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Self-excited oscillation is a method to make a resonator vibrate at its natural frequency. This is widely used in vibration sensors such as mass sensors, atomic force microscopes and stiffness sensors. Self-excited oscillation is mainly produced by positive velocity feedback and time-delayed displacement feedback including phase-shifted displacement feedback. In this paper, we consider phase-shifted displacement feedback using a phase shifter. We perform nonlinear analysis to clarify the finite steady-state amplitude of the self-excited oscillation by considering nonlinear damping without a Laplace transform in the frequency domain, and formulate the effect of the phase shifter using an ordinary differential equation. We apply the method of multiple scales to the third-order ordinary differential equations expressing the coupling between the resonator and the phase shifter. This analytically reveals the parameter range of the phase shifter that produces self-excited oscillation in the resonator and the steady-state amplitude depending on the phase shift. We conduct an experiment using a cantilever as the resonator and produce self-excited oscillation via the feedback signal based on a phase shifter in a digital computer. The theoretically predicted characteristics of the self-excited oscillation agree well with the experimental ones.

Prediction of the response of nonlinear dynamical systems under interacting parametric and external excitations is important in designing systems such as sensors, amplifiers or energy harvesters, to achieve the desired performance. This paper concerns the nonlinear forced Mathieu equation, with linear damping and a 2:1 ratio between the parametric and external excitation frequencies. The Method of Varying Amplitudes (MVA) is employed to derive approximate analytical expressions for the response of the system. Both single-term and double-term solutions are developed: it is seen that, employing the double-term approximation, the MVA can accurately predict the response of the system over a wide range of frequencies and system parameters, showing a maximum of 0.2% deviation from numerical results obtained by direct integration of the equation of motion. This is in contrast with most of the available theoretical approaches such as the conventional Method of Multiple Scales, which can predict the response accurately only for a narrow range of system parameters and excitation frequencies. Furthermore, it is seen that the response is bounded, and analytical expressions for the frequency and amplitude of the upper bound are developed: this is unlike other methods which predict unbounded response, unless nonlinear damping is considered. Analytical expressions for the response are developed, and results are verified with numerical results obtained from direct integration of the equation of motion. Numerical examples are presented, showing good agreement with results obtained by the MVA.

In this paper, we investigated the realization of all possible internal resonance conditions between the first three modes of an electrostatically actuated, straight, clamped-hinged microbeam. We first obtained the static displacement and the first three natural frequencies around the deflected shape of the beam by using Galerkin based reduced order model and the finite element method. We then found all six possible commensurable relations between these frequencies as a function of two non-dimensional parameters ( $$\alpha _1$$ and $$\alpha _2 V_{\mathrm{dc}}^2$$ ) that depend on beam dimensions, material properties, and external forcing. Furthermore, we also examined the governing equations to check the feasibility of dynamic modal interaction. We found that although dynamical modal coupling was theoretically possible for all resonance conditions upon external excitation, the 3:1 internal resonance between the first two modes seemed to be the most experimentally feasible. Hence, we carried out a detailed forced vibrations analysis corresponding to this condition by solving the governing nonlinear equations using numerical time integration and the method of multiple scales. The system exhibited dynamically rich internal resonance behavior that can be controlled with the two non-dimensional parameters ( $$\alpha _1$$ and $$\alpha _2 V_{\mathrm{dc}}^2$$ ) and external damping. Specifically, we observed saddle node bifurcations, jump conditions, and internal loops in the amplitude response curves as a function of excitation frequency and amplitude. Overall, our work provides a systematic methodology and simple rules for in-depth exploration of internal resonance in microbeams. The findings of this paper could also assist in the development of sensors based on internal resonance.

In this study, we investigate the nonlinear resonant response of a magnetorheological (MR) damper–stay cable system with time delay, and the one-to-one internal resonance is considered. Based on Hamilton’s principle, the motion equations of the MR damper–cable system are derived, and the Galerkin method is applied to obtain the discrete model. Then, the method of multiple scales is applied to determine the modulation equations and the second-order solution of the nonlinear response of the MR damper–cable system. Following, the equilibrium solution and dynamic solution of the modulation equations are examined via the Newton–Raphson method and shooting method. The results show that the equilibrium solution may undergo Hopf bifurcation, resulting in the periodic solution. Moreover, the effects of the time delay and the inclination angle on the resonant response of the MR damper–cable system are investigated as well as those of the damper position. The numerical results show that the time delay increases the amplitudes of in-plane and out-of-plane modes and results in the more remarkable hardening behavior and relatively poor mitigation performance of the MR damper. However, the large time delay may suppress the complex chaotic modulation motion of the MR damper–cable system.

A two-degree-of-freedom model for a two-beam structure with nonlinear joints is presented and used to investigate the nonlinear responses of the system to a primary resonance of its first two modes in the presence of three-to-one internal resonance. By using the equilibrium conditions between the beams and the joints, the joint dynamic characteristics included linear and nonlinear torsional stiffness are introduced into the model of the system. The first two global mode functions of the two-beam structure, which can be obtained by the global mode method, are used to formulate the nonlinear dynamic model of the system and the specific linear torsional stiffness that result in three-to-one internal resonance is mainly considered. The method of multiple scales is employed to obtain the governing equations of the amplitudes and phases for the two-degree-of-freedom nonlinear dynamical system under the three-to-one internal resonance condition. Based on the modulation equations obtained by the method of multiple scales, the approximation solutions are derived and compared with those obtained by the numerical integration method. Through the cases of primary resonance of the first two modes with the three-to-one internal resonance condition, the frequency–response and force-response curves are plotted for investigating the nonlinear dynamic behavior in the two-beam structure connected with nonlinear joints.

Taking into account damping effects, the propagation-distance dependence of the amplitude of nonlinear guided waves arising from an internal resonance was investigated. In the analysis of Lamb-wave propagation, we introduced nonlinear quadratic effects derived from both geometric and material nonlinearities, referred to as classical nonlinearities, and damping effects by expressing the Lame constants as complex numbers. Assuming a small detuning from the phase matching condition with respect to wavenumber and frequency, and using the method of multiple scales and self-adjointness of the governing equations with boundary conditions, the amplitude equations describing the nonlinear coupling of the two wave modes are derived as solvability conditions. This derivation also provides both physical and mathematical insights into both the phase and group velocity matching conditions. Numerical results obtained from these amplitude equations show that because of the nonlinear coupling an internal resonance between the two modes can occur under the above matching conditions. Without detuning, one mode increases monotonically in amplitude while the other decreases monotonically. Nevertheless, their changes gradually decrease with propagation distance, and each amplitude converges to a finite value. However, with wavenumber and frequency detuning, both amplitudes increase and decrease cyclically with propagation distance, resulting in a weakening of this resonance. Furthermore, results with damping effects show an attenuation of both amplitudes with propagation distance, which limits the distance resonance waves can propagate.

The primary and subharmonic simultaneous resonance of Duffing oscillator with fractional-order derivative is studied. Firstly, the approximately analytical solution of the resonance is obtained by the method of multiple scales, and the correctness and satisfactory precision of the analytical solution are verified by numerical simulation. Then, the amplitude–frequency curve equation and phase–frequency curve equation are derived from the analytical solution. The stability condition of the steady-state response is obtained by Lyapunov’s first method, and the state switching between two stable periodic orbits is demonstrated. Finally, the effects of nonlinear factor on the system response are analyzed, and the difference between stiffness softening and stiffness hardening system is demonstrated. The influence of fractional-order term on the system is analyzed in depth, and the effect mechanism of fractional-order term is revealed, i.e., the focus and intensity of effect are determined by the order and coefficient of the fractional-order derivative, respectively.

Nonlinear vibrations of viscoelastic thin cylindrical shells are studied in this paper. The viscoelastic properties are modeled using the Kelvin–Voigt fractional-order constitutive relationship. Based on the nonlinear Love thin shell theory, the structural dynamics of the cylindrical shell is modeled by using the Newton’s second law, and the Galerkin method is used to discretize the nonlinear partial differential equations into the set of nonlinear ordinary differential equations. The method of multiple scales is used to solve the nonlinear ordinary differential equations, and the amplitude–frequency and phase–frequency equations are extracted. The obtained results are verified with available investigations, and the effects of fractional parameters, excitation, and nonlinearity on the amplitude–frequency and phase–frequency responses of the viscoelastic cylindrical shells are outlined.

To better understand the dynamic behaviors of cable-stayed bridges, this study investigates the dynamic behaviors of a cable-stayed shallow arch subjected to two external harmonic excitations using the analytical approach. First, dimensionless planar vibration equations of the system are obtained by applying the Hamilton principle, and three ordinary differential equations of the arch and the two cables are obtained by using the Galerkin discretization method. Second, modulation equations involving the amplitude and phase of the dynamic response of the system are derived by applying the method of multiple scales. Third, three simultaneous resonance cases are considered. Finally, parametric study results are illustrated through frequency responses, amplitude responses, phase plane and bifurcation diagrams. Chaos phenomenon is also detected and presented. To validate the developed analytical solutions, numerical simulations are conducted by applying the Runge–Kutta method to integrate the original ordinary differential equations. The results demonstrate that acceptable consistency is reached in the results obtained from the analytical solutions and the Runge–Kutta method in the three simulated cases. The obtained results show that the system’s dynamic responses in the three simulated cases exhibit similarities in their frequency and amplitude responses, while some qualitative differences exist in the phase plane portraits (e.g., period-1, period-2, period-3 solutions) and their bifurcation diagrams.

The multipulse homoclinic orbits and chaotic dynamics of a reinforced composite plate with the carbon nanotubes (CNTs) under combined in-plane and transverse excitations are studied in the case of 1 : 1 internal resonance. The method of multiple scales is adopted to derive the averaged equations. From the averaged equations, the normal form theory is applied to reduce the equations to a simpler normal form associated with a double zero and a pair of pure imaginary eigenvalues. The energy-phase method proposed by Haller and Wiggins is utilized to examine the global bifurcations and chaotic dynamics of the CNT-reinforced composite plate. The analytical results demonstrate that the multipulse Shilnikov-type homoclinic orbits and chaotic motions exist in the system. Homoclinic trees are constructed to illustrate the repeated bifurcations of multipulse solutions. In order to verify the theoretical results, numerical simulations are given to show the multipulse Shilnikov-type chaotic motions in the CNT-reinforced composite plate. The results obtained here imply that the motion is chaotic in the sense of the Smale horseshoes for the CNT-reinforced composite plate.

Bifurcation analysis of the nonlinear vibration of an inextensible cantilever beam is analyzed by using the nonlinear normal mode concept. Two flexural modes of the cantilever beam, one in each transverse plane is considered. Two degrees-of-freedom nonlinear model for the vibration in the transverse direction is obtained by the discretization of the governing equation using Galerkins method based on the eigenmodes in each direction. The method of multiple scales is used to derive two first-order nonlinear ordinary differential equations governing the modulation of the amplitude and the phase of the dominant mode for the case of 1:1 internal resonance. The bifurcation diagrams are computed considering the frequency of excitation and the magnitude of the excitation as the control parameters. The stability of the fixed point is determined by examining the eigenvalues of the Jacobian matrix. The results show that a saddle-node-type bifurcation of the solution can occur under certain parameter conditions.

Bifurcations in the Thomas cyclic system leading from simple dynamics into chaotic regimes are considered. In particular, the existence of only one non-trivial fixed point of the system in parameter space implies that this point attractor may only be destabilized via a Hopf bifurcation as the single system parameter is varied. Saddle-node, transcritical and pitchfork bifurcations are precluded. The periodic orbit immediately following the Hopf bifurcation is constructed analytically by the method of multiple scales, and its stability is determined from the resulting normal form and verified by numerical simulations. The dynamically rich range of parameters past the Hopf bifurcation is next systematically explored. In particular, the period-doubling sequences there are found to be more complex than noted previously, and include period-three-like windows for instance. As the system parameter is decreased below these period-incrementing bifurcations, various additional features of the subsequent crises are also carefully tracked. Finally, we consider the effect of delay on the system, leading to the suppression of both the Hopf bifurcation as well as all of the subsequent complex dynamics. In modern terminology, this is an example of Amplitude Death, rather than Oscillation Death, as the complex system dynamics is quenched, with all of the variables settling to a fixed point of the original system.

The aim of this study is to present a general model corresponding to some structural elements such as beam, string, bar and rod. The structural element material may be either elastic or Kelvin-Voigt type visco-elastic. So far, the general models with constant coefficients have considered in the literature. The considered general model is with variable coefficients. For solving general model with variable coefficients, a different solution procedure combining method of multiple scales (MMS) and finite difference method (FDM) are presented. This technique provides an advantage in the numerical solution of the structural element model containing any discontinuity and in its dynamical analysis by perturbation method. Furthermore, two problems including discontinuity are considered to indicate accuracy of the method presented. The comparison of the numerical results obtained from the proposed method and classical method are introduced.

Specific solutions of the nonlinear Schr\"odinger equation, such as the Peregrine breather, are considered to be prototypes of extreme or freak waves in the oceans. An important question is, whether these solutions also exist in the presence of gusty wind. Using the method of multiple scales, a nonlinear Schr\"odinger equation is obtained for the case of wind forced weakly nonlinear deep water waves. Thereby, the wind forcing is modeled as a stochastic process. This leads to a stochastic nonlinear Schr\"odinger equation, which is calculated for different wind regimes. For the case of wind forcing which is either random in time or random in space, it is shown that breather type solutions such as the Peregrine breather occur even in strong gusty wind conditions., Comment: 30 pages, 15 figures

For the electrically actuated microbeam subjected to a combination of DC and AC voltage loadings, we studied the nonlinear dynamical responses and internal resonance analytically. The flexible boundary condition is considered. The modal interaction due to three-to-one internal resonance between the first and second modes is highlighted. The method of multiple scales is employed to get the modulation equation, which describes the amplitude and phases of the involving modes. Then, the equilibrium solutions of the modulation equation are calculated and their stability is examined. The frequency and force response curves reflecting the primary resonance (the first mode) are presented. We find that the first and second modes in the proposed model are coupled, and hence the energy transfer can occur between the involving modes. Moreover, it is found that the response of the system may encounter Hopf bifurcation for some specific parameters. As for simulation, the Galerkin scheme is applied to verify the analytical results in terms of time history, phase-plane portrait, Fourier spectrum, and Poincare section. The simulation results are in good agreement with the analytical ones.

When a bi-stable oscillator undergoes a supercritical Hopf bifurcation due to a galloping instability, intra-well limit cycle oscillations of small amplitude are born. The amplitude of these oscillations grows as the flow speed is increased to a critical speed at which the dynamic trajectories escape the potential well. The goal of this paper is to obtain a simple yet accurate analytical expression to approximate the escape speed. To this end, three different analytical approaches are implemented: (i) the method of harmonic balance, (ii) the method of multiple scales using harmonic and elliptic basis functions, and (iii) the Melnikov criterion. All methods yielded an identical expression for the escape speed with only one key difference which lies in the value of a constant that changes among the different methods. A comparison between the approximate analytical solutions and a numerical integration of the equation of motion demonstrated that the escape speed obtained via the multiple scales method using the elliptic functions and the Melnikov criterion are in excellent agreement with the numerical simulations. On the other hand, the first-order harmonic balance technique and the multiple scales using harmonic functions provide analytical estimates that significantly underestimate the actual escape speed. Using the Melnikov criterion, the influence of parametric and additive noise on the escape speed was also studied.

I have applied multiple-scale perturbation theory to a generalized complex PT-symmetric Mathieu equation in order to find the stability boundaries between bounded and unbounded solutions. The analysis suggests that the non-Hermitian parameter present in the equation can be used to control the shape and curvature of these boundaries. Although this was suggested earlier by several authors, analytic formulas for the boundary curves were not given. This paper is a first attempt to fill this gap in the theory.

In this present work, the dynamic behavior of a pneumatic artificial muscle (PAM) has been studied by varying different muscle parameters and air pressure into it. The governing equation of motion has been derived using Newton’s law of motion to study the various responses in the system at principal parametric resonance condition. The temporal equation of motion contains various nonlinear parameters with forced and nonlinear parametric excitation. Then, the second-order method of multiple scales is used to find the approximate solutions and to study the dynamic stability and bifurcations of the system. The results are found to be in good agreement with the solutions obtained by solving the temporal equation of motion numerically. The instability regions by varying different system parameters have been plotted. The time responses and phase portraits have been plotted to study the system behavior with the nonlinearity. The influences of the different system parameters in the amplitude for the muscle have also been studied with the help frequency responses. In order to verify the solution, the basin of attraction has also been plotted. The obtained results will be very useful for designing the desired PAM use in different rehabilitation robotics and exoskeleton system.

The nonlinear dynamics of functionally graded (FG) microbeams have been studied based on the von Karman nonlinear theory, the modified couple stress and Euler–Bernoulli beam theories. The internal damping of materials is taken into account using the Kelvin–Voigt model. The coupled nonlinear mode equations of FG microbeams are obtained using Hamilton's principle and Galerkin's method. The primary resonance and internal resonance are investigated by means of the method of multiple scales and the Runge–Kutta numerical method. The effects of the length scale parameter, volume fraction exponent and internal damping constant on the nonlinear vibration are discussed using the numerical simulation. The numerical results show that periodic, period-n, and chaotic oscillations can be displayed by changing the length scale parameter, volume fraction exponent and internal damping constant. Boundary conditions can also change the nonlinear vibration behavior of FG microbeams. In the present study, the second natural frequency is approximately equal to three times the first one for the FG clamped–clamped microbeam, and a three-to-one internal resonance is detected using the time response curves. The present analysis is validated by comparing the numerical results with existing results and very good agreement is obtained.

The history of formulation of the efficient method for studying the nonlinear dynamic response of structures, damping features of which depend on natural frequencies of vibrations, is presented. This technique is the modified version of the method of multiple scales, the efficiency of which is illustrated by the examples of nonlinear vibrations of suspension bridges and plates subjected to the conditions of the internal resonances.

Global bifurcations and multi-pulse chaotic motions of cantilevered pipes conveying time-varying fluid under external excitation are investigated. The method of multiple scales and Galerkin’s approach are utilized on the partial differential governing equation to yield the four-dimensional averaged equation with 1:2 internal resonance and primary parameter resonance. Based on the averaged equations, the normal form theory is adopted to derive the explicit expressions of normal form associated with a double zeroes and a pair of purely imaginary eigenvalues. Then the energy-phase method is employed to analyze the chaotic dynamics by identifying the existence of the multi-pulse Silnikov-type orbits in the perturbed phase space. The homoclinic trees which describe the repeated bifurcations of multi-pulse solutions are demonstrated in both Hamiltonian and dissipative perturbation. The diagrams indicate a gradual breakup of homoclinic tree in the system as the dissipative factor grows. Numerical simulations are performed to show the multi-pulse jumping orbits and Silnikov-type chaotic behaviors may occur. The influence of the external excitation and the flow velocity of the cantilevered pipe on the dynamics of the system is discussed simultaneously in numerical results. The global dynamics also exhibits the existence of the chaos in the sense of Smale horseshoes for the cantilevered pipe conveying time-varying fluid under external excitation.

This paper presents a non-linear piezoelectric size-dependent Rayleigh's beam model, based on modified couple stress theory, a non-classical continuum theory that is sensitive to the size effect. Hamilton principle is employed to determine the partial differential governing equations and the boundary conditions. Free periodic vibrations are studied, with the goal of investigating the modes of vibration of piezoelectric small-scale beams in the non-linear regime, a study that appears to be lacking in the literature. The method of multiple scales and a shooting method are applied to solve the equations of motion; the stability of the solutions computed by the shooting method is investigated. A hinged-hinged beam is chosen as an example to delineate the non-linear size dependent free vibration behaviour. For the first time in this specific problem, secondary branches due to bifurcations are found. An important novelty of the work is that these secondary branches are determined using the shooting method, which does not suffer from the drawbacks of popular alternatives, as perturbation methods and the method of harmonic balance.

Nonlinear transverse vibrations of in-plane accelerating viscoelastic plates are analytically and numerically investigated in the presence of principal parametric and 3:1 internal resonance. Due to the axial acceleration, the plate tension varies in the transporting direction. For a range of mean velocity, the natural frequency of the second mode is almost equal to three times that of the first mode, and that leads to a possible 3:1 internal resonance. The governing equation and the corresponding boundary conditions are derived from the generalized Hamilton principle. The method of multiple scales is applied to reveal that the steady-state responses have two types: trivial and nontrivial (two-mode) solutions. The stabilities of the steady-state responses are determined via the Routh–Hurwitz criterion. The analytical investigations demonstrate the effects of viscous damping coefficient, the viscoelastic coefficient, and the moving speed fluctuation amplitude on the stability of zero solutions and the amplitude of the nontrivial solutions. A differential quadrature scheme is developed to solve numerically the governing equation under the given boundary conditions. The numerical results agree well with the approximate analytical results.

A machining tool can be subject to different kinds of excitations. The forcing may have external sources (such as rotating imbalance, misalignment of the workpiece or ultrasonic excitation), or it can arise from the cutting process itself (e.g., periodic chip formation). We investigate the classical one-degree-of-freedom tool vibration model, a delay-differential equation with quadratic and cubic nonlinearity, and periodic forcing. The method of multiple scales is used to derive the slow flow equations. Stability and bifurcation analysis of equilibria of the slow flow equations is presented. Analytical expressions are obtained for the saddle-node and Hopf bifurcation points. Bifurcation analysis is also carried out numerically. Sub- and supercritical Hopf, cusp, fold, generalized Hopf (Bautin), Bogdanov–Takens bifurcations are found. Limit cycle continuation is performed using MatCont. Local and global bifurcations are studied and illustrated with phase portraits and direct numerical integration of the original equation.

Taut cables such as those used in cable-stayed bridges are prone to exert a large amplitude response due to support motion. Support motion generally involve longitudinal and transverse components with respect to the cord. Such motions induce parametric and external excitation of the cable. For a certain excitation frequency, multimodal interaction due to simultaneous parametric and primary resonance can be expected. Previous studies have focused on changes in the boundary curves of the primary region of the parametric resonance affected by forced vibrations due to primary resonance. The hysteresis regions have not been determined; they have only been analyzed using the frequency-amplitude curves of the response. Moreover, the influence of the longitudinal and transverse displacement ratio has not been discussed thus far. This paper presents a complete formulation of the continuum equations of a cable model that is excited by support motion. A nonlinear discretized model that includes quadratic and cubic nonlinearities is obtained using the Galerkin method. Further, the analytical solution is determined using the method of multiple scales (MMS). By obtaining the expressions for the amplitudes and phases, the mathematical conditions are set for the amplitude of the parametric and primary resonance from which analytical expressions for the boundary curves of the interaction resonance region are derived. Local stability analysis is conducted for the steady-state response, and direct numerical integration is used for validating the frequency-amplitude curves obtained using the MMS. The solutions are verified using two independent numerical models. It is shown that the ratio of the transverse and longitudinal components of support motion significantly affects the resonance region, and several different response solutions can be obtained. It is also shown that different values of the mechanical cable parameters affect the resonance region and vibration amplitude.

In this paper, the nonlinear forced vibration of functionally graded (FG) Timoshenko microbeams under thermal effects and parametric excitation is studied using von Karman nonlinear theory, Hamilton's principle and the modified couple stress theory. The coupled nonlinear differential equations of transverse vibration modes are obtained by the static condensation and Galerkin scheme. The nonlinear algebraic equations of steady-state responses are presented based on the method of multiple scales, and solved using Newton iterative method. Finally, the internal resonance and principal resonance are proposed, and the influence of system parameters, involving the material parameter, length scale parameter, frequency of parametric excitation and thermal loading on the frequency-response is investigated. The reliability of presented Timoshenko microbeam model is confirmed by comparison with previously published results.

This study focuses on the analysis of the linear and nonlinear dynamics of a reduced nonlinear coupling model of cable-stayed bridge consisting of two cables and a shallow arch, considering the effect of geometric nonlinearity of cables and the shallow arch. The planar 1:2:2 internal resonance among three first modes of cables and the shallow arch is firstly investigated with the external harmonic excitation applied on the shallow arch. Partial differential equations that govern motion of the system are derived. The piecewise mode function of the shallow arch, satisfying continuous and mechanical conditions simultaneously, is newly derived and chosen as the trial function in Galerkin’s method rather than commonly trigonometric function. Moreover, Galerkin’s method is applied to get the corresponding ordinary differential equations of the system and the method of multiple scales is used to obtain modulation equations for exploring the nonlinear dynamic behaviors. Lastly, the frequency- and force-response curves are given to investigate the planar dynamic behaviors of the system. Some interesting and novel conclusions have been obtained.

An investigation is presented for primary resonance and internal resonance of two-degree-of-freedom (TDOF) viscoelastic system with some complex nonlinear coupled terms under randomly disordered periodic excitations. The assumed viscoelastic damping depending on the past history of motion is chosen as the form of convolution integrals over an exponentially decaying kernel function. Then, the steady-state responses of the TDOF system are discussed in detail through the method of multiple scales (MMS), which is employed to derive the determined and stochastic differential equations of amplitude and phase modulation for the internal and external resonance modes. Further, the numerical simulation method is absorbed to test the effectiveness and accuracy of the theoretical analysis solutions for steady-state moments with the changing excitation amplitude. The appearance of (random) jump, (random) saturation, and double-jumping can be explored by the development of steady-state moments with different excitation amplitudes and frequency, which also show the resonance bandwidth increases with the excitation amplitude. Besides, viscoelastic damping parameters have great influence on the steady-state responses under the different detuning conditions, which accelerate or delay the appearance of the above mentioned phenomena. Numerical simulation results of phase portraits tell that the increase of the intensity of the random excitation leads to the steady-state solutions changing from a limit cycle to a diffused limit cycle, thus, such multi-valued steady-state responses produce jump from one stable branch to another also response to the previous analysis conclusion.

We study the chaotic vibration of a rotor with an asymmetric and non-linear stiffness. The system of motion equations of the rotor in the fixed coordinate system is derived and converted to the normal form by the method of multiple scales. On the basis of the normal form, we derive the conditions under which the chaotic vibration of the Shilnikov-type in the rotor can occur. Finally, the numerical simulations are presented to verify the above-mentioned conditions.

The analysis on the nonlinear dynamics of the circular truss antenna is an important problem for its design and control since its large flexible structure. This paper investigates the nonlinear dynamic behaviors of a beam-ring structure modeling the circular truss antenna subjected to the periodic thermal excitation. Based on describing the displacements and the nonlinear strains of the beam-ring structure, the kinetic energy and potential energy are calculated for the system. Using Hamilton's principle, the nonlinear partial differential governing equations of motion and the boundary conditions are derived for the beam-ring structure. Applying Galerkin's approach, the nonlinear partial governing differential equations of motion for the beam-ring structure are truncated into a two-degree-of-freedom system of the ordinary differential equation including the quadratic nonlinearities. The four-dimensional averaged equation is obtained for the case of the primary resonance and 1:2 internal resonance by using the method of multiple scales. From the averaged equation, numerical simulations are presented to investigate the frequency–response curves and the influences of the thermal excitation on the nonlinear dynamic responses of the beam-ring structure. The numerical results demonstrate that there exist the complex nonlinear dynamic phenomena of the beam-ring structure. The finding phenomena of this paper are helpful for controlling the nonlinear vibrations of the antenna.

An axially moving printing web with variable density in a printing process causes a geometric nonlinear vibration, and a nonlinear vibration system is established using the von Karman nonlinear plate theory and the D’Alembert principle. The time and displacement variables are separated using the Galerkin method. The ordinary differential equation of a web is solved using the method of multiple scales. The amplitude–frequency response equation of a moving web is obtained. The time histories, phase–plane portraits, and amplitude–frequency curves of the system are obtained by numerical calculations. The influence of different dimensionless speeds and variable density coefficients on the nonlinear vibration characteristics of the printing web is analyzed. The results show that the overprinting accuracy can be ensured by making a reasonable choice of web speed in the stable region.

Made available in DSpace on 2019-10-04T12:37:12Z (GMT). No. of bitstreams: 0 Previous issue date: 2019-03-01 Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) Nowadays, a novelty of devices that use magnetic restoring forces to generate oscillations has increased substantially. These kinds of devices have been commonly used to energy harvesting area. Therefore, in this paper, numerical and analytical analyses of a non-ideal magnetic levitation system are carried out. The mathematical modeling of the magnetic levitation device is developed and examined considering an electrodynamical shaker to base-excite the main system, which is a non-ideal excitation. The magnetic levitation system has the form of a Duffing oscillator; thus, the nonlinear analysis is required to investigate the energy harvesting potential of this nonlinear system. The novelty here is the use of the shaker to the excitation which is non-ideal. The method of multiple scales is applied to investigate the modes of vibration of the coupled system, which will remark the non-ideality and nonlinear phenomena of the system. The average harvested power is described by through expressions related to the coupling between the mechanical and electrical domains. Moreover, it was developed an expression for the excitation frequency where the maximum harvested power is obtained. The results were obtained based on the numerical method of Runge-Kutta of fourth order with fixed step whose results are shown through phase planes, Poincare maps and parametrical variation. Such results showed multiple existence of behaviors (periodic, quasiperiodic and chaos), coexistence of attractors in a high sensibility of the initial conditions and interesting results of the maximum average power, obtaining high and continuous amount of energy in periodic and chaotic regions. Univ Tecnol Fed Parana, Dept Elect, Ave Monteiro Lobato, BR-84016210 Ponta Grossa, SP, Brazil Sao Paulo State Univ, Fac Mech Engn Bauru, BR-17033360 Bauru, SP, Brazil Univ Fed Sao Joao del Rei, Dept Math & Phys, BR-36420000 Ouro Branco, MG, Brazil Sao Paulo State Univ, Fac Mech Engn Bauru, BR-17033360 Bauru, SP, Brazil CNPq: 447539/2014-0

In this paper the dynamics of differential equation with two linearly state-dependent delays is considered, with particular attention focused on non-resonant double Hopf bifurcation. Firstly, we identify the sufficient and necessary conditions of the double Hopf bifurcation by formal linearization, linear stability analysis and Hopf bifurcation theorem. Secondly, in the first time the method of multiple scales (MMS) is employed to classify the dynamics in the neighborhoods of two kinds of non-resonant double Hopf bifurcation points, i.e. Cases III and Ib, in the bifurcation parameters plane. Finally, numerical simulation is executed qualitatively to verify the previously analytical results and demonstrate the rich phenomena, including the stable equilibrium, stable periodic solutions, bistability and stable quasi-periodic solution and so on. It also implies that MMS is simple, effective and correct in higher co-dimensional bifurcation analysis of the state-dependent DDEs. Besides, the other complicated dynamics, such as the switch between torus and phase-locked solution, period-doubling and the route of the break of torus to chaos are also found in this paper.

In this article, the propagation of nonlinear shear horizontal waves for some comparisons between the heterogeneous and homogeneous plates is considered. It is assumed that one plate is made of up hyper-elastic, heterogeneous, isotropic, and generalized neo-Hookean materials, and the other consists of hyper-elastic, homogeneous, isotropic, and generalized neo-Hookean materials. Using the known solitary wave solutions, called bright and dark solitary wave solutions, to the nonlinear Schrodinger equation, these comparisons are made in terms of the heterogeneous and nonlinear effects. All numerical results, based on the asymptotic analyses in which the method of multiple scales is used, are graphically presented for the lowest dispersive symmetric branch of both dispersion relations.

A mechanical system composed of two weakly coupled oscillators under harmonic excitation is considered. Its main part is a vibro-impact unit composed of a linear oscillator with an internally colliding small block. This block is coupled with the secondary part being a damped linear oscillator. The mathematical model of the system has been presented in a non-dimensional form. The analytical studies are restricted to the case of a periodic steady-state motion with two symmetric impacts per cycle near 1:1 resonance. The multiple scales method combined with the sawtooth-function-based modelling of the non-smooth dynamics is employed. A conception of the stability analysis of the periodic motions suited for this theoretical approach is presented. The frequency–response curves and force–response curves with stable and unstable branches are determined, and the interplay between various model parameters is investigated. The theoretical predictions related to the motion amplitude and the range of stability of the periodic steady-state response are verified via a series of numerical experiments and computation of Lyapunov exponents. Finally, the limitations and extensibility of the approach are discussed.

Using a discrete version of the method of multiple scales, the problem of fluid flow through an arbitrary periodic network of one-dimensional channels is homogenized to give Darcy flow. A simple, low-dimensional, linear algebra problem gives the effective permeability in terms of the channel conductances. The results are illustrated with an investigation of the statistical properties of effective permeability for a variety of random networks, as well as an evaluation of the correlation between permeability and network tortuosity.

Due to the complementary direct and inverse piezoelectric effects, piezoelectric materials have been proposed for various critical applications. More recently, there has been increased attention toward extending their realm to produce higher power or to detect large strains, which can potentially force them to exhibit several nonlinear phenomena. Modeling these nonlinear phenomena is still in its infancy, and universally agreed nonlinear constitutive relations are still under development. The vital step in understanding the experimental response and in proposing corresponding nonlinear constitutive relations involves identification of the nonlinear material parameters. In this chapter, we consider a unimorph subjected to (a) direct and (b) parameter excitations, and discuss several aspects of mathematical modeling of the nonlinear piezoelectric behavior and propose parameter identification schemes using the approximate solution of the governing equations obtained from the method of multiple scales.

A nonlinear Euler–Bernoulli model of slender piezoelectric beams is employed to investigate parametric resonance motions driven by a pulsating voltage with a DC component. The beam model is based on 3D electric charge conservation and 1D reduction of the momentum balance laws assuming as space coordinate the arclength along the beam base line as space coordinate. The 3D constitutive relationships for a piezoelectric material are specialized according to the Euler–Bernoulli ansatz of transverse unshearability. The nonlinear piezoelastic problem of the parametrically excited beam is directly attacked by the method of multiple scales up to the fifth nonlinear order overcoming modal projection drawbacks and severe restrictions on the oscillation amplitude range. The transition curves, separating regions of stable and unstable trivial solutions in the voltage frequency–amplitude plane, are obtained for various modes of a PVDF beam. The onset of parametric resonances and the post-critical large motion are investigated upon variations of meaningful parameters such as the prestress DC voltage amplitude. The analytical outcomes are compared with those provided by the finite element code ABAQUS which yields the numerical solution of the full 3D nonlinear piezoelectric problem. Moreover, the frequency–response curves to within fifth order are computed applying the pseudo-arclength method to the algebraic frequency–response equation. The frequency–response curves of the lowest few modes reveal new aspects of the parametric resonance motions which can be exploited in applications such as dynamic morphing of thin surfaces.

There are typically several perturbation methods for approaching the solution of weakly nonlinear vibrations (where the nonlinear terms are “small” compared to the linear ones): the Method of Strained Parameters, the Naive Singular Perturbation Method, the Method of Multiple Scales, the Method of Harmonic Balance and the Method of Averaging. The Straightforward Expansion Perturbation Method (SEPM) applied to weakly nonlinear vibrations does not usually yield to correct solutions. In this manuscript, we provide mathematical proof of the inaccuracy of the SEPM in general cases. Nevertheless, we also provide a sufficient condition for the SEPM to be successfully applied to weakly nonlinear vibrations. This mathematical formalism is written in the syntax of the first-order formal language of Set Theory under the methodology framework provided by the Category Theory.

International audience; This paper investigates a problem of passive mitigation of vibratory instabilities caused by two unstable modes by means of a single nonlinear energy sink (NES). For this purpose, a linear four-degree-of-freedom (DOF) primary structure having two unstable modes (reproducing the typical dynamic behavior of a friction system) and undergoing, as it is linear, unbounded motions when it is unstable, is coupled to a NES. In this work, the NES involves an essentially cubic restoring force and a linear damping force. We are interested in characterizing analytically the response regimes resulting from the coupling of the two unstable linear modes of the primary structure and the nonlinear mode of the NES. To this end, from a suitable rescaling of the governing equations of the coupled system in which the dynamics of the primary structure is reduced to its unstable modal coordinates, the complexification-averaging method is applied. The resulting averaged system appears to be a fast-slow system with four fast variables and two slow ones related to the two unstable modes of the primary structure. The critical manifold of the averaged dynamics is obtained through the geometric singular perturbation theory and appears as a two-dimensional parametric surface (with respect to two of the four fast variables) which evolves in the whole six-dimensional variable space. The asymptotic analysis reveals that the NES attachment can produce some bounded responses and suggests that the system may have simultaneous stable attractors. Numerical simulations complement the study, highlighting a possible competition between stable attractors and allowing us to investigate their basins of attraction. In each considered situation, a good agreement has been observed between theoretical results and numerical simulations, which validates the proposed asymptotic analysis.

In this article, the non-linear anti-symmetric shear motion for some comparative studies between the non-homogeneous and homogeneous plates, having two free surfaces with stress-free, is considered. Assuming that one plate contains hyper-elastic, non-homogeneous, isotropic, and generalized neo-Hookean materials and the other one consists of hyper-elastic, homogeneous, isotropic, and generalized neo-Hookean materials. Using the method of multiple scales, the self-modulation of the non-linear anti-symmetric shear motion in these plates, as the non-linear Schrodinger (NLS) equations, can be given. Using the known solitary wave solutions, called bright and dark solitary wave solutions, to NLS equations, these comparative studies in terms of the non-homogeneous and non-linear effects are made. All numerical results, based on the asymptotic analyses, are graphically presented for the lowest anti-symmetric branches of both dispersion relations, including the deformation fields of plates.

Recent studies in passively-isolated systems have shown that mode coupling is desirable for best vibration suppression, thus refuting the long-standing rule of mode decoupling. However, these studies have ignored the non-linearities in the isolators. In this work, we consider stiffness nonlinearity from pneumatic isolators and study the nonlinear free undamped vibrations of a passively-isolated ultra-precision manufacturing (UPM) machine. Experimental analysis is conducted to guide the mathematical formulation. The system comprises linearly and nonlinearly coupled in-plane horizontal and rotational motion of the UPM machine with quadratic nonlinear stiffness from the isolators. We present closed-form expressions using the method of multiple scales for two cases viz. the non-resonant case and the bounded internal resonance case. We validate our theoretical findings through direct numerical simulations. For the non-resonant case, we show that the system behaves similar to a linear system. However, for the nearly internal resonance case, we demonstrate strong energy exchange between the modes stemming from nonlinear mode coupling. We further study the effect of nonlinear mode coupling on the vibration isolation performance and demonstrate that mode coupling is not always desirable.

In this paper, we introduce a three-component Schnakenberg model, whose key feature is that it has a solution consisting of N spikes that undergoes Hopf bifurcations with respect to N distinct modes nearly simultaneously. This results in complex oscillatory dynamics of the spikes, not seen in typical two-component models. For parameter values beyond the Hopf bifurcations, we derive reduced equations of motion which consist of coupled ordinary differential equations (ODEs) of dimension 2N for spike positions and their velocities. These ODEs fully describe the slow-time evolution of the spikes near the Hopf bifurcations. We then apply the method of multiple scales to the resulting ODEs to derive the long-time dynamics. For a single spike, we find that its long-time motion consists of oscillations near the steady state whose amplitude can be computed explicitly. For two spikes, the long-time behavior can be either in-phase or out-of-phase oscillations. Both in-phase and out-of-phase oscillations are stable, coexist for the same parameter values, and the fate of motion depends solely on the initial conditions. Further away from the Hopf bifurcation points, we offer numerical experiments indicating the existence of highly complex oscillations.

The history of formulation of the efficient method for studying the nonlinear dynamic response of structures, damping features of which depend on natural frequencies of vibrations, is presented. This technique is the modified version of the method of multiple scales.

The chapter deals with the analytical investigation of the physical pendulum mounted on the spring-damper suspension. The pendulum exhibits three degrees of freedom in plane motion. Three types of external loading and viscous damping are considered. The method of multiple scales is used to solve the initial value problem defined through Lagrange formalism. The proposed analytical method allows for the prediction of resonance conditions. The amplitude-frequency response curves have been determined for the external resonance, and their stability has been assessed using the Routh–Hurwitz criterion. The modulation equations of the amplitudes and phases are the basis for studying the impact of the chosen parameters on the internal resonance 1:2.

The nonlinear dynamic response of carbon nanotube (CNT)/polymer nanocomposite beams to harmonic base excitations is investigated asymptotically via the method of multiple scales. The hysteresis associated with the CNT/polymer interfacial frictional sliding is described by a 3D mesoscopic theory reduced via a uniaxial strain assumption for a beam in pure plane bending. Such reduction leads to a Bouc–Wen-like hysteretic moment–curvature relationship. The generalized memory-dependent constitutive law is developed asymptotically and, subsequently, introduced in two archetypal cases of nonlinear beam models. A beam model is tailored for axially restrained, extensible beams (e.g., hinged–hinged beams) for which the dominant geometric nonlinearity is associated with the multiplicative effect of the tension with the bending curvature. The second model is valid for inextensible beams (e.g., cantilever beams) dominated by inertia and curvature nonlinearities. The piece-wise integration of the moment–curvature relationship yields an exponential law which is treated asymptotically to obtain the quadratic and cubic curvature contributions. The ensuing asymptotic equations of motion in the unknown deflection field are discretized according to the Galerkin method employing the eigenmode directly excited near its primary resonance to thus obtain a piece-wise reduced-order model (ROM). The method of multiple scales applied to the ROM yields the asymptotic response together with the frequency response functions for the lowest mode. A parametric study unfolds rich nonlinear dynamic responses in terms of behavior charts highlighting regions of hardening and softening behavior, regions of single-valued stable behavior and regions of multi-valued multi-stable behavior. Such richness of responses is caused by the unusual and unique combination of material and geometric nonlinearities.

This paper examines the role of stiffness nonlinearity on a periodic one-dimensional chain with multiple local resonators. The cells of the chain consist of lumped masses connected through nonlinear springs. Each cell is embedded with multiple local resonators having different parameters. In one case the local resonators are assumed to be linear and in another case they are nonlinear. The dispersion equation for the system is derived analytically by the method of multiple scales (MMS). The results are validated via comparison with those in the literature and numerically via Matlab. The nonlinearity shows enhancement in the bandgap regions, especially with increasing number of local resonators.

A new multimodal theory is developed analytically using the method of multiple scales to investigate the dynamic behavior of arbitrarily sagged and inclined cables oscillating around a catenary static profile. Fully non-condensed kinematics are adopted to solve the eigenvalue problem and the enhanced modal properties obtained at this stage are used as first inputs to the resolution of the nonlinear time dependent problem leading to original results where the contribution of longitudinal vibration is better captured and the modal coupling is accurately described in both space and frequency domains.