupled pendulums

The motion of two identical pendulums connected by a linear elastic spring is studied. The pendulums move in a fixed vertical plane in a homogeneous gravity field. The nonlinear problem of orbital stability of such a periodic motion of the pendulums is considered under the assumption that they vibrate in the same direction with the same amplitude. (This is one of the two possible types of nonlinear normal vibrations.) An analytic investigation is performed in the cases of small vibration amplitude or small rigidity of the spring. In a special case where the spring rigidity and the vibration amplitude are arbitrary, the study is carried out numerically. Arbitrary linear and nonlinear vibrations in the case of small rigidity (the case of sympathetic pendulums) were studied earlier [1, 2].     

Stochastic responses of viscoelastic system with real-power stiffness under randomly disordered periodic excitations

The stochastic dynamic responses of viscoelastic systems with real-power exponents of stiffness term subjects to randomly disordered periodic excitations are studied. The assumed viscoelastic damping depends on the past history of motion via convolution integrals over an exponentially decaying kernel function. The multiple scales method is used to derive the stochastic different equations of modulation of amplitude and phase. The changes of the shape of resonance curves are obtained with real-power exponents of stiffness term and viscoelastic parameters, and then, the numerical simulation method was used to verify the accuracy of the theoretical analysis results. Theoretical analysis and numerical simulations show that as the intensity of the random excitation increases, the steady-state solution changes from a limit cycle to a diffused limited cycle. Under some conditions, the system may have two steady-state solutions and the phenomenon of jumps will happen to them under the random excitations.     

Transient response of a moving spherical shell in an acoustic medium

A ring-stiffened spherical shell is submerged in an acoustic medium. The shell is thin and elastic. The acoustic medium is inviscid, irrotational and compressible. The center of mass of the shell is subjected to a translational acceleration which is an arbitrary function of time. The absolute displacements of the shell are expressed in terms of the relative displacements and the displacement of the base of the shell, base being defined as the rigid ring placed at the equator. The motion of the acoustic medium is governed by the wave equation. The transient response of the shell is investigated numerically. The results are compared with the results of the in-vacuo response. The effects of the plane wave approximation and the base velocity on the transient response of the shell are studied. The numerical results show that the plane wave approximation accurately predicts the response of the shell in the acoustic medium for short times after excitation. The displacements of the shell in fluid are larger than those in vacuo. But when the base of the shell is restrained from translating, the displacements in fluid are smaller than those in vacuo. Therefore, base translation has a very significant effect on the transient response of the shells submerged in an acoustic medium.     

Nonlinear dynamics and performance analysis of modified snap-through vibration energy harvester with time-varying potential function

Vibration energy harvesting has emerged as a promising method to harvest energy for small-scale applications. Enhancing the performance of a vibration energy harvester(VEH) incorporating nonlinear techniques, for example, the snap-through VEH with geometric non-linearity, has gained attention in recent years. A conventional snap-through VEH is a bi-stable system with a time-invariant potential function, which was investigated extensively in the past. In this work, a modified snap-through VEH wit...     

Nonlinear vibration of a rotating laminated composite circular cylindrical shell: traveling wave vibration

In this paper, the large-amplitude (geometrically nonlinear) vibrations of rotating, laminated composite circular cylindrical shells subjected to radial harmonic excitation in the neighborhood of the lowest resonances are investigated. Nonlinearities due to large-amplitude shell motion are considered using the Donnell’s nonlinear shallow-shell theory, with account taken of the effect of viscous structure damping. The dynamic Young’s modulus which varies with vibrational frequency of the laminated composite shell is considered. An improved nonlinear model, which needs not to introduce the Airy stress function, is employed to study the nonlinear forced vibrations of the present shells. The system is discretized by Galerkin’s method while a model involving two degrees of freedom, allowing for the traveling wave response of the shell, is adopted. The method of harmonic balance is applied to study the forced vibration responses of the two-degrees-of-freedom system. The stability of analytical steady-state solutions is analyzed. Results obtained with analytical method are compared with numerical simulation. The agreement between them bespeaks the validity of the method developed in this paper. The effects of rotating speed and some other parameters on the nonlinear dynamic response of the system are also investigated.     

Stochastic Response of Offshore Structures via Statistical Cubicization

This paper deals with the stochastic response of single-degree-of-freedom structures with polynomial restoring force excited by random Morisons forces with mean current. The problem is recast by expressing the excitation by means of a cubic polynomial of the wave elevation, which in turn is assumed as a stationary zero-mean Gaussian process, whose spectral density is given by the output of a cascade of two second order linear filters having a Gaussian white noise as primary excitation. Thus, Itôs stochastic differential calculus becomes applicable, and the solution is pursued with a moment equation approach by using a suitable closure scheme. The results of the applications compare well with digital simulation.     

A new study of chaotic behavior and the existence of Feigenbaum’s constants in fractional-degree Yin–Yang Hénon maps

hing dynamics in a square tank are numerically investigated when the tank is subjected to horizontal, narrowband random ground excitation. The natural frequencies of the two predominant sloshing modes are identical and therefore 1:1 internal resonance may occur. Galerkin’s method is applied to derive the modal equations of motion for nonlinear sloshing including higher modes. The Monte Carlo simulation is used to calculate response statistics such as mean square values and probability density functions (PDFs). The two predominant modes exhibit complex phenomena including “autoparametric interaction” because they are nonlinearly coupled with each other. The mean square responses of these two modes and the liquid elevation are found to differ significantly from those of the corresponding linear model, depending on the characteristics of the random ground excitation such as bandwidth, center frequency and excitation direction. It is found that the direction of the excitation is a significant factor in predicting the mean square responses. The frequency response curves for the same system subjected to equivalent harmonic excitation are also calculated and compared with the mean square responses to further explain the phenomena. Changing the liquid level causes the peak of the mean square response to shift. Furthermore, the risk of the liquid overspill from the tank is discussed by showing the three-dimensional distribution charts of the mean square responses of liquid elevations.     

Complex response analysis of a non-smooth oscillator under harmonic and random excitations

It is well-known that practical vibro-impact systems are often influenced by random perturbations and external excitation forces, making it challenging to carry out the research of this category of complex systems with non-smooth characteristics. To address this problem, by adequately utilizing the stochastic response analysis approach and performing the stochastic response for the considered non-smooth system with the external excitation force and white noise excitation, a modified conducting process has proposed. Taking the multiple nonlinear parameters, the non-smooth parameters, and the external excitation frequency into consideration, the steady-state stochastic P-bifurcation phenomena of an elastic impact oscillator are discussed. It can be found that the system parameters can make the system stability topology change. The effectiveness of the proposed method is verified and demonstrated by the Monte Carlo(MC) simulation.Consequently, the conclusions show that the process can be applied to stochastic non-autonomous and non-smooth systems.     

Exploiting the subharmonic parametric resonances of a buckled beam for vibratory energy harvesting

The potential of harvesting vibratory energy via a bistable beam subjected to subharmonic parametric excitations is investigated. The vibrating structure is a buckled beam with two stable equilibria separated by a potential barrier. The beam is subjected to a superposition of a static axial load beyond its buckling load and a harmonic axial excitation whose frequency is around twice the frequency of the buckled beam’s first vibration mode. A macro-fiber composite patch is attached to one side of the beam to convert the strain energy resulting from the beam’s oscillation into electricity. The study considers two regimes of excitations: an amplitude sweep and a frequency sweep. In the first regime, the amplitude of excitation is quasi-statically varied while the excitation frequency is tuned at twice the natural frequency of the first vibration mode. In the second regime, the excitation frequency is swept forward and backward around the subharmonic resonant frequency while the amplitude of excitation is kept constant. A theoretical model which governs the electromechanical coupling of the transverse vibrations of the beam and the output voltage is used to monitor the response as the excitation parameters are changed. An experimental setup is also built and a series of tests is performed to validate the theoretical findings. It is shown that, depending on the amplitude and frequency of excitation, the harvester can perform small-amplitude periodic intra-well motion, intra- and inter-well chaotic motions, as well as periodic inter-well motions. Experimental results also show that, as compared to the classical linear resonance, utilizing the sub-harmonic resonance of a bistable energy harvesters can result in a broadband frequency response.