Convergence of Galerkin truncation for dynamic response of finite beams on nonlinear foundations under a moving load

The present paper investigates the convergence of the Galerkin method for the dynamic response of an elastic beam resting on a nonlinear foundation with viscous damping subjected to a moving concentrated load. It also studies the effect of different boundary conditions and span length on the convergence and dynamic response. A train–track or vehicle–pavement system is modeled as a force moving along a finite length Euler–Bernoulli beam on a nonlinear foundation. Nonlinear foundation is assumed to be cubic. The Galerkin method is utilized in order to discretize the nonlinear partial differential governing equation of the forced vibration. The dynamic response of the beam is obtained via the fourth-order Runge–Kutta method. Three types of the conventional boundary conditions are investigated. The railway tracks on stiff soil foundation running the train and the asphalt pavement on soft soil foundation moving the vehicle are treated as examples. The dependence of the convergence of the Galerkin method on boundary conditions, span length and other system parameters are studied.     

Vibration of Timoshenko beam on hysteretically damped elastic foundation subjected to moving load

The vibration of beams on foundations under moving loads has many applications in several fields, such as pavements in highways or rails in railways. However, most of the current studies only consider the energy dissipation mechanism of the foundation through viscous behavior; this assumption is unrealistic for soils. The shear rigidity and radius of gyration of the beam are also usually excluded. Therefore, this study investigates the vibration of an infinite Timoshenko beam resting on a hysteretically damped elastic foundation under a moving load with constant or harmonic amplitude. The governing differential equations of motion are formulated on the basis of the Hamilton principle and Timoshenko beam theory, and are then transformed into two algebraic equations through a double Fourier transform with respect to moving space and time. Beam deflection is obtained by inverse fast Fourier transform. The solution is verified through comparison with the closed-form solution of an Euler-Bernoulli beam on a Winkler foundation. Numerical examples are used to investigate:(a) the effect of the spatial distribution of the load, and(b) the effects of the beam properties on the deflected shape, maximum displacement, critical frequency, and critical velocity. These findings can serve as references for the performance and safety assessment of railway and highway structures.     

Nonlinear response of shear deformable beams on tensionless nonlinear viscoelastic foundation under moving loads

In this paper, a boundary element method is developed for the geometrically nonlinear response of shear deformable beams of simply or multiply connected constant cross-section, traversed by moving loads, resting on tensionless nonlinear three-parameter viscoelastic foundation, undergoing moderate large deflections under general boundary conditions. The beam is subjected to the combined action of arbitrarily distributed or concentrated transverse moving loading as well as to axial loading. To account for shear deformations, the concept of shear deformation coefficients is used. Three boundary value problems are formulated with respect to the transverse displacement, to the axial displacement and to a stress functions and solved using the Analog Equation Method, a Boundary Element based method. Application of the boundary element technique yields a system of nonlinear Differential-Algebraic Equations, which is solved using an efficient time discretization scheme, from which the transverse and axial displacements are computed. The evaluation of the shear deformation coefficient is accomplished from the aforementioned stress function using only boundary integration. Analyses are performed to illustrate, wherever possible, the accuracy of the developed method, to investigate the effects of various parameters, such as the load velocity, load frequency, shear deformation, foundation nonlinearity, damping, on the beam displacements and stress resultants and to examine how the consideration of shear and axial compression affects the response of the system.     

Nonlinear dynamics of higher-dimensional system for an axially accelerating viscoelastic beam with in-plane and out-of-plane vibrations

In this paper, the bifurcations and chaotic motions of higher-dimensional nonlinear systems are investigated for the nonplanar nonlinear vibrations of an axially accelerating moving viscoelastic beam. The Kelvin viscoelastic model is chosen to describe the viscoelastic property of the beam material. Firstly, the nonlinear governing equations of nonplanar motion for an axially accelerating moving viscoelastic beam are established by using the generalized Hamilton’s principle for the first time. Then, based on the Galerkin’s discretization, the governing equations of nonplanar motion are simplified to a six-degree-of-freedom nonlinear system and a three-degree-of-freedom nonlinear system with parametric excitation, respectively. At last, numerical simulations, including the Poincare map, phase portrait and Lyapunov exponents are used to analyze the complex nonlinear dynamic behaviors of the axially accelerating moving viscoelastic beam. The bifurcation diagrams for the in-plane and out-of-plane displacements via the mean axial velocity, the amplitude of velocity fluctuation and the frequency of velocity fluctuation are respectively presented when other parameters are fixed. The Lyapunov exponents are calculated to identify the existence of the chaotic motions. From the numerical results, it is indicated that the periodic, quasi-periodic and chaotic motions occur for the nonplanar nonlinear vibrations of the axially accelerating moving viscoelastic beam. Observing the in-plane nonlinear vibrations of the axially accelerating moving viscoelastic beam from the numerical results, it is found that the nonlinear responses of the six-degree-of-freedom nonlinear system are much different from that of the three-degree-of-freedom nonlinear system when all parameters are same.     

Polaron on a one-dimensional lattice: II. A moving polaron

Continuum-limit equations for moving polarons on a one-dimensional lattice with a harmonic interaction potential between adjacent particles and a simple nonlinear potential with a cubic nonlinearity are derived for the first time; for some particular cases, their solutions are obtained. For a harmonic lattice in the continuum limit, a system of integrable nonlinear partial differential equations is derived. A one-soliton solution to this system describes a polaron moving with a constant velocity. The speed of this polaron is uniquely related to its amplitude, with its values ranging from zero to the speed of sound. For a nonlinear lattice, the resulting system of differential equations is integrable at a certain ratio of the problem parameters. The one-soliton solution to this system, as in the harmonic case, describes a polaron moving with a constant velocity. At arbitrary values of the lattice parameters, the nonlinear lattice was studied by numerical methods. It turned out that, in the entire range of parameters, the nonlinear lattice gives rise to moving polarons, with the speed of the polaron being determined by the competition between the electron-photon interaction parameter α and the nonlinearity parameter β. At α ? β, the behavior of the polaron is very close to the dynamics on the harmonic lattice. In the opposite case, the dynamic nonlinearity begins to dominate, giving rise to dynamics inherent to solitons, so that speed of the polaron can exceed the speed of sound. In a certain range of α and β, numerical calculations revealed a family of polaron-type stable solutions, the envelope of which can have several peaks. The numerical and exact analytical solutions are in very good agreement for a sufficiently large radius of the polaron, when the system of equations obtained in the continuum approximation has a solution.     

An adaptive harmonic balance method for predicting the nonlinear dynamic responses of mechanical systems—Application to bolted structures

Aeronautical structures are commonly assembled with bolted joints in which friction phenomena, in combination with slapping in the joint, provide damping on the dynamic behavior. Some models, mostly nonlinear, have consequently been developed and the harmonic balance method (HBM) is adapted to compute nonlinear response functions in the frequency domain. The basic idea is to develop the response as Fourier series and to solve equations linking Fourier coefficients. One specific HBM feature is that response accuracy improves as the number of harmonics increases, at the expense of larger computational time. Thus this paper presents an original adaptive HBM which adjusts the number of retained harmonics for a given precision and for each frequency value. The new proposed algorithm is based on the observation of the relative variation of an approximate strain energy for two consecutive numbers of harmonics. The developed criterion takes the advantage of being calculated from Fourier coefficients avoiding time integration and is also expressed in a condensation case. However, the convergence of the strain energy has to be smooth on tested harmonics and this constitutes a limitation of the method. Condensation and continuation methods are used to accelerate calculation. An application case is selected to illustrate the efficiency of the method and is composed of an asymmetrical two cantilever beam system linked by a bolted joint represented by a nonlinear LuGre model. The practice of adaptive HBM shows that, for a given value of the criterion, the number of harmonics increases on resonances indicating that nonlinear effects are predominant. For each frequency value, convergence of approximate strain energy is observed. Emergence of third and fifth harmonics is noticed near resonances both on vibratory responses and on approximate strain energy. Parametric studies are carried out by varying the excitation force amplitude and the threshold value of the adaptive algorithm. Maximal amplitudes of vibration and frequency response functions are plotted for three different points of the structure. Nonlinear effects become more predominant for higher force amplitudes and consequently the number of retained harmonics is increased.     

Micro-vibration suppression of equipment supported on a floor incorporating magneto-rheological elastomer core

In this study, magneto-rheological elastomers (MREs) are adopted to construct a smart sandwich beam for micro-vibration control of equipment. The micro-vibration response of a smart sandwich beam with MRE core which supports mass-concentrated equipment under stochastic support-motion excitations is investigated to evaluate the vibration suppression capability. The dynamic behavior of MREs as a smart viscoelastic material is characterized by a complex modulus dependent on vibration frequency and controllable by external magnetic fields. A frequency-domain solution method for the stochastic micro-vibration response of the smart sandwich beam supporting mass-concentrated equipment is developed based on the Galerkin method and random vibration theory. First, the displacements of the beam are expanded as series of spatial harmonic functions and the Galerkin method is applied to convert the partial differential equations of motion into ordinary differential equations. With these equations, the frequency-response function matrix of the beam–mass system and the expression of the velocity response spectrum are then obtained, with which the root-mean-square (rms) velocity response in terms of the one-third octave frequency band can be calculated. Finally, the optimization problem of the complex modulus of the MRE core is defined by minimizing the velocity response spectrum and the rms velocity response of the sandwich beam, through altering the applied magnetic fields. Numerical results are given to illustrate the influence of MRE parameters on the rms velocity response and the response reduction capacity of the smart sandwich beam. The proposed method is also applicable to response analysis of a sandwich beam with arbitrary core characterized by a complex shear modulus and subject to arbitrary stochastic excitations described by a power spectral density function, and is valid for a wide frequency range.     

Nonlinear response of superparamagnetic particles to a sudden change of a high constant magnetic field

For a system of superparamagnetic particles in a high external constant magnetic field, a technique for calculating the nonlinear response to a sudden change in the field direction and magnitude is proposed. A set of momentary equations for the averaged spherical harmonics, which is derived from the Fokker-Planck equation for the magnetization-orientation distribution function is the basis of this technique. As an example, the nonlinear response of a system of particles with anisotropy of the easy-axis type is examined. For this case, a solution to the momentary equations is obtained by using matrix continued fractions. The magnetization relaxation time and the spectrum of the relaxation function are calculated for typical values of anisotropy, dissipation, and nonlinearity parameters. It is shown that the magnetization kinetics is essentially dependent on these parameters.     

Nonlinear resonant behavior of microbeams over the buckled state

The present study investigates the nonlinear resonant behavior of a microbeam over its buckled (non-trivial) configuration. The system is assumed to be subjected to an axial load along with a distributed transverse harmonic load. The axial load is increased leading the system to lose the stability via a pitchfork bifurcation; the postbuckling configuration is obtained and the nonlinear resonant response of the system over the buckled state is examined. More specifically, the nonlinear equation of motion is obtained employing Hamilton’s principle along with the modified couple stress theory. The continuous system is truncated into a system with finite degrees of freedom; the Galerkin scheme is employed to discretize the nonlinear partial differential equation of motion into a set of ordinary differential equations. This set of equations is solved numerically employing the pseudo-arclength continuation technique; first a nonlinear static analysis is performed upon this set of equations so as to obtain the onset of buckling (supercritical pitchfork bifurcation) and the buckled configuration of the microbeam. The frequency-response and force-response curves of the system are then constructed over the buckled configurations. A comparison is made between the frequency-response curves obtained by means of the modified couple stress and the classical theories. The effect of different system parameters on the frequency-response and force-response curves is also examined.     

Numerical analysis of isolation of the vibration due to moving loads using pile rows

The isolation of the vibration due to moving loads using pile rows embedded in a poroelastic half-space is investigated in this study. Based on Biot's theory and integral transform method, the free field solution for a moving load applied on the surface of a poroelastic half-space and the fundamental solution for a harmonic circular patch load applied in the poroelastic half-space are derived first. Using Muki and Sternberg's method and the fundamental solution for the circular patch load as well as the obtained free field solution for the moving load, the second kind of Fredholm integral equations in the frequency domain describing the dynamic interaction between pile rows and the poroelastic half-space is developed. Numerical solution of the frequency domain integral equations and numerical inversion of the Fourier transform yield the time domain response of the pile–soil system. Comparison of our results with some known results shows that our results are in a good agreement with existing ones. Numerical results of this study show that velocity of moving loads has an important impact on the vibration isolation effect of pile rows. The same pile row has a better vibration isolation effect for the lower speed moving loads than for the higher speed ones. Also, the optimal length of piles for higher speed moving loads is shorter than that for lower speed moving loads. Moreover, stiff pile rows tend to produce a better vibration isolation effect than flexible pile rows do.     

Stability and bifurcation of an axially moving beam tuned to three-to-one internal resonances

This study analyzed the nonlinear vibration of an axially moving beam subject to periodic lateral force excitations. Attention is paid to the fundamental and subharmonic resonances, since the excitation frequency is close to the first two natural frequencies of the system. The incremental harmonic balance (IHB) method was used to evaluate the nonlinear dynamic behaviour of the axially moving beam. The stability and bifurcations of the periodic solutions for given parameters were determined by the multivariable Floquet theory using Hsu’s method. The solutions obtained from the IHB method agreed very well with those obtained from numerical integration. Furthermore, numerical examples are given to illustrate the effects of the three-to-one internal resonance on the response of the system.     

M-shaped asymmetric nonlinear oscillator for broadband vibration energy harvesting: Harmonic balance analysis and experimental validation

Over the past few years, nonlinear oscillators have been given growing attention due to their ability to enhance the performance of energy harvesting devices by increasing the frequency bandwidth. Duffing oscillators are a type of nonlinear oscillator characterized by a symmetric hardening or softening cubic restoring force. In order to realize the cubic nonlinearity in a cantilever at reasonable excitation levels, often an external magnetic field or mechanical load is imposed, since the inherent geometric nonlinearity would otherwise require impractically high excitation levels to be pronounced. As an alternative to magnetoelastic structures and other complex forms of symmetric Duffing oscillators, an M-shaped nonlinear bent beam with clamped end conditions is presented and investigated for bandwidth enhancement under base excitation. The proposed M-shaped oscillator made of spring steel is very easy to fabricate as it does not require extra discrete components to assemble, and furthermore, its asymmetric nonlinear behavior can be pronounced yielding broadband behavior under low excitation levels. For a prototype configuration, linear and nonlinear system parameters extracted from experiments are used to develop a lumped-parameter mathematical model. Quadratic damping is included in the model to account for nonlinear dissipative effects. A multi-term harmonic balance solution is obtained to study the effects of higher harmonics and a constant term. A single-term closed-form frequency response equation is also extracted and compared with the multi-term harmonic balance solution. It is observed that the single-term solution overestimates the frequency of upper saddle-node bifurcation point and underestimates the response magnitude in the large response branch. Multi-term solutions can be as accurate as time-domain solutions, with the advantage of significantly reduced computation time. Overall, substantial bandwidth enhancement with increasing base excitation is validated experimentally, analytically, and numerically. As compared to the 3 dB bandwidth of the corresponding linear system with the same linear damping ratio, the M-shaped oscillator offers 3200, 5600, and 8900 percent bandwidth enhancement at the root-mean-square base excitation levels of 0.03g, 0.05g, and 0.07g, respectively. The M-shaped configuration can easily be exploited in piezoelectric and electromagnetic energy harvesting as well as their hybrid combinations due to the existence of both large strain and kinetic energy regions. A demonstrative case study is given for electromagnetic energy harvesting, revealing the importance of higher harmonics and the need for multi-term harmonic balance analysis for predicting the electrical power output accurately.     

Multi-frequency excitation of stiffened triangular plates for large amplitude oscillations

Free and forced vibrations of triangular plate are investigated. Diverse types of stiffeners were attached onto the plate to suppress the undesirable large-amplitude oscillations. The governing equation of motion for a triangular plate, based on the von Kármán theory, is developed and the nonlinear ordinary differential equation of the system using Galerkin approach is obtained. Closed-form expressions for the free undamped and large-amplitude vibration of an orthotropic triangular elastic plate are presented using the two well-known analytical methods, namely, the energy balance method and the variational approach. The frequency responses in the closed-form are presented and their sensitivities with respect to the initial amplitudes are studied. An error analysis is performed and the vibration behavior, as well as the accuracy of the solution methods, is evaluated. Different types of the stiffened triangular plates are considered in order to cover a wide range of practical applications. Numerical simulations are carried out and the validity of the solution procedure is explored. It is demonstrated that the two methods of energy balance and variational approach have been quite straightforward and reliable techniques to solve those nonlinear differential equations. Subsequently, due to the importance of multiple resonant responses in engineering design, multi-frequency excitations are considered. It is assumed that three periodic forces are applied to the plate in three specific positions. The multiple time scaling method is utilized to obtain approximate solutions for the frequency resonance cases. Influences of different parameters, namely, the position of applied forces, geometry and the number of stiffeners on the frequency response of the triangular plates are examined.     

Internal resonance of axially moving laminated circular cylindrical shells

The nonlinear vibrations of a thin, elastic, laminated composite circular cylindrical shell, moving in axial direction and having an internal resonance, are investigated in this study. Nonlinearities due to large-amplitude shell motion are considered by using Donnell’s nonlinear shallow-shell theory, with consideration of the effect of viscous structure damping. Differently from conventional Donnell’s nonlinear shallow-shell equations, an improved nonlinear model without employing Airy stress function is developed to study the nonlinear dynamics of thin shells. The system is discretized by Galerkin’s method while a model involving four degrees of freedom, allowing for the traveling wave response of the shell, is adopted. The method of harmonic balance is applied to study the nonlinear dynamic responses of the multi-degrees-of-freedom system. When the structure is excited close to a resonant frequency, very intricate frequency–response curves are obtained, which show strong modal interactions and one-to-one-to-one-to-one internal resonance phenomenon. The effects of different parameters on the complex dynamic response are investigated in this study. The stability of steady-state solutions is also analyzed in detail.     

The response of a beam to a transient pressure wave load

The dynamic behaviour of beam structures under pressure waves is investigated. The propagation of the bending waves under a moving single load is first studied for three types of beam: a Bernoulli-Euler beam, a beam with shear deflection and a Timoshenko beam. Then the responses of the Bernoulli-Euler and the Timoshenko beam are studied under moving pressure wave excitation. The results are presented as dynamic amplification factors (DAF). The influence of the load parameters (load shape, propagation speed, pressure wave duration, etc.) and the beam parameters (slenderness, damping, etc.) is discussed. The load shape (symmetrical, asymmetrical) and the propagation speed strongly influence the response. The results are compared with available approximate solutions for the corresponding lumped element, single degree of freedom model of the structure.     

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.     

弛豫媒质中有限束超声波的非线性传播畸变理论

文中提出弛豫媒质中有限束非线性声波方程,并采用微扰法求得由非线性传播畸变产生的高次谐波的一般解.研究表明,对高斯型声波,其谐波畸变解可以解析给出,而且其径向分布始终维持高斯函数.虽然其频散量大小会影响各次谐波的振幅,但其相速的变化却仍与对在频率的小振幅波相同.文中还用Blackstock算子将所得的结果应用于任何吸收-频散媒质,包括只能从经验得到其吸收与频率关系的一些生物媒质.     

Effective nonlinear AC response to graded cylindrical composites

The perturbation method is used to study the effective nonlinear response of graded cylindrical composites with power-law gradient inclusions under a sinusoidal alternating current (AC) external field of finite frequency ω. In dilute limit, the local potentials and the formulae of effective nonlinear AC responses are derived at the fundamental frequency and the third harmonics. Furthermore, the general effective nonlinear responses are given and compared with the effective nonlinear AC responses of fundamental frequency and the third harmonics, and the relationships between the nonlinear effective AC response and the general effective nonlinear response are also determined.     

Bending vibration of axially loaded Timoshenko beams with locally distributed Kelvin-Voigt damping

Utilizing the Timoshenko beam theory and applying Hamilton's principle, the bending vibration equations of an axially loaded beam with locally distributed internal damping of the Kelvin-Voigt type are established. The partial differential equations of motion are then discretized into linear second-order ordinary differential equations based on a finite element method. A quadratic eigenvalue problem of a damped system is formed to determine the eigenfrequencies of the damped beams. The effects of the internal damping, sizes and locations of damped segment, axial load and restraint types on the damping and oscillating parts of the damped natural frequency are investigated. It is believed that the present study is valuable for better understanding the influence of various parameters of the damped beam on its vibration characteristics.     

Vibration of vehicle–pavement coupled system based on a Timoshenko beam on a nonlinear foundation

This paper focuses on the coupled nonlinear vibration of vehicle–pavement system. The pavement is modeled as a Timoshenko beam resting on a six-parameter foundation. The vehicle is simplified as a spring–mass–damper oscillator. For the first time, the dynamic response of vehicle–pavement coupled system is studied by modeling the pavement as a Timoshenko beam resting on a nonlinear foundation. Consequently, the shear effects and the rotational inertia of the pavement are included in the modeling process. The pavement model is assumed to be a linear-plus-cubic Pasternak-type foundation. Furthermore, the convergent Galerkin truncation is used to obtain approximate solutions to the coupled vibratory response of the vehicle–pavement coupled system. The dynamic responses of the vehicle–pavement system with the asphalt pavement on soft soil foundation are investigated via the numerical examples. The numerical results show that the calculation for the coupled vibratory response needs high-order modes. Moreover, the coupling effects between the pavement and the vehicle are numerically examined by using the convergent modal truncation. The physical parameters of the vehicle–pavement system such as the shear modulus are compared for determining their influences on the coupled vibratory response.