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This study uses the piezoelectric technology to collect vibration energy from the fixed-fixed nonlinear elastic beams attached with the piezo-patch between the two ends. Both single elastic steel sheet (SESS) and double elastic steel sheet (DESS) systems are investigated and correlated. To simulate the power generation of the vibration energy harvester (VEH) of both the SESS and the DESS in different engineering elements, the simple harmonic external force generated by a shaker at the location of the piezo-patch is used as the source. With this, more vibration converted electric energy is derived from the transverse deformation and flapping from the DESS than the SESS beam. The equation of a nonlinear Euler–Bernoulli beam is coupled with the electric energy equation of the piezo-patch to simulate the SESS VEH system. The flapping force from the DESS VEH system can be considered the concentrated external load applied on the SESS beam model. The method of multiple scales (MOMS) is employed to analyze this nonlinear problem. The fixed points plots and the numerical results confirm this theory presented for the two beam systems, which can be used for evaluating similar engineering systems. Experiments are also performed in this study. The Taguchi method is used to analyze the optimum locations of the shaker and piezo-patch, as well as the confidence level of the factors. The method of nonlinear analysis presented in this study demonstrates its accuracy compared with the linear case. The transverse DESS VEH model proposed is proved to be feasible and more effective than the SESS system.

In this paper, the nonlinear free and forced vibrations analysis of in-plane bi-directional functionally graded (IBFG) rectangular plate with temperature-dependent properties is studied for the first time. For this purpose, with the aid of von Karman nonlinearity strain–displacement relations, the partial differential equations of motion are developed based on the first-order shear deformation theory (FSDT). Then, the nonlinear partial differential equations are transformed into the time-dependent nonlinear ordinary differential equations by applying the Galerkin method. The primary and super harmonic resonances are analyzed by the method of multiple scales (MMS). The material properties are assumed to be temperature-dependent and graded in the thickness direction according to the power-law distribution. The effects of some system parameters, such as vibration amplitude, volume fraction indexes, length-to-thickness ratio, temperature and aspect ratio on the nonlinear frequency and also frequency responses curve, are discussed in detail. To validate the analysis, the results of this paper are compared with the published data and good agreements are found.

This paper presents the superharmonic and subharmonic resonances of spiral stiffened functionally graded (SSFG) cylindrical shells under harmonic excitation. The stiffeners are considered to be externally or internally added to the shell. Also, it is assumed that the material properties of the stiffeners are continuously graded in the thickness direction. In order to model the stiffeners, the smeared stiffener technique is used. Within the context of the classical plate theory of shells, the von Kármán nonlinear equations are derived for the shell and stiffeners based on Hooke’s law and the relations of stress-strain. Using Galerkin’s method, the equation of motion is discretized. The superharmonic and subharmonic resonances are analyzed by the method of multiple scales. The influence of the material parameters and various geometrical properties on the superharmonic and subharmonic resonances of SSFG cylindrical shells is investigated. Considering these results, the hardening nonlinearity behavior and jump value of cylindrical shell is less and more than others, when the angle of stiffeners is [Formula: see text] and [Formula: see text], respectively.

In this paper, we study the buckling of an incompressible hyperelastic rectangular layer due to compression with sliding end conditions. The combined series-asymptotic expansions method is used to derive two-coupled nonlinear ordinary differential equations (ODEs) governing the leading order of the axial strain W and shear strain G. Linear analysis yields the critical stress values of buckling. For the nonlinearly coupled system, by introducing a small parameter, the approximate analytical solutions for post-buckling deformations are obtained by using the method of multiple scales. The amplitude of buckling is expressed explicitly by the aspect ratio, the incremental dimensionless engineering stress and the mode of buckling. To the authors' best knowledge, it is the first time that such an analytical formula is obtained within the framework of two-dimensional field equations for nonlinearly elastic materials (including both geometric and material nonlinearity). Numerical computations of the coupled system are also carried out. Good agreements between the numerical and analytical solutions are found when the amplitudes of buckling are moderate. Finally, some energy analysis regarding material failure is made.

This paper deals with the nonlinear response of electrostatically actuated cantilever beam microresonators near-half natural frequency. A first-order fringe correction of the electrostatic force, viscous damping, and Casimir effect are included in the model. Both forces, electrostatic and Casimir, are nonlinear. The dynamics of the resonator is investigated using the method of multiple scales (MMS) in a direct approach of the problem. The reduced order model (ROM) method, based on Galerkin procedure, is used as well. Steady-state motions are found. Numerical simulations are conducted for uniform microresonators. The influences of damping, actuation, and fringe effect on the resonator response are found.

The nonlinear dynamic behavior of a rigid disc-rotor supported by active magnetic bearings (AMB) is investigated, without gyroscopic effects. The vibration of the rotor is modeled by a coupled second order nonlinear ordinary differential equations with quadratic and cubic nonlinearities. Their approximate solutions are sought applying the method of multiple scales in the case of primary resonance. The Newton–Raphson method and the pseudo-arclength path-following algorithm are used to obtain the frequency response curves. Choosing the Hopf bifurcations as the initial points and applying the shooting method and the pseudo-arclength path-following algorithm, the periodic solution branches are obtained. At the same time, the Floquet theory is used to determine the stability of the periodic solutions. A detailed bifurcation analysis of the dynamic (periodic and chaotic) solutions of the averaged equations is presented. The three types of primary Hopf bifurcations are found for the first time in the rotor-AMB system. It is shown that the limit cycles undergo cyclic fold, period doubling bifurcations, and intermittent chaotic attractor, whereas the chaotic attractors undergo explosive bifurcation and boundary crises. In the regime of multiple coexisting solutions, multiple stable equilibriums, periodic solutions and chaotic solutions are the most interesting phenomena observed.

This paper focuses on the dynamical behavior of a Lotka–Volterra competitive system with nonlocal delay. We first establish the conditions of Turing bifurcation occurring in the system. According to it and by using multiple scale method, the amplitude equations of the different Turing patterns are obtained. Then, we observe when these patterns (spots pattern and stripes pattern) arise in the Lotka–Volterra competitive system. Finally, some numerical simulations are given to verify our theoretical analysis.

Nonlinear magneto-elastic combined resonance of parametric and forced excitations is investigated for a rotating circular plate with a variable speed under alternating load. According to the magneto-elastic vibration equations of a conductive rotating thin circular plate, the axisymmetric vibration differential equations of the rotating circular plate under transverse magnetic field are obtained through the application of the Galerkin integral method. The method of multiple scales is applied to solve the differential equations of the circular plate under alternating magnetic field, and the resonance states of the system under combined parametric and forced excitations are obtained by analyzing secular terms. The respective amplitude–frequency response equations are also derived, as well as the necessary and sufficient conditions of the system to make it stable. A numerical method is adopted to acquire amplitude–frequency response curves, bifurcation diagrams of amplitude and the variation pattern of amplitude with magnetic induction intensity and radial force. The influence of parameter variation on stability of the system is also investigated. Based on the global bifurcation diagram of the system, the influence of the change of bifurcation parameters on the system dynamics is discussed.

The nonlinear magnetoelastic vibration equations and electromagnetic field equations of a conductive thin cylindrical shell in magnetic fields are derived. The nonlinear principal resonances and dynamic stabilities of the cylindrical shell simply supported in a transverse magnetic field are investigated. Approximate analytical solution and bifurcation equations of the system with principal resonances are obtained by using the method of multiple scales. The stabilities and singularities of the steady-state solutions are analyzed and the stability criterion is given. The transition sets and bifurcation figures of unfolding parameters are also obtained. The variations of the resonance amplitudes with respect to the detuning parameter, the magnetic induction intensity, and the amplitude of excitations are presented. The corresponding phase trajectories in moving phase planes are given. The stabilities of solutions, characteristics of singular points, and bifurcation are analyzed. The impacts of electromagnetic and mechanical parameters on dynamic behaviors are discussed in detail.

The nonlinear dynamic responses of a string-beam coupled system subjected to harmonic external and parametric excitations are studied in this work in the case of 1:2 internal resonance between the modes of the beam and string. First, the nonlinear governing equations of motion for the string-beam coupled system are established. Then, the Galerkin's method is used to simplify the nonlinear governing equations to a set of ordinary differential equations with four-degrees-of-freedom. Utilizing the method of multiple scales, the eight-dimensional averaged equation is obtained. The case of 1:2 internal resonance between the modes of the beam and string — principal parametric resonance-1/2 subharmonic resonance for the beam and primary resonance for the string — is considered. Finally, nonlinear dynamic characteristics of the string-beam coupled system are studied through a numerical method based on the averaged equation. The phase portrait, Poincare map and power spectrum are plotted to demonstrate that the periodic and chaotic motions exist in the string-beam coupled system under certain conditions.

Based on the Maxwell equations, the electromagnetic constitutive relations and boundary condition, the electrodynamic equation and the electromagnetic force expressions in an electromagnetic field are derived. Using the principle of virtual work, the basic set of equations for nonlinear electromagnetic elasticity vibration expressed by the displacement of a thin plate in a longitudinal and a transverse magnetic field is obtained, respectively. In addition, we study the nonlinear principal resonance and the solution stability of a thin plate with two opposite sides simply supported and subjected to a mechanical live load and in a constant transverse magnetic field. By the method of multiple scales, the amplitude frequency response equation and the approximate analytic solution in steady motion are also derived. According to the characteristic of singularity and the Lyapunov stability theory, the stability of the solution is analyzed and the critical condition of stability is determined. Finally, by means of numerical calculations, the amplitude frequency response curves, time history response plots and phase charts of the magnetoelasticity vibration are obtained.

In this paper, the principal resonance is investigated for a cantilever with a contact end. The cantilever is modeled as an Euler–Bernoulli beam, and the contact is modeled by the Derjaguin–Müller–Toporov theory. The problem is formulated as a linear nonautonomous partial-differential equation with a nonlinear autonomous boundary condition. The method of multiple scales is applied to determine the steady-state response. The equation of response curves is derived from the solvability condition of eliminating secular terms. The stability of steady-state responses is analyzed by using the Lyapunov-linearized stability theory. Numerical examples are presented to highlight the effects of the excitation amplitude, the damping coefficient, and the coefficients related to the contact.

In this paper, higher-dimensional periodic and chaotic oscillations for a parametrically excited viscoelastic moving belt with multiple internal resonances are investigated for the first time. The external damping and internal damping of the material for the viscoelastic moving belt are considered simultaneously. First, the nonlinear governing equation of planar motion for the viscoelastic moving belt with the external damping is given. Then, the transverse nonlinear oscillations of the viscoelastic moving belt are considered. The method of multiple scales and the Galerkin approach are applied directly to the governing partial differential equation of motion for the viscoelastic moving belt to obtain an eight-dimensional averaged equation for the case of 1:2:3:4 internal resonances for the first-, the second-, the third- and the fourth-order modes and primary parametric resonance of the first-order mode. Finally, numerical method is used to investigate higher-dimensional periodic and chaotic motions of the viscoelastic moving belt. The results of numerical simulation demonstrate that there exist the period, period 2, period 4, multiple period and chaotic motions of the viscoelastic moving belt. The multipulse chaotic motions of the viscoelastic moving belt are observed from numerical simulations.

The Shilnikov-type multipulse orbits and chaotic dynamics for a simply supported rectangular thin plate under combined parametric and external excitations are studied in this paper for the first time. The rectangular thin plate is subjected to spatially and temporally varying transversal and in-plane excitations, simultaneously. The formulas of the rectangular thin plate are derived from the von Kármán equation and Galerkin's method. The method of multiple scales is used to find the averaged equation in the case of 1:2 internal resonance. Based on the averaged equation, the theory of normal form is used to obtain the explicit expressions of normal form associated with a double zero and a pair of purely imaginary eigenvalues using the Maple program. The dissipative version of the energy-phase method is utilized to analyze the multipulse global bifurcations and chaotic dynamics of a parametrically and externally excited rectangular thin plate. The global dynamical analysis indicates that there exist the multipulse jumping orbits in the perturbed phase space of the averaged equation for a parametrically and externally excited rectangular thin plate. These results show that the chaotic motions of the multipulse Shilnikov-type can occur for a parametrically and externally excited rectangular thin plate. Numerical simulation results are presented to verify the analytical predictions. It is also found from the results of numerical simulation that the Shilnikov-type multipulse orbits exist for a parametrically and externally excited thin plate.

The feedback controllers are designed to modify the amplitude of limit cycles in van der Pol oscillator and generalized van der Pol oscillator. Bifurcation control equations of weakly nonlinear systems are obtained by using the method of multiple scales. Gain-amplitude curves of controlled systems are drawn. Based on numerical study, the brief results of controlling amplitude of limit cycle are given for strongly nonlinear system.

This paper is concerned with the problem of occurrence of combination resonances in parametrically excited doubly curved panels. The dynamic instability of doubly curved panels, subjected to non-uniform in-plane harmonic loading is investigated. Sander's first-order shear deformation theory is used to model the doubly curved panels, considering the effects of transverse shear deformation and rotary inertia. The theory can be reduced to Love's and Donnell's theories by means of tracers. Analytical expressions for the instability regions are obtained at Ω = ωm+ ωn(Ω is the excitation frequency and ωmand ωnare the natural frequencies of the system), by using the method of multiple scales. It is shown that besides the principal instability region at Ω =2ω1, where ω1is the fundamental frequency, other cases of Ω = ωm+ ωnwhich are related to other modes, can be of major importance and yield a significantly enlarged instability region. The results show that under localized edge loading, combination resonance zones are as important as simple resonance zones. The effects of edge loading, curvature, shallowness ratio, edge length to thickness ratio, aspect ratio, boundary conditions and the static load factor on dynamic instability regions are considered.

The multipulse Shilnikov orbits and chaotic dynamics for the nonlinear nonplanar oscillations of a cantilever beam are studied in this paper for the first time. The cantilever beam studied here is subjected to a harmonic axial excitation and two transverse excitations at the free end. The nonlinear governing equations of nonplanar motion with parametric and external excitations are obtained. The Galerkin procedure is applied to the partial differential governing equations to obtain a two-degree-of-freedom nonlinear system under combined parametric and forcing excitations. The resonant case considered here is principal parametric resonance-1/2 subharmonic resonance for the first mode and fundamental parametric resonance-primary resonance for the second mode. The parametrically and externally excited system is transformed to the averaged equation by using the method of multiple scales. From the averaged equation, the theory of normal form is used to find their explicit formulas. Based on normal form obtained above, the dissipative version of the energy-phase method is utilized to analyze the multipulse global bifurcations and chaotic dynamics for the nonlinear nonplanar oscillations of the cantilever beam. The energy-phase method is further improved to ensure the equivalence of topological structure for the phase portraits. The analysis of global dynamics indicates that there exist the multipulse jumping orbits in the perturbed phase space of the averaged equation for the nonlinear nonplanar oscillations of the cantilever beam. These results show that the multipulse Shilnikov type chaotic motions can occur for the nonlinear nonplanar oscillations of the cantilever beam. Numerical simulations are given to verify the analytical predictions. It is also found from the results of numerical simulation in three-dimensional phase space that the multipulse Shilnikov type orbits exist for the nonlinear nonplanar oscillations of the cantilever beam.

It is well known that complicated stochastic evolution systems near a change of stability are often dominated by slow modes. We rigorously approximate the evolution of the system by stochastic ordinary differential equations describing the amplitudes of the dominating modes on a slow time-scale.In this paper we focus on equations with quadratic nonlinearity and give applications to the Rayleigh–Bénard problem and a model from surface growth perturbed by fractional noise.The unique existence of global solutions is not necessary.

The galloping or vortex-induced vibration of transmission lines will lead to a periodic excitations to the masts of the cross-rope suspension tower (CRST). The mast of the CRST is modeled as a straight beam with an elastic support subjected to a pulsating axial force on the top, which will change the stiffness of the mast, thereby resulting in produce harmonic excitation and instability. The dynamic characteristics of the system are investigated, which show that the bending frequency of the CRST decreases linearly with increase in axial static load, while it increases nonlinearly with the increase in boundary stiffness. Then, the method of multiple scales is adopted to analyze the vibration. It is found that the wind load on the mast brings primary resonance, but has no effects on instability. In addition, the steady state solution of the primary resonance is obtained by the polar form of the reduced amplitude modulation equations (RAMEs), with the effects of the following parameters on the vibration amplitude of the mast studied: the prestressing load in the guy, magnitude of the dynamic force, detuning parameter and wind load. Finally, the instability regions of two cases ([Formula: see text] near [Formula: see text] and [Formula: see text] near [Formula: see text]) are studied by the Cartesian form of the RAMEs, with focus on the influence of the axial harmonic load produced by the galloping of the transmission lines on the instability area. It is observed that the magnitude of excitation frequency of the dynamic force in the range of instability region becomes larger until the spring stiffness is increased up to a certain value.

Nonlinear resonance response of an electrostatic-actuated nanobeam is controlled by using a delayed axial electrostatic force with near-half the natural frequency. A graphene sensor pasted on the surface of the nanobeam is used to extract the vibration voltage signal. An axial-delayed capacitive controller is designed to produce delayed axial force to control the nonlinear vibration of the nanobeam. The vibration voltage signal from the graphene sensor is input to the axial-delayed capacitive controller to attenuate the nonlinear vibration of the nanobeam. The dynamic response of the resonator is investigated by using the method of multiple scales directly. The sufficient conditions of guaranteeing the system stability and the saddle-node bifurcation are studied. The attenuation ratio is defined as the ratio of the peak amplitude of the nonlinear vibration system with control to that without control. A critical feedback gain is given, which can shift the frequency–amplitude curves from the nonlinear vibration to a linear vibration. An optimal method in which the attenuation ratio is taken as objective function and the aforementioned sufficient conditions as the constraint conditions is given to calculate the optimal feedback gains. Numerical simulations are conducted for uniform nanobeams.

Predicting the nonlinear vibration responses of a Z-shaped folding wing during the morphing process is a prerequisite for structural design analysis. Therefore, the present study focuses on the nonlinear dynamical characteristics of a Z-shaped folding wing. The folding wing is divided into three carbon fiber composite plates connected by rigid hinges. The nonlinear dynamic equations of the system are derived using Hamilton’s principle based on the von Kármán equations and classical laminate plate theory. The mode shape functions of the system are then obtained using finite element analysis. Galerkin’s approach is employed to discretize the partial differential governing equations into a two-degree-of-freedom nonlinear system. The case of 1:1 inner resonance is considered. The method of multiple scales is employed to obtain the averaged equations of the system. Finally, numerical simulation is performed to investigate the nonlinear dynamical characteristics of the system. Bifurcation diagrams and wave-form diagrams illustrate the different motions of the Z-shaped folding wing, including periodic and chaotic motions under given conditions. The influence of transverse excitations on the bifurcations and chaotic motion of the Z-shaped folding wing is investigated numerically.

This paper studies the vibration of a nonlinear 3D-string fixed at both ends and supported by a nonlinear elastic foundation. Newton’s second law is adopted to derive the equations of motion for the string resting on an elastic foundation. Then, the method of multiple scales (MOMS) is employed for the analysis of the nonlinear system. It was found that 1:3 internal resonance exists in the first and fourth modes of the string when the wave speed in the transverse direction is [Formula: see text] and the elasticity coefficient of the foundation is [Formula: see text]. Fixed point plots are used to obtain the frequency responses of the various modes and to identify internal resonance through observation of the amplitudes and mode shapes. To prevent internal resonance and reduce vibration, a tuned mass damper (TMD) is applied to the string. The effects of various TMD masses, locations, damper coefficients ([Formula: see text]), and spring constants ([Formula: see text]) on overall damping were analyzed. The 3D plots of the maximum amplitude (3D POMAs) and 3D maximum amplitude contour plots (3D MACPs) are generated for the various modes to illustrate the amplitudes of the string, while identifying the optimal TMD parameters for vibration reduction. The results were verified numerically. It was concluded that better damping effects can be achieved using a TMD mass ratio [Formula: see text]–0.5 located near the middle of the string. Furthermore, for damper coefficient [Formula: see text], the use of spring constant [Formula: see text]–13 can improve the overall damping.

Global dynamics of supercritical pipes conveying pulsating fluid considering superharmonic resonance of the second mode with 1:2 internal resonance are investigated. The governing partial differential equations in the supercritical regime are obtained based on the nontrivial equilibrium configuration of the pipes conveying fluid and then transformed into a discretized nonlinear gyroscopic system via assumed modes and Galerkin’s method. The method of multiple scales and canonical transformation are applied to reduce the equations of motion to the near-integrable Hamiltonian standard form. 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 subsequently interpreted in terms of the physical motion of such gyroscopic system. Two types of nonlinear normal modal motion and the chaotic pattern conversion between the locked simple bidirectional traveling wave motion and the complex bidirectional traveling wave motion are discussed.

A two-degree-of-freedom model of a nonlinear vibration absorber is considered in this paper. Both the global bifurcations and chaotic dynamics of the nonlinear vibration absorber are investigated. The nonlinear equations of motion of this model are derived. The method of multiple scales is used to find the averaged equations. Based on the averaged equations, the theory of normal form is used to obtain the explicit expressions of normal form associated with a double zero and a pair of pure imaginary eigenvalues by Maple software program. The fast and slow modes may simultaneously exist in the averaged equations. On the basis of the normal form, the global bifurcation and the chaotic dynamics of the nonlinear vibration absorber are analyzed by a global perturbation method developed by Kovacic and Wiggins. The chaotic motion of this model is also found by numerical simulation.

Analytical study of the two first period-doubling bifurcations in a three-dimensional system is reported. The multiple scales method is first applied to construct a higher-order approximation of the periodic orbit following Hopf bifurcation. The stability analysis of this periodic orbit is then performed in terms of Floquet theory to derive the critical parameter values corresponding to the first and second period-doubling bifurcations. By introducing suitable subharmonic components in the first order of the multiple scale analysis the two critical parameter values are obtained simultaneously solving analytically the resulting system of two algebraic equations. Comparisons of analytic predictions to numerical simulations are also provided.

The Galerkin procedure is used to discretize the nonlinear partial differential equation and boundary conditions governing the flutter of a simply supported panel in a supersonic stream. These equations have repeated natural frequencies at the onset of flutter. The method of multiple scales is used to derive five first-order nonlinear ordinary-differential equations governing the modulation of the amplitudes and phases of the excited modes. Then, the modulation equations are used to calculate the equilibrium solutions and their stability, and hence to identify the excitation parameters that suppress flutter and those that lead to complex motions. A combination of a shooting technique and Floquet theory is used to calculate limit cycles and their stability. The numerical results indicate the existence of a sequence of period-doubling bifurcations that culminates in chaos. The complex motions are characterized by using phase planes, power spectra, Lyapunov exponents, and dimensions.

Nonlinear dynamics of a sandwich plate with viscoelastic soft core in supersonic flow are investigated by considering the in-plane periodic loading. Using Reddy’s third-order shear deformation theory and von Karman’s nonlinearity, the governing partial differential equations are derived by Hamilton’s principle and then truncated into a set of ordinary differential equations by Galerkin method. The critical speed for flutter is discussed by employing the linear theory. Further, the method of multiple scales is used in the nonlinear dynamical analysis of the truncated system. The Poincaré map is numerically calculated to identify dynamical behaviors. The bifurcation diagrams are presented for varying the dimensionless in-plane loading fluctuation amplitude and the Mach number related dimensionless parameter while other parameters are unchanged.

The response of a parametrically excited two-degree-of-freedom system with quadratic and cubic nonlinearities and a nonsemisimple one-to-one internal resonance is investigated. The method of multiple scales is used to derive four first-order differential equations governing the modulation of the amplitudes and phases of the two modes for the cases of fundamental and principal parametric resonances. Bifurcation analysis of the case of fundamental parametric resonance reveals that the quadratic nonlinearities qualitatively change the response of the system. They change the pitchfork bifurcation to a transcritical bifurcation. Cyclic-fold, Hopf bifurcations of the nontrivial constant solutions, and period-doubling sequences leading to chaos are induced by these quadratic terms. The effects of quadratic nonlinearities for the case of principal parametric resonance are discussed.

This paper investigates the nonlinear breathing vibrations and chaos of a circular truss antenna under changing thermal environment with 1:2 internal resonance for the first time. A continuum circular cylindrical shell clamped by one beam along its axial direction on one side is proposed to replace the circular truss antenna composed of the repetitive beam-like lattice by the principle of equivalent effect. The effective stiffness coefficients of the equivalent circular cylindrical shell are obtained. Based on the first-order shear deformation shell theory and the Hamilton’s principle, the nonlinear governing equations of motion are derived for the equivalent circular cylindrical shell. The Galerkin approach is utilized to discretize the nonlinear partial governing differential equation of motion to the ordinary differential equation for the equivalent circular cylindrical shell. The case of the 1:2 internal resonance, primary parametric resonance and 1/2 subharmonic resonance is taken into account. The method of multiple scales is used to obtain the four-dimensional averaged equation. The frequency-response curves and force-response curves are obtained when considering the strongly coupled of two modes. The numerical results indicate that there are the hardening type and softening type nonlinearities for the circular truss antenna. Numerical simulation is used to investigate the influences of the thermal excitation on the nonlinear breathing vibrations of the circular truss antenna. It is demonstrated from the numerical results that there exist the bifurcation and chaotic motions of the circular truss antenna.

This study investigated the effect of a pendulum tuned mass damper (PTMD) on the vibration of a slender two-dimensional (2D) rigid body with 1:2 internal resonance. Focus is placed on the damping effect of various parameters of the PTMD on preventing the internal resonance of the system. The instruments used include fixed points plots, time response and Poincaré maps, which were compared for confirmation of accuracy. The Lagrange's equation is employed to derive the equations of motion for the system. The method of multiple scales (MOMS) is applied to analyzing this nonlinear vibration model. The internal resonance conditions of the rigid body in vibration are obtained by the eigen-analysis. Moreover, a 3D internal resonance contour plot (3D-IRCP) aided by various amplitude analysis tables is proposed for identification of the parameter combinations of the PTMD for preventing internal resonance. This approach enables the designers to evaluate the effectiveness of various parameter combinations of the PTMD prior to the design process. The present study indicates that without changing the main configuration, the vibration amplitudes in the main body can be greatly reduced under certain parameter combinations of the PTMD.

The work investigates steady-state responses of a pipe conveying fluid with a harmonic component of flow speed superposed on a constant mean value in the supercritical regime. If the flow speed exceeds a critical value, the straight configuration of the pipe becomes unstable and bifurcates into two stable curved configurations. The transverse motion measured from each of the curved equilibrium configurations is governed by a nonlinear integro-partial-differential equation. The Galerkin method is employed to discretize the governing equation into a set of coupled nonlinear ordinary differential equations with gyroscopic terms. For the pipes in the supercritical regime, the method of multiple scales is used to determine the steady-state in the vicinity of two-to-one resonance. In the presence of the internal resonance, the subharmonic, the superharmonic and the summation, and the difference resonances exist due to the pulsating fluid. The amplitude–frequency relationships are established with the emphasis on the effects of the viscosity, the pulsating amplitude, the nonlinearity, and the mean flow speed. Some nonlinear phenomena such as the appearance of the peak or jumps pertaining to modal interaction are demonstrated. The numerical integration results are in agreement with the analytical predictions.

In this paper, a general and systematic scheme is provided to research strong and weak resonances derived from delay-induced various double Hopf bifurcations in delayed differential systems. The method of multiple scales is extended to obtain a common complex amplitude equation when the double Hopf bifurcation with frequency ratio k1:k2 occurs in the systems under consideration. By analyzing the complex amplitude equation, we give the conditions of the strong and weak resonances respectively in the analytical expressions. The weak resonances correspond to the codimension-two double Hopf bifurcations since the amplitudes and the phases may be decoupled, but the strong resonances to the codimension-three double Hopf bifurcations in the system. It is seen that the weak resonances happen in the system even for a lower-order ratio, i.e. k1 + k2 ≤ 4. As applications, two examples are displayed. Three cases of the delay-induced resonance with 1:2, 1:3, 1:5 and [Formula: see text] are discussed in detail and the corresponding normals are represented. Thus, the relative dynamical behaviors can be easily classified in the physical parameter space in terms of nonlinear dynamics. The results show the provided conditions may be used to determine that a resonance is strong or weak.

This study examines how the vibration absorbers influence the stability of nonlinear flow-solid interaction systems. A novel approach is proposed for the analysis of dynamic stability of the two-dimensional nonlinear system using the internal resonance contour plot (IRCP) and flutter speed contour plot (FSCP). The system considered is a planar rigid-body with plunge and pitch vibrations. The two ends of the body are supported by cubic nonlinear springs with quadratic damping. The vibration absorber attached beneath is also considered as a rigid body, with the mass and position of the absorber adjusted for optimization of vibration reduction. The method of multiple scales (MOMS) is employed to obtain a fixed point solution. The P-method is also adopted to obtain the eigen plot and Poincaré Map for comparison. Finally, both the IRCP with FSCP are cross-referenced to provide guide-lines for identifying the optimal location for the absorber with regard to stability and vibration reduction without the need to alter the framework of the main body.

This paper investigates the multi-pulse global heteroclinic bifurcations and chaotic dynamics for nonlinear, nonplanar oscillations of the parametrically excited viscoelastic moving belts by using an extended Melnikov method in the resonant case. Applying the method of multiple scales, the Galerkin's approach and the theory of normal form, the explicit normal form is obtained for the case of 1:1 internal resonance and primary parametric resonance. Studies are performed for the heteroclinic bifurcations of the unperturbed system and for the characteristics of the hyperbolic dynamics of the dissipative system, respectively. The extended Melnikov method is used to investigate the Shilnikov type multi-pulse bifurcations and chaotic dynamics of the viscoelastic moving belt. Based on the investigation, the geometric structure of the multi-pulse orbits is described in the four-dimensional phase space. Numerical simulations show that the Shilnikov type multi-pulse chaotic motions can occur. Furthermore, numerical simulations lead to the discovery of the new shapes of chaotic motion. Overall, both theoretical and numerical studies suggest that chaos for the Smale horseshoe sense in motion exists.

A perturbation algorithm Multiple Scales Modified Lindstedt–Poincare (MSMLP), combination of method of Multiple Scales and modified Lindstedt–Poincare is proposed for the solution of Quintic Duffing equation which combines the advantages of both the methods. Solution obtained by the MSMLP method is compared with the Multiple Scales method and accurate closed form approximate solution of the Quintic Duffing equation. The proposed method produces better results for a wide range of amplitude values of oscillations and strong nonlinearities. Numerical simulation has been performed in MATHEMATICA 7.0.

An analysis on nonlinear oscillations and chaotic dynamics is presented for a simply-supported symmetric cross-ply composite laminated rectangular thin plate with parametric and forcing excitations in the case of 1:3:3 internal resonance. Based on Reddy's third-order shear deformation plate theory and the von Karman-type equations, the nonlinear governing partial differential equations of motion for the composite laminated rectangular thin plate can be established via the Hamilton's principle. Such partial differential equations are further discretized by the Galerkin method to form a three-degree-of-freedom coupled nonlinear system including the cubic nonlinear terms. The method of multiple scales is then employed to derive a set of averaged equations. Through the stability analysis, the steady-state solutions of the averaged equations are provided. An illustrative case of 1:3:3 internal resonance and fundamental parametric resonance, 1/3 subharmonic resonance is considered. Numerical simulation is applied to investigate the intrinsically nonlinear behavior of the composite laminated rectangular thin plate. With certain external load excitations, the simulation results demonstrate that the nonlinear dynamical system of the composite laminated plate exhibits different kinds of periodic and chaotic motions.

Global bifurcations and Shilnikov type multipulse chaotic dynamics for a nonlinear vibration absorber are investigated by using the energy-phase method for the first time. A two-degree-of-freedom model of a nonlinear vibration absorber is considered. After the nonlinear nonautonomous equations of this model are given, the method of multiple scales is used to derive four first-order nonlinear ordinary differential equations governing the modulation of the amplitudes and phases of the two interacting modes in the presence of 1:1 internal resonance and primary resonance. Using several coordinate transformations to transform the modulation equation into a standard form, we can apply the energy-phase method to show the existence of the multipulse chaotic dynamics by identifying Shilnikov-type multipulse orbits in the perturbed phase space. We are able to obtain the explicit restriction on the damping, forcing excitation and the detuning parameters, under which the multipulse chaotic dynamics is expected. These multipulse orbits represent the repeated departure from purely vertical oscillations for the nonlinear vibration absorber. Numerical simulations also indicate that there exist different forms of the multipulse chaotic responses and jumping phenomena for the nonlinear vibration absorber.