Search

This article considers the influence of an electric field on the instability of the interface between two fluids, of which one is an ideal electrically conducting fluid, and another is an ideal dielectric fluid. Our study has been carried out under the assumption of small viscosity, which allows to use the boundary layer theory. The boundary layer structure is investigated using the method of multiple scales. Instability boundaries near the threshold of the instability are found analytically.

The fluid-conveying pipes made of polymer-like materials are widely applied in engineering fields. However, the fractional dynamics of fluid–solid interaction remain unknown. In this work, the fractional dynamics of the pipes subjected to the excitation of supporting foundation are studied. A new nonlinear, fractional-order dynamic model is presented. The method of multiple scales is adopted directly to solve the model for the case of primary resonances. Numerical results are presented to show the effects of fractional order, foundation vibration, and other physical parameters on the steady-state response and stability.

This paper investigates principal parametric resonance of axially accelerating hyperelastic beam. Hyperelasticity is integrated into axially moving material for the first time. Based on the continuum mechanics theory, the coupled nonlinear partial differential equations of motion are derived from the extended Hamilton’s principle. The model equations are simplified into a single integro-differential equation, which governs the transverse vibration of the hyperelastic beam. The method of multiple scales is used to solve the integro-differential equation to obtain the nonlinear response of the principal parametric resonance. The effect of the material parameter(i.e. in plane Poisson’s ratio) on the type of the nonlinear vibration behavior and amplitude of the nontrivial solutions of steady-state have been investigated. Further, the couple nonlinear governing equations are solved by Galerkin’s method. Comparison between the analytical results and the results of the Galerkin’s method are made and good agreement are found.

We theoretically investigated the cumulative nonlinear guided waves caused by internal resonance, using the method of multiple scales (MMS), which can construct better approximations to the solutions of perturbation problems. In this study, we consider nonlinearity only on the boundary instead of material nonlinearity or geometric nonlinearity. We showed nonlinear effects on the amplitudes of a lower mode and a higher mode depending on the propagation length. Also, we examined effects of wavenumber detuning from a phase matching condition of the two modes. If the wavenumber detuning is exactly equal to zero, the mechanical energy of the lower mode is transferred through nonlinear coupling to the energy of the higher mode, unilaterally. However, if a wavenumber detuning is not equal to zero, amplitude of the two modes change in a cyclic fashion during wave propagation. The amount of this amplitude variation and its cycle length are determined by the eigenfunctions of the two modes, the nonlinear parameter and the wavenumber detuning.

We present a detailed bifurcation analysis of the state-dependent delay model of rotary drilling considering only the axial and torsional modes. This analysis is presented for the general case of independent natural frequencies of these two modes. The regenerative effect accompanying axial vibrations gives rise to a delayed model with the delay determined by the torsional oscillations. It is observed that steady drilling loses stability through a Hopf bifurcation. The nature of bifurcation is ascertained by the method of multiple scales for the general values of system parameters. Analytical results suggest that both supercritical and subcritical bifurcations exist for different choices of operating and system parameters. These analytical findings are further confirmed by numerical simulations. Possible unfoldings of the dynamics near the codimension-2 point, guided by numerical simulations and analytical results for the codimension-1 Hopf branches, are also presented. We find two different scenarios at the primary codimension-2 point viz. both Hopf branches having supercritical bifurcation, and one branch being supercritical while the other being subcritical. Our numerical simulations suggest that the dynamics near the codimension-2 point is dominated by the low-frequency limit cycles in both the scenarios.

Nonlinear vibrations of fractional viscoelastic plate with Kelvin–Voigt fractional order constitutive relationship is investigated in this paper. Based on the Kirchhoff hypothesis for thin shells and von Karman’s assumption, the structural dynamic of the plate is modeled by using the Newton’s second law. The nonlinear coupled equations of motion are obtained by introducing the Airy stress function and the Galerkin method is used to discretize the partial differential equations. Analytical solutions for fully simply supported and clamped plate are obtained by using the method of multiple scales and finally the equations of amplitude–frequency and phase–frequency are obtained for primary, super-harmonic and sub-harmonic resonance. The obtained amplitude–frequency and phase–frequency equations are used for studying the effects of excitation, fractional parameters and nonlinearity on the frequency responses of the fractional viscoelastic plate and finally some conclusions are outlined.

In this paper, we explored the nonlinear dynamics behavior and vibration suppression of a nonlinear electromechanical oscillator system under harmonic and parametric excitation. The model comprises of an electrical part coupled to mechanical part and displayed by a coupled nonlinear ordinary differential equations. The analytical up to second order approximate solutions are sought applying the method of multiple scales method. We utilized the time-series and method of averaging to analyze the response and stability of the solutions at the worst resonance cases. We checked the results of perturbation solution through numerical simulations and the effects of different system parameters have been reported. Comparison between analytical and numerical solutions is obtained. Also, the numerical results are obtained using MAPLE and MATLAB algorisms.

In the present study, the nonlinear aeroelastic and sensitivity analysis of high aspect ratio wings subjected to a transverse follower force are discussed. A nonlinear structural model of wings is extracted and coupled with an incompressible unsteady aerodynamic model. The governing equations of motions are obtained via Hamilton’s principle and Galerkin method. Utilizing the method of multiple-scales, analytical approximate flutter response of the system is obtained. For validation, the analytical solution is compared with numerical solution and good agreement is observed. The time history of the tip displacement and tip twist solution are plotted for different airspeeds. Effects of follower force and its spanwise location and also the wing geometric characteristics on the flutter margin are discussed. Moreover, flutter margin sensitivity to different design parameters is analyzed. Results indicate that increasing the wing chord makes the system unstable. Furthermore, according to the analytical solution, effects of the wing chord and mass per unit length on the flutter margins are more important than the other system parameters.

In this paper, we analyze the complicated nonlinear dynamics of rotor-active magnetic bearings (rotor-AMB) with 16-pole legs and the time varying stiffness. The magnetic force with 16-pole legs is obtained by applying the electromagnetic theory. The governing equation of motion for rotor-active magnetic bearings is derived by using the Newton’s second law. The resulting dimensionless equation of motion for the rotor-AMB system is expressed as a two-degree-of-freedom nonlinear system including the parametric excitation, quadratic and cubic nonlinearities. The averaged equation of the rotor-AMB system is obtained by using the method of multiple scales when the primary parametric resonance and 1/2 subharmonic resonance are taken into account. From the frequency-response curves, it is found that there exist the phenomena of the soft-spring type nonlinearity and the hardening-spring type nonlinearity in the rotor-AMB system. The effects of different parameters on the nonlinear dynamic behaviors of the rotor-AMB system are investigated. The numerical results indicate that the periodic, quasi-periodic and chaotic motions occur alternately in the rotor-AMB system.

In this paper, a nonlinear random model is built to describe the dynamics of planetary gear trains (PGTs), in which the time-varying mesh stiffness, tooth profile modification (TPM), tooth contact loss, and random tooth profile error are considered. A stochastic method based on the method of multiple scales (MMS) is extended to analyze the statistical property of the dynamic performance of PGTs. By the proposed multiple-scales based stochastic method, the distributions of the dynamic transmission errors (DTEs) are investigated, and the lower and upper bounds are determined based on the 3σ principle. Monte Carlo method is employed to verify the proposed method. Results indicate that the proposed method can be used to determine the distribution of the DTE of PGTs high efficiently and allow a link between the manufacturing precision and the dynamical response. In addition, the effects of tooth profile modification on the distributions of vibration amplitudes and the probability of tooth contact loss with different manufacturing tooth profile errors are studied. The results show that the manufacturing precision affects the distribution of dynamic transmission errors dramatically and appropriate TPMs are helpful to decrease the nominal value and the deviation of the vibration amplitudes.

The present work proposes an analytical model for vibration analysis of partially cracked rectangular plates coupled with fluid medium. The governing equation of motion for the isotropic plate based on the classical plate theory is modified to accommodate a part through continuous line crack according to simplified line spring model. The influence of surrounding fluid medium is incorporated in the governing equation in the form of inertia effects based on velocity potential function and Bernoulli's equations. Both partially and totally submerged plate configurations are considered. The governing equation also considers the in-plane stretching due to lateral deflection in the form of in-plane forces which introduces geometric non-linearity into the system. The fundamental frequencies are evaluated by expressing the lateral deflection in terms of modal functions. The assessment of the present results is carried out for intact submerged plate as to the best of the author's knowledge the literature lacks in analytical results for submerged cracked plates. New results for fundamental frequencies are presented as affected by crack length, fluid level, fluid density and immersed depth of plate. By employing the method of multiple scales, the frequency response and peak amplitude of the cracked structure is analyzed. The non-linear frequency response curves show the phenomenon of bending hardening or softening and the effect of fluid dynamic pressure on the response of the cracked plate.

The van der Pol(VDP) oscillator is an essential nonlinear oscillator, which has been extensively studied. However, the results of delayed generalized VDP oscillator are relatively less. The issue of bifurcation control for a delayed extended VDP oscillator is fully considered in this paper. Firstly, a nonlinear time-delayed feedback controller is carefully designed to control the dynamics for the proposed oscillator, and the results of stability of such oscillator are obtained by means of stability switches theory. Secondly, a general explicit formula is derived to establish the properties of Hopf bifurcation by utilizing the method of multiple scales. Moreover, the impact of feedback gain on the bifurcation point for the controlled oscillator is demonstrated numerically. Finally, the effectiveness of theoretical results is verified via a numerical example.

A machining tool can be subject to different kinds of excitations. The forcing may have external sources (such as rotating imbalance or misalignment of the workpiece) or it can arise from the cutting process itself (e.g. chip formation). We investigate the classical tool vibration model which is a delay-differential equation with a quadratic and cubic nonlinearity and periodic forcing. The method of multiple scales gave an excellent approximation of the solution. The resonance curves found here are similar to those for the Duffing-equation, having a hardening characteristic. We found subcritical Hopf and saddle-node bifurcations.

We investigate metastable dynamical systems subject to non-stationary forcing as they appear in molecular dynamics for systems driven by external fields. We show, that if the strength of the forcing is inversely proportional to the length of the slow metastable time scales of the unforced system, then the effective behavior of the forced system on slow time scales can be described by a low-dimensional reduced master equation. Our construction is explicit and uses the multiscale perturbation expansion method called two-timing, or method of multiple scales. The reduced master equation—a Markov state model—can be assembled by constructing two equilibrium Markov state models; one for the unforced system, and one for a slightly perturbed one.

In the present work, vibration suppression of a single degree of freedom nonlinear spring-mass primary (main) system subjected to simultaneous external harmonic and parametric excitation by using traditional and non-traditional active nonlinear vibration absorber (ANVA) is carried out. The traditional and non-traditional ANVA consists of a PZT (lead zirconate titanate) stack actuator in series with a nonlinear spring and a damper. In the traditional ANVA, the damper is connected to the main system whereas, in non-traditional ANVA, the damper is connected to the base of the main system. The governing nonlinear equations of motion for the system have been derived using weighted modal matrix approach considering novel actuation force by time delay in displacement, velocity and acceleration feedback of the main system by the PZT stack actuator and the nonlinear spring in series combination. The method of multiple scales (MMS) is used to obtain the reduced equations which are solved using Newton’s method. The responses of the system for the primary, principal parametric and simultaneous primary and principal parametric resonance conditions have been studied. The results are compared with those obtained by using the harmonic balance method (HBM), numerical method and with previous literature. From these results, two resonating peaks in the frequency response of the main system are shown by using the reduced equation of the MMS which was not reported in earlier literature. It is shown that for the active case the vibration of the undamped main system is completely reduced at the specified operating frequency for both the linear and nonlinear systems. Also, for a wider range of operating frequencies, the vibration of the main system is reduced by actively tuning the absorber for various mass ratios. Further, the efficacy of the active force is observed in reducing the instability regions and improving the critical value of excitations in the trivial state.

In the commented paper, the authors considered a two-dimensional airfoil with nonlinear stiffness and damping, and derived the amplitude modulation equation (AME) of the system limit cycle oscillation around its bifurcation. Based on the variation equation of the AME, they conducted the stability analysis. Unfortunately, the obtained variation equation associated with the AME is incorrect.

A novel dual proof mass accelerometer is proposed by introducing mode localization in two electrostatically coupled resonators. The levering mechanism is utilized to amplify the inertial force applied axially to the two weakly coupled resonators. The dynamic model considering the electrostatic and mechanical nonlinearities is established and solved by the method of multiple scales, and also is validated by the harmonic balance method (HBM) coupled with the asymptotic numerical method (ANM). The linear and nonlinear sensitivities depicted as the difference of relative shift of amplitude ratio are investigated as well as the main influencing factors. It has been found that linear sensitivity is up to 4 orders of magnitude higher than that expressed by the difference of relative frequency shift. Also, the nonlinear sensitivity is further increased by 1.47 times comparing to the linear sensitivity. Moreover, the resolution is also greatly improved when the sensor is driven beyond its critical amplitude. Finally, the effect of the electrostatic coupling on the sensing performances is explored, and the optimal coupling voltage is theoretically identified at the limit of mode aliasing.

In this paper, the one-to-one interaction of a cable-stayed arch structure under the cable’s primary resonance is investigated. Based on the coupling condition at the arch tip, the partial differential equations governing the planar motion of the system are derived using the extended Hamiltonian principle, while with the application of the Galerkin method, these equations are transformed into a set of ordinary equations. Applying the method of multiple scales to these ordinary equations, the first approximated solutions and solvability condition are obtained. The one-to-one interaction between the cable and the arch is investigated under simultaneous internal and external resonances for an actual cable-stayed arch structure. Based on the shooting method and the pseudo-arclength algorithm, the dynamic solutions of the system are obtained, and a period-doubling route to chaos is analyzed. The effects of the cable’s initial tension, inclination angle, the arch’s rise-to-span ratio and intersection angle between the cable and the arch are explored, and the results show that the interaction response mainly depends on specific parameters.

An investigation is made into the capillary–gravity waves which arise on the interface of two ideal fluids and which are due to the resonant interaction between two harmonics of the motion. We consider the particular case when the upper fluid moves with speed V and the lower with speed − ρ V where ρ is the relative density of the fluids. This case is significant because previous analyses are invalid when the parameters have these special values. The method of multiple scales is employed to derive a pair of nonlinear partial differential equations which describe the evolution of the interface. These differ from the more usual nonlinear Schrodinger equations in that the roles of space and time are reversed. It is found that these equations possess a large number of solutions corresponding to sinusoidal waves. These waves are shown to be unstable.

The nonlinear harmonic response of an autoparametric system comprised of a linear oscillator with a vertically attached flexural beam is investigated and the capability of the beam as a vibration absorber is assessed. A weak torsional spring is used for constraining the rotation of the beam giving rise to an almost non-flexural rotational mode with a low frequency. The system parameters are also tuned to enforce the zero-to-one-to-one internal resonance condition. The Lagrange’s formulation accompanied by the assumed-mode method is used to derive the discretized equations based on the order three nonlinear Euler–Bernoulli beam theory. An analytical solution is developed based on the method of multiple scales where the generalized coordinate corresponding to the non-flexural rotational mode is approximated by higher order perturbation expansion than the other coordinates, due to much larger contribution of the non-flexural rotation to the response. Comprehensive response and bifurcation analysis are performed using analytical and direct numerical solutions. The results are obtained for vertically-aligned and also initially inclined beams and various interesting behaviors are recognized for different non-dimensional system parameters. Different types of bifurcations such as the Pitch-fork, Hopf, Period-doubling and symmetry breaking bifurcations are observed in the solution of slow-flow equations and some of them are found to be beneficial for vibration absorption in a limited range of excitation amplitudes and frequencies.

In this paper, the dynamic behavior of a rotor system with electromechanically coupled boundary condition under periodic axial load is studied, where the boundary condition is derived from a ring-shaped piezoelectric damper which is developed for vibration control of rotor system. This damper is based on the piezoelectric shunt damping technique, which can produce much or little damping performance depends on the selection of shunt circuit parameters. By using assumed mode method and Lagrange equation, the equations of motion are derived. Actually these equations can also be derived from a general forced parametrically excited gyroscopic system. Thus, to analyze such system, a method of multiple scales is developed firstly, where a general procedure is proposed to establish solvability conditions. Subsequently, this procedure is applied to obtain the analytical instability boundaries and forced vibration responses. The analytical results show that the additional combination instability regions are created due to the introduction of shunt circuit. When the parametrically excited eccentric rotor is rotating, the parametric vibrations are superimposed on the unbalanced responses. As the resistance value of shunt circuit is increasing, both parametric vibrations and unbalance vibrations can be significantly suppressed. For the unbalanced responses, both the resonance and anti-resonance phenomena can be observed; whereas for the parametric vibrations, only the resonance phenomena exist. These phenomena indicate that we may achieve great vibration control performance for the rotor parametric vibrations by introducing the piezoelectric shunt damping, as long as the circuit parameters are well adjusted. To validate the obtained analytical expressions, the numerical methods are applied. Specifically, the discrete state transition matrix method (DSTM) is applied to validate the analytical instability boundaries and the Runge-Kutta method is conducted to verify the analytical frequency response functions.

In this work, the method of multiple scales is applied to analysis of cosmological dynamics. The method is used to construct solutions to the dynamic equations of the Universe filled with a scalar field in the Friedman-Robertson-Walker metric. A general scheme is described for choosing small dimensionless parameters of the expansion of model functions and applying the method itself to the equations of cosmological dynamics. Solutions are given that are constructed for two different types of a small parameter - a small field value and a small slow roll parameter.

The nonlinear dynamic features of compression roller batteries were investigated together with their nonlinear response to primary resonance excitation and to internal interactions between modes. Starting from a parametric nonlinear model based on a previously developed Lagrangian formulation, asymptotic treatment of the equations of motion was first performed to characterize the nonlinearity of the lowest nonlinear normal modes of the system. They were found to be characterized by a softening nonlinearity associated with the stiffness terms. Subsequently, a direct time integration of the equations of motion was performed to compute the frequency response curves (FRCs) when the system is subjected to direct harmonic excitations causing the primary resonance of the lowest skew-symmetric mode shape. The method of multiple scales was then employed to study the bifurcation behavior and deliver closed-form expressions of the FRCs and of the loci of the fold bifurcation points, which provide the stability regions of the system. Furthermore, conditions for the onset of internal resonances between the lowest roller battery modes were found, and a 2:1 resonance between the third and first modes of the system was investigated in the case of harmonic excitation having a frequency close to the first mode and the third mode, respectively.

Nonlinear vibration absorbers have been extensively investigated for passive vibration control, motion isolation, and synchronous energy harvesting. This paper studies the exact nonlinear dynamics of a simply-supported beam carrying a nonlinear spring-inerter-damper energy absorber for primary resonance vibration reduction. The nonlinear governing equations of the system are derived from the energy method by considering the midplane stretching, structural discontinuity, and nonlinear boundary conditions at the spring-inerter-damper location of the beam and directly solved using the method of multiple scales. The nonlinear frequency correction factor, frequency response function, peak nonlinear frequency response, and bifurcation frequency are obtained and investigated for various system parameters. The influence of the location, spring stiffness, inertial mass, and damping of the nonlinear vibration absorber on the beam dynamics, including natural frequency, mode shape, and nonlinear frequency response, are studied. The stiffness and mass of the nonlinear vibration absorber are optimally tuned to minimize the peak nonlinear frequency response of the beam. The results show that ignoring the nonlinear boundary conditions at the spring-inerter-damper location could lead to serious underestimation of the nonlinear frequency responses. The nonlinear stiffness of the vibration absorber enhances the system nonlinearity but has no contribution to the peak nonlinear frequency response of the beam. Increasing the damping of the vibration absorber could effectively mitigate the beam vibration. When the nominal frequency of the absorber is tuned to be close to the natural frequency of the beam, the beam vibration is mostly reduced.

Weakly nonlinear propagation of plane progressive pressure waves in an initially quiescent liquid uniformly containing many spherical microbubbles is theoretically investigated, especially focusing on an initial small polydispersity of both the bubble radius and the number density of bubbles (i.e., void fraction), which appears in a field far from the sound source. Nonlinear waves in polydispersed bubbly liquids are classified into a form of three cases of nonlinear wave equation describing long-range propagation of waves. Using the method of multiple scales with perturbation expansions and the scaling relations of some nondimensional ratios, from the set of basic equations based on a two-fluid model, (i) for a low-frequency long wave, the Korteweg–de Vries–Burgers (KdVB) equation is derived and (ii) for a moderately high-frequency short wave, (ii-a) the NLS (nonlinear Schrodinger)-I (or LG (Landau–Ginzburg)-I) equation for a weak polydisperse medium and (ii-b) the NLS-II (or LG-II) equation in a strong polydisperse medium are derived in a unified manner. For all cases, polydispersity contributes to the advection effect of waves and induces variable coefficients into the KdVB, NLS-I, and NLS-II equations. Furthermore, the KdVB equation includes an inhomogeneous term owing to the polydispersity and the NLS-II equation includes second-order nonlinearity of polydispersity. The polydisperse effect is finally clarified quantitatively by focusing on the advection coefficients with a help of numerical examples.

In this paper primary and secondary resonance of carbon nanotube conveying magnetic nanofluid and subjected to a longitudinal magnetic field resting on viscoelastic foundation with different boundary conditions is investigated. To investigate the small scale effects, stress driven nonlocal integral model has been used and to show the more correctness of stress driven nonlocal integral model response, in studying the behavior of carbon nanotube with different boundary conditions, its results are compared with strain gradient model. The governing partial differential equations are derived from the Bernoulli–Euler beam theory utilizing the von Karman strain–displacement relations. Using the Galerkin method, the governing equations are reduced to a nonlinear ordinary differential equation. The nonlinear natural frequencies are obtained from the perturbation method and the divergence and flutter instability due to the increase in nanofluid velocity is investigated. Then the frequency response for primary, subharmonic and superharmonic resonance is obtained using the method of multiple scales. Finally, the effects of length small scale parameters, longitudinal magnetic field, magnetic nanofluid and boundary conditions on nonlinear free and forced vibration of carbon nanotube are investigated. As the most important results, as the intensity of the magnetic field increases, the critical flow velocity increases and divergence and flutter occur later. But the critical flow velocity decreases with increasing the intensity of the magnetic field for a carbon nanotube conveying magnetic nanofluid. In forced vibration, increasing the intensity of the magnetic field increases the amplitude of the response for all boundary conditions in primary and secondary resonance.

High-voltage transmission line possesses a typical suspended cable structure that produces ice in harsh weather. Moreover, transversely galloping will be excited due to the irregular structure resulting from the alternation of lift force and drag force. In this paper, the nonlinear dynamics and internal resonance of an iced cable under wind excitation are investigated. Considering the excitation caused by pulsed wind and the movement of the support, the nonlinear governing equations of motion of the iced cable are established using a three-degree-of-freedom model based on Hamilton’s principle. By the Galerkin method, the partial differential equations are then discretized into ordinary differential equations. The method of multiple scales is then used to obtain the averaged equations of the iced cable, and the principal parametric resonance-1/2 subharmonic resonance and the 2:1 internal resonance are considered. The numerical simulations are performed to investigate the dynamic response of the iced cable. It is found that there exist periodic, multi-periodic, and chaotic motions of the iced cable subjected to wind excitation.

In this paper, the profile and stability of alternating wave solution, which arises as a bifurcated periodic solution of equivariant Hopf bifurcation with amazing properties, are investigated for a Stuart-Landau system consisting of three oscillators. The method of multiple scales is used to compute the normal form equation up to fifth order. The Floquet theory is introduced because it is difficult to directly analyze the stability of the alternating wave solution. By applying a time-varying complex coordinate transformation which does not change the stability of the solution of normal form that represents the alternating wave, the multipliers that completely determine the stability of alternating wave solution are explicitly solved. As a result, the criteria on parameters such that stable alternating wave solutions can be observed are provided. Based on studies through examples, we show that the proposed scheme of analysis is effective and some results on how parameters influence the stability of the alternating wave solution can be summarized. Our analysis confirms Golubitsky's assertion that the alternating wave solution will not be stable immediately after the equivariant Hopf bifurcation. We also find that a large time delay and a complex nonlinear gain will enhance the stability of alternating wave solution.

This paper considers the laminar to turbulent transition of pressure-gradient driven Poiseuille flow in a plane channel system rotating with some angular speed under the influence of the Coriolis force. The nonlinearities are taken into account in the fundamental equations of motion for the perturbations grown enough to cause laminar to turbulent transition in rotating channel flow. For the values of Reynolds number slightly greater than the critical values above which the perturbations may grow, the time and length scales are chosen for the slow variations in the amplitude parameter B of a nonlinear two-dimensional perturbation wave in plane channel flow rotating as a whole. By the method of multiple scales, the nonlinear complex evolution equation, called here the Ginzburg–Landau–Stewartson–Stuart equation, is derived for scalar function B dependent on slow and long scale variables of both time and space. The coefficients of this equation are calculated analytically and computationally, which contain all the parameters causing laminar to turbulent transition along with the departure parameter from the neutral stable surface toward the growing side. The laminar to turbulent transition can be analyzed by the solution of this nonlinear equation, which is dependent on its coefficients containing the non-dimensional flow parameters.

The stability of a two-dimensional gravity-driven thin viscous Newtonian fluid with broken time-reversal-symmetry draining down a uniformly heated inclined plane is discussed. The presence of the odd part of the Cauchy stress tensor with an odd viscosity coefficient brings new characteristics in fluid flow. A theoretical model is implemented, which captures the dependence of the surface tension on temperature, and the model also allows for variation in the density of the liquid with a thermal difference. The coupled effect of odd viscosity, variable density, and surface tension has been investigated both analytically and numerically. A nonlinear evolution equation of the free surface is derived by the method of systematic asymptotic expansion. A linear stability analysis is carried out, which yields the critical conditions for the onset of instability in long-wave perturbations. New interesting results illustrating how the critical Reynolds number depends on the odd viscosity as well as other various dimensionless parameters have been obtained. In addition, a weakly nonlinear stability analysis is performed based on the method of multiple scales from which a complex Ginzburg–Landau equation is obtained. It is observed that the film not only has supercritical stable and subcritical unstable zones, but also unconditional stable and explosive zones. It has been also shown that the spatial uniform solution corresponding to the sideband disturbance may be stable in the unstable region. Employing the Crank–Nicolson method in a periodic domain, the spatiotemporal evolution of the model has been analyzed numerically for different values of the odd viscosity as well as other flow parameters. Nonlinear simulations are found to be in good agreement with the linear and weakly nonlinear stability analysis. The results are conducive to the further development of related experiments.

A gravity-driven, thin, incompressible liquid film flow on a non-uniformly heated, slippery inclined plane is considered within the framework of the long-wave approximation method. A mathematical model incorporating variation in surface tension with temperature has been formulated by coupling the Navier–Stokes equation, governing the flow, with the equation of energy. For the slippery substrate, the Navier slip boundary condition is used at the solid–liquid interface. An evolution equation is formed in terms of the free surface, which includes the effects of inertia, thermocapillary as well as slip length. Using the normal mode approach, linear stability analysis is carried out and a critical Reynolds number is obtained, which reflects its dependence on the Marangoni number M n as well as slip length δ . This depicts that δ and M n both have the destabilization effect on the flow field. The linear study also reveals that the inertia force has a negligible effect compare to the thermocapillary or slip. In addition, the study highlights a critical Marangoni number at which the instability sets in when the thermocapillary stress attains a critical value. The method of multiple scales is used to investigate the weakly nonlinear stability analysis of the flow. The study interprets that the variation of M n and δ have substantial effects on different stable/unstable zones. It also shows that within a considered parametric domain, the unconditional stable zone completely vanishes for any value of M n , when the slip length δ attains a critical value. The study also divulges that in the subcritical unstable (supercritical stable) zone the threshold amplitude ( ζ a ) decreases (increases) with the increment of M n and δ . Further, we discussed the spatial uniform solution of the complex Ginzburg–Landau equation for sideband disturbances. Employing the Crank–Nicolson method, the nonlinear evolution equation of the free surface is solved numerically in a periodic domain, considering the sinusoidal initial perturbation of small amplitude. The nonlinear simulations are found to be in good agreement with the linear and weakly nonlinear stability analysis. The evolution of the maximum h max and minimum h min thickness amplifies, for small change of M n and δ . Further, it shows that the influence of the thermocapillary force amplifies the destabilizing nature of δ . The traveling wave solution confirms the existence of a fixed point for the considered parametric domain, chosen from the experimental data. Finally, the Hopf bifurcation of the fixed point exhibits that the nonlinear wave speed at the transcritical point increases as δ increases but decreases as M n increases. The noteworthy result which arises from the study is that a transcritical Hopf bifurcation exists if the slip length δ > max 1 6 M n − 1 3 , 1 2 M n − 2 3 − M n .

The main objective of the present study is to analyze the nonlinear vibration behavior of fractional Kelvin–Voigt viscoelastic beams on a nonlinear elastic foundation under harmonic excitation. Attention is concentrated on the primary, superharmonic, and subharmonic resonances with the fractional Kelvin–Voigt constitutive model. To describe some viscoelastic material behaviors such as biomaterials, intelligent and polymeric materials, the fractional Kelvin–Voigt constitutive equation is used. Governing equations are derived using Hamilton’s principle based on the Euler–Bernoulli beam theory, with nonlinear elastic foundation and Von Karman’s nonlinearities due to stretching. These nonlinear partial differential equations are reduced into nonlinear ordinary differential equations by the Galerkin projection technique. The method of multiple scales is utilized to obtain the response of beams under hard and soft excitations. Results are verified by the available literature. A parametric analysis is conducted to determine the influence of the fractional Kelvin–Voigt viscoelastic model on primary and secondary resonances. The numerical results illustrate that effect of the fractional Kelvin–Voigt model on the frequency-response and amplitude-response is remarkable. Therefore, the obtained results provide a useful benchmark for further nonlinear analysis of fractional Kelvin–Voigt viscoelastic beams with elastic foundations.

This paper deals with the frequency–amplitude response of superharmonic resonance of second order (order two) of electrostatically actuated Micro-Electro-Mechanical System (MEMS) cantilever resonators. The structure of MEMS resonators consists of a cantilever resonator over a parallel ground plate, with a given gap in between, and under AC voltage. This resonance results from hard excitations and AC voltage frequency near one-fourth of the natural frequency of the resonator. The forces acting on the resonator are the nonlinear electrostatic force to include fringe effect, and a linear damping force. In order to solve the dimensionless partial differential equation of motion along with boundary and initial conditions, two types of models are developed, namely Reduced Order Models (ROMs), and Boundary Value Problem (BVP) model. The BVP model is essentially a finite difference model with a discretization in time only. ROMs are developed using one through five modes of vibration. The Method of Multiple Scales (MMS), numerical integrations using MATLAB, as well as a continuation and bifurcation analysis are used to solve the ROMs. The BVP model, resulting from using finite differences for time derivatives, is also numerically integrated. Five modes of vibration ROM is found to make accurate predictions in all amplitudes. A softening effect of the response is predicted. The response consists of a bifurcation with a bifurcation point of amplitude one fourth of the gap, and a stable branch in larger frequencies with a pull-in instability end point at three fourths of the gap. The bifurcation point shifts to lower frequencies as the voltage and/or fringe effect increase, and/or damping decreases. If damping increases, the branches coalesce, peak amplitude decreases, and a linear behavior is experienced.

The nonlinear vibration response of the pipe conveying fluid with a breathing transverse crack, under primary and secondary excitations is investigated. Attention is concentrated on the superharmonic resonance which is the most sensitive phenomena for small crack detection. The crack is modeled with a transverse partial cut of the pipe wall thickness. The equivalent bending stiffness of the cracked pipe is derived based on an energy method and the breathing crack model is used to simulate the transition state between the fully open and the fully closed states. Based on Von Karman’s nonlinear geometric assumption and Euler–Bernoulli beam theory, the nonlinear geometric partial differential equations due to stretching effect, have been derived. The fractional viscoelastic model and the plug flow assumptions are used to model the damping and fluid flow. The coupled nonlinear partial differential equations are reduced into nonlinear ordinary differential equations by the Galerkin method. The method of multiple scales is adopted to analyze steady-state solutions for the primary and superharmonic resonances. A parametric sensitivity analysis is carried out and the effects of different parameters, namely the geometry and location of the crack, nonlinear geometric parameter, fluid flow velocity, mass ratio, fractional derivative order and retardation time on the frequency-response solution are examined. In general, this paper shows that the superharmonic response amplitude measurement is useful tool for damage detection of pipes conveying fluid in the early stages.

As a stable limit cycle oscillation occurred after subcritical Hopf bifurcation, galloping was preferred in scattered wind energy harvesting. For practical applications across seasons and regions, environmental adaptability of galloping piezoelectric energy harvester on atmospheric temperature and wind speed is investigated. A multi-field coupled geometric and aerodynamic nonlinear distributed-parameter model is established. The method of multiple scales is used to derive the analytical solutions of the nonlinear coupled continuous model. Wind-tunnel tests of the piezoelectric galloping energy harvesters are performed for model validation. The empirical coefficients of nonlinear piezoelectric parameters varied by the temperature are identified. The optimal load resistance tailored for the atmospheric temperature and airflow speed is determined. With the environment adaptable load resistance, the onset speed of energy harvesting is minimized to be lower than 1 m/s for the galloping to occur. High temperature benefits galloping energy harvesting begin at low wind speed. Excellent environmental adaptability of the piezoelectric galloping energy harvester is realized in atmospheric temperature of − 40 ° C to 50 ° C and breeze to gale wind speed of 1–24 m/s.

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

In this paper, principal parametric resonance of rotating Euler–Bernoulli beams with varying rotating speed is investigated. The model contains the geometric nonlinearity due to the von Karman strain–displacement relationship and the centrifugal forces due to the rotation. The rotating speed of the beam is considered as a mean value which is perturbed by a small harmonic variation. In this case, when the frequency of the periodically perturbed value is twice the one of the axial mode frequencies, the principal parametric resonance occurs. The direct method of multiple scales is implemented to study on the dynamic instability produced by the principal parametric resonance phenomenon. A closed-form relation which determines the stability region boundary under the condition of the principal parametric resonance is derived. Numerical simulation based on the fourth-order Runge–Kutta method is established to validate the results obtained by the method of multiple scales. The numerical analysis is applied on the discretized equations of motion obtained by the Galerkin approach. After validation of the results, a comprehensive study is adjusted for demonstrating the damping coefficient and the mode number influences on the critical parametric excitation amplitude and the parametric stability region boundary. A discussion is also provided to illustrate the advantages and disadvantages of the fourth-order Runge–Kutta method in comparison with the method of multiple scales.

s Irregular instability boundaries of axially accelerating beams with 1:3 internal resonance are analytically and numerically investigated in this paper. The distributed parameter is due to the small simple harmonic axial speed. The viscoelastic characteristic of the beam is described by the Kelvin–Voigt model in which the material time derivative is used. A linear partial-differential equation with the variable coefficient and the relevant boundary conditions governing the transverse motion is presented. The effects of the nonhomogeneous boundaries are highlighted. By the method of multiple scales, the solvability conditions in summation and principal parametric resonances are established by some different manipulations in the process of the classical multiple scales method. The Routh–Hurwitz criterion is used to determine the instability boundaries. The effects of viscoelastic coefficient and the viscous damping coefficient are examined on the instability boundaries. Irregular instability boundaries appeared when the 1:3 internal resonance is introduced. It is shown that the numerical calculations by the differential quadrature scheme can verify the approximate analytical results.

The continuous model and nonlinear dynamic responses of a circular mesh antenna subjected to the thermal excitation in the space environment are investigated for the first time. A continuum cantilever circular cylindrical short shell, which is clamped at one side of the shell along the axial direction, is proposed to take place of the circular mesh antenna composed of the repetitive beamlike lattice by the principle of equivalent effect. Based on the first-order shear deformation shell theory and von Karman nonlinear strain-displacement relationship, the nonlinear governing equations of motion are derived by using the Hamilton’s principle. The Galerkin approach is used to transform the governing nonlinear partial differential equations into a set of nonlinear ordinary differential equations. The method of multiple scales is utilized to obtain the four-dimensional averaged equation when the 1:1 internal resonance is taken into account. The numerical results, which include the time histories, phase plots, and frequency spectrum, are obtained for the mesh antenna. The influences of the thermal excitation and the damping coefficient on the nonlinear dynamics are analyzed for the mesh antenna.

Suppression of synchronization is a central issue of the Internet congestion control. Current desynchronization scheme suffers from the breakdown of fairness. We propose a desynchronization scheme by introducing state feedback controllers in such a way that the sum of all components of the eigenvectors of the linearized system around the equilibrium is zero. With the aid of the method of multiple scales, the conditions on the values of controller parameters such that the desynchronous oscillation arises are obtained and two different patterns of desynchronization, i.e., discrete wave and anti-synchronization are observed, depending on the parity of number of sources. Numerical simulations are carried out to validate the theoretical analysis and the effectiveness of the proposed scheme is confirmed.

In this paper, the combination of the cubic nonlinearity and time delay is proposed to improve the performance of a piecewise-smooth (PWS) system with negative stiffness. Dynamical properties, feedback control performance and symmetry-breaking bifurcation are mainly considered for a PWS system with negative stiffness under nonlinear position and velocity feedback control. For the free vibration system, the homoclinic-like orbits are firstly derived. Then, the amplitude-frequency response of the controlled system is obtained analytically in aspect of the Lindstedt–Poincare method and the method of multiple scales, which is also verified through the numerical results. In this regard, a softening-type behavior, which directly leads to the multi-valued responses, is illustrated over the negative position feedback. Especially, the five-valued responses in which three branches of them are stable are found. And complex multi-valued characteristics are also observed in the force-amplitude responses. Furthermore, for explaining the effectiveness of feedback control, the equivalent damping and stiffness are also introduced. Sensitivity of the system response to the feedback gain and time delay is comprehensively considered and interesting dynamical properties are found. Relatively, from the perspective of suppressing the maximum amplitude and controlling the resonance stability, the selection of the feedback parameters is discussed. Finally, the symmetry-breaking bifurcation and chaotic motion are considered.

This paper investigates analytically and numerically the effect of initial offset imperfection on the mechanical behaviors of microbeam-based resonators. Symmetry breaking of DC actuation, due to different initial offset distances of microbeam to lower and upper electrodes, is concerned. For qualitative analysis, time-varying capacitors are introduced and a lumped parameter model, considering nonlinear electrostatic force and midplane stretching of microbeam, is adopted to examine the system statics and dynamics. The Method of Multiple Scales (MMS) is applied to determine the primary resonance solution under small vibration assumption. Meanwhile, the Finite Difference Method (FDM) combined with Floquet theory is utilized to generate frequency response curves for medium- and large-amplitude vibration simulations. Static bifurcation, phase portrait and Hamiltonian function are firstly investigated to examine the system inherent behaviors. Besides, basins of attraction are briefly depicted to grasp the effects of initial offset and AC excitation on the system global dynamics. Then, variation of equivalent natural frequency versus DC voltage is analyzed. Results show that initial offset may induce complex frequency rebound phenomenon as well as a separate frequency branch under secondary pull-in condition. In what follows, emergences of softening, linear and hardening vibration are classified through discussing a key parameter obtained from the frequency response equation. New linear behavior induced by initial offset imperfection is found, which exhibits much higher sensitivity to DC voltage. Medium- and large-amplitude in-well motions are also investigated, indicating the existence of alternations of softening and hardening behaviors. Finally, lumped parameters are deduced via Galerkin procedure, and case studies are provided to illustrate the effectiveness of the whole analysis.

We study the bifurcation characteristics of a lumped-parameter model of rotary drilling with 1:1 internal resonance between the axial and the torsional modes which leads to the largest stability thresholds. For this special case, the two-degree-of-freedom model for the drill-string reduces to an effectively single-degree-of-freedom system facilitating further analysis. The regenerative effect of the cutting action due to the axial vibrations is incorporated through a delayed term in the cutting force with the delay depending on the torsional oscillations. This state dependency of the delay introduces nonlinearity in the current model. Steady drilling loses stability via a Hopf bifurcation, and the nature of the bifurcation is determined by an analytical study using the method of multiple scales. We find that both subcritical and supercritical Hopf bifurcations are present in this system depending on the choice of operating parameters. Hence, the nonlinearity due to the state-dependent delay term could both be stabilizing or destabilizing in nature, and the self-interruption nonlinearity is essential to capture the global behavior. Numerical bifurcation analysis of a global axial–torsional model of rotary drilling further confirms the analytical results from the method of multiple scales. Further exploration of the rotary drilling dynamics unravels more complex phenomena including grazing bifurcations and possibly chaotic solutions.

Transverse vibrations of axially moving beams with multiple concentrated masses have been investigated. It is assumed that the beam is of Euler–Bernoulli type, and both ends have simply supports. Concentrated masses are equally distributed on the beam. This system is formulated mathematically and then sought to find out approximate solutions. In case of three-to-one internal resonance, analytical solutions are derived by means of method of multiple scales (a perturbation method). It is assumed that axial velocity of the beam is harmonically varying around a mean-constant velocity. Steady-state vibration characteristics are investigated from the amplitude-phase modulation equations. Then, the effects of both magnitude and number of the concentrated masses on nonlinear vibrations are investigated numerically in detail.

Local and global resonances under the condition of 3:1 internal resonance of a super-critically axially moving beam, subjected to a harmonic exciting force, are investigated in the present work. The governing equation is derived from the generalized Hamilton's principle and discreted into a multiple-degrees-of-freedom system by the Galerkin's method. In the super-critical regime, the axially moving beam becomes a bistable system with two symmetrical non-trivial equilibrium configurations. Based on the transformation around one of them, natural frequencies and the condition of internal resonance are obtained. By employing the method of multiple scales, resonances for first-two modes and harmonics under the condition of internal resonance are discussed analytically. Total displacement at the middle of the beam is composed by them and confirmed by direct numerical method. Internal resonance is found to have a big effect on the phase angle of and the amplitude. Coupling ship between the first-two modes is verified to be produced by the cubic nonlinearity and the 3:1 commensurability together. The effect of moving speed acting on the internal resonance is discussed and an energy transmission region is found. Different with the internal resonance in the sub-critical regime, most of the transferred energy is absorbed by the quadratic nonlinearity in the super-critical regime. The critical excitation of the local response is predicted by the analytical method and certified by simulations. The global response for the primary resonance has two stable focal points. However, the global response for the secondary resonance only has one stable focal point for the non-trivial equilibrium configuration is counteracted.