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Nonlinear interactions in physical systems make analyzing the dynamics challenging. Multiple-scale analyses are asymptotic perturbation techniques useful for analyzing nonlinear systems that possess scale-separation between the relevant physical scales. Graphene is a two-dimensional lattice of carbon atoms which exhibits many unique electrical properties, such as a viscous, hydrodynamic regime where the electrons become strongly interacting. We consider both the Dirac fluid and Fermi liquid regimes in gated graphene, and we investigate the one-dimensional propagation of electronic solitons. By leveraging the scale-separation between the wave-propagation time scale and nonlinear-interaction time scale, we utilize a multiple-scale analysis to derive a Korteweg-de Vries (KdV)-Burgers equation governing the wave's evolution. We numerically solve the KdV-Burgers equation and analyze the viscous decay and entropy production. Finally, we propose experimental realizations of these effects to measure the shear viscosity of graphene.Surface water waves also possess nonlinear interactions and permit nonlinear, periodic waves of permanent form known as Stokes waves in intermediate- and deep-water. Wind forcing causes wave growth and decay, but it can also influence wave shape. To study the effect of wind on the shape of these nonlinear Stokes waves, we again utilize scale-separation between the wave-propagation time scale and the nonlinear-interaction time scale. We then analytically solve the resulting system for three different wind-induced surface pressure profiles (Jeffreys, Miles, and generalized Miles) to calculate the wind-induced changes to the wave's shape statistics, growth rate, and phase speed. These results are constrained by existing large eddy simulations and are consistent with prior laboratory experiments.Shallow water supports surface waves known as solitary waves that balance nonlinearity and dispersion. By applying the same Jeffreys-type surface pressure forcing, we analyze the effect of wind on shallow-water wave shape. Another multiple-scale analysis yields a KdV-Burgers equation for the wave's profile, and we solve this numerically to investigate the wave's skewness and asymmetry resulting from the solitary waves's wind-induced, bound, dispersive tail. Extending this analysis to a planar, sloping bathymetry instead yields a variable-coefficient KdV-Burgers equation. Numerically solving this equation reveals the interaction of wind-forcing and shoaling on wave growth, width change, and rear-shelf generation.

The purpose of the current study was to develop an accurate model to investigate the nonlinear resonances in an axially functionally graded beam rotating with time-dependent speed. To this end, two important features including stiffening and Coriolis effects are modeled based on nonlinear strain relations. Equations governing the axial, chordwise, and flapwise deformations about the determined steady-state equilibrium position are obtained, and the rotating speed variation is considered as a periodic disturbance about this equilibrium condition. Multi-mode discretization of the equations is performed via the spectral Chebyshev approach and the method of multiple scales for gyroscopic systems is employed to study the nonlinear behavior. After determining the required polynomial number based on convergence analysis, results obtained are verified by comparing to those found in literature and numerical simulations. Moreover, the model is validated based on simulations carried out by commercial finite element software. Properties of the functionally graded material and the values of average rotating speed leading to 2:1 internal resonance in the system are found. Time and steady-state responses of the system under primary and parametric resonances caused by the time-dependent rotating speed are investigated when the system is tuned to 2:1 internal resonance. A comprehensive study on the time response, frequency response, and stability behavior shows that the rotating axially functionally graded beam exhibits a complicated nonlinear behavior under the effect of the rotating speed fluctuation frequency, damping coefficient, and properties of the functionally graded material.

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.

A low-dimensional model of a top-tensioned riser under excitations from vortices and time-varying tension is proposed, where the van der Pol wake oscillator is used to simulate the loading caused by the vortex shedding. The governing partial differential equations describing the fluid–structure interactions are formulated and multi-mode approximations are obtained using the Galerkin projection method. The one mode approximation is applied in this study and two different resonances are investigated by employing the method of multiple scales. They are the 1:1 internal resonance between the structure and wake oscillator (also known as ‘lock-in’ phenomenon) and the combined 1:1 internal and 1:2 parametric resonances. Bifurcations under the varying nondimensional shedding frequency for different mass–damping parameters are investigated and the results of multiple-scale analysis are compared with direct numerical simulations. Analytical responses are calculated using the continuation method and their stability is determined by examining the eigenvalues of the corresponding characteristic equations. Effects of the system parameters including the amplitude of the tension variation, vortex shedding frequency and mass–damping parameter on the system bifurcations have been investigated. The analytical approach has allowed to probe bifurcations occurring in the system and to identify stable and unstable responses. It is shown that the combined resonances can induce large-amplitude vibration of the structure. Counter-intuitively, the amplitude of such responses increases rapidly as the amplitude of the tension variation grows. Comparisons between the analytical and numerical results confirm that the span of the system vibration can be accurately predicted analytically with respect to the obtained response amplitudes of responses. The proposed multi-mode approximation and presented findings of this study can be used to enhance design process of top tension risers.

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.

This paper is devoted to investigating the pattern dynamics of Lotka–Volterra cooperative system with nonlocal effect and finding some new phenomena. Firstly, by discussing the Turing bifurcation, we build the conditions of Turing instability, which indicates the emergence of Turing patterns in this system. Then, by using multiple scale analysis, we obtain the amplitude equations about different Turing patterns. Furthermore, all possible pattern structures of the model are obtained through some transformation and stability analysis. Finally, two new patterns of the system are given by numerical simulation.

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.

In this paper, a Duffing-type energy harvester with time delay circuit under narrow-band random excitation is studied by using the method of multiple scales. The analytical expressions of the steady-state responses near the principal resonance and their stable conditions are derived. Results show that the phenomenon of stochastic jump is found, and the operational bandwidth of the nonlinear energy harvester can be extended. The effects of frequency detuning, noise intensity, time delay, piezoelectric coupling, and time constant ratio on the dynamical behaviors are discussed. From the viewpoint of improving system performance and optimizing harvester design, a proper choose of time delay and feedback gain is proposed by the power conversion efficiency and the RMS voltage. It indicates that the negative feedback gain is favorable for an energy harvester to provide more electrical output power and miniaturize design. Moreover, the Monte Carlo simulations are given to verify the validity of the theoretical method.

In this study, time-delayed feedback control is investigated for an elastically mounted rectangular prism undergoing subcritical galloping in the transverse direction, when subjected to wind excitation. The mathematical model of the galloping system under consideration is established by using the quasi-steady aerodynamic theory. The control performance in terms of the galloping onset speed of the time-delayed displacement, velocity and acceleration feedback is investigated via linear stability analysis, respectively. Subsequently, the method of multiple scales is implemented for nonlinear analysis in order to derive the analytical expression of the vibration amplitude of the galloping system and determine the criticality curve that is the boundary of the subcritical and supercritical bifurcation regions. The results show that the hybrid objective of increasing the galloping onset speed, changing the Hopf bifurcation behavior from subcritical to supercritical and reducing the amplitude of limit-cycle oscillations can be achieved by means of delayed acceleration feedback. This study provides an analytical tool and procedure for time-delayed feedback control design of such a kind of flow–structure interaction system.

Vortex-induced vibrations are one of the major factors in fatigue failure of power transmission lines and can be mitigated using vibration absorbers in the form of Stockbridge dampers. Since power transmission lines play an important role in modern infrastructure, a thorough understanding of the nonlinear dynamical interactions between conductors, dampers, and wind forces is crucial. Although different nonlinear models exist for conductor vibration with attached dampers or under wind force, no work combines all these nonlinearities in a single model and examines the dynamics of the conductor along with dampers. In an attempt to fill this gap, this work combines the nonlinearities from the mid-plane stretching of the conductor, equivalent cubic stiffness of the Stockbridge damper, and fluctuating lift force modeled as a Van der Pol oscillator in a single model to investigate the nonlinear vortex-induced vibrations. In this work, the conductor is modeled as a simply supported beam and the Stockbridge damper as a mass–spring–damper–mass system with a combination of cubic and linear stiffness. The governing equations of motion are solved analytically using the method of multiple scales for the case of primary resonance between the fluctuating lift-force and conductor. Analytical findings are further validated by comparing against the numerical integration of a reduced-order system, and the results show an excellent match. The analysis is extended by conducting a parametric study to investigate the effect of different system parameters on the frequency response curves. These findings are promising and further provide a direction to design an optimal vibration absorber.

In this study, the large amplitude vibration of the cable-stayed beam subjected to external excitation is investigated. Emphasis is focused on the one-to-one resonant interaction between hybrid and hybrid/local modes, which may be activated in the veering and crossover regions. The Hamilton’s principle and multi-mode discretization are applied to obtain the discrete model governing the in-plane vibration of the cable-stayed beam. Then, the method of multiple scales is applied to solve the equation of motion, and the displacement of the cable-stayed beam and corresponding modulation equations are determined. In the following, the equilibrium solution of the modulation equations and the associated stability are examined to discuss the periodic motion of the cable-stayed beam. Whereas the shooting method is applied to investigate the dynamic solution to qualitatively illustrate the non-periodic motion. Numerical simulations are performed to verify the periodic solution and examine the nonlinear dynamics of the cable-stayed beam. Particular attention is placed on the chaotic dynamic of the cable-stayed beam in unstable region. It is shown that the equilibrium and dynamic solutions may undergo different bifurcations, i.e., Hopf, torus and cyclic-fold bifurcations.

In contrast to the approach of coupling a nonlinear oscillator that represents the lift force with the cylinder’s equation of motion to predict the amplitude of vortex-induced vibrations, we propose and show that the displacement can be directly predicted by a nonlinear oscillator without a need for a force model. The advantages of the latter approach include reducing the number of equations and, subsequently, the number of coefficients to be identified to predict displacements associated with vortex-induced vibrations. The implemented single-equation model is based on phenomenological representation of different components of the transverse force as required to initiate the vibrations and to limit their amplitude. Three different representations for specific flow and cylinder parameters yielding synchronization for Reynolds numbers between 104 and 114 are considered. The method of multiple scales is combined with data from direct numerical simulations to identify the parameters of the proposed models. The variations in these parameters with the Reynolds number, reduced velocity or force coefficient over the synchronization regime are determined. The predicted steady-state amplitudes are validated against those obtained from high-fidelity numerical simulations. The capability of the proposed models in assessing the performance of linear feedback control strategy in reducing the vibrations amplitude is validated with performance as determined from numerical simulations.

Micro-electro-mechanical system (MEMS) gyroscopes are an important sort of inertial sensor for identifying parameters of spinning structures, such as the spinning speed and angular deviation, based on the Coriolis effect. In this paper, the nonlinear mechanism of MEMS vibratory ring gyroscopes is analyzed by applying a fully coupled nonlinear model, in which the gyroscopic coupling and geometrically and structurally nonlinear couplings are all taken into account. The coupled differential equations governing the drive and sense motions are established via the Lagrangian equations. Numerical simulation is conducted, and the key nonlinear components and energy transfer behaviors between the drive and sense modes are elucidated. It is revealed that the cubic rigidity nonlinearity is another significant factor leading to the coupling between the drive and sense modes other than the gyroscopic coupling. Perturbation analysis is also carried out by using the method of multiple scales. The nonlinear frequency–amplitude responses of the drive and sense vibrations are obtained, and comprehensive parametric studies are performed. The significant effects of system damping, excitation amplitude, drive amplitude and spinning speed on the responses are discussed, which will facilitate to improve the nonlinear performance and sensitivity of the gyroscope.

Avoidance of resonance in fluctuation of milling tool is vital for reaching excellence quality and performance of the cutting operation. The cutting tool in resonance condition vibrates with considerable magnitude that causes to increase milling tool wear and manufacturing prices. Analytical study of primary resonances and bifurcation behavior of a micro-milling process, including structural nonlinearities, gyroscopic moment, rotary inertia, velocity-dependent process damping, static and dynamic chip thickness, is chief aim of this article. The milling tool is modeled as a 3-D spinning cantilever beam that is motivated by cutting forces. To get the analytical solution for frequency response function and bifurcation behavior of the system under primary resonances, the method of multiple scales is operated on converted ordinary differential equations that are obtained by applying assumed modes method on nonlinear partial differential equations of tool vibration. The effects of different process parameters and nonlinear terms on the frequency response of the tool tip oscillations are examined. In addition, the effects of detuning parameter and damping ratio on the bifurcation and behavior of the limit cycle under primary resonances are examined. The results shows that these parameters are the bifurcation parameters and Neimark, symmetry breaking, flip, and period-3 bifurcations occur when the detuning parameter is varied.

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.

An analytical solution for the problem of an immersed flexible and vertical heavy rod subjected to a vertical top motion is developed using the multiple scales method directly applied to the partial differential equations of motion. The obtained results show good agreement with a numerical solution obtained using the finite element method for a study case. The analytical solution is then used to carry out some sensitivity studies. The effects of the structural nonlinearities, hydrodynamic and structural damping terms are investigated. It is shown that the nonlinearities play a role in defining the frequency of the top motion that causes the maximum amplitude of response, but not the value of the amplitude itself. In turn, the major role played by the hydrodynamic damping in defining the response amplitude is addressed. It is also shown that the structural damping have an important effect on the response amplitude even in the case of small damping ratio. This occurs due to the combined effect of the structural with the hydrodynamic damping. Finally, it is pointed out that small differences in the structural damping ratio can lead to significant differences in the response amplitude.

Forced vibration of the cutting system with a three-dimensional composite cutter bar is investigated. The composite cutter bar is simplified as a rotating cantilever shaft which is subjected to a cutting force including regenerative delay effects and harmonic exciting items. The nonlinear curvature and inertia of the cutter bar are taken into account based on inextensible assumption. The effects of the moment of inertia, gyroscopic effect, and internal and external damping are also considered, but shear deformation is neglected. Equation of motion is derived based on Hamiltonʼs extended principle and discretized by the Galerkin method. The analytical solutions of the steady-state response of the cutting system are constructed by using the method of multiple scales. The response of the cutting system is studied for primary and superharmonic resonances. The effects of length-to-diameter ratio, damping ratio, cutting force coefficients, ply angle, rotating speed, and internal and external damping are investigated. The results show that nonlinear curvature and inertia imposed a significant effect on the dynamic behavior of the cutting process. The equivalent nonlinearity of the cutting system shows hard spring characteristics. Multiple solutions and jumping phenomenon of typical Duffing system are found in forced response curves.

In this work, theoretical and experimental analysis of a piezoelectric energy harvester with parametric base excitation is presented under combination parametric resonance condition. The harvester consists of a cantilever beam with a piezoelectric patch and an attached mass, which is positioned in such a way that the system exhibits 1:3 internal resonance. The generalized Galerkin’s method up to two modes is used to obtain the temporal form of the nonlinear electromechanical governing equation of motion. The method of multiple scales is used to reduce the equations of motion into a set of first-order differential equations. The fixed-point response and the stability of the system under combination parametric resonance are studied. The multi-branched non-trivial response exhibits bifurcations such as turning point and Hopf bifurcations. Experiments are performed under various resonance conditions. This study on the parametric excitation along with combination and internal resonances will help to harvest energy for a wider frequency range from ambient vibrations.

Recent studies in passively isolated systems have shown that mode coupling is desirable for better vibration suppression, thus refuting the long-standing rule of modal decoupling. However, these studies have ignored the nonlinearities in the isolators. In this work, we consider stiffness nonlinearity from pneumatic isolators and study the nonlinear forced damped 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 an analytical study using the method of multiple scales and the method of harmonic balance for different cases of external resonances, viz., the primary and the secondary resonances (superharmonic and combined resonances) with 1 : 2 internal resonance between the modes. We further validate our analytical findings using direct numerical integration and observe an excellent match. On extending our analysis, we observe the existence of subcritical, supercritical, and s-shaped bifurcation depending on the location of the isolators and the case of external resonance. Also, the saturation and quasi-saturation phenomenon are observed for the case of resonances close to the higher natural frequency and combined resonance, respectively. A parametric study is conducted to examine the effect of different parameters on the dynamics of the system, and consecutively the critical parameters of the system are identified for different cases of external resonance.

The objective of this manuscript is to elucidate the role of topology on the nonlinear response of resonators. The resonators are composed of two beams and designed to exhibit two-to-one internal resonances. The manuscript explores the sensitivity to variations in the topological parameters, i.e., the relative orientation, and the ratio of lengths and heights, on the modal frequencies of the structure. The nonlinear governing equations are solved using the method of multiple scales when the system is excited near the first and second resonances. The occurrence of bifurcations and modal interactions has been shown both numerically and experimentally and related to the topological parameters. The effect of the topology on modal saturation has also been studied.

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.

Pull-in and snap-through are two representative bifurcation phenomena in electrically actuated microcomponents, based on which various microdevices are designed and applied in many sensing and actuating fields. In this paper, the nonlinear mechanisms of pull-in and snap-through in a two-electrode actuated microbeam due to asymmetric bias voltages are qualitatively identified. With the considerations of midplane stretching of clamped–clamped microbeam and nonlinear electrostatic force, a continuous dynamic equation of motion is introduced after which a generalized one-degree-of-freedom (1-DOF) model derived using the differential quadrature method is deduced. By the application of the singularity theory on the static equation, transient sets are theoretically obtained which separate two-parameter space of cubic stiffness and DC voltage into seven regions. Associated bifurcation diagrams show that initially straight microbeam can exhibit snap-through motion as well as pull-in behavior when loading proper bias voltages on two electrodes. Besides, detailed division on snap-through region is numerically done and the snap-through direction is discussed and then verified. In what follows, primary resonant frequency closely related to the static solution is examined which show various attractive scenarios versus bias voltage ratio. Moreover, the method of multiple scales is utilized to derive the primary resonant response for small vibration estimation. Combined with theoretical and numerical results, dynamic snap-through motion is observed which implies that dynamic saddle-node bifurcation and interwell jumping due to energy enhancement can both increase the possibility of dynamic snap-through. During dynamic snap-through procedure, microbeam may exhibit cross-well transient chaos.

We investigate the control of friction-induced vibration in a precision motion stage under the effect of the LuGre friction dynamics. We consider a lumped parameter model of the precision motion stage with PID and linear time-delayed state feedback control acting in the direction of the motion of the stage. Linear stability analysis reveals the criticality of integral gain in the stability and, accordingly, the existence of multiple stability lobes and codimension-2 Hopf points for a given choice of system parameters. The nature of the bifurcation is determined by an analytical study using the method of multiple scales and harmonic balance. We observe the existence of both subcritical and supercritical Hopf bifurcations in the system, depending on the choice of control parameters. Hence, the nonlinearity due to dynamic friction model could both be stabilizing or destabilizing in nature, and therefore, stick-slip nonlinearity is essential to capture the global behavior of the system dynamics. Furthermore, numerical bifurcation analysis of the system reveals the existence of period-doubling bifurcation near the Hopf points. We observe complicated solutions such as period-4, quasi-periodic, large-amplitude stick-slip limit cycles along with chaotic attractor in the system.

In this paper, we investigate the nonlinear vibration of a metamaterial structure that consists of a rotating cantilever beam attached to a periodic array of spring–mass–damper subsystems deployed for vibration suppression. The full nonlinear model of the system is developed. The nonlinear response due to a primary resonance excitation is investigated, and the capability of the metastructure to suppress vibration is examined. The mass of the resonators (absorbers) comes at the expense of the host structure’s mass itself, which makes the total mass of the system conservative. Free and forced vibration analyses are performed. We first use the method of multiple scales to analyze the nonlinear behavior of rotating beams. The perturbation solutions are validated against their numerical counterparts. Results show the presence of a critical rotational speed at which the beam undergoes bifurcation and starts to flutter. The addition of the absorbers is observed to slightly reduce this critical speed. Nevertheless, the amplitude of limit-cycle oscillations beyond bifurcation is found to decrease when equipping the rotating beam with local absorbers. The results demonstrate the capability of the metamaterial structure as an efficient damping treatment. Furthermore, we show that careful placement of the absorbers along the cantilever beam (close to the tip) enables further vibration mitigation.

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.

This paper proposes a two-to-one internal resonance to widen the bandwidth of vibratory energy harvesters. To describe the improved characteristic, an electromagnetic spring-pendulum harvester is designed. Approximate analytical solutions of the electromechanical coupled system are carried out by introducing the method of multiple scales, and the frequency response relationships of the displacement and the current are obtained. The character of broadband harvesting performance is examined, the two peaks and double jump phenomena for variation of design parameters were observed. The effect of key control parameters on the harvesters bandwidth is considered, and the nonlinear behaviors of the harvester are validated via numerical results.

The present paper deals with the investigation of dynamic responses of a rotating pre-deformed blade in four cases of resonance, including two subharmonic resonances and two combination resonances. The dimensionless gas excitation amplitude is assumed to share the same order with the dimensionless vibration displacement. Four cases of resonance are confirmed by examining the secular terms. The theoretical analysis framework is established for each resonance case based on the method of multiple scales. The original dynamic system is integrated numerically by the Runge–Kutta method. The frequency components and phases obtained from fast Fourier transform of the numerical response are used to verify the theoretical results. For the purpose of contrast, modulation equations are also integrated numerically. In all four resonance cases, the theoretical results agree well with the numerical simulation. Parameter studies are conducted to clarify the effects of system parameters on the perturbation curves. Various results are obtained for the rotating blade. A quasi-saturation phenomenon occurs in both combination resonances of summed type and difference type, and the corresponding limit value of the second-mode response can be reduced by decreasing the external detuning parameter. The quasi-saturation phenomenon of rotating blade only appears with high gas pressure. The subharmonic resonance of second mode and the combination resonance of summed type are hard to excite in practice compared with the other two cases.

The nonlinearities of geometric nature that is characteristic for pendulum-type systems are expressed by the trigonometric functions. In order to apply the method of multiple scales in time domain to solve problems concerning such systems, the trigonometric functions of the generalised coordinates are usually approximated by a few terms of their Taylor series. In the paper we apply the polynomial approximation in quadratic means. In contrast to the approximation by Taylor series, the proposed manner approximates the trigonometric functions not around a given point but on the given interval. Quality and accuracy of the solutions obtained using the multiple scales method based on such approach have been tested. The steady state responses in the main resonance have been also examined and compared with their counterparts obtained using the method of multiple scales based on the Taylor series.

A dynamic rolling force model with multi-parametric excitations and multi-frequency external excitations caused by entry thickness fluctuation of strip was established. Based on the dynamic rolling force, a nonlinear vertical vibration model with multi-parametric excitations and multi-frequency external excitations was established. The method of multiple scales was used to solve the amplitude-frequency characteristic equation of primary resonance of the nonlinear vibration system of a rolling mill. The transition set and the topology structure of systematic global bifurcation were obtained by using the singularity theory. Finally, primary resonance characteristics of the system under entry thickness fluctuation of strip were analyzed by using actual parameters of the rolling mill. The global bifurcation curves with the change of amplitude and frequency of entry thickness fluctuation of strip were obtained by using numerical simulation, and many dynamic behaviors were found such as single-cycle motion, multi-cycle motion and chaotic motion, which can provide a theoretical reference for further restraining the vibration of a rolling mill.

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.