Snap-through behaviors and nonlinear vibrations of a bistable composite laminated cantilever shell:an experimental and numerical study

The snap-through behaviors and nonlinear vibrations are investigated for a bistable composite laminated cantilever shell subjected to transversal foundation excitation based on experimental and theoretical approaches.An improved experimental specimen is designed in order to satisfy the cantilever support boundary condition,which is composed of an asymmetric region and a symmetric region.The symmetric region of the experimental specimen is entirely clamped,which is rigidly connected to an electromagnetic shaker,while the asymmetric region remains free of constraint.Different motion paths are realized for the bistable cantilever shell by changing the input signal levels of the electromagnetic shaker,and the displacement responses of the shell are collected by the laser displacement sensors.The numerical simulation is conducted based on the established theoretical model of the bistable composite laminated cantilever shell,and an off-axis three-dimensional dynamic snap-through domain is obtained.The numerical solutions are in good agreement with the experimental results.The nonlinear stiffness characteristics,dynamic snap-through domain,and chaos and bifurcation behaviors of the shell are quantitatively analyzed.Due to the asymmetry of the boundary condition and the shell,the upper stable-state of the shell exhibits an obvious soft spring stiffness characteristic,and the lower stable-state shows a linear stiffness characteristic of the shell.

Nonlinear vibrations of a composite circular plate with a rigid body

The influence of weights is usually ignored in the study of nonlinear vibrations of plates.In this paper,the effect of structure weights on the nonlinear vibration of a composite circular plate with a rigid body is presented.The nonlinear governing equations are derived from the generalized Hamilton's principle and the von Kármán plate theory.The equilibrium configurations due to weights are determined and validated by the finite element method(FEM).A nonlinear model for the vibration around the equilibrium configuration is established.Moreover,the natural frequencies and amplitude-frequency responses of harmonically forced vibrations are calculated.The study shows that the structure weights introduce additional linear and quadratic nonlinear terms into the dynamical model.This leads to interesting phenomena.For example,considering weights increases the natural frequency.Furthermore,when the influence of weights is considered,the vibration response of the plate becomes asymmetrical.

Effect of porosity on active damping of geometrically nonlinear vibrations of a functionally graded magneto-electro-elastic plate

This paper investigates the effect of porosity on active damping of geometrically nonlinear vibrations(GNLV)of the magneto-electro-elastic(MEE)functionally graded(FG)plates incorporated with active treatment constricted layer damping(ATCLD)patches.The perpendicularly/slanted reinforced 1-3 piezoelectric composite(1-3 PZC)constricting layer.The constricted viscoelastic layer of the ATCLD is modeled in the time-domain using Golla-Hughes-Mc Tavish(GHM)technique.Different types of porosity distribution in the porous magneto-electro-elastic functionally graded PMEE-FG plate graded in the thickness direction.Considering the coupling effects among elasticity,electrical,and magnetic fields,a three-dimensional finite element(FE)model for the smart PMEE-FG plate is obtained by incorporating the theory of layer-wise shear deformation.The geometric nonlinearity adopts the von K arm an principle.The study presents the effects of a variant of a power-law index,porosity index,the material gradation,three types of porosity distribution,boundary conditions,and the piezoelectric fiber’s orientation angle on the control of GNLV of the PMEE-FG plates.The results reveal that the FG substrate layers’porosity significantly impacts the nonlinear behavior and damping performance of the PMEE-FG plates.

Nonlinear phenomena in vibrations of embedded carbon nanotubes conveying viscous fluid

Various nonlinear phenomena such as bifurcations and chaos in the responses of carbon nanotubes(CNTs)are recognized as being major contributors to the inaccuracy and instability of nanoscale mechanical systems.Therefore,the main purpose of this paper is to predict the nonlinear dynamic behavior of a CNT conveying viscousfluid and supported on a nonlinear elastic foundation.The proposed model is based on nonlocal Euler–Bernoulli beam theory.The Galerkin method and perturbation analysis are used to discretize the partial differential equation of motion and obtain the frequency-response equation,respectively.A detailed parametric study is reported into how the nonlocal parameter,foundation coefficients,fluid viscosity,and amplitude and frequency of the external force influence the nonlinear dynamics of the system.Subharmonic,quasi-periodic,and chaotic behaviors and hardening nonlinearity are revealed by means of the vibration time histories,frequency-response curves,bifurcation diagrams,phase portraits,power spectra,and Poincarémaps.Also,the results show that it is possible to eliminate irregular motion in the whole range of external force amplitude by selecting appropriate parameters.

Stability and nonlinear vibrations of a flexible pipe parametrically excited by an internal varying flow density

Pipes are often used to transport multiphase flows in many engineering applications.The total fluid flow density inside a pipe may vary with time and space.In this paper,a simply supported pipe conveying a variable density flow is modeled theoretically,and its stability and nonlinear vibrations are investigated in detail.The variation of the flow density is simulated using a mathematical function.The equation governing the vibration of the pipe is derived according to Euler-Bernoulli beam theory.When the internal flow density varies with time,the pipe is excited parametrically.The stability of the pipe is determined by Floquet theory.Some simple parametric and combination resonances are determined.For a higher mass ratio(mean flow mass/pipe structural mass),higher flow velocity,or smaller end axial tension,the pipe becomes unstable more easily due to wider parametric resonance regions.In the subcritical flow velocity regime,the vibrations of the pipe are periodic and quasiperiodic for simple and combination resonances,respectively.However,in the supercritical regime,the vibrations of the pipe exhibit much richer dynamics including periodic,multiperiodic,quasiperiodic,and chaotic behaviors.

Nonlinear Vibrations and Stability of an Axially Moving Plate Immersed in Fluid

In this paper,the nonlinear forced vibrations and stability of an axially moving large deflection plate immersed in fluid are investigated.Based on von Karman's large deflec・tion plate theory and taking into consideration the influence of fluid-strueture interaction,axial moving and axial tension,nonlinear dynamic equations are obtained by applying D'Alembert's principle.These dynamic equations are further discretized into ordinary differential equations via the Galerkin method.The frequency-response curves of system are obtained and examined.Then numerical method is used to analyze the bifurcation behaviors of immersed plate.Results show that as the parameters vary,the system displays periodic,multi-periodic,quasi-periodic and even chaotic motion.Through the analysis on global dynamic characteristics of fluid-strueture interaction system,rich and varied nonlinear dynamic characteristics are obtained,and various ways that lead to chaotic motion of the system are further revealed.

Nonlinear transverse vibrations of a slightly curved beam carrying a concentrated mass

In this study, a slightly curved Euler Bernoulli beam carrying a concentrated mass was handled. The beam was resting on an elastic foundation and simply supported at both ends. Effects of the concentrated mass on nonlin- ear vibrations were investigated. Sinusoidal and parabolic type functions were used as curvature functions. Equations of motion have cubic nonlinearities because of elongations during vibrations. Damping and harmonic excitation terms were added to the equations of motion. Method of mul- tiple scales, a perturbation technique, was used for solving integro-differential equation analytically. Natural frequen- cies were calculated exactly for different mass ratios, mass locations, curvature functions, and linear elastic foundation coefficients. Amplitude-phase modulation equations were found by considering primary resonance case. Effects of nonlinear terms on natural frequencies were calculated. Frequency-amplitude and frequency-response graphs were plotted. Finally effects of concentrated mass and chosen curvature function on nonlinear vibrations were investigated.

Transverse nonlinear vibrations of a circular spinning disk with a varying rotating speed

We analyze the transverse nonlinear vibrations of a rotating flexible disk subjected to a rotating point force with a periodically varying rotating speed. Based on Hamilton’s principle, the nonlinear governing equations of motion (coupled equations among the radial, tangential and transverse displacements) are derived for the rotating flexible disk. When the in-plane inertia is ignored and a stress function is introduced, the three nonlinearly coupled partial differential equations are reduced to two nonlinearly coupled partial differential equations. According to Galerkin’s approach, a four-degree-of-freedom nonlinear system governing the weakly split resonant modes is derived. The resonant case considered here is 1:1:2:2 internal resonance and a critical speed resonance. The primary parametric resonance for the first-order sin and cos modes and the fundamental parametric resonance for the second-order sin and cos modes are also considered. The method of multiple scales is used to obtain a set of eight-dimensional nonlinear averaged equations. Based on the averaged equations, using numerical simulations, the influence of different parameters on the nonlinear vibrations of the spinning disk is detected. It is concluded that there exist complicated nonlinear behaviors including the periodic, period-n and multi-pulse type chaotic motions for the spinning disk with a varying rotating speed. It is also found that among all parameters, the damping and excitation have great influence on the nonlinear responses of the spinning disk with a varying rotating speed.

Nonlinear Vibrations of Timoshenko Beams with Various Boundary Conditions

This paper is concerned with the effects of boundary conditions on the large-amplitude free vi-brations of Timoshenko beams. The effects of nonlinear terms on the frequency of Timoshenko beams with simply supported ends (supported-supported, SS), clamped ends (clamped-clamped, CC) and one end simply supported and the other end clamped (clamped-supported, CS) are discussed in detail. Given a spe-cific vibration amplitude, the change of nonlinear frequency according to the effects of boundary conditions is always in the following descending order: SS, CS, and CC. It is found that the slenderness ratio has a significant influence on the nonlinear frequency. For slender beams, the nonlinear effects of bending curva-ture and shear strain are negligible regardless of the boundary conditions. For short beams and especially for those of large amplitude vibrations, however, the nonlinear effects of bending curvature and shear strain become noticeable in the following ascending order: SS, CS, and CC.

Energy flow characteristics of friction-induced nonlinear vibrations in a water-lubricated bearing-shaft coupled system

Based on the energy flow theory of nonlinear dynamical system,the stabilities,bifurcations,possible periodical/chaotic motions of nonlinear water-lubricated bearing-shaft coupled systems are investigated in this paper.It is revealed that the energy flow characteristics around the equlibrium point of system behaving in the three types with different friction-para-mters.(a)Energy flow matrix has two negative and one positive energy flow factors,constructing an attractive local zero-energy flow surface,in which free vibrations by initial disturbances show damped modulated oscillations with the system tending its equlibrium state,while forced vibrations by external forces show stable oscillations,(b)Energy flow matrix has one negative and two positive energy flow factors,spaning a divergence local zero-energy flow surface,so that the both free and forced vibrations are divergence oscillations with the system being unstable,(c)Energy flow matrix has a zero-energy flow factor and two opposite factors,which constructes a local zero-energy flow surface dividing the local phase space into stable,unstable and central subspace,and the simulation shows friction self-induced unstable vibrations for both free and forced cases.For a set of friction parameters,the system behaves a periodical oscillation,where the bearing motion tends zero and the shaft motion reaches a stable limit circle in phase space with the instant energy flow tending a constant and the time averaged one tending zero.Numerical simulations have not found any possible chaotic motions of the system.It is discovered that the damping matrices of cases(a),(b)and(c)respectively have positive,negative and zero diagonal elements,resulting in the different dynamic behavour of system,which gives a giderline to design the water-lubricated bearing unit with expected performance by adjusting the friction parameters for applications.

Nonlinear Vibrations of Two-Span Composite Laminated Plates with Equal and Unequal Subspan Lengths

The nonlinear transverse vibrations of ordered and disordered twodimensional(2D)two-span composite laminated plates are studied.Based on the von Karman’s large deformation theory,the equations of motion of each-span composite laminated plate are formulated using Hamilton’s principle,and the partial differential equations are discretized into nonlinear ordinary ones through the Galerkin’s method.The primary resonance and 1/3 sub-harmonic resonance are investigated by using the method of multiple scales.The amplitude-frequency relations of the steady-state responses and their stability analyses in each kind of resonance are carried out.The effects of the disorder ratio and ply angle on the two different resonances are analyzed.From the numerical results,it can be concluded that disorder in the length of the two-span 2D composite laminated plate will cause the nonlinear vibration localization phenomenon,and with the increase of the disorder ratio,the vibration localization phenomenon will become more obvious.Moreover,the amplitude-frequency curves for both primary resonance and 1/3 sub-harmonic resonance obtained by the present analytical method are compared with those by the numerical integration,and satisfactory precision can be obtained for engineering applications and the results certify the correctness of the present approximately analytical solutions.

Nonlinear Flap-Wise Vibration Characteristics ofWind Turbine Blades Based onMulti-Scale AnalysisMethod

This work presents a novel approach to achieve nonlinear vibration response based on the Hamilton principle.We chose the 5-MW reference wind turbine which was established by the National Renewable Energy Laboratory(NREL),to research the effects of the nonlinear flap-wise vibration characteristics.The turbine wheel is simplified by treating the blade of a wind turbine as an Euler-Bernoulli beam,and the nonlinear flap-wise vibration characteristics of the wind turbine blades are discussed based on the simplification first.Then,the blade’s large-deflection flap-wise vibration governing equation is established by considering the nonlinear term involving the centrifugal force.Lastly,it is truncated by the Galerkin method and analyzed semi-analytically using the multi-scale analysis method,and numerical simulations are carried out to compare the simulation results of finite elements with the numerical simulation results using Campbell diagram analysis of blade vibration.The results indicated that the rotational speed of the impeller has a significant impact on blade vibration.When the wheel speed of 12.1 rpm and excitation amplitude of 1.23 the maximum displacement amplitude of the blade has increased from 0.72 to 3.16.From the amplitude-frequency curve,it can be seen that the multi-peak characteristic of blade amplitude frequency is under centrifugal nonlinearity.Closed phase trajectories in blade nonlinear vibration,exhibiting periodic motion characteristics,are found through phase diagrams and Poincare section diagrams.

Stabilities Analysis of Electromechanical Nonlinear Vibration of Electric Machine

An electromechanical nonlinear model of rotor system of electric machine is built.Respondance curves in parameter excited nonlinear vibration of this system caused by electromagnetic forces are investigated.Further more,the analysis reveals the effects of various electromagnetic and mechanical parameters on resonances, and some valuable results are obtained.The analytical result of this paper provides electric machine with the condition of 1/2 subharmonic resonance under the electromechanical electromagnetic forces.Electromagnetic forces apparently affect the stability zone, and both linear term and nonlinear term can excite parametric resonance.The revealed dynamic phenomena provide some new theories and active methods for the fault recognition of electric machine and the defination of stability range,and the theoretical bases for qualitatively controlling the stable operating state of rotors.

A SIMPLIFIED CALCULATING METHOD OF NONLINEAR FREQUENCY OF CABLE NET UNDER MEAN WIND LOAD

The cable net supported glass curtain wallas the most advanced technique in dot point supported glass curtain wall, is widely used in China. Because of its large deflection and high nonlinearity under wind load, the dynamic performance of the cable net is greatly different from that of the conventional linear structures. The continuous membrane theory is used to construct the nonlinear vibration differential equation of the cable net, and the harmonic balance method is used to solve the analytic formula of the nonlinear frequency. In order to verify the accuracy of the above analytic formula, the results of the formula and the nonlinear FEM time-history method are compared and found to be in good agreement. Furthermore, the nonlinear vibration differential equation and the nonlinear frequency obtained in this paper are the basis for the wind-induced response analysis of a cable net under fluctuating wind load.

Dynamic Stability Analysis for Helicopter Rotor/Fuselage Coupled Nonlinear Systems

In order to accurately predict the dynamic instabilities of a helicopterrotor/fuselage coupled system, nonlinear differential equations are derived and integrated in thetime domain to yield responses of rotor blade flapping, lead-lag and fuselage motions to simulatethe behavior of the system numerically. To obtain quantitative instabilities, Fast Fourier Transform(FFT) is conducted to estimate the modal frequencies, and Fourier series based moving-blockanalysis is employed in the predictions of the modal damping in terms of the response time history.Study on the helicopter ground resonance exhibits excellent correlation among the time-domain (TD)analytical results, eigenvalues and wind tunnel test data, thus validating the methodology of thepaper. With a large collective pitch set, the predictions of regressive lag modal damping from TDanalysis correlate with the experimental data better than from eigen analysis. TD analysis can beapplied in the dynamic stability analysis of helicopter rotor/fuselage coupled systems incorporatedwith nonlinear blade lag dampers.

Global Dynamic Characteristic of Nonlinear Torsional Vibration System under Harmonically Excitation

Torsional vibration generally causes serious instability and damage problems in many rotating machinery parts. The global dynamic characteristic of nonlinear torsional vibration system with nonlinear rigidity and nonlinear friction force is investigated. On the basis of the generalized dissipation Lagrange＇s equation, the dynamics equation of nonlinear torsional vibration system is deduced. The bifurcation and chaotic motion in the system subjected to an external harmonic excitation is studied by theoretical analysis and numerical simulation. The stability of unperturbed system is analyzed by using the stability theory of equilibrium positions of Hamiltonian systems. The criterion of existence of chaos phenomena under a periodic perturbation is given by means of Melnikov＇s method. It is shown that the existence of homoclinic and heteroclinic orbits in the unperturbed system implies chaos arising from breaking of homoclinic or heteroclinic orbits under perturbation. The validity of the result is checked numerically. Periodic doubling bifurcation route to chaos, quasi-periodic route to chaos, intermittency route to chaos are found to occur due to the amplitude varying in some range. The evolution of system dynamic responses is demonstrated in detail by Poincare maps and bifurcation diagrams when the system undergoes a sequence of periodic doubling or quasi-periodic bifurcations to chaos. The conclusion can provide reference for deeply researching the dynamic behavior of mechanical drive systems.

HOPF BIFURCATION OF A NONLINEAR RESTRAINED CURVED PIPE CONVEYING FLUID BY DIFFERENTIAL QUADRATURE METHOD

This paper proposes a new method for investigating the Hopf bifurcation of a curved pipe conveying fluid with nonlinear spring support.The nonlinear equation of motion is derived by forces equilibrium on microelement of the system under consideration.The spatial coordinate of the system is discretized by the differential quadrature method and then the dynamic equation is solved by the Newton-Raphson method.The numerical solutions show that the inner fluid velocity of the Hopf bifurcation point of the curved pipe varies with different values of the parameter, nonlinear spring stiffness.Based on this,the cycle and divergent motions are both found to exist at specific fluid flow velocities with a given value of the nonlinear spring stiffness.The results are useful for further study of the nonlinear dynamic mechanism of the curved fluid conveying pipe.

A state-of-the-art review on low-frequency nonlinear vibration isolation with electromagnetic mechanisms

Vibration isolation is one of the most efficient approaches to protecting host structures from harmful vibrations,especially in aerospace,mechanical,and architectural engineering,etc.Traditional linear vibration isolation is hard to meet the requirements of the loading capacity and isolation band simultaneously,which limits further engineering application,especially in the low-frequency range.In recent twenty years,the nonlinear vibration isolation technology has been widely investigated to broaden the vibration isolation band by exploiting beneficial nonlinearities.One of the most widely studied objects is the"three-spring"configured quasi-zero-stiffness(QZS)vibration isolator,which can realize the negative stiffness and high-static-low-dynamic stiffness(HSLDS)characteristics.The nonlinear vibration isolation with QZS can overcome the drawbacks of the linear one to achieve a better broadband vibration isolation performance.Due to the characteristics of fast response,strong stroke,nonlinearities,easy control,and low-cost,the nonlinear vibration with electromagnetic mechanisms has attracted attention.In this review,we focus on the basic theory,design methodology,nonlinear damping mechanism,and active control of electromagnetic QZS vibration isolators.Furthermore,we provide perspectives for further studies with electromagnetic devices to realize high-efficiency vibration isolation.

Analytical and numerical studies on the nonlinear dynamic response of orthotropic membranes under impact load

Orthotropic membrane components and structures are widely used in building structures, instruments and meters, electronic engineering, space and aeronautics, etc., because of their light weights. However, the same lightweight combined with low stiffness make membranes prone to vibration under dynamic loads, and in some cases the vibration may lead to structural failure. Herein, the undamped nonlinear vibration response of pretension rectangular orthotropic membrane structures subjected to impact loading is studied by analytical and numerical methods. The analytical solution is obtained by solving the governing equations by the Bubnov-Galerkin method and the Lindstedt-Poincare perturbation method. Numerical analysis has also been carried out based on the same theoretical model. The analytical and numerical results have been compared and analyzed, and the influence of various model parameters on membrane vibration discussed. The results obtained herein provide some theoretical basis for the vibration control and dynamic design of orthotropic membrane components and structures.

Nonlinear vibration of functionally graded graphene platelet-reinforced composite truncated conical shell using first-order shear deformation theory

In this study,the first-order shear deformation theory(FSDT)is used to establish a nonlinear dynamic model for a conical shell truncated by a functionally graded graphene platelet-reinforced composite(FG-GPLRC).The vibration analyses of the FG-GPLRC truncated conical shell are presented.Considering the graphene platelets(GPLs)of the FG-GPLRC truncated conical shell with three different distribution patterns,the modified Halpin-Tsai model is used to calculate the effective Young’s modulus.Hamilton’s principle,the FSDT,and the von-Karman type nonlinear geometric relationships are used to derive a system of partial differential governing equations of the FG-GPLRC truncated conical shell.The Galerkin method is used to obtain the ordinary differential equations of the truncated conical shell.Then,the analytical nonlinear frequencies of the FG-GPLRC truncated conical shell are solved by the harmonic balance method.The effects of the weight fraction and distribution pattern of the GPLs,the ratio of the length to the radius as well as the ratio of the radius to the thickness of the FG-GPLRC truncated conical shell on the nonlinear natural frequency characteristics are discussed.This study culminates in the discovery of the periodic motion and chaotic motion of the FG-GPLRC truncated conical shell.