Abstract: The dynamic response of parametrically excited, axially moving viscoelastic belts is investigated in this paper. Results are compared to previous work in which the partial, not material, time derivative was used in the viscoelastic constitutive relation. It is found that this added steady state dissipation greatly affects both the existence and amplitudes of nontrivial limit cycles. The discrepancy increases with increasing translation speed. To limit the comparison to the additional physics included in the model, the solution procedure of Zhang and Zu [1,2], who applied the method of multiple scales to the governing equations prior to discretization, is retained. The excitation here is provided by physically stretching the belt. In this case, viscoelastic behavior and excitation frequency also affects the amplitude of the tension fluctuations.

Abstract: In this paper,the problems of superharmonic resonance of time-delay systems are studied.The famous method of multiple scales is successfully extended to the time-delay systems,which provides great convenience for the study of the systems with time lag.As to the complex forced oscillation problem of a class of time-delay systems with damping and general force, a uniformly valid asymptotic expansion is obtained according to the distinct circumstance of superharmonic resonance,and an approximate analytic formula which is very simple and explicit is also given.By using the formula,the approximate analytic solutions for a great number of resonance problems of timedelay oscillation systems,especially the amplitude,frequency,period and phase can be got conveniently.The conclusions of related literature become simple corollaries of the paper.

Abstract: This paper presents an analysis of the chaotic motion and its control for the nonlinear nonplanar oscillations of a cantilever beam subjected to a harmonic axial excitation and transverse excitations at the free end. A new method of controlling chaotic motion for the nonlinear nonplanar oscillations of the cantilever beam, refereed as to the force control approach, is proposed for the first time. The governing nonlinear equations of nonplanar motion under combined parametric and external excitations are obtained. The Galerkin procedure is applied to the governing equation to obtain a two-degree-of-freedom nonlinear system under combined parametric and forcing excitations for the in-plane and out-of-plane modes. The work is focused on the case of 2:1 internal resonance, principal parametric resonance-1/2 subharmonic resonance for the in-plane mode and fundamental parametric resonance-primary resonance for the out-of-plane mode. The method of multiple scales is used to transform the parametrically and externally excited system to the averaged equations which have a constant perturbation force. Based on the averaged equations obtained here, numerical simulation is utilized to discover the periodic and chaotic motions for the nonlinear nonplanar oscillations of the cantilever beam. The numerical results indicate that the transverse excitation in the z direction at the free end can control the chaotic motion to a period n motion or a static state for the nonlinear nonplanar oscillations of the cantilever beam. The methodology of controlling chaotic motion by using the transverse excitation is proposed. The transverse excitation in the z direction at the free end may be thought about to be an open-loop control. For the problem investigated in this paper, this approach is an effective methodology of controlling chaotic motion to a period n motion or a static state for the nonlinear nonplanar oscillations of the cantilever beam.

Abstract: Non-linear coupled vertical and torsional vibrations of suspension bridges are investigated. Method of Multiple Scales, a perturbation technique, is applied to the equations to find approximate analytical solutions. The equations are not discretized as usually done, rather the perturbation method is applied directly to the partial differential equations. Free and forced vibrations with damping are investigated in detail. Amplitude and phase modulation equations are obtained. The dependence of non-linear frequency on amplitude is described. Steady-state solutions are analyzed. Frequency–response equation is derived and the jump phenomenon in the frequency–response curves resulting from non-linearity is considered. Effects of initial amplitude and phase values, amplitude of excitation, and damping coefficient on modal amplitudes, are determined.

Abstract: The present investigation deals with nonlinear dynamic behavior of a parametrically excited simply supported rectangular symmetric cross-ply laminated composite thin plate for the first time. The governing equation of motion for rectangular symmetric cross-ply laminated composite thin plate is derived by using von Karman equation. The geometric nonlinearity and nonlinear damping are included in the governing equations of motion. The Galerkin approach is used to obtain a two-degree-of-freedom nonlinear system under parametric excitation. The method of multiple scales is utilized to transform the second-order non-autonomous differential equations to the first-order averaged equations. Using numerical method, the averaged equations are analyzed to obtain the steady state bifurcation responses. The analysis of stability for steady state bifurcation responses in laminated composite thin plate is also given. Under certain conditions laminated composite thin plate may have two or multiple steady state bifurcation solutions. Jumping phenomenon occurs in the steady state bifurcation solutions. The chaotic motions of rectangular symmetric cross-ply laminated composite thin plate are also found by using numerical simulation. The results obtained here demonstrate that the periodic, quasi-periodic and chaotic motions coexist for a parametrically excited fore-edge simply supported rectangular symmetric cross-ply laminated composite thin plate under certain conditions.

Abstract: The steady-state transverse responses and the stability of an axially accelerating viscoelastic string are investigated. The governing equation is derived from the Eulerian equation of motion of a continuum, which leads to the Mote model for transverse motion. The Kirchhoff model is derived from the Mote model by replacing the tension with the averaged tension over the string. The method of multiple scales is applied to the two models in the case of principal parametric resonance. Closed-form expressions of the amplitudes and the existence conditions of steady-state periodical responses are presented. The Lyapunov linearized stability theory is employed to demonstrate that the first (second) non-trivial steady-state response is always stable (unstable). Numerical calculations show that the two models are qualitatively the same, but quantitatively different. Numerical results are also presented to highlight the effects of the mean axial speed, the axial-speed fluctuation amplitude, and the viscoelastic parameters.

Abstract: An approach for implementing an active nonlinear vibration absorber is presented. The strategy uses the saturation phenomenon that is exhibited by multi-degree-of-freedom systems with cubic nonlinearities possessing one-to-one internal resonance. The proposed technique consists of introducing a second-order controller and coupling it to the plant through a sensor and an actuator, where both the feedback and control signals are cubic. Once the structure is forced near its resonances, the oscillatory response is suppressed through the saturation phenomenon. We present theoretical results of the application of the proposed vibration absorber. The structure consists of a cantilever beam, the feedback signal is generated by a strain gage, and the actuation is achieved through piezoceramic patches. The equations of motion are developed and analyzed through perturbation techniques and numerical simulation. We use the method of multiple scales to obtain an approximate solution of these equations and investigate the vibration stability. There are two cases of fixed points. In the first case, the response amplitude is symmetric about the origin and divided into two branches with increasing magnitudes for decreasing and increasing the natural frequency ω and the coefficient of external excitation f respectively. In the second case, the response amplitudes are symmetric about the origin for variation of all parameters but the symmetry disappeared for increasing detuning parameter σ.

Abstract: Using the method of multiple scales, the nonlinear instability problem of two superposed dielectric fluids is studied. The applied electric filed is taken into account under the influence of external modulations near a point of bifurcation. A time varying electric field is superimposed on the system. In addition, the viscosity and variable gravity force are considered. A generalized equation governing the evolution of the amplitude is derived in marginally unstable regions of parameter space. A bifurcation analysis of the amplitude equation is carried out when the dissipation due to viscosity and the control parameter are both assumed to be small. The solution of a nonlinear equation in which parametric and external excitations are obtained analytically and numerically. The method of generalized synchronization is applied to determine the equations that describe the modulation of the amplitude and phase. These equations are used to determine the steady state equations. Frequency response curves are presented graphically. The stability of the proposed solution is determined applying Liapunov's first method. Numerical solutions are presented graphically for the effects of the different equation parameters on the system stability, response and chaos.

Abstract: In this paper, a theoretical investigation of nonlinear vibrations of a 2 degrees of freedom system when subjected to saturation is studied. The method has been especially applied to a system that consists of a DC motor with a nonlinear controller and a harmonic forcing voltage. Approximate solutions are sought using the method of multiple scales. It is shown that the closed-loop system exhibits different response regimes. The nature and stability of these regimes are studied and the stability boundaries are obtained. The effects of the initial conditions on the response of the system have also been investigated. Furthermore, the second-order solution is presented and the corresponding results are compared with those of the first-order solution. It is shown that by increasing the amplitude of the excitation voltage, the higher-order term in the solution becomes significant and causes a drift in the response. In order to verify the obtained theoretical results, they are compared with the corresponding numerical results. Good agreement between the two sets of results is observed.

Abstract: We investigate the axisymmetric dynamics of forced closed spherical shells. The nonlinear equations of motion are formulated using a variational approach and surface analysis. First, we revisit the linear eigenvalue problem. Then, using the method of multiple scales, we assess the possibility of the activation of two-to-one internal resonances between the different types of modes. Lastly, we examine the shell’s nonlinear responses to an axisymmetric primary-resonance excitation and analyze their bifurcations.Copyright © 2005 by ASME

Abstract: The steady-state transverse responses and the stability of an axially accelerating viscoelastic string are investigated. The governing equation is derived from the Eulerian equation of motion of a continuum, which leads to the Mote model for transverse motion. The Kirchhoff model is derived from the Mote model by replacing the tension with the averaged tension over the string. The method of multiple scales is applied to the two models in the case of principal parametric resonance. Closed-form expressions of the amplitudes and the existence conditions of steady-state periodical responses are presented. The Lyapunov linearized stability theory is employed to demonstrate that the first (second) nontrivial steady-state response is always stable (unstable). Numerical calculations show that the two models are qualitatively the same, but quantitatively different. Numerical results are also presented to highlight the effects of the mean axial speed, the axial speed fluctuation amplitude, and the viscoelastic parameters.Copyright © 2005 by ASME

Abstract: The trivial equilibrium of a nonlinear autonomous system with time delay may become unstable via a Hopf bifurcation of multiplicity two, as the time delay reaches a critical value. This loss of stability of the equilibrium is associated with two coincident pairs of complex conjugate eigenvalues crossing the imaginary axis. The resultant dynamic behaviour of the corresponding nonlinear non-autonomous system in the neighbourhood of the Hopf bifurcation is investigated based on the reduction of the infinite-dimensional problem to a four-dimensional centre manifold. As a result of the interaction between the Hopf bifurcating periodic solutions and the external periodic excitation, a primary resonance can occur in the forced response of the system when the forcing frequency is close to the Hopf bifurcating periodic frequency. The method of multiple scales is used to obtain four first-order ordinary differential equations that determine the amplitudes and phases of the phase-locked periodic solutions. The first-order approximations of the periodic solutions are found to be in excellent agreement with those obtained by direct numerical integration of the delay-differential equation. It is also found that the steady state solutions of the nonlinear non-autonomous system may lose their stability via either a pitchfork or Hopf bifurcation. It is shown that the primary resonance response may exhibit symmetric and asymmetric phase-locked periodic motions, quasi-periodic motions, chaotic motions, and coexistence of two stable motions.

Abstract: In this paper we study the unsteady heat conduction due to a sudden temperature step in the external surfaces of a solid slab. In order to estimate the temperature profile in the solid, we applied the multiple-scale perturbation technique by identifying the “early” and “late” transient regimes for small values of the Biot number, Bi. In this sense, we have re-visited the classical lumped method, incorporating this particular case as an asymptotic limit, which is fully described by the “late” regime for small values of Bi. Once the temperature distribution is analytically predicted, this solution is compared against the exact solution and with other analytical results obtained by using regular perturbation techniques, for different values of the Biot number Bi. Observing a good agreement between the corresponding comparisons, we obtain a very simple and useful formula to predict the nondimensional temperature of the solid slab.

Abstract: The stationary Korteweg-de Vries equation perturbed by two different damping terms will be discussed. The solution is oscillatory with slowly varying amplitude, wave length and mean value. The multiple scales method is applied and equations for the slowly varying quantities are be derived.

Abstract: In this paper the non-linear interaction of a three-degrees-of-freedom structural model subjected to external excitation is examined. The non-linearity of the system results in different critical regions of internal resonance and this has a significant effect on the response. We use the method of multiple scales to determine a first-order uniform asymptotic expansion of the solution, including the modulation equations governing the dynamics of the amplitudes and phases of the interacting modes. Then, we investigate behavior and stability of the equilibrium and dynamic solutions as the forcing amplitude and resonance detunings are slowly varied. The solutions are stable when σ3 = 0.01 and the response amplitude a3 has a saddle node bifurcation for minimum value of the left branch for all other variations. The first and second modes reach minimum values at about σ3 = 0 while the third mode reaches the maximum value at the same point. Each mode is separated into two branches for decreasing and increasing µ2 and β4 respectively.

Abstract: Non-linear stability theories for the characterization of Newtonian film flow down an infinite vertical rotating cylinder is given. A generalized non-linear kinematic model is derived to represent the physical system and is solved by the long-wave perturbation method in a two-step procedure. In the first step, the normal mode method is used to characterize the linear behaviours. In the second step, an elaborated non-linear film flow model is solved by using the method of multiple scales to characterize flow behaviours at various states of subcritical stability, subcritical instability, supercritical stability, and supercritical explosion. The modelling results indicate that by increasing the rotation speed, ω, and decreasing the radius of cylinder, R, the film flow becomes less stable, generally.

Abstract: The response of two nonlinear oscillators subject to a large delay is investigated by using a two-time multiple scale analysis. The first problem has been proposed by Johnson and Moon [13] as a model for machine-tool vibrations. The second problem has been formulated by Minorsky [22] in his study of a delayed control. For both problems, we derive slow time amplitude equations and show analytically that the delay is responsible for a secondary bifurcation to quasiperiodic oscillations. For Minorsky’s equation, higher order bifurcations are further analyzed for very large delays.© 2005 ASME

Abstract: The paper presents both of the linear and nonlinear stability theories for characterization of Newtonian film flows down on the inner surface of a rotating in.nite vertical cylinder. After showing the insufficiency of the linear model in characterizing certain flow behaviors, a generalized nonlinear kinematic model is then derived to represent the physical system. The model is solved by the long wave perturbation method in a two-step procedure. In the first step, the normal mode method is used to characterize the linear behaviors. The amplitude growth rates and the threshold conditions are characterized subsequently and summarized as the by-products of the linear solutions. In the second step, an elaborated nonlinear film flow model is solved by using the method of multiple scales to characterize flow behaviors at various states of sub-critical stability, sub-critical instability, supercritical stability, and supercritical explosion. The modeling results indicate that by increasing the rotation speed, X, and the radius of cylinder, R, the film flow will make the flow system more stable.

Abstract: In this paper, the large amplitude free vibration of a doubly clamped microbeam is considered. The effects of shear deformation and rotary inertia on the large amplitude vibration of the microbeam are investigated. To this end, first Hamilton’s principle is used in deriving the partial differential equation of the microbeam response under the mentioned conditions. Then, implementing the Galerkin’s method the partial differential equation is converted to an ordinary nonlinear differential equation. Finally, the method of multiple scales is used to determine a second order perturbation solution for the obtained ODE. The results show that nonlinearity acts in the direction of increasing the natural frequency of the doubly clamped microbeam. Shear deformation and rotary inertia have significant effects on the large amplitude vibration of thick and short microbeams.Copyright © 2005 by ASME

Abstract: Abstrcat The second order differential equation to govern longitudinal motion of reentry vehicle and aerodynamic coefficient models resulting from symmetry considerations in the body axis system are described. The result is used to construct phase planes, which reveal the general global nature of motion-spiral points, saddle points, Hopf bifurcation, limit cycles and domains of initial conditions leading to oscillatory motion and divergence. An asymptotic approximation to the solution of the governing equation is obtained by using MMS(method of multiple scales);This result provides expressions for the amplitudes and frequencies of limit cycles. In the meantime,the type of Hopf bifurcation is also discussed。Finally,the effects on the changes of statical pitch moment have also been analyzed.

Abstract: The dynamic stability and complicated motions of a vessel in regular sea are investigated when the frequency in the pitch is nearly twice the frequency in the roll. We use the Method of Multiple Scales to determine four first-order nonlinear ordinary differential equations that govern the modulation of the amplitudes and phases when the forcing frequency is near either the pitch or the roll natural frequency. These equations are used to determine the steady-state solutions and their stability. Force-response and frequency-response curves are generated. Coexistence of multiple solutions is found in the presence of quadratic nonlinearities. The linear solution is symmetric. The pitch mode has multi-valued solutions and the roll mode has a single-valued solution for the case of primary resonance of the pitch mode. Both pitch and roll modes have multivalued solutions in small intervals for the case of primary resonance of the roll mode.