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Global bifurcations and multi-pulse chaotic motions of flexible multi-beam structures derived from an L-shaped beam resting on a vibrating base are investigated considering one to two internal resonance and principal resonance. Base on the exact modal functions and the orthogonality conditions of global modes, the PDEs of the structure including both nonlinear coupling and nonlinear inertia are discretized into a set of coupled autoparametric ODEs by using Galerkin’s technique. The method of multiple scales is applied to yield a set of autonomous equations of the first order approximations to the response of the dynamical system. A generalized Melnikov method is used to study global dynamics for the “resonance case”. The present analysis indicates multi-pulse chaotic motions result from the existence of Silnikov’s type of homoclinic orbits and the critical parameter surface under which the system may exhibit chaos in the sense of Smale horseshoes are obtained. The global results are finally interpreted in terms of the physical motion of such flexible multi-beam structure and the dynamical mechanism on chaotic pattern conversion between the localized mode and the coupled mode are revealed.

The dynamic modelling of a simply-supported thin laminated plate subject to in-plane excitation is established based on the classic shear theory and von Karman nonlinear theory. The method of multiple scales is used to determine an approximate solution for the system. According to solvability conditions, the nonlinear modulation equations arising from the principal parametric resonances are obtained and two possible nontrivial solutions are performed. To analyze the nonlinear dynamic response of the plate embedded with auxetic layers, 5-layered sandwich plate, in which two auxetic elastic layers are alternatively sandwiched between three positive Poisson’s ratio (PPR) elastic ones, is presented. The natural frequency of model ( m , n ) shows an increase with respect to the absolute value of Poisson’s ratio. Particularly, the amplitude-frequency responses of the laminated plate subject to principal parametric resonance are analyzed for different values of Poisson’s ratio. Moreover, it can be found that for model ( m , n ), there must be some certain value or interval of negative Poisson’s ratio (NPR), which, results in zero response effect, in other words, the in-plane excitation will be ineffective for this model when the Poisson’s ratio just lies at such a value or interval. Furthermore, it can also be observed that the certain interval of Poisson’s ratio becomes wider with the increase of damping.

Nonlinear modal interactions have recently become the focus of intense research in micro-resonators for their use to improve oscillator performance and probe the frontiers of fundamental physics. Understanding and controlling nonlinear coupling between vibrational modes is critical for the development of advanced micromechanical devices. This article aims to theoretically investigate the influence of antisymmetry mode on nonlinear dynamic characteristics of electrically actuated microbeam via considering nonlinear modal interactions. Under higher-order modes excitation, two nonlinear coupled flexural modes to describe microbeam-based resonators are obtained by using Hamilton’s principle and Galerkin method. Then, the Method of Multiple Scales is applied to determine the response and stability of the system for small amplitude vibration. Through Hopf bifurcation analysis, the bifurcation sets for antisymmetry mode vibration are theoretically derived, and the mechanism of energy transfer between antisymmetry mode and symmetry mode is detailed studied. The pseudo-trajectory processing method is introduced to investigate the influence of external drive on amplitude and bifurcation behavior. Results show that nonlinear modal interactions can transit vibration energy from one mode to nearby mode. In what follows, an effective way is proposed to suppress midpoint displacement of the microbeam and to reduce the possibility of large deflection. The quantitative relationship between vibrational modes is also obtained. The displacement of one mode can be predicted by detecting another mode, which shows great potential of developing parameter design in MEMS. Finally, numerical simulations are provided to illustrate the effectiveness of the theoretical results.

The method of multiple scales is adopted to investigate the dynamic response of a nonlinear Vibration Absorber (VA) whose constitutive behavior is governed by hysteresis with pinching. The asymptotic analysis is first devoted to study the response of the absorber to harmonic excitations and to evaluate its sensitivity to the main constitutive parameters. The frequency response obtained in closed form allows to carry out the stability analysis together with a parametric study leading to behavior charts characterizing multi-valued softening/hardening responses or single-valued, quasi-linear responses. A two-degree-of-freedom model of a primary nonlinear structure endowed with the hysteretic vibration absorber is investigated to explore transfers of energy from the structure to the absorber resulting into optimal vibration amplitude reduction. The asymptotic solution is proved to be in good agreement with the numerical solution obtained via continuation. The asymptotic approach is embedded into a differential evolutionary algorithm to obtain a multi-parameter optimization procedure by which the optimal hysteresis parameters are found.

The dynamical modeling, multi-pulse orbits and chaotic dynamics of cantilevered pipe conveying pulsating fluid with harmonic external force are studied using the energy-phase method for the first time. The nonlinear geometric deformation of the pipe and the Kelvin constitutive relation of the pipe material are considered. The nonlinear governing equation of motion for cantilevered pipe conveying pulsating fluid is determined using Hamilton principle. The four-dimensional averaged equation in the case of primary parametric resonance-1/2 subharmonic resonance and 1:2 internal resonance is obtained by directly using the method of multiple scales and the Galerkin approach to the partial differential governing equation of motion for cantilevered pipe conveying pulsating fluid. From the averaged equation, the normal form theory is used to find the explicit formulas of normal form. Based on normal form obtained here, the energy-phase method is utilized to analyze the multi-pulse global bifurcations and chaotic dynamics of cantilevered pipe conveying pulsating fluid. The analysis of global dynamics indicates that the multi-pulse jumping orbits exist in the perturbed phase space of the averaged equation. Based on the averaged equation, the chaotic motions and the multi-pulse orbits for the cantilevered pipe are found by using numerical simulations. The results described above indicate the existence of chaos in the Smale horseshoe sense for cantilevered pipe conveying pulsating fluid. It has been concluded that it is dangerous to induce the chaotic vibrations of the pipes conveying pulsating fluid because the amplitudes of the chaotic vibrations are larger than those of the periodic motions.

The nonlinear dynamics of a flexible space antenna supported by a serial of two rigid arms mounted on a heavy satellite is studied under the conditions of 2:1 or 3:1 internal resonances. The nonlinear dynamic equations of the in-plane vibration of the antenna system of three degrees of freedom are derived first via the method of assumed modes. Then, the resonance and non-resonance regions are determined in terms of the dimensionless parameters of a rigid arm and a torsional spring in the antenna system. Afterwards, the approximate analytic solutions of frequency-amplitude response of the antenna system under 2:1 or 3:1 internal resonances are derived by using the method of multiple scales. Furthermore, the topologic description for the bifurcation of each internal resonance with respect to unfolding parameters is studied by means of the theory of singularity. Finally, the accuracy of the approximate analytic solutions is validated via the numerical solutions. The study shows that the internal resonances exist over a wide range of the detuning parameter. In addition, the 3:1 internal resonance exhibits much more complicated bifurcation topology than the 2:1 internal resonance. The approximate analytic solutions of internal resonances and their bifurcation analysis results provide an essential reference for designing the antenna system of concern.

We investigated the cubic–quintic nonlinear response in a parametrically excited simply supported beam subjected to a spring force in the axial direction. Taking into account the cubic and quintic geometric nonlinearities of curvature of the beam, the governing equation of the parametrically excited beam was derived based on Hamilton’s principle. The fifth-order approximate solution was analytically obtained using the method of multiple scales; with this calculation, the third-order nonlinear normal mode was also obtained. Its associated amplitude revealed saddle-node bifurcation and a hysteresis in the frequency response curve, which could not be predicted using the third-order approximate solution for the governing equation that included only cubic nonlinearity. Experimental results taken using a simple apparatus qualitatively verify the theoretically predicted nonlinear features in the parametric resonance caused by the cubic–quintic geometric nonlinearity of the beam.

The nonlinear responses and stability of double-layered nanoplate embedded in the elastic medium are investigated in the presence of 3:1 internal resonance. By the nonlocal theory, the method of multiple scales is employed to obtain the analytical nonlinear frequency-response relations. Two different external primary resonance conditions, i.e. the first and the second modes being directly excited, are considered. The influences of the small scale effect and viscous damping on the nonlinear vibration are explored in details. From the results, the frequency-response curves for the two primary resonance cases present complete different characteristics. It should be noted that the response curves are closed loops for the resonance of the second mode, which implies the steady-state response just exists in a finite frequency range. The regions of multi-values appear for both cases and the stability of the response is determined. When the first mode is directly excited, the impact of the viscidity of nanoplate and small scale effect on the frequency range of unstable response is rather significant. Furthermore, when the second mode is directly excited, a novel phenomenon, i.e. the frequency range for the closed loops of response diminished enormously as the increase of the viscidity of nanoplate, can be observed.

This study investigates the vibration of rotating elastic rings that are dynamically excited by an arbitrary number of space-fixed discrete stiffnesses with periodically fluctuating stiffnesses. The rotating, elastic ring is modeled using thin-ring theory with radial and tangential deformations. Primary and combination instability regions are determined in closed-form using the method of multiple scales. The ratio of peak-to-peak fluctuation to average discrete stiffness is used as the perturbation parameter, so the resulting perturbation analysis is not limited to small mean values of discrete stiffnesses. The natural frequencies and vibration modes are determined by discretizing the governing equations using Galerkin's method. Results are demonstrated for compliant gear applications. The perturbation results are validated by direct numerical integration of the equations of motion and Floquet theory. The bandwidths of the instability regions correlate with the fractional strain energy stored in the discrete stiffnesses. For rings with multiple discrete stiffnesses, the phase differences between them can eliminate large amplitude response under certain conditions.

During the on-orbit service of an deployable antenna, the resonance of the antenna will affect the accuracy of direction. The nonlinear dynamics of a rigid-flexible space antenna is studied under the conditions of two-to-one or three-to-one internal resonance analytically and experimentally. The nonlinear dynamic equations for the planar vibration of the antenna structure with two degrees of freedom are derived via the assumed modes method. Then, the resonance parameter planes are obtained according to the length ratio, the mass ratio and the stiffness of the torsional spring in the antenna system. Afterwards, the method of multiple scales is utilized to obtain the approximate solutions under two-to-one or three-to-one internal resonance. Furthermore, the bifurcation characteristics of the nonlinear normal modes of the antenna system are investigated as well. The results show that more than one nonlinear normal mode exist over a wide range of the detuning parameter. To validate the accuracy of the approximate solution, a numerical solution and an experimental investigation are presented, respectively. The results show that both the numerical and the experimental results agree well with the analytical one.

The nonlinear axial-lateral-torsional free vibration of the rotating shaft is analyzed by employing the Rayleigh beam theory. The effects of lateral, axial and torsional deformations, gyroscopic forces and rotary inertia are taken into account, but the shear deformations are neglected. In the new developed dynamic model, the nonlinearities are originated from the stretching of beam centerline, nonlinear curvature and twist and inertial terms which leads to the coupling between the axial, lateral and torsional deformations. The deformed configuration of the cross section of the beam is represented by the axial and lateral deformations, also the geometry of the beam in the deformed configuration is represented by Euler angles. A system of coupled nonlinear differential equations is obtained which is examined by the method of multiple scales and the nonlinear natural frequencies are determined. The accuracy of the solutions is inspected by comparing the free vibration response of the system with the numerical integration of the governing equations. The effect of the spin speed and radius-to-length ratio of the rotating shaft on the free vibrational behavior of the system is inspected. The study demonstrates the effect of axial-lateral-torsional coupling on the nonlinear free vibrations of the rotating shaft.

The global bifurcations and multi-pulse orbits of an aero-thermo-elastic functionally graded material (FGM) truncated conical shell under complex loads are investigated with the case of 1:2 internal resonance and primary parametric resonance. The method of multiple scales is utilized to obtain the averaged equations. Based on the averaged equations obtained, the normal form theory is employed to find the explicit expressions of normal form associated with a double zero and a pair of pure imaginary eigenvalues. The energy-phase method developed by Haller and Wiggins is used to analyze the multi-pulse homoclinic bifurcations and chaotic dynamics of the FGM truncated conical shell. The analytical results obtained here indicate that there exist the multi-pulse Shilnikov-type homoclinic orbits for the resonant case which may result in chaos in the system. Homoclinic trees which describe the repeated bifurcations of multi-pulse solutions are found. The diagrams show a gradual breakup of the homoclinic tree in the system as the dissipation factor is increased. Numerical simulations are presented to illustrate that for the FGM truncated conical shell, the multi-pulse Shilnikov-type chaotic motions can occur. The influence of the structural-damping, the aerodynamic-damping, and the in-plane and transverse excitations on the system dynamic behaviors is also discussed by numerical simulations. The results obtained here mean the existence of chaos in the sense of the Smale horseshoes for the FGM truncated conical shell.

Nonlinear responses of a single pile with soil-structure interaction under primary resonance and two-to-one internal resonance are investigated. Considering the effect of soil-structure interaction and geometric nonlinearity, the Hamilton principle is applied to derive the equation of motion of the pile subjected to a lateral excitation and axial load. Then, the effective nonlinear coefficient and the frequency-response equation are presented by employing the multimode discretization and the method of multiple scales. Moreover, the nonlinear frequencies are obtained for the analysis of the nonlinear dynamic characteristics of the pile. The equilibrium and dynamic solutions of modulation equations are examined by using the Newton-Raphson, shooting, and continuation methods. The nonlinear responses of the pile are explored by means of the backbone curves and the frequency (force)-response curves. Meanwhile, the effect of the axial load on the nonlinear dynamic characteristics of the pile is discussed. The numerical results show that the nonlinear responses of the pile exhibit some rich and complex nonlinear phenomena.

A theoretical and experimental nonlinear analysis of cellular response/displacement to ultrasound excitations is presented. Linear cell models can predict the resonant frequency ( f R ∼ 5 MHz ) , but only a nonlinear analysis can reveal the amount of mechanical energy that couples into the cell and the bifurcation behavior of the cell when it is excited near resonance. The cell dynamics is described by the nonlinear viscoelastic constitutive behavior of the cytoplasm, nucleus and their respective membranes, in the presence of a fluid with an oscillating pressure field. The method of multiple scales is used to derive the amplitude of oscillation of the cytoplasm and nucleus as a function of frequency. A major finding is the existence of multiple solutions for a range of sub-resonant frequencies. At positive detuning ( f > f R ) , the mechanical energy that couples into the cell is small, it is higher at resonance but significantly higher at sub-resonant frequencies in the multiplicity range. Experimentally it was shown when 3.5 MHz is approached sub- and supra-resonance and 6.5 MHz is approached sub-resonance, gene expression was statistically higher than that when stimulated directly. Thus, there exists an optimal range of frequencies for ultrasound treatment – in the region of multiplicity where deformation and thus mechanical energy coupling is maximized. The ultrasound protocol must be designed to operate at the solution associated with the higher mechanical energy – thus the start-up conditions should be in the domain of attraction of the high energy solution.

Aeroelastic flutter characteristics and dynamic response of a composite laminated circular cylindrical shell under combined action of radial harmonic excitation, compressive in-plane force and aerodynamic pressure are studied in this paper. The first-order piston theory is employed to model the aerodynamics pressure. Partial differential equations governing the vibrations of the cylindrical shell are derived based on the Hamilton’s principle and the Donnell’s nonlinear shell theory. The Galerkin’s method is adopted to discretize the partial differential governing equations to a set of nonlinear ordinary differential equations. The four-dimensional averaged equation is obtained by applying the method of multiple scales under the case of 1:2 internal resonance. The critical free stream static pressure originating flutter of the shell is determined by solving the eigenvalue problem. The phase portrait and time history diagrams are presented to demonstrate the character of the limit cycle oscillation of the shell. The influence of different geometrical parameters, such as the radius, length and thickness of the shell, on the flutter characteristics of the composite laminated circular cylindrical shell are discussed in details. The influences of the amplitudes of the in-plane and transverse excitations on the frequency–response curves and force-response curves are also investigated.

A theoretical solution is formulated to analyze the vibration behaviors of circular diaphragm-type piezoactuators based on the Hamilton’s principle and Rayleigh-Ritz method, which are particular suitable for modeling the deflection of multilayer structures. Each of the actuator three layers is considered as an individual layer in the modeling. The energy associated with the solution includes the kinetic energy of the actuator, the elastic potential energy of the various layers, the electric potential energy in the piezodisc, and the work done by the force of electric filed. The transverse displacement is separated into a time dependence term and a mode shape term, then the vibrational governing equation is derived using the functional variation, and is approximately solved through the method of multiple scales. Moreover, added mass loads are introduced to the diaphragm center for the sake of decreasing the resonant frequency, where many MEMS devices, such as gas micropumps and ejectors, have a higher working efficiency. The proposed analytical solution is validated numerically via the finite element method (FEM) and experimentally via measurements; the theoretical results are found to be in good agreement with the FEM results as well as with the experimental results. Furthermore, the effects of mass loads, geometric dimensions and material properties of the piezoactuator on the resonant frequency are discussed.

This work is motivated by the self-powered component of biomedical nano-robotic device which is expected to move in arterial blood vessels. The transverse vibrations and steady-state responses of axially moving viscoelastic piezoelectric two-dimensional nanostructures are investigated based on the nonlocal viscoelasticity thin plate theory. The constitutive relations of viscoelastic piezoelectric nanoplate containing the thermal effect are performed and the governing partial differential equations of the problem model are derived using the Hamilton's principle. The natural frequencies are numerically determined via the Galerkin method, the complex mode method, as well as the finite element method for comparison. Moreover, the theoretical calculations are compared with those in previous literature to verify effects in nonlocal nanoscale framework and average speed on natural frequency. Afterwards, the instable behaviors of axially non-uniformly moving viscoelastic piezoelectric nanoplate characterized as a sine variation about the constant average speed are addressed using the method of multiple scales. The analyses are mainly focused on the boundaries of instable regions in combination parametric resonance and principal parametric resonance. The non-dimensional numerical results imply the existences of nonlocal nanoscale parameter and average speed contribute to reduce the rigidity, and further produce the coupled vibrations and flutter instabilities for complex frequencies. Additionally, the instable regions of combination and principal parametric resonances decrease with increases in the biaxial compression, change of temperature, positive electric voltage and viscoelastic coefficient. The dynamic responses of axially moving viscoelastic piezoelectric nanoplate under the coupling of thermo-electro-mechanical multi-fields are expected to play significant roles in designing biomedical nano-robots.

Here, the large amplitude axisymmetric vibration of a circular plate having a circumferential crack is studied considering the simply supported and clamped boundary conditions. Exact modal functions are generated by satisfying the boundary conditions and constraints along the crack. The inplane or membrane forces resulting from the large deflection are derived from the Berger's formulation. Next, the Galerkin's method is used to transform the differential equation into the Duffing equation with cubic nonlinearities. The amplitude and phase curves are generated using the Method of multiple scales. Natural frequency variations of axisymmetric modes are studied for a change in crack depth and crack position, considering both boundary conditions. Some natural frequencies are also compared with the finite element results and they are found to be in good agreement. For clamped boundary conditions, the natural frequency of a cracked plate is shown to approach that of an uncracked plate for a particular crack position independent of different crack depths. The amplitude and phase curves are shown for different crack parameters. Finally, the phase plane plot variations are presented for different crack depths, positions and boundary conditions.

In the super-critical regime, steady-state responses of an axially moving beam are analyzed subjected to parametric combined with forced excitations. By employing the method of multiple scales, the primary resonance is investigated. Steady-state resonances exist unless the parametric frequency and the external frequency are commensurable. Natural modes are triggered when the parametric frequency is close to two times of or just the natural frequency. For the case of the first one, the combined excitation deduces a response curve with twin resonance peaks. Distance of them is determined by the parametric excitation, and the widths are depended on the external force. Double jumping is found in the response curve. For the case of the second one, the combined excitation produces a simple resonance in the form of a typical forced vibration. The response curve is superimposed by each of the excitations. With the numerical method, the incommensurability of excitation frequencies is found to produce beats and quasi-periodicity vibrations.

The dynamical stability and transverse vibration of the cantilever beam with oscillating length are analyzed in this study. The differential equation of motion with time-dependent coefficients is discretized by the Galerkin method, and then the method of multiple scales for multi-degree of freedom is applied to investigate the parametric resonances of the cantilever beam with oscillating length. The effects of the oscillation amplitude and frequency on the parametric resonance regions and the tip responses are discussed. Tip responses simulation by Runge–Kutta method confirms the parametric resonance regions obtained by multiple scales method. In addition, the ‘jump’ phenomenon on the tip response of the axially oscillating deploying cantilever beam is also discussed.

In this paper, we study the longitudinal linear and nonlinear free vibration response of a single walled carbon nanotube (CNT) embedded in an elastic medium subjected to different boundary conditions. This formulation is based on a large deformation analysis in which the linear and nonlinear von Karman strains and their gradient are included in the expression of the strain energy and the velocity and its gradient are taken into account in the expression of the kinetic energy. Therefore, static and kinetic length scales associated with both energies are introduced to model size effects. The governing motion equation along with the boundary conditions are derived using Hamilton's principle. Closed-form solutions for the linear free vibration problem of the embedded CNT rod are first obtained. Then, the nonlinear free vibration response is investigated for various values of length scales using the method of multiple scales.

This paper presents the vibration and stability analyses of an unbalanced rotor mounted on high-static-low-dynamic-stiffness supports. The stiffness of the supports is modeled as symmetric of cubic order. Then a second-order multiple scales method is used for studying the primary resonance of the system. The types of singular points are investigated and phase-plane of the system is plotted using analytical and numerical methods. The difference between analytical and numerical solutions is less than 2 percent.

A novel idea that applies the multiple scale analysis to a discretized decoupled system of gyroscopic continua is introduced and an axial moving string is treated as an example. First, the invariant manifold method is applied to the discretized ordinary differential equations of the axially moving string. Complex gyroscopic mode functions that agree well with true analytical results are obtained. The gyroscopic modes are subsequently used for the discretized ordinary differential equations with gyroscopic and nonlinear coupling terms that yield a gyroscopically decoupled system. Further the method of multiple scales is used to obtain the equations at a slow scale. This novel procedure is compared to solutions obtained by directly applying the classical multiple scale analysis to the gyroscopically coupled system without decoupling. The modal decoupled system analysis yields better frequency with comparing to the classic method. The proposed methodology provides a novel alternative for nonlinear dynamic analysis of gyroscopic continua.

Viscoelastic phenomena widely exist in MEMS materials, which may have certain effects on quasi-static behaviors and transition mechanism of nonlinear jumping phenomena. The static and dynamic behaviors of a doubly clamped viscoelastic microbeam actuated by one sided electrode are investigated in detail, based on a modified couple stress theory. The governing equation of motion is introduced here, which is essentially nonlinear due to its midplane stretching effect and electrostatic force. Through quasi-static analysis, the equilibrium position, pull-in voltage and pull-in location of the system are obtained with differential quadrature method and finite element method. The equivalent geometric nonlinear parameter is presented to explain the influence of the scale effect on the pull-in location. Different from elastic material, there are two kinds of pull-in voltages called as instantaneous pull-in voltage and the durable pull-in voltage in viscoelastic system. Then, Galerkin discretization and the method of multiple scales are applied to determine the response and stability of the system for small vibration amplitude. A new perturbation method to deal with viscoelastic term is presented. Theoretical expressions about the parameter spaces of linear-like vibration, hardening-type vibration and softening-type vibration are then deduced. The influence of viscoelasticity and scale effect on nonlinear dynamic behavior is studied. Results show that the viscoelasticity can reduce the effective elastic modulus and make the system tend to softening-type vibration; the scale effect can increase effective elastic modulus and make the system tend to hardening-type vibration. And most of all, simulation results of case studies are used to realize parameter optimization. Then parameter conditions of linear-like vibration, which is desired for many applications, are obtained. In this paper, the results of multi-physical field coupling simulation are used to verify the theoretical analysis.

In this paper, nonlinear dynamics of an unbalanced composite spinning shaft are studied. Extensional–flexural–flexural–torsional equations of motion are derived via utilizing the three-dimensional constitutive relations of the material and Hamilton’s principle. The gyroscopic effects, rotary inertia and coupling due to material anisotropy are included, while the shear deformation is neglected. To analyze the rotor dynamic behavior, the full form of the equations without any simplification assumption (e.g., stretching or shortening assumption) is used. The method of multiple scales is applied to the discretized equations. An analytical expression as a function of the system parameters describing the forced vibration of a spinning composite shaft in the neighborhood of the primary resonance is obtained. The discretization is done with both one and two modes, and the results are compared. It is shown that although the excitation is tuned in the neighborhood of the first mode, one-mode discretization is not sufficient and it leads to inaccurate results. It shows the necessity of employing at least two modes in discretization due to the coupling in the equations. The effects of the external damping, eccentricity and the lamination angle on the vibration amplitude are investigated. In addition, the effect of the extensional–torsional coupling on the frequency response curves is investigated. To validate the perturbation results, numerical simulation is used.

The method of multiple scales is one of the common perturbation techniques for exploring nonlinear ordinary differential equations. Sitnikov has presented a mathematics model of a negligible mass body oscillation perpendicular to a plane in which two heavy bodies with equal mass orbit on itself Keplerian ellipse. This problem is well known as the Sitnikov problem. In this investigation, the method of multiple scales is applied to the Sitnikov problem and it presented an analytical response that is consistent with numerical solution. At first step, the circular form of the Sitnikov equation is approximated by MMS and then the obtained dynamic model from first step is employed to investigate the elliptical form of the Sitnikov equation. Some initial conditions and eccentricities of the primary bodies are studied and results are compared with numerical solution. Results show that estimated analytical solution can help to discover the Sitnikov equation behavior.

The nonlinear harmonic response of a cantilever hard-coating plate which is made of a layer of anisotropic hard-coating material and isotropic metal substrate is investigated based on the theory of high-order shear deformation of plate. Firstly, based on the theories of von Karman and Reddy’s three-order shear deformation, the nonlinear dynamic equations of hard-coating plate are built by Hamilton variation principle. Secondly, to obtain nonlinear governing equation of hard-coating plate under transverse load, these equations are discretized in Galerkin method. The system averaged equations with 1:3 internal resonances are obtained by the method of multiple scales, and the multi-periodic responses behavior of cantilever hard-coating plate under transverse loading could be presented. Finally, the vibration response experiment of hard-coating plate is conducted, and the multi-periodic responses are also present for the hard-coating plate with three-to-one internal resonance. Besides, through the vibration response experiment of uncoated titanium alloy plate, the damping characteristic of hard coating is further analyzed.

A novel nonlinear dynamic double-cable-stayed shallow-arch model of cable-stayed bridge is established and the in-plane 1:1:1 internal resonance between three first modes of shallow arch and two cables under both external primary and subharmonic resonance is investigated, respectively. The Galerkin discretization and the method of multiple scales are applied to obtain the modulation equations of the dynamic system. The stable equilibrium solutions of the modulation equations are examined by Newton-Raphson method. Numerical simulations are carried out to investigate the dynamic behavior of the new dynamic system and Runge-kutta method is also used to solve the ordinary differential equations to verify the results. The results show the rich nonlinear phenomena and some new conclusions are also drawn.

The nonlinear characteristics of an autoparametric vibration system are investigated. This system consists of a base structure and a cantilever beam with a tip mass. The dynamic equations for the system are derived using the extended Hamilton's principle. The method of multiple scales (MMS) is used to determine an approximate analytical solution of the nonlinear governing equations and, hence, analyze the stability and bifurcation of the system. Compared with the numerical simulation, the first-order MMS is not sufficient. A Lagrangian-based approach is proposed to perform a second-order analysis, which is applicable to a large class of nonlinear systems. The effects of the amplitude and frequency of the external force, damping and frequency of the attached cantilever beam, and the tip mass on the nonlinear responses of the autoparametric vibration system are determined. The results show that this system exhibits many interesting nonlinear phenomena including saturation, jumps, hysteresis and different kinds of bifurcations, such as saddle-node, supercritical pitchfork and subcritical pitchfork bifurcations. Power spectra, phase portraits and Poincare maps are employed to analyze the unstable behavior and the associated Hopf bifurcation and chaos. Depending on the application of such a system, its dynamical behaviors could be exploited or avoided.

The properties of discrete breathers and modulational instability in a discrete $$\phi ^{4}$$ nonlinear lattice which includes the next-nearest-neighbor coupling interaction are investigated analytically. By using the method of multiple scales combined with a quasi-discreteness approximation, we get a dark-type and a bright-type discrete breather solutions and analyze the existence conditions for such discrete breathers. It is found that the introduction of the next-nearest-neighbor coupling interactions will influence the existence condition for the bright discrete breather. Considering that the existence of bright discrete breather solutions is intimately linked to the modulational instability of plane waves, we will analytically study the regions of discrete modulational instability of plane carrier waves. It is shown that the shape of the region of modulational instability changes significantly when the strength of the next-nearest-neighbor coupling is sufficiently large. In addition, we calculate the instability growth rates of the $$q=\pi $$ plane wave for different values of the strength of the next-nearest-neighbor coupling in order to better understand the appearance of the bright discrete breather.

We study coarse pattern formation in a cellular automaton modelling a spatially-extended stochastic neural network. The model, originally proposed by Gong and Robinson (Phys Rev E 85(5):055,101(R), 2012), is known to support stationary and travelling bumps of localised activity. We pose the model on a ring and study the existence and stability of these patterns in various limits using a combination of analytical and numerical techniques. In a purely deterministic version of the model, posed on a continuum, we construct bumps and travelling waves analytically using standard interface methods from neural field theory. In a stochastic version with Heaviside firing rate, we construct approximate analytical probability mass functions associated with bumps and travelling waves. In the full stochastic model posed on a discrete lattice, where a coarse analytic description is unavailable, we compute patterns and their linear stability using equation-free methods. The lifting procedure used in the coarse time-stepper is informed by the analysis in the deterministic and stochastic limits. In all settings, we identify the synaptic profile as a mesoscopic variable, and the width of the corresponding activity set as a macroscopic variable. Stationary and travelling bumps have similar meso- and macroscopic profiles, but different microscopic structure, hence we propose lifting operators which use microscopic motifs to disambiguate them. We provide numerical evidence that waves are supported by a combination of high synaptic gain and long refractory times, while meandering bumps are elicited by short refractory times.

This paper deals with the amplitude-frequency response of soft AC electrostatically actuated MEMS clamped circular plates. Alternating Current (AC) frequency is near half natural frequency of the plate. This results in primary resonance of the system. Two methods are used and compared, namely the Method of Multiple Scales (MMS), and Reduced Order Model (ROM). Both methods are in excellent agreement for amplitudes less than 0.4 of the gap. For amplitudes between 0.4 and 1 of the gap only seven terms ROM accurately predicts the behavior of the system. The effects of the voltage and damping parameters on the frequency response are reported.

The interaction between nonlinearity and seasonal forcing in childhood infectious diseases often leads to multiyear cycles with large amplitude. Regular biennial cycles in particular were observed in measles reports throughout the world. The objective of this paper is to understand the mechanism of such biennial cycles, especially the conditions under which the large amplitude biennial oscillation might appear. It is proposed that such biennial cycles are caused by parametric resonance, which might occur when varying the parameter at a frequency close to twice the natural frequency of the system. The analysis is carried out by solving an SIR model semi-analytically using method of multiple scales (MMS). This analysis shows how parametric resonance occurs due to the interaction between nonlinearity and periodic forcing. Using the MMS solution, the boundary between the resonance region and the non-resonance region in the parameter space is obtained. The effects of different parameters on the triggering of parametric resonance are studied, such as transmission rate, recovery rate, birth rate and amplitude of seasonality. The effects of stochasticity on the onset of parametric resonance are also studied.

Despite the shear amount of research studies on nonlinear flexural dynamics of cantilever beams, very few efforts address the practical geometry involving a constant thickness and linearly-varying width. This stems from the nature of the associated linear eigenvalue problem which cannot be easily solved in closed form. In this paper, we present a closed-form solution to this particular linear eigenvalue problem in the form of a general Meijer-G differential equation for which a solution is readily available in the shape of the Meijer-G functions. Using this approach, the exact linear modal frequencies and shapes are obtained and used in the discretization of the nonlinear partial-differential equation describing the dynamics of the system. The discretized system of ordinary-differential equations is then solved using the method of multiple scales to obtain an approximate analytical solution describing the primary resonance behavior of a given vibration mode. An analytical expression for the modal effective nonlinearity is obtained and used to analyze the influence of the beam's tapering on the nonlinear primary resonance behavior of the response (softening/hardening). Results are then compared to a finite element (FE) solution of the linear eigenvalue problem in which the modal shapes obtained using the FE method are fit into a set of orthogonal polynomial functions and used to discretize the nonlinear problem. It is shown that, while the modal frequencies obtained using the FE method approximate those obtained analytically with negligible error (less than 1%), there is a substantial error in the resulting estimates of the modal effective nonlinearity. This indicates that, even negligible errors in the approximate solution of the linear problem, can propagate to become significant when analyzing the nonlinear problem further reinforcing the importance of the exact solution.

The nonlinear vibrations of composite laminated plates with time-dependent base excitation and boundary conditions are investigated. According to the von Kármán nonlinear plate theory, the dynamic equations of motion of the laminated plates are established. The nonlinear partial differential equations are transformed to the nonlinear ordinary differential ones using the Bubnov-Galerkin’s method. The primary resonance and the primary parametric resonance of the laminated plate with time-dependent boundary conditions are investigated by means of the method of multiple scales. The validity of the present theoretical method is verified by comparing the amplitude–frequency relationship curves acquired from the present theoretical method with those calculated from the numerical simulation. The amplitude–frequency characteristic curves and the displacement time histories for different ply angles of the composite laminated plate are analyzed. The effects of the viscous damping factor and the transverse displacement excitation on the amplitude–frequency relationship curves are also studied. The present results are helpful for the nonlinear dynamical analysis and design of the composite laminated plate with time-dependent boundary conditions.

We obtain analytical approximations for period-1 rotations of both vertically and horizontally excited pendulum using Galerkin projections with elliptic functions (GP), elliptic averaging (EA), method of multiple scales (MMS) and harmonic balance (HB). The application of GP and EA has been extended to parametrically excited pendulum for the first time in this paper, while the results from MMS and HB have been adapted from the existing literature. We compare these approximations with the numerical solution to ascertain their accuracy and identify the correct approximate solution to be used for determining other properties like stability of the solution. This comparison has been made for two different forcing frequencies: one closer to the natural frequency of the pendulum while the other at a higher frequency. We find that the appearance of the period-1 rotation at smaller amplitudes of forcing as a saddle-node bifurcation is best captured by GP and EA, but the approximation for the initial angular displacement and velocity as well as the root- mean-square error over the entire time period using GP and EA deteriorates at larger forcing amplitudes. In the large amplitude regime, HB with one-term approximation gives the best result for lower forcing frequency, while MMS results in the best approximation for the higher frequency. Inclusion of more terms in the harmonic balance approximation leads to an increased accuracy in the entire frequency and amplitude range. This observation holds for both vertical and horizontal excitation. We have also used these approximations to ascertain the stability of period-1 rotation and find that the HB approximation results in the most consistent prediction of stable parameter regimes. Hence, a multi-term HB analysis, which is the simplest and straight forward technique among the methods studied in this paper, leads to the best approximation for period-1 rotation of parametric pendula.

We explore the dominant mechanism leading to mode-locking in real strings used in plucked stringed instruments like sitar and veena which are reported to have large number of nearly harmonic overtones. Strings used in most musical instruments have flexural rigidity and/or dissipation of energy and/or nonrigid end supports which generate inharmonic overtones. However, instruments like sitar and veena have a finite-sized bridge over which the strings wrap and unwrap during their vibrations. This wrapping generates quadratic nonlinearities in the system which facilitates locking of the higher modes to the fundamental mode, thereby countering the inharmonicity. To account for the flexural rigidity/nonrigid supports and dissipation, we introduce a frequency detuning and damping in the discretized modal equations. Method of multiple scales is used to obtain the equations governing the evolution of the modal amplitudes and frequencies. These equations are then exploited to obtain conditions for harmonic mode-locking of periodic solutions. We obtain all possible branches of harmonically resonant solutions and discuss their nature for two- and three-mode system. It has been shown that the inharmonicity introduced by different sources in a real string can be countered by mode-locking facilitated by the presence of a finite bridge in musical instruments like sitar. Analytical expression for mode-locked periodic solutions and the corresponding basins of attraction are obtained for the two-mode undamped system. For the damped system, the amplitudes are observed to decay such that the frequencies oscillate around the mean harmonic frequency. We also observe possibly chaotic modulations around the harmonically mode-locked solutions for the three-mode system.

Global dynamics of an autoparametric beam structure derived from a flexible L-shaped beam subjected to base excitation with one-to-two internal resonance and principal resonance are investigated. Hamilton’s principle is employed to obtain the nonlinear partial differential governing equations of the multi-beam structure. A linear theoretical analysis is implemented to derive the modal functions, and the orthogonality conditions are established. The analytical modal functions obtained are then adopted to truncate the partial differential governing equations into a set of coupled nonlinear ordinary differential equations via the Galerkin’s procedure. The method of multiple scales is applied to yield a set of autonomous equations of the first-order approximations to the response of the dynamical system. The Energy-Phase method is used to study the global bifurcation and multi-pulse chaotic dynamics of such autoparametric system. The present analysis indicates that the chaotic dynamics results from the existence of Silnikov’s type of homoclinic orbits and the parameter set for which the system may exhibit chaotic motions in the sense of Smale horseshoes are predicted analytically. Numerical simulations are performed to validate the theoretical results.

Interactions of activator and inhibitor can be widely applied in different fields. As a result, dynamical behaviors of an activator–inhibitor model with different sources are investigated. We obtained the condition of Turing bifurcation by using linear stability analysis, and the amplitude equation by employing multiple scale analysis. Moreover, the stability of amplitude equation was presented. It turned out that the model can show spot patterns, stripe patterns and the coexistence of spot and stripe patterns. Numerical simulations well validate our theoretical analysis. Our results highlight the relationship between pattern formation and variation of biological tissues.

Bolted joint structures are prone to bolt loosening under environmental and operational vibrations, which may severely affect the structural integrity. This paper presents a bolt looseness recognition method based on the subharmonic resonance analysis. The bolted joint structure was simplified to a two-degree-of-freedom nonlinear model, and a multiple timescale method was used to explain the phenomenon of the subharmonic resonance and conditions for the generation of subharmonics. Numerical simulation predictions for the generation of the subharmonics and conditions for the subharmonics can be found with respect to the excitation frequency and the excitation amplitude. Experiments were performed on a bolt-joint aluminum beam, where the damage was simulated by loosening the bolts. Two surface-bonded piezoelectric transducers were utilized to generate continuous sinusoidal excitation and to receive corresponding sensing signals. The experimental results demonstrated that subharmonic components would appear in the response spectrum when the bolted structure was subjected to the excitation of twice its natural frequency. This subharmonic resonance method was found to be effective on bolt looseness detection.

In this paper, we study the buckling and post-buckling of a compressible hyperelastic slab under both axial compressive force and distributed force on two lateral surfaces. The combined series-asymptotic expansions method is used to obtain the simplified model equations. We further assume that the lateral load is uniformly distributed and is a constant. We discuss three cases: (I) axial compression dominant, (II) lateral compression dominant and (III) uniform compression. In case (I), the lateral stress is given as a constant and no buckling bifurcation will occur when the axial stress is not imposed. In this case, the lateral stress may be compressive or tensile stress. In case (II), the axial compressive stress is given as a constant and no buckling bifurcation will occur when the lateral compressive stress is not imposed. In case (III), the slab is uniformly compressed at two lateral boundaries and two ends of the slab. For all three cases, with sliding-sliding end conditions at two ends, we obtain the approximate expressions of the critical stress values and mode numbers from linear bifurcation analysis and obtain the approximate analytical post-buckling solutions by using the method of multiple scales. We find both supercritical and subcritical bifurcations may occur depending on the material constants, slenderness ratio of the slab and bifurcation modes. The analytical post-buckling solutions are compared with the numerical solutions of the model equations and good agreements are found.

In the past few years, the energy harvester of piezoelectric beam has been extensively studied. In this paper, the bistable piezoelectric plate was adopted as the structure of energy harvester. The model consists of four parts including the substructure layer, the piezoelectric film, the protective layer and the magnet, which the material of the substructure layer and the piezoelectric layer are respectively carbon fiber and PVDF. The boundary conditions of the plate are simply supported by two opposite sides, which the other two sides are free. The base of the plate is subjected to the harmonic excitation. Based on von Karman large deformation theory and Hooke law, the partial differential equation for nonlinear vibration of the bistable piezoelectric system is derived by using Hamilton's principle. The Galerkin method is employed to discrete the partial differential equations to the ordinary differential equations. Then, the method of multiple scales is applied to perturbation analysis to obtain the nonlinear averaged equations in the polar form. Frequency-response curves of the piezoelectric plate are obtained by numerical simulations. The effects of the excitation amplitude, the damping coefficient, the piezoelectric parameters, the nonlinear parameters and the magnetic distance on nonlinear vibration of the bistable piezoelectric plate are studied.

Multi-pulse homoclinic orbits and chaotic dynamics of a supercritical composite panel with free layer damping treatment in subsonic flow are investigated considering one to two internal resonance. Inviscid potential flow theory is employed to exhibit the aerodynamic pressure and Kelvin’s model is used to describe the viscoelastic property of the free damping layer. By Hamilton’s principle, the governing equation of the composite panel in the subcritical regime is derived. In the supercritical regime, the buckling configuration is solved analytically and the PDE is obtained by introducing a displacement transformation for nontrivial equilibrium configuration. Then the governing equation in the first supercritical region is transformed into a discretized nonlinear gyroscopic system via assumed modes and then Galerkin’s method. The method of multiple scales and canonical transformation are applied to reduce the equations of motions to the near-integrable Hamiltonian standard forms. The Energy-Phase method is employed to demonstrate the existence of chaotic dynamics by identifying the existence of multi-pulse jumping orbits in the perturbed phase space. The global solutions are finally interpreted in terms of the physical motion of the gyroscopic continua and the dynamical mechanism of chaotic pattern conversion between the forward traveling wave motion and the complex bidirectional traveling wave motion are discussed.

Wave propagation in a nonlinear periodic material is investigated, by considering an infinite chain of two-mass unit cells with cubic stiffness nonlinearity. The chain is analysed using the method of multiple scales, predicting the dispersion shift in the band structure due to nonlinear self-interaction. The solution further reveals modest higher harmonic generation within the limits of the solution approach, proportional to the strength of nonlinearity and energy level in the chain. The possibility for controlling the higher harmonic generation by changing the distribution of the cubic nonlinearity is investigated. The predictions based on the analytical model are verified by numerical simulations, which also explores the limits of the infinite, analytical model.

Nonlinear principal parametric resonance and stability are investigated for rotating circular plate subjected to parametric excitation resulting from the time-varying speed in the magnetic field. According to the conductive rotating thin circular plate in magnetic field, the magnetoelastic parametric vibration equations of a conductive rotating thin circular plate are deduced by the use of Hamilton principle with the expressions of kinetic energy and strain energy. The axisymmetric parameter vibration differential equation of the variable-velocity rotating circular plate is obtained through the application of Galerkin integral method. Then, the method of multiple scales is applied to derive the nonlinear principal parametric resonance amplitude-frequency equation. The stability and the critical condition of stability of the plate are discussed. The influences of detuning parameter, rotation rate, and magnetic induction intensity are investigated on the principal parametric resonance behavior. The result shows that stable and unstable solutions exist when detuning parameter is negative, and the resonance amplitude can be weakened by changing the magnetic induction intensity.

The mechanical–thermal noise (MTN) equivalent rotation rate (Ωn) is computed by using the linear approximation of the system response and the nonlinear “slow” system. The slow system, which is obtained using the method of multiple scales, is used to identify the linear single-valued response of the system. The linear estimate of the noise equivalent rate fails as the drive direction stroke increases. It becomes imperative in these conditions to use a more complex nonlinear estimate of the noise equivalent rate developed here for the first time in literature. The proposed design achieves a high performance regarding noise equivalent rotation rate.

Recently, it has been shown that nonlinear vibratory energy harvesters possessing a tri-stable potential function are capable of harvesting energy efficiently over a wider range of frequencies in comparison to harvesters with a double-well potential function. However, the effect of the design parameters of the harvester on the dynamic response and the effective bandwidth of such devices remains uninvestigated. To fill this void, this paper establishes an analytical approach to characterize the effective frequency bandwidth of harvesters that possess a hexic potential energy function. To achieve this goal, the method of multiple scales is utilized to construct analytical solutions describing the amplitude and stability of the intra- and inter-well dynamics of the harvester. Using these solutions, critical bifurcations in the parameter's space are identified and used to define an effective frequency bandwidth of the harvester. The influence of the electric parameters, namely, the time constant ratio (ratio between the period of the mechanical system and the time constant of the harvesting circuit) and the electromechanical coupling, on the effective frequency bandwidth is analyzed. Experimental studies performed on the harvester are presented to validate some of the theoretical findings.