Large-amplitude nonlinear vibrations of a Mooney–Rivlin rectangular membrane

An analysis of the linear and nonlinear vibration response and stability of a pre-stretched hyperelastic rectangular membrane under harmonic lateral pressure and finite initial deformations is presented in this paper. Geometric nonlinearity due to finite deformations and material nonlinearity associated with the hyperelastic constitutive law are taken into account. The membrane is assumed to be made of an isotropic, homogeneous, and incompressible Mooney–Rivlin material. The results for a neo-Hookean material are obtained as a particular case and a comparison of these two constitutive models is carried out. First, the exact solution of the membrane under a biaxial stretch is obtained, being this initial stress state responsible for the membrane stiffness. The equations of motion of the pre-stretched membrane are then derived. From the linearized equations, the natural frequencies and mode shapes of the membrane are analytically obtained for both materials. The natural modes are then used to approximate the nonlinear deformation field using the Galerkin method. A detailed parametric analysis shows the strong influence of the stretching ratios and material parameters on the linear and nonlinear oscillations of the membrane. Frequency–amplitude relations, resonance curves, and bifurcation diagrams, are used to illustrate the nonlinear dynamics of the membrane. The present results are compared favorably with the results evaluated for the same membrane using a nonlinear finite element formulation.     

Free and forced vibration analysis of a nonlinear system with cyclic symmetry: Application to a simplified model

This work program is devoted to studying the nonlinear dynamics of a structure with cyclic symmetry under conditions of geometric nonlinearity, through the use of the harmonic balance method (HBM). In order to study the influence of nonlinearity due to the large deflection of blades, a simplified model has been developed. This approach leads to a system of linearly coupled, second-order nonlinear differential equations, in which nonlinearity appears via cubic terms. Periodic solutions, in both the free and forced cases, are sought by applying HBM coupled with an arc-length continuation method. Solution stability has been investigated using Floquet's theorem. In addition to featuring similar and nonsimilar nonlinear modes, the unforced system is known to contain localized nonlinear modes that arise from branching point bifurcation at certain vibration amplitudes. In the forced case, these nonlinear modes give rise to a complex dynamic behavior. Many bifurcations can take place, thus leading to strong or weak localization that may or may not be stable. In this study, special attention has been paid to the influence of excitation on dynamic responses. Several cases of excitation have been analyzed herein: localized excitation, and low-engine-order excitation. In the case of low-engine-order excitation, sensitivity of the response to a perturbation of this excitation type has been investigated, and it has been shown that for a localized, or sufficiently detuned excitation, several solutions can coexist, some of which are represented by closed curves in the Frequency-Amplitude domain. These various solutions overlap when increasing the force amplitude, leading to forced nonlinear localization. Because closed curves are not tied up with the basic nonlinear solution, they can easily be overlooked. In this study, they have been calculated using a sequential continuation with the force amplitude as a parameter.     

Experimental Investigation of Nonlinear Vibration Isolator with Fluidic Actuators (NLVIFA)

This paper elaborates a nonlinear fluidic low frequency vibration isolator designed with the characteristics of quasi-zero stiffness (QZS). The existing model of QZS vibration isolator enhances amplitude of vibration and attenuating vibration frequencies. This concern with displacement plays a vital role in the performance and instability of oblique spring setup reduces the isolator performance in horizontal non-nominal loads, in this accordance; this paper associates double acting hydraulic cylinder (fluidic actuators in short) in oblique and helical coil spring. An approximate expression of unique analytical relationship between the stiffness of vertical spring and bulk modulus of the fluid is derived for Quasi – Zero Stiffness Non-Linear Vibration Isolator with Fluidic Actuators (NLVIFA in short) system and the force transmissibility is formulated and damping ratio are discussed for characteristic analysis. Modal analysis carried out and compared with analytical results and an experimental prototype is developed and investigated. The performance of the NLVIFA reduces the external embarrassment more at low frequencies and the series of experimental studies showing that the soft nonlinearity causes limitation in the resonant frequency thereupon the isolation will be enhanced and NLVIFA greatly outperform some other type of nonlinear isolators.     

Predicting phase shift effects for vibrating fluid-conveying pipes due to Coriolis forces and fluid pulsation

Knowing the influence of fluid flow perturbations on the dynamic behavior of fluid-conveying pipes is of relevance, e.g., when exploiting flow-induced oscillations of pipes to determine the fluids mass flow or density, as done with Coriolis flow meters (CFM). This could be used in the attempts to improve accuracy, precision, and robustness of CFMs. A simple mathematical model of a fluid-conveying pipe is formulated and the effect of pulsating fluid flow is analyzed using a multiple time scaling perturbation analysis. The results are simple analytical predictions for the transverse pipe displacement and approximate axial shift in vibration phase. The analytical predictions are tested against pure numerical solution using representative examples, showing good agreement. Fluid pulsations are predicted not to influence CFM accuracy, since proper signal filtering is seen to allow the determination of the correct mean phase shift. Large amplitude motions, which could influence CFM robustness, do not appear to be induced by the investigated fluid pulsation. Pulsating fluid of the combination resonance type could, however, influence CFMs robustness, if induced pipe motions go unnoticed and uncontrolled during CFM operation by feedback control. The analytical predictions offer an immediate insight into how fluid pulsation affects phase shift, which is a quantity measured by CFMs to estimate the mass flow, and lead to hypotheses for more complex geometries, i.e. industrial CFMs. The validity of these hypotheses is suggested to be tested using laboratory experiments, or detailed computational models taking fluid-structure interaction into account.     

Dynamics of a vertical riser with weak structural nonlinearity excited by wakes

In this paper we investigate the effect of a weak structural nonlinearity on the dynamical behaviour of a vertical offshore riser subjected to vortex-induced vibration (VIV). Coupling of the riser dynamics with the flow of the surrounding fluid is achieved by attaching a wake oscillator to a reduced model of the structure, which is obtained through the application of the invariant manifold technique for the derivation of nonlinear normal modes.By comparing the free responses of the linear and the nonlinear structure, it was found that the structural nonlinearity has a stiffening effect on the oscillation of the riser, which becomes more pronounced when the internal flow is incorporated into the model. Consequently, in the coupled system, the response is considerably modified for the structure as well as for the fluid variable.     

Nonlinear Dust Acoustic Waves in Strongly Coupled Dusty Plasmas with Positively Charged Dust Particles

The nonlinear propagation of dust acoustic waves is investigated in four-component plasmas consisting of positively charged dust grains, trapped ions, nonthermal electrons, and photoelectron due to ultraviolet irradiation.We use generalized viscoelastic hydrodynamic model for strongly coupled dust grain. In the weak nonlinearity limit, a modified Kadomstev–Petviashvili(KP) equation and a modified KP-Burger equation, which have a damping term coming from nonadiabatic charge variation, have been derived in the kinetic regime and hydrodynamic regime, respectively. With the increasing of UV photon flux, the hydrodynamic regime changes to kinetic regime. The approximate analytical line soliton and shock solutions are investigated in the kinetic regime and hydrodynamic regime, respectively.     

Modelling nonlinearity in piezoceramic transducers: From equations to nonlinear equivalent circuits

Quadratic nonlinear equations of a piezoelectric element under the assumptions of 1D vibration and weak nonlinearity are derived by the perturbation theory. It is shown that the nonlinear response can be represented by controlled sources that are added to the classical hexapole used to model piezoelectric ultrasonic transducers. As a consequence, equivalent electrical circuits can be used to predict the nonlinear response of a transducer taking into account the acoustic loads on the rear and front faces. A generalisation of nonlinear equivalent electrical circuits to cases including passive layers and propagation media is then proposed. Experimental results, in terms of second harmonic generation, on a coupled resonator are compared to theoretical calculations from the proposed model.     

A general solution procedure for vibrations of systems with cubic nonlinearities and nonlinear/time-dependent internal boundary conditions

The subject of this paper is the development of a general solution procedure for the vibrations (primary resonance and nonlinear natural frequency) of systems with cubic nonlinearities, subjected to nonlinear and time-dependent internal boundary conditions—this is a commonly occurring situation in the vibration analysis of continuous systems with intermediate elements. The equations of motion form a set of nonlinear partial differential equations with nonlinear, time-dependent, and coupled internal boundary conditions. The method of multiple timescales, an approximate analytical method, is applied directly to each partial differential equation of motion as well as coupled boundary conditions (i.e. on each sub-domain and the corresponding internal boundary conditions for a continuous system with intermediate elements) which ultimately leads to approximate analytical expressions for the frequency-response relation and nonlinear natural frequencies of the system. These closed-form solutions provide direct insight into the relationship between the system parameters and vibration characteristics of the system. Moreover, the suggested solution procedure is applied to a sample problem which is discussed in detail.     

On the response of a harmonically excited two degree-of-freedom system consisting of a linear and a nonlinear quasi-zero stiffness oscillator

There are many systems which consist of a nonlinear oscillator attached to a linear system, examples of which are nonlinear vibration absorbers, or nonlinear systems under test using shakers excited harmonically with a constant force. This paper presents a study of the dynamic behaviour of a specific two degree-of-freedom system representing such a system, in which the nonlinear system does not affect the vibration of the forced linear system. The nonlinearity of the attachment is derived from a geometric configuration consisting of a mass suspended on two springs which are adjusted to achieve a quasi-zero stiffness characteristic with pure cubic nonlinearity. The response of the system at the frequency of excitation is found analytically by applying the method of averaging. The effects of the system parameters on the frequency-amplitude response of the relative motion are examined. It is found that closed detached resonance curves lying outside or inside the continuous path of the main resonance curve can appear as a part of the overall amplitude-frequency response. Two typical situations for the creation of the detached resonance curve inside the main resonance curve, which are dependent on the damping in the nonlinear oscillator, are discussed.     

A study of vibroacoustic coupling between a pump and attached water-filled pipes

This paper presents a model for the vibroacoustical behavior of a pump coupled with water-filled pipes. Coupling between (a) the pump and the inlet and outlet pipes, and (b) the pipe wall and the fluid contained in the pipe, is investigated through analytical modeling and numerical simulation. In the model, the pump is represented by a rigid body supported by multiple elastic mounts, and the inlet and outlet pipes by two semi-infinite water-filled pipes. The vibration characteristics of the coupled system under the excitation of mechanical forces and fluid-borne forces at the pump are calculated. The results enhance our understanding about how the input mechanical and fluid excitation energy at the pump is transmitted to the pipes and how to relate the piping vibroacoustical response to the excitations at the pump. This study assists in predicting dynamic stress in pipes for given excitations at the pump, and in developing methods to identify the nature (fluid or mechanical) of the excitation forces at the pump using the vibration and dynamic pressure measurements on the pump/pipe system.     

Nonlinear nonuniform torsional vibrations of bars by the boundary element method

In this paper a boundary element method is developed for the nonuniform torsional vibration problem of bars of arbitrary doubly symmetric constant cross-section taking into account the effect of geometrical nonlinearity. The bar is subjected to arbitrarily distributed or concentrated conservative dynamic twisting and warping moments along its length, while its edges are supported by the most general torsional boundary conditions. The transverse displacement components are expressed so as to be valid for large twisting rotations (finite displacement-small strain theory), thus the arising governing differential equations and boundary conditions are in general nonlinear. The resulting coupling effect between twisting and axial displacement components is considered and torsional vibration analysis is performed in both the torsional pre- or post-buckled state. A distributed mass model system is employed, taking into account the warping, rotatory and axial inertia, leading to the formulation of a coupled nonlinear initial boundary value problem with respect to the variable along the bar angle of twist and to an “average” axial displacement of the cross-section of the bar. The numerical solution of the aforementioned initial boundary value problem is performed using the analog equation method, a BEM based method, leading to a system of nonlinear differential-algebraic equations (DAE), which is solved using an efficient time discretization scheme. Additionally, for the free vibrations case, a nonlinear generalized eigenvalue problem is formulated with respect to the fundamental mode shape at the points of reversal of motion after ignoring the axial inertia to verify the accuracy of the proposed method. The problem is solved using the direct iteration technique (DIT), with a geometrically linear fundamental mode shape as a starting vector. The validity of negligible axial inertia assumption is examined for the problem at hand.     

Three-dimensional nonlinear vibration of gear pairs

This work investigates the three-dimensional nonlinear vibration of gear pairs where the nonlinearity is due to portions of gear teeth contact lines losing contact (partial contact loss). The gear contact model tracks partial contact loss using a discretized stiffness network. The nonlinear dynamic response is obtained using the discretized stiffness network, but it is interpreted and discussed with reference to a lumped-parameter gear mesh model named the equivalent stiffness representation. It consists of a translational stiffness acting at a changing center of stiffness location (two parameters) and a twist stiffness. These four parameters, calculated from the dynamic response, change as the gears vibrate, and tracking their behavior as a post-processing tool illuminates the nonlinear gear response. There is a gear mesh twist mode where the twist stiffness is active in addition to the well-known mesh deflection mode where the translational stiffness is active. The twist mode is excited by periodic back and forth axial movement of the center of stiffness in helical gears. The same effect can occur in wide facewidth spur gears if tooth lead modifications or other factors such as shaft and bearing deflections disrupt symmetry about the axial centers of the mating teeth. Resonances of both modes are shown to be nonlinear due to partial and total contact loss. Comparing the numerical results with gear vibration experiments from the literature verifies the model and confirms partial contact loss nonlinearity in experiments.     

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.     

Localization in the model of contacting media with specific nonlinearity and interface interaction

In this paper, we consider the new model of nonlinear contacting media based on nonlinear Schrodinger equation with point potential and term, which is depended stepwise on field amplitude. Such a model theoretically describes a change in properties of the boundary regions along the interface between a Kerr-type crystal with cubic nonlinearity and a nonlinear medium characterized by abruptly change in dielectric constant depending on field amplitude. The short-range local interaction between wave and interface is taken into account by point potential in nonlinear Schrodinger equation. We obtain two new types of localized states characterized by composite structure consisting of three parts of the field distributions. We find exact and approximate solutions of dispersion equations. We described new properties of the spectrum of localized states arising as a result of the interaction of the wave with the interface and the presence of threshold field of the switching between the medium constants. All results are obtained in an analytical form. The proposed theory can be used to describe the propagation features of intense light beams localized along media interfaces in nonlinear optics, and to describe Bose-Einstein condensates with cubic nonlinearity.     

Dynamic analysis of an optical fiber coupler in telecommunications

This paper studies the vibration of an optical fiber coupler which is used in telecommunications subjected to a half sine shock. The emphasis is focused on analyzing the vibration response of the optical fibers inside the coupler and examining the influence of various coupler parameters on the vibration of the optical fibers, since their dynamic behavior is a critical factor in optical fiber communications. A simplified model of the optical fiber coupler is proposed, which consists of a beam and a string representing the substrate and the bundle of the optical fibers of the coupler, respectively. The beam and the string are bonded at four points using adhesive material, and therefore the boundary conditions for their equations of motion are coupled, which increases the complexity of the problem. For the string, two models are developed— the linear model assumes that the tension in the string is constant, while the non-linear one takes into account large transverse deflection and tension variation. With each model, both analytical study and numerical simulations for the vibration of the system under a half shock are carried out. Furthermore, numerical results are compared between the two models. Finally, parametric study leads to conclusions which are of practical importance to the design of optical fiber couplers.     

Consideration of piezoceramic actuator nonlinearity in the active isolation of deterministic vibration

There is an increasing need to effectively control micro-vibration in such fields as metrology, optics and micro-electronics. This paper describes the design of an adaptive feedforward strategy for vibration isolation of harmonic disturbance using a piezoelectric actuator with hysteretic behavior. A nonlinear analytical model of the piezoelectric actuator including a ferroelectric-like behavior is built using a Preisach model of hysteresis. Pre-multiplication of a single-frequency reference signal by the nonlinear model of the stack is investigated in order to effectively compensate the actuator nonlinearity. It is observed that a simple linear model of the stack is sufficient in the adaptation of a filtered-X LMS feedforward controller to effectively compensate the actuator nonlinearity, provided the reference signal has frequency components at the disturbance frequency and its higher harmonics.     

一种光电载荷非线性隔振装置的研究

针对光电载荷对隔振性能的需求,提出一种采用菱形连杆机构作为负刚度组件,具有高静、低动刚度特点的非线性隔振器（简称菱形HSLDS隔振器）。采用静力学分析方法,建立了隔振器数学模型,研究了刚度参数设定以及非线性调节方法;利用谐波平衡法（HBM）求解动力学方程,分析了各参数对隔振性能的影响关系;采用动力学仿真软件ADAMS及实物样机对理论模型与结论进行了验证。测试结果表明:菱形HSLDS隔振器具有较方便的参数调整能力,零位刚度及刚度非线性可通过拉簧参数与连杆参数进行设定、优化,隔振的刚度非线性优化程度受主隔振器阻尼以及零位刚度参数影响。相比于传统线性隔振器,菱形HSLDS具有显著的非线性隔振优势,可较好地满足光电载荷隔振需求。     

Modelling of structures with developing discontinuity

Discontinuity is one of the most important factors which contribute to the nonlinearity, challenging both methods of nonlinear analysis and its application for the analysis of interacting mechanical processes. The problem becomes more complicated when the discontinuity is developing under the influence of dynamic processes in the structures. This paper presents the methodology for analysis and simulation of the systems with the developing discontinuity based on the combination of analytical technique and the Matlab–Simulink computation. As an example, model of a cracked bar subjected to longitudinal excitation is used to analyse a nonlinear response as a way to monitor structural health. Different strategies of monitoring are compared based on the tracking the qualitative transformation of the bar?s vibration.     

The dynamic characteristics of a valve-less micropump

The aim of this paper is to investigate the dynamic characteristics of a valve-less micropump. A dynamic mathematical model of the micropump based on a hydraulic analogue system and a simulation method using AMESim software are developed. By using the finite-element analysis method, the static analysis of the diaphragm is carried out to obtain the maximum deflection and volumetric displacement. Dynamic characteristics of the valve-less micropump under different excitation voltages and frequencies are simulated and tested. Because of the discrepancy between simulation results and experimental data at frequencies other than the natural frequency, the revised model for the diaphragm maximum volumetric displacement is presented. Comparison between the simulation results based on the revised model and experimental data shows that the dynamic mathematical model based on the hydraulic analogue system is capable of predicting dynamic characteristics of the valve-less micropump at any excitation voltage and frequency.     

BIFURCATIONS AND CHAOS OF AN IMMERSED CANTILEVER BEAM IN A FLUID AND CARRYING AN INTERMEDIATE MASS

The concern of this work is the local stability and period-doubling bifurcations of the response to a transverse harmonic excitation of a slender cantilever beam partially immersed in a fluid and carrying an intermediate lumped mass. The unimodal form of the non-linear dynamic model describing the beam-mass in-plane large-amplitude flexural vibration, which accounts for axial inertia, non-linear curvature and inextensibility condition, developed in Al-Qaisia et al. (2000Shock and Vibration7 , 179-194), is analyzed and studied for the resonance responses of the first three modes of vibration, using two-term harmonic balance method. Then a consistent second order stability analysis of the associated linearized variational equation is carried out using approximate methods to predict the zones of symmetry breaking leading to period-doubling bifurcation and chaos on the resonance response curves. The results of the present work are verified for selected physical system parameters by numerical simulations using methods of the qualitative theory, and good agreement was obtained between the analytical and numerical results. Also, analytical prediction of the period-doubling bifurcation and chaos boundaries obtained using a period-doubling bifurcation criterion proposed in Al-Qaisia and Hamdan (2001 Journal of Sound and Vibration244, 453-479) are compared with those of computer simulations. In addition, results of the effect of fluid density, fluid depth, mass ratio, mass position and damping on the period-doubling bifurcation diagrams are studies and presented.