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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.

This study presents an analytical solution for the nonlinear forced vibration response of bridged carbon nanotube (CNT)-based mass sensors subjected to different types of thermal loading and external harmonic excitations. The structure undergoes a temperature change through its thickness when considering three different distribution patterns of the temperature, namely uniform, linear, and nonlinear. The CNT is modeled as a nonlocal doubly clamped beam that complies with Euler–Bernoulli beam theory and Eringen’s nonlocal elasticity theory. The von Karman geometric nonlinearity is adopted to account for the mid-plane stretching of end-constrained beams. The extended Hamilton’s principle is used to derive the nonlinear governing equations of motion, boundary conditions, and continuity conditions accounting for the location of the deposited particle. The method of multiple scales (MMS) is employed to perform the nonlinear dynamical analysis of the nanobeam. Analytical expressions of the nonlinear dynamic response of the system exposed to thermal loadings in the case of primary resonance are presented. An important contribution of this work is the provision for closed-form expressions of the resonance frequency and the critical amplitude of the considered system. Simultaneously, validation of the results by MMS is achieved by Galerkin’s procedure and Runge–Kutta method. Then, the effects of the thermal loads and some key parameters on the nonlinear resonance frequency and amplitude shifts are discussed, and some meaningful conclusions are drawn.

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.

In this paper, an efficient perturbation algorithm combining the method of Multiple Scales and Modified Lindstedt–Poincare Techniques is proposed to solve the equation of Van der Pol oscillator with very strong nonlinearity. This algorithm combines the advantages of both methods. Solution of Van der Pol equation by the Multiple Scales Modified Lindstedt–Poincare (MSMLP) method is compared with the Multiple Scales method and numerical solution using MATLAB 7.8. The convergence criterion for the solution by Multiple Scales and MSMLP methods is discussed and shown that Multiple Scales method fails the convergence criterion for large values of small parameter, while MSMLP method satisfies the convergence criterion for both small and large values. Numerical simulation has been performed in MATLAB 7.8 for different values of small parameter to prove the efficiency and accuracy of the proposed method.

Nonlinear coupling between modes through internal resonance has many applications in MEMS resonators. The primary objective of this work is to systematically study all possible internal resonance conditions between the first three modes and the feasibility of mode interaction in an electrostatically excited, straight, clamped-clamped beam. The static displacement and the first three natural frequencies of the beam are obtained for an applied DC voltage by using Galerkin based reduced order model and finite element method. All six possible commensurable relations between the first three frequencies of the beam are obtained by sweeping the non-dimensional parameters ($$\alpha _1$$ and $$\alpha _2 V_\mathrm{dc}^2$$) which depend on beam dimensions, material properties and external forcing. It is also demonstrated by a further examination of the dynamical equations that only one resonance condition is capable of exhibiting modal coupling when externally excited. A detailed analysis is carried out for this feasible resonance condition by altering $$\alpha _1$$ and $$\alpha _2 V_\mathrm{dc}^2$$ and solving the relevant nonlinear coupled equations by using both numerical time integration and the method of multiple scales. From this study, we observe that the lower mode is automatically excited after driving amplitude for the higher mode reaches a critical value—a sign of mode interaction initiation. We also find that amplitude of the higher mode is saturated after interaction with the lower mode for a combination of $$\alpha _1$$ and $$\alpha _2 V_\mathrm{dc}^2$$. Moreover, we see that the frequency bandwidth of modal interaction increases with excitation amplitude for all combinations of $$\alpha _1$$ and $$\alpha _2 V_\mathrm{dc}^2$$. Finally, the role of external damping on the amplitude-frequency response curve of both modes during the mode interaction is also investigated.

In this paper, we introduce a nonlinear vibrations analysis and dynamic responses of a vertical conveyor system under multi excitation forces. By adding the nonlinear proportional derivative controller (NPD) to the vibrating motion of the vertical vibration conveyor, the energy was transferred between uncontrolled and controlled system. We calculate the approximate solutions of the vibrating system utilizing the method of multiple scales. In addition, we investigate the stability at worst resonance cases using phase plane technique, equations of frequency response, and averaging method. The vertical vibration conveyor behavior was studied numerically at the values of its different parameters. The results exhibit the efficiency of NPD control unit to avoid the oscillations of the vertical conveyor system. Numerical simulations have been carried out using MAPLE and MATLAB software's to ensure the fidelity of our results. Comparisons are made between analytical and numerical results. Also, findings of the present work are discussed in details and compared with published works.

The present work extends a recent two-dimensional analytical model for sound generation and transmission in an axial-flow outlet guide vanes row, by taking into account vane camber and stagger, as well as cut-on/cut-off or cut-off/cut-on transitions of inter-vane channel modes. It is aimed at demonstrating that such transitions may have a significant role in turbomachinery aeroacoustics, typically by changing the balance of upstream and downstream scattered waves. The model is based on a mode-matching procedure and generates a uniformly valid description of the sound field. Previous implementations neglecting the vane curvature have shown significant limitations at high frequencies compared to numerical simulations performed with a finite element method code. The extension presented in this paper introduces curvature by a slowly-varying duct formalism. Comparison with numerical simulations show substantial improvements at the cost of a reasonably higher computation time., Ce travail étend un modèle analytique bidimensionnel développé récemment pour la génération et la transmission du bruit au travers d’aubes de redresseur, par la prise en compte de la cambrure et du calage des aubes, ainsi que des transitions de modes coupé/passant ou passant/coupé dans les canaux inter-aubes. L’objectif est de démontrer qu’un tel phénomène de transition peut jouer un rôle significatif sur l’aéroacoustique des turbomachines, en modifiant la répartition des ondes diffractés à l’amont et à l’aval des aubes. Le modèle est basé sur la technique de raccordement modal et fournit une description uniformément valide du champ sonore. De précédentes implémentations négligeant la courbure des aubes ont montrées des limitations importantes à haute fréquence en se comparant à des résultats issus de simulations numériques par méthode des éléments finis. L’extension présentée dans ce travail introduit la courbure par une approche lentement variable. Les comparaisons avec la simulation numérique montrent une nette amélioration des résultats au prix d’une augmentation raisonnable du temps de calcul.

Аналіз електроенцефалограм є невід’ємна складова діагностики неврологічних захворюваннях. З огляду на їхню просторово-часову залежність традиційні підходи у клінічній практиці пов’язані із аналізом амплітуд сигналів із визначеним інтервалом його дискретизації. При цьому зовсім не враховуються залежності сигналів у просторі, що притаманне реальним змінюванням електроенцефалограм. Мета роботи полягає у розробці підходу, що надає можливість під час аналізу електроенцефалограм здорових і хворих пацієнтів урахувати просторову і часову залежність цих сигналів для підвищення якості діагностування. На ґрунті аналізу підходів до спектрального аналізу нестаціонарних медичних сигналів, зокрема, електроенцефалограм, запропоновано застосування системи ортонормованих вейвлет - функцій Морле, які, на відміну від інших розповсюджених вейвлет-функцій, здатні забезпечити отримання спектрів сигналів, що аналізуються, як у часі, так й у просторовому вимірі. Отримано спектральне подання ЕЕГ-сигналів із застосуванням вейвлет-функцій Морле. Виконано кратномасштабний аналіз електроенцефалограм для здорових і хворих пацієнтів, що надає можливість підвищити якість спектрального аналізу сигналів у просторово-часовому вимірі. Застосування вейвлет-функцій до аналізу ЕЕГ-сигналів пацієнтів, хворих на епілепсію надало можливість суттєво підвищити якість діагностики шляхом відокремлення низькочастотних і високочастотних компонент сигналу із застосуванням методів фільтрації завад. Аналіз ЕЕГ-сигналів із застосуванням вейвлет-перетворень є ефективний метод надійної ідентифікації біоелектричного стану головного мозку як здорових пацієнтів так і хворих на епілепсію. Моніторинг та аналіз пацієнта упродовж тривалішого часу може надати більше інформації про розвиток епілепсії. Вейвлет-перетворення у сполученні із штучними нейронними мережами надає можливість реалізувати класифікатор на грунті розподілу енергії складових сигналу ЕЕГ. Визначення активності окремих компонентів сигналів ЕЕГ, а також матеріальність процесів, що мають місце у джерелах цих хвиль, може стати предметом подальших досліджень. The analysis of electroencephalograms is an integral part of the diagnosis of neurological diseases. Given their spatial-temporal dependence, traditional approaches in clinical practice are associated with the analysis of signal amplitudes with a defined interval of its discretization. At the same time, the dependence of signals in space, which is inherent in real changes of electroencephalograms, is not taken into account at all. The purpose of the work is to develop an approach that makes it possible to take into account the spatial and temporal dependence of these signals during the analysis of electroencephalograms of healthy and sick patients to improve the quality of diagnosis. Based on the analysis of approaches to the spectral analysis of non-stationary medical signals, in particular, electroencephalograms, it is proposed to use a system of orthonormal wavelets - Morle functions, which, unlike other widespread wavelet functions, are capable of obtaining the spectra of the analyzed signals, both in time and in the spatial dimension. A spectral presentation of EEG signals was obtained using Morle wavelet functions. Multiple-scale analysis of electroencephalograms for healthy and sick patients was performed, which provides an opportunity to improve the quality of spectral analysis of signals in the space-time dimension. The application of wavelet functions to the analysis of EEG signals of patients with epilepsy made it possible to significantly improve the quality of diagnosis by separating low-frequency and high-frequency components of the signal using noise filtering methods. The analysis of EEG signals using wavelet transformations is an effective method of reliable identification of the bioelectrical state of the head brains of both healthy patients and patients with epilepsy. Monitoring and analyzing the patient over a longer period of time can provide more information about the development of epilepsy. Wavelet transformation in combination with artificial neural networks provides an opportunity to implement a classifier based on the energy distribution of the components of the EEG signal. Determining the activity of individual components of EEG signals, as well as the materiality of the processes taking place in the sources of these waves, may become the subject of further research.

International audience; The transition from regular to apparently chaotic motions is often observed in nonlinear flows. The purpose of this article is to describe a deterministic mechanism by which several smaller scales (or higher frequencies) and new phases can arise suddenly in a nonlinear flow under the impact of forcing terms. This phenomenon is illustrated in the context of Hamilton-Jacobi equations. It is derived from a multiscale and multiphase analysis of nonlinear differential equations involving stiff oscillating source terms. We introduce an original method allowing to get the existence of solutions during long times, as well as asymptotic descriptions and reduced models. This is achieved by implementing three main tools: a blow-up procedure (extending the normal form method), a Wentzel-Kramers-Brillouin approximation (of super-critical type) and the Hadamard's global inverse function theorem which can be applied here due to transparency conditions (coming from a complete integrability condition).

The present paper investigates the effect of oscillatory flow on columnar crystal growth in undercooled melt by means of the method of multiple scales. The resulting analytical solution shows that the oscillatory flow alternately accelerates and decelerates the columnar crystal growth. The effect of changes in the acceleration and frequency of the oscillatory flow on the temperature distribution are revealed. For low frequency and low acceleration, the interface temperature rises rapidly immediately after nucleation but then decreases. For high frequency, there is no significant variation in interface temperature with increasing frequency. For high acceleration, the interface temperature oscillates with the oscillatory flow. The undulation of the interface temperature produces local undercooling near the interface, leading to new nucleation.

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.

Free and forced vibration analysis of a geometrically rotating shaft supported on ball bearings has been studied using the numerical method and compared with the results obtained experimentally. This study is concerned with vibration analysis of geometrically nonlinear rotating model with a rigid disk. The shaft has been designed under the frame of Euler–Bernoulli beam theory with additional effects such as rotary inertia, gyroscopic effect, higher-order large deformations, and rotor mass unbalance in order to replicate an equivalent practical model of rotor-bearing system. The mathematical expressions have been derived to demonstrate the nonlinear free and forced vibrations of the rotating shaft coupled with rigid disk in two transverse planes. Solutions of the nonlinear equation are being obtained using method of multiple scales as well as numerical methods. Effects of rotor parameters such as bearing stiffness and damping coefficient are examined with help of this nonlinear mathematical model. The obtained results are portrayed for a better understanding free and forced vibration analysis with time response, FFT, phase portrait, and Poincare’s map. The present outcomes enable an understanding on how the system dynamics influenced with the variations in the values of different parameters.

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.

Piezoelectric bimorph actuators are used in a variety of applications, including micro positioning, vibration control, and micro robotics. The nature of the aforementioned applications calls for the dynamic characteristics identification of actuator at the embodiment design stage. For decades, many linear models have been presented to describe the dynamic behavior of this type of actuators; however, in many situations, such as resonant actuation, the piezoelectric actuators exhibit a softening nonlinear behavior; hence, an accurate dynamic model is demanded to properly predict the nonlinearity. In this study, first, the nonlinear stress–strain relationship of a piezoelectric material at high frequencies is modified. Then, based on the obtained constitutive equations and Euler–Bernoulli beam theory, a continuous nonlinear dynamic model for a piezoelectric bending actuator is presented. Next, the method of multiple scales is used to solve the discretized nonlinear differential equations. Finally, the results are compared with the ones obtained experimentally and nonlinear parameters are identified considering frequency response and phase response simultaneously. Also, in order to evaluate the accuracy of the proposed model, it is tested out of the identification range as well.

Recent focus has been given to spectro-spatial analysis of nonlinear metamaterials since they can predict interesting nonlinear phenomena not accessible by spectral analysis (i.e., dispersion relations). However, current studies are limited to a nonlinear chain with single linear resonator or linear chain with nonlinear resonator. There is no work that examines the combination of nonlinear chain with nonlinear resonators. This paper investigates the spectro-spatial properties of wave propagation through a nonlinear metamaterials consisting of nonlinear chain with multiple nonlinear local resonators. Different combinations of softening and hardening nonlinearities are examined to reveal their impact on the traveling wave features and the band structure. The method of multiple scales is used to obtain closed-form expressions for the dispersion relations. Our analytical solution is validated via the numerical simulation and results from the literature. The numerical simulation is based on spectro-spatial analysis using signal processing techniques such as spatial spectrogram, wave filtering, and contour plots of 2D Fourier transform. The spectro-spatial analysis provides a detailed information about wave distortion due to nonlinearity and classify the distortion into different features. The observations suggest that nonlinear chain with multiple nonlinear resonators can affect the waveform at all wavelength limits. Such nonlinear metamaterials are suitable for broadband vibration control and energy harvesting, as well as other applications such as acoustic switches, diodes, and rectifiers, allowing wave propagation only in a pre-defined direction.

The present work investigates the super-harmonic resonances combined with 2:1 internal resonance of a rotating blade subjected to strong gas pressure. The dynamic model includes the effect of the initial curved axis of the blade induced by the thermal gradient. The dimensionless gas excitation amplitude is assumed to be the same magnitude of the dimensionless vibration displacement. The vibration of the blade in the plane of rotation and the plane perpendicular to it are described by a set of coupled ordinary differential equations with quadratic and cubic nonlinearities. The steady-state response of the rotating blade is calculated via the method of multiple scales. The stabilities of the steady-state responses are determined via Lyapunov theory. Parametric studies are performed to clarify the influences of system parameters on dynamic response and unstable regions. The various typical phenomena including jump, hysteresis and saturation are observed in the dynamic model. The stable and unstable regions of the solution are analyzed in the plane of external detuning parameter and excitation amplitude. The evaluation of the blade dynamic response is revealed in the unstable region. The theoretical results obtained via the method of multiple scales coincide with the numerical solutions.

Internal resonance is a common phenomenon in super-critically axially moving beams. It can lead to an energy exchange among associated modes and produce a large-amplitude response. However, the effect of the internal resonance on the dynamics of system is not clear. Therefore, in the present paper, we investigated the nonlinear dynamics of a super-critically axially moving beam with two-to-one internal resonance. By applying a proper transporting speed, the two-to-one internal resonance condition of the system is established. The method of multiple scales is employed to solve the governing equation so as to obtain the nonlinear response of the system. To analyze the effect of the quadratic and cubic nonlinearities, the perturbation solution is expanded up to three orders. The primary resonance of the first and second modes is investigated. Their steady-state solutions are solved, and the stability of these solutions is examined. Response of the system is demonstrated via the frequency response and force-response curves. Results show that jumping, saturation, and hysteresis phenomenon may occur. Moreover, the modulated motion can be found in the case of primary resonance of the first mode. The approximate analytical results are verified by the numerical results.

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.

Zero-Reynolds-number instability in a three-layer Stokes flow of viscous fluid with an inhomogeneous layer thickness in a two-dimensional region with a free boundary is investigated. The method of multiple scales is applied for constructing an asymptotic expansion of the solution of the boundary-value problem for the Stokes equations. The stability of the system of first-approximation equations is analyzed using the Fourier method, and it is concluded that the most significant increase in the zero-Reynolds-number instability occurs in a region of waves whose lengths are comparable with the thickness of the middle layer. In contrast to the case of a constant layer thickness, the instability parameters are variables. The mechanism of formation of geological folds is studied.

Dynamic stability characteristics of variable stiffness composite plates under periodic in-plane loads are investigated here using a shear-deformable sixteen-node plate bending element. The modal transformation technique is used to reduce the equations of motion for the flexural vibration problem of a rectangular panel into a set of Mathieu-Hill equations. The diagonal and off-diagonal dominance of the parametric coefficient matrix is studied for the representative cases of in-plane loading, i.e., periodic compression/shear in presence or absence of static compression for the isotropic and composite plates. The multi-modal response (participating and accompanying modes) of the shear panels under periodic edge traction is demonstrated through the time history analysis. Then, the method of multiple scales is used to identify the dynamic instability regions corresponding to the simple (single mode flexural vibration) or combination (two mode flexural vibrations) type of resonances in various (isotropic and laminated composite made of straight or curvilinear fibers) shear panels, for which results are not available in the literature.

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.

This paper focuses on the nonlinear dynamics near internal resonance of a truncated FGM conical shell. The FGM conical shell is subjected to the aerodynamic load and the in-plane excitation along the meridian direction. Material properties depend on the temperature and the constituent phases of the truncated FGM conical shell. The volume fractions are modified in the thickness direction based on a power-law function continuously and smoothly. The first-order piston theory is applied for the supersonic aerodynamic pressure. Based on the first-order shear deformation theory, von-Karman type nonlinear geometric assumptions, Hamilton principle and Galerkin method, the nonlinear equations of motion for the truncated FGM conical shell are derived. The averaged equations of the truncated FGM conical shell are obtained under the situation of 1:1 internal resonance and 1/2 subharmonic resonance by using the method of multiple scales. The frequency-response curves, the force-response curves, the bifurcation diagrams, the phase portraits, the time history diagrams, and the Poincare maps are obtained by using numerical calculations. The influences of the Mach number, the exponent of volume fraction and the in-plane excitation on the nonlinear resonant behaviors of the truncated FGM conical shell are investigated.

We use Whitham’s averaged Lagrangian method extended with the multiple-scale formalism to derive a sixth-order nonlinear Schrodinger equation for the complex amplitude of the envelope of the slowly modulated wave trains whose evolution is governed by the nonlinear Klein–Gordon equation with polynomial nonlinearity. Such a high-order nonlinear Schrodinger equation is obtained as a variational problem for the Lagrangian averaged over the rapid oscillations of the carrier wave train. As compared to classical Whitham’s approach, we take into account the derivatives of the complex amplitude with respect to slow timescales and long coordinate in the averaged Lagrangian. The coefficients of the high-order nonlinear Schrodinger equation derived in this work identically coincide with those derived by us earlier by the method of multiple scales applied directly to the original nonlinear Klein–Gordon equation. Finally, we consider the sine-Gordon equation as a partial case of the nonlinear Klein–Gordon equation and demonstrate one example of the numerical solution to the corresponding sixth-order nonlinear Schrodinger equation in the form of the envelope quasi-soliton.

The study of both linear and nonlinear structural vibrations routinely circles the concise yet complex problem of choosing a set of coordinates which yield simple equations of motion. In both experimental and mathematical methods, that choice is a difficult one because of measurement, computational, and interpretation difficulties. Often times, researchers choose to solve their problems in terms of linear, undamped mode shapes because they are easy to obtain; however, this is known to give rise to complicated phenomena such as mode coupling and internal resonance. This work considers the nature of mode coupling and internal resonance in systems containing non-proportional damping, linear detuning, and cubic nonlinearities through the method of multiple scales as well as instantaneous measures of effective damping. The energy decay observed in the structural modes is well approximated by the slow-flow equations in terms of the modal amplitudes, and it is shown how mode coupling enhances the damping observed in the system. Moreover, in the presence of a 3:1 internal resonance between two modes, the nonlinearities not only enhance the dissipation, but can allow for the exchange and transfer of energy between the resonant modes. However, this exchange depends on the resonant phase between the modes and is proportional to the energy in the lowest mode. The results of the analysis tie together interpretations used by both experimentalists and theoreticians to study such systems and provide a more concrete way to interpret these phenomena.

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.

This work deals with the amplitude–frequency response of coaxial parametric resonance of electrostatically actuated double-walled carbon nanotubes (DWCNTs). Nonlinear forces acting on the DWCNT are intertube van der Waals and electrostatic forces. Soft alternating current (AC) excitation and small viscous damping are assumed. In coaxial vibration, the outer and inner carbon nanotubes move synchronously (in-phase). Euler–Bernoulli beam model is used for DWCNTs of high length-to-diameter ratio. Modal coordinates are used for decoupling the linearized differential equations of motion without damping. The reduced-order model (ROM) method is used in this investigation. All ROMs using one through five modes of vibration (terms) are developed in terms of modal coordinates. ROM using one term is solved and frequency–amplitude response predicted by using the method of multiple scales (MMS). All other ROMs using two through five terms are numerically integrated using MATLAB in order to simulate time responses of the structure and also solved using AUTO-07P, a software package of continuation and bifurcation, in order to predict the frequency–amplitude response. All models and methods are in agreement at lower amplitudes, while in higher amplitudes only ROM with five terms provides reliable results. The effects of voltage and damping on the amplitude–frequency response of electrostatically actuated DWCNTs are reported. It is shown that increasing voltage and/or decreasing damping results in a larger range of frequencies for which pull-in occurs.

Micro-milling is one of the micro-manufacturing techniques used for creating micro-scale features. In this article, a size-dependent formulation based on the strain gradient elasticity theory is developed to analyze the micro milling tool vibration. A new cutting forces formulation in rotating frame is presented in this paper. Considering structural nonlinearities , gyroscopic moment, rotary inertia , process damping and size effect, nonlinear equations of tool motion are derived using non-classical Timoshenko beam theory and Hamilton's principal. Partial differential governing equations of the tool are converted to ordinkary differential equations by using assumed modes method. Then the method of multiple scales is used to obtain the analytical solution for tool vibrations. The proposed approach is applied to investigate the chatter instability observed in micro-milling operations. To verify the presented model, simulated stability lobe diagrams are compared with the results obtained from experimental tests and literature. According to the results, neglecting size effect, gyroscopic and rotary terms in the tool model causes significant errors in prediction of the chatter in micro-milling process.

In this paper, the nonlinear breathing vibrations of an eccentric rotating composite laminated circular cylindrical shell are studied for the first time, which is subjected to the lateral and temperature excitations. Based on Donnell thin shear deformation theory, von Karman-type nonlinear relation and Hamilton’s principle, the nonlinear partial differential governing equations of motion are established for the eccentric rotating composite laminated circular cylindrical shell. The nonlinear partial differential governing equations of motion are discretized into a set of coupled nonlinear ordinary differential equations of motion by Galerkin approach. Based on the case of 1:2 internal resonance and principal parametric resonance-1/2 subharmonic resonance, the method of multiple scales is employed to obtain four-dimensional nonlinear averaged equations. Considering the effects of different parameters, for example, the eccentricity ratio, the geometric parameters and the excitations, the nonlinear dynamic behaviors and the jumping phenomena are exhibited for the frequency-response curves and the amplitude-response curves. The periodic and chaotic motions of the eccentric rotating composite laminated circular cylindrical shell are found when the rotating speed corresponds to the internal resonant point at the certain conditions for different lateral and temperature excitations.

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.

Basic single-degree-of-freedom mechanical models of force control are presented to achieve desired contact forces between actuators and objects. Nonlinear governing equations are constructed for both collocated and non-collocated force sensor configurations. The models take into account large time delays in the feedback loop, which may occur, for example, in case of human or remote force control. The corresponding stability charts are compared for the collocated and non-collocated cases. The bifurcations at the stability boundaries are analyzed in the presence of the relevant nonlinearity that is originated in the saturation of the actuation. The stability properties as well as the nonlinear vibrations for the two sensor locations are compared also from the viewpoint of the achievable maximal proportional gains. The bifurcation calculations are done with the method of multiple scales and normal form calculations. The results on global dynamic properties are supported with numerical simulations, and the two force control strategies are discussed from application viewpoint.

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.

It is common that a spinning turbine blade may be subjected to pitching or yawing motion about other axes, as in aircrafts and water-borne vehicles. Such phenomenon can also occur in ground installed rotating machines subjected to various types of floor disturbances. This paper presents an investigation into the response and stability of a spinning turbine blade subjected to pitching and yawing motions. The partial differential equation that governs the motion of the system is developed using Hamilton’s principle. Time domain conversion using Galerkin method is carried out. The simultaneous differential equations are in the form of forced Mathieu equations. These equations are solved analytically by the method of multiple scales, assuming small pitching to spinning ratio. System response and the expressions for stability boundaries are obtained. The time domain equations are also solved numerically and compared with the analytical results.

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.

Objectives: The work is to learn the impingement of tangential periodic magnetic field on the interface in the middle of two superposed ferrofluids with the rotation through a nonlinear perturbation analysis. Methods or Statistical Analysis: A nonlinear instability due to streaming superposed ferrofluids stressed by rotation and a periodic tangential magnetic field is examined by employing the mode of multiple scales. The fluids are taken to be incompressible. Using the boundary conditions, the solutions of the linearized equations of motion lead to the linear dispersion equation. Findings: From this work, it is clearly seen the unfolding of amplitude of waves was influenced by an equation of nonlinear Schrodinger. Applications or Improvements: The rotation and a periodic tangential magnetic field have enormous important effects over the stability of the system. Keywords: Dispersion Relation and Schrodinger Equation, Ferrofluids, Linear Dispersion Relation, Method of Multiple Scales, Rotation

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.

The performance of bimorph cantilever energy harvester subjected to horizontal and vertical excitations is investigated. The energy harvester is simulated as an inextensible piezoelectric beam with the Euler–Bernoulli assumptions. A horizontal base excitation along the axis of the beam is converted into the parametric excitation. The governing equations include geometric, inertia and electromechanical coupling nonlinearities. Using the Galerkin method, the electromechanical coupling Mathieu–Duffing equation is developed. Analytical solutions of the frequency response curves are presented by using the method of multiple scales. Some analytical results are obtained, which reveal the influence of different parameters such as the damping, load resistance and excitation amplitude on the output power of the energy harvester. In the case of parametric excitation, the effect of mechanical damping and load resistance on the initiation excitation threshold is studied. In the case of combination of parametric and direct excitations, the dynamic characteristics and performance of the nonlinear piezoelectric energy harvesters are studied. Our studies revealed that the bending deformation generated by direct excitation pushes the system out of axial deformation and overcomes the limitation of initial threshold of parametric excitation system. The combination of parametric and direct excitations, which compensates and complements each other, can be served as a better solution which enhances performance of energy harvesters.

In this paper, the linear and nonlinear free vibrations of functionally graded rotating composite Timoshenko beams reinforced by carbon nanotubes are presented. The formulation is based on the assumptions of Timoshenko beam theory in addition to consideration of the nonlinear von Karman strain–displacement relationship. The effective material properties of carbon nanotube reinforced composites are determined employing the Mori–Tanaka micromechanics model and the extended mixture rule. For the carbon nanotube reinforced composite beams, uniform distribution and four types of functionally graded distribution patterns of single-walled carbon nanotube reinforcements are considered. A differential transform method is applied on the nondimensionalized equations of motion to release the flapping modeshapes and the associated natural frequencies. The direct method of multiple scales is implemented to derive the effective nonlinearity and the corresponding nonlinear natural frequency. The accuracy of the present outcomes is validated by the comparison with the results given in the literature. The numerical results are presented in both tabular and graphical forms to investigate the effects of nanotube volume fractions, distribution types of the carbon nanotubes, and rotation speed on linear and nonlinear free vibration characteristics of carbon nanotube reinforced composite beam. The results demonstrate the important role of carbon nanotube distribution profile on linear and nonlinear free vibration features.

In this paper, analytical analysis of the vibrational tristable energy harvester with a RL resonant circuit is presented. The analytical solutions of the steady-state response displacement and the steady-state output voltage are derived via the method of multiple scales. The influence mechanism of the excitation amplitude and frequency, the electromechanical coupling coefficient, the damping and the detuning parameters on the dynamic response characteristics and the output voltage is studied. In order to enhance the energy harvesting performance, the appropriate choice of the excitation amplitude and the electromechanical coupling coefficient is discussed.

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.

The nonlinear saturation principle and 1:2 internal resonance are used in the design of the piezoelectric autoparametric vibration absorber for vibration suppression and energy harvesting. A novel prototype is manufactured. The proposed theoretical model is verified by the experimental data. Synergy optimization is performed based on the approximate analytical solutions obtained using the harmonic balance method and the method of multiple scales. By leveraging the Routh-Hurwitz criterion, the system’s stability boundary is analytically determined and confirmed by the numerical simulation of the original full coupled governing equations. The system undergoes periodic motion, aperiodic motion and chaos successively from the stable region to the center of the unstable region. The unstable region is shown to be adjustable and minimized by the maximal electrical damping to improve the vibration suppression. The design proposed for the minimal base (main structure) displacement at resonance locates in the stable zone. The optimized piezoelectric autoparametric absorber not only effectively mitigates the vibration of the main structure but also harvests large electric power at resonance or near resonance. Such a system is shown to perform well in the environment with small external noises.

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 work, the nonlinear dynamic analysis of a parametrically base excited cantilever beam based piezoelectric energy harvester is carried. The system consists of a cantilever beam with piezoelectric patches in bimorph configuration and attached mass at an arbitrary position. The attached mass is placed in such a way that the system exhibits 3:1 internal resonance. The governing spatio-temporal equation of motion is discretized to its temporal form by using generalized Galerkin’s method. To obtain the steady state voltage response and stability of the system, Method of multiple scales is used to reduce the resulting equation of motion into a set of first-order differential equations. The response and stability of the system under principal parametric resonance conditions has been studied. The parametric instability regions are shown for variation in different system parameters such as excitation amplitude and frequency, damping and load resistance. Bifurcations such as turning point, pitch-fork and Hopf are observed in the multi-branched non-trivial response. By tuning the attached mass an attempt has been made to harvest the electrical energy for a wider range of frequency. Such kind of smart self-sufficient systems may find application in powering low power wireless sensor nodes or micro electromechanical systems.

In this present work, the pneumatic artificial muscle is modelled as a single degree of freedom system and the governing nonlinear equation of motion has been derived to study the responses of the system under simultaneous simple and principal parametric resonance conditions. The equation has been developed considering the pneumatic muscle force as a function of the static and the time varying pressure, dimensions and material properties of the artificial muscle. The second order method of multiple scales has been used to find the approximate solutions and to study the dynamic stability and bifurcations of the system. The influences of the various system parameters in the amplitude for the muscle have also been studied with the help time responses, regions of parametric instability and frequency responses. The applications of the present study are illustrated through numerical examples. This work will help the designer or researchers working in this field to determine the safe operating range of the system parameters for different applications of pneumatic artificial muscle.

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.

In this paper, instability boundaries of axially accelerating plates with internal resonance are investigated for the first time. The relation between the acceleration and the longitudinally varying tensions are introduced. The governing equation and the corresponding boundary conditions are derived from the generalized Hamilton principle. The effects of internal resonances and the nonhomogeneous boundary conditions on the instability boundaries are highlighted. By the method of multiple scales, the modified solvability conditions in principal parametric and internal resonances are established. The Routh-Hurwitz criterion is introduced to determine the instability boundaries. The effects of the viscoelastic coefficient and the viscous damping coefficient on the instability boundaries are examined. Abnormal instability boundaries are detected when the internal resonance is introduced. The phenomenon of local zigzag and V-shape boundaries are explained from the viewpoint of modal interactions. The numerical calculations of the differential quadrature schemes about the first four complex frequencies, the first four complex modes, and the stability boundaries are used to confirm the results of the analytical method.

We establish a nonlinear non-conservative mathematical framework for the acoustic-electro-elastic dynamics of the response of a piezoelectric disk to high-level acoustic excitation in the context of ultrasound acoustic energy transfer. Nonlinear parameter identification is performed to estimate the parameters representing nonlinear piezoelectric coefficients. The identification is based on exploiting the vibrational response of the disk operating in the thickness mode under dynamic actuation. The nonlinearly coupled electro-elastic governing equations, for the piezoelectric receiver subjected to acoustic excitation, are derived using the generalized Hamilton’s principle. The method of multiple scales is used to obtain an approximate solution that forms the basis for parameter identification. The identified coefficients are then experimentally validated. The effects of varying these coefficients on the nonlinear response, optimal resistive electrical loading, and power generation characteristics of the receiver are investigated.