Abstract: Parameters optimization for energy pumping is considered. The case of strongly nonhomogeneous nonlinear system with 2 dof is analyzed. Damping in strongly nonlinear energetic sink as well as in the linear primary structure are taken into account. In particular efficiency of energy pumping is studied by using an analytical expression. This analytical form is obtained owing to the multiple scale analysis by using the complex variables. Analytical results are confirmed by numerical ones. An experimental verification based on a reduced-scale building model is also considered.

Abstract: The present work deals with the non-linear vibration of a harmonically excited single link roller-supported flexible Cartesian manipulator with a payload. The governing equation of motion of this system is developed using extended Hamilton's principle, which is reduced to the second-order temporal differential equation of motion, by using generalized Galerkin's method. This equation of motion contains both cubic non-linearities of geometric and inertial type in addition to linear forced and non-linear parametric excitation terms. Method of multiple scales is used to solve this non-linear equation and study the stability and bifurcations of the system. Influence of amplitude of the base excitation and mass ratio on the steady state response of the system is investigated for both simple and subharmonic resonance conditions. Critical bifurcation points are determined from the fixed-point responses and periodic, quasi-periodic responses are also found for different system parameters. The results obtained using the perturbation analysis are compared with the previously published experimental work and are found to be in good agreement. This work will be useful for the designer of a flexible manipulator.

Abstract: The non-linear integro-differential equations of motion for a slender cantilever beam subject to axial narrow-band random excitation are investigated. The method of multiple scales is used to determine a uniform first-order expansion of the solution of equations. According to solvability conditions, the non-linear modulation equations for the principal parametric resonance are obtained. Firstly, The largest Lyapunov exponent which determines the almost sure stability of the trivial solution is quantificationally resolved, in which, the modified Bessel function of the first kind is introduced. Results show that the increase of the bandwidth facilitates the almost sure stability of the trivial response and stabilizes the system for a lower acceleration oscillating amplitude but intensifies the instability of the trivial response for a higher one. Secondly, the first and second order non-trivial steady state response of the system is obtained by perturbation method and the corresponding amplitude–frequency curves are calculated when the bandwidth is very small. Results show that the effective non-linearity of whether the amplitude expectation of the first order steady state response or the amplitude expectation of the second order steady state response is of the hardening type for the first mode, whereas for the second mode the effective non-linearity of whether the amplitude expectation of the first order steady state response or the amplitude expectation of the second order steady state response is of the softening type. Finally, the stochastic jump and bifurcation is investigated for the first and second modal parametric principal resonance. The basic jump phenomena indicate that, under the conditions of system parameters with a smaller bandwidth, the most probable motion is around the non-trivial branch of the amplitude response curve, whereas with a higher bandwidth, the most probable motion is around the trivial one of the amplitude response curve. However, the stochastic jump is sometimes more sensitive to the change of the bandwidth, in other words, a small change of bandwidth may induce a series of stochastic jump and bifurcation.

Abstract: The principal resonance of a Duffing oscillator with delayed state feedback under narrow-band random parametric excitation is studied by using the method of multiple scales and numerical simulations. The first-order approximations of the solution, together with the modulation equations of both amplitude and phase, are derived. The effects of the frequency detuning, the deterministic amplitude, the intensity of the random excitation and the time delay on the dynamical behaviors, such as stability and bifurcation, are studied through the largest Lyapunov exponent. Moreover, the appropriate choice of the feedback gains and the time delay is discussed from the viewpoint of vibration control. It is found that the appropriate choice of the time delay can broaden the stable region of the trivial steady-state solution and enhance the control performance. The theoretical results are well verified through numerical simulations.

Abstract: In this paper we study, both theoretically and experimentally, the nonlinear vibration of a shallow arch with one end attached to an electro-mechanical shaker. In the experiment we generate harmonic magnetic force on the central core of the shaker by controlling the electric current flowing into the shaker. The end motion of the arch is in general not harmonic, especially when the amplitude of lateral vibration is large. In the case when the excitation frequency is close to the nth natural frequency of the arch, we found that geometrical imperfection is the key for the nth mode to be excited. Analytical formula relating the amplitude of the steady state response and the geometrical imperfection can be derived via a multiple scale analysis. In the case when the excitation frequency is close to two times of the nth natural frequency two stable steady state responses can exist simultaneously. As a consequence jump phenomenon is observed when the excitation frequency sweeps upward. The effect of geometrical imperfection on the steady state response is minimal in this case. The multiple scale analysis not only predicts the amplitudes and phases of both the stable and unstable solutions, but also predicts analytically the frequency at which jump phenomenon occurs.

Abstract: 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

Abstract: Global bifurcations and chaos in modal interactions of an imperfect circular plate with one-to-one internal resonance are investigated. The case of primary resonance, in which an excitation frequency is near natural frequencies, is considered. The damping force is not included in the analysis. The method of multiple scales is used to obtain an autonomous system from a non-autonomous system of ordinary differential equations governing non-linear oscillations of an imperfect circular plate. The Melnikov's method for heteroclinic orbits of the autonomous system is used to obtain the criteria for chaotic motion. It is shown that the existence of heteroclinic orbits in the unperturbed system implies chaos arising from breaking of heteroclinic orbits under perturbation. The validity of the result is checked numerically. It is also observed numerically that chaos can appear due to breaking of invariant tori under perturbation.

Abstract: The response of two-degree-of-freedom systems with quadratic and quartic nonlinearities to a principal parametric resonance in the presence of two-to-one internal resonance is investigated. The method of multiple scales is used to construct a first-order uniform expansion yielding four first-order nonlinear ordinary differential (averaging) equations governing the modulation of the amplitudes and phases of the two modes. These equations are used to determine the steady state (fixed point) solutions and their stability. Bifurcations of the fixed points are investigated. Numerical solutions are carried out and graphical representations of the results are presented. The effect of the different parameters on the system response is studied. It is found that each mode has a single-valued curve and there exist zones of multivalued with the increasing and decreasing of some parameters. Both modes lose stability with the varying of some parameters. The region of definition decreases with the increasing and decreasing of some parameters.

Abstract: The stability of nonlinear dynamical system of relative rotation is studied. Firstly, the dynamics equation of relative rotation autonomous nonlinear dynamical system with commonly damped force and forced excitation is deduced. Secondly, the stability of relative rotation nonlinear dynamical system is studied. For the nonlinear dynamical system, it is proved that the closed orbit bifurcation can occur under some conditions. Finally, The approximate solution of the equation under forced excitation is obtained by the method of multiple scales.

Abstract: The method of multiple scales is modified to nonlinear analysis in rotor systems. Amplitude equations for forward and backward whirling modes are directly derived and the method makes it easier to understand resonance mechanism. As an example, we analyze near the major critical speed the nonlinear dynamics of a horizontally supported Jeffcott rotor and show that nonlinear and gravity effects cause the backward whirling mode in addition to the forward one. Some experiments are performed and the validity of the theoretical results is confirmed.

Abstract: The natural frequencies of an axially moving beam were determined by using the method of multiple scales. The method of second-order multiple scales could be directly applied to the governing equation if the axial motion of the beam is assumed to be small. It can be concluded that the natural frequencies affected by the axial motion are proportional to the square of the velocity of the axially moving beam. The results obtained by the perturbation method were compared with those given with a numerical method and the comparison shows the correctness of the multiple-scale method if the velocity is rather small.

Abstract: We analyse the two-dimensional, gravitationally-driven spreading of fluid through a porous medium overlying a horizontal impermeable boundary from which fluid can drain freely at one end. Under the assumption that none of the intruding fluid is retained within the pores in the trail of the current, the motion of the current is described by the dipole self-similar solution of the first kind derived by Barenblatt and Zel’dovich (1957). We show that small perturbations of arbitrary shape imposed on this solution decay in time, indicating that the self-similar solution is linearly stable. We use the connection between the perturbation eigenfunctions and symmetry transformations of the self-similar solution to demonstrate that variables can always be specified in terms of which the rate of decay of the perturbations is maximised. Unsaturated flow can be modelled by assuming that a constant fraction of the fluid is retained within the pores by capillary action in the trail of the current. It has been shown (Barenblatt and Zel’dovich, 1998; Ingerman and Shvets, 1999) that in this case, the motion of the current is described by a self-similar solution of the second kind characterised by an anomalous exponent. We derive leading-order analytic expressions for the anomalous exponent and the self-similar quantities valid for small values of the fraction of fluid retained using direct asymptotic analysis and by using a novel application of the method of multiple scales. The latter offers a number of advantages and permits the evolution of the current to be clearly connected with its initial conditions in a way not possible with conventional approaches. We demonstrate that the theoretical predictions provided by these expressions are in excellent agreement with results from the numerical integration of the governing equations.

Abstract: The chaotic dynamics of parametrically excited, simply supported laminated composite piezoelectric rectangular plates are analyzed, The plates are forced by transverse loads. It is assumed that different layers are perfectly bonded to each other with piezoelectric actuator patches embedded in them. Firstly, based on von Karman-type equations and third-order shear deformation laminate theory of Reddy, the nonlinear equations of motions of the laminated composite piezoelectric rectangular plates are derived. Here, we consider the piezoelectric parametric loads and in-plane parametric loads acting in both x-direction and y-direction. Then, the Galerkin’s approach is applied to convert partial differential equations to the ordinary differential equations. The method of multiple scales is used to obtain the averaged equations. Finally, based on the averaged equations, periodic and chaotic motions of the plates are found by using numerical simulation. The numerical results show the existence of periodic and chaotic motions in averaged equations. The chaotic responses are sensitive to initial conditions especially to forcing loads and the parametric excitation.Copyright © 2007 by ASME

Abstract: The response of the Maglev system with delayed position feedback control under the sub-harmonic excitation of the flexible guideway is investigated. The dynamical model is linearized at the equilibrium. Employing time delay as its bifurcation parameter, the condition under which the Hopf bifurcation may occur is investigated. Center manifold reduction is applied to get the Poincar normal form of the nonlinear system with guideway disturbance so that we can study the relation between periodic solution and system parameter. The sub-harmonic resonant periodic solution of the normal form is calculated based on the method of multiple scales, and we get the bifurcation equation of the free oscillation. The existence condition of the free oscillation in the solution is analyzed. Relationship between periodic solution and control and excitation parameters is also investigated. Finally numerical method is applied to study how system and excitation parameters affect the system response. It was shown that the critical time delay to keep the response of the system stable is less than that without perturbation. Time delay can not only suppress sub_harmonic resonance, but also control the appearance of the chaos. Control parameter can govern the emergence of the free oscillation and affect the amplitude of the forced oscillation. So carefully selecting the system parameters can restrain the oscillation effectively.

Abstract: SUMMARY A trajectory-based solution of Maxwell's equations is derived using the method of multiple scales. This time-domain technique utilizes an asymptotic expansion in terms of the ratio of the wave front length-scale to the length-scale of the heterogeneity. The approach provides an alternative to an expansion in terms of frequency and allows one to model electromagnetic wave propagation over a wide frequency band. At the lower end of the frequency band, the trajectory-based solution reduces to a previously derived diffusive solution. Similarly, at higher frequencies one obtains the ‘delta-like’ solution associated with hyperbolic wave propagation. However, the solution is also valid at intermediate frequencies which cannot be characterized as either diffusive or hyperbolic. A numerical illustration demonstrates the importance of both conduction and displacement currents at frequencies between 10 and 100 MHz. The amplitudes computed using the trajectory-based approach compare well with analytic results for a homogeneous whole-space. Using the technique I am able to model observations from a broad-band (3–300 kHz) experiment at the Richmond Field Station in California. In addition, ground penetrating radar waveforms in the 5–200 MHz range, gathered at the Boise Hydrogeophysical Research Site, are matched using the results of a radar velocity tomogram.

Abstract: A nonlinear dynamic model of laminated disk affixed with piezoelectric materials was established while considering the nonlinear constitutive relations of piezoelectric materials by means of Hamilton's principle and Rayleigh-Ritz method.With the use of the method of multiple scales,the standing wave primary resonance of the laminated disk was investigated.The results show that the primary resonance has multi-solutions and jump phenomena when excitation frequency varies from low value to high value.The steady-state primary resonance can be single or multiple(two) when the excitation frequency changes.The realization of primary resonance lies on the stability conditions and initial conditions.To the laminated disk using piezoelectric materials with soft character,the reasonable area of detuning parameter was proposed according to the analytical results.Numerical results verify the solution of analytical analysis.

Abstract: Usage of self-excitation as an excitation method for cantilever-probe in atomic force microscopy (AFM) has been proposed in order to improve the low quality factor Q in liquid environments. For realization of non-contact mode AFM, it is necessary to reduce the amplitude of the self-excited cantilever-probe. In this study, the self-excited oscillation of the cantilever-probe is generated by the angular velocity feedback. In addition, the small steady state amplitude is achieved by nonlinear feedback proportional to the squared deflection angle and the angular velocity. Regarding the micro cantilever-probe as a micro cantilever beam, we show the equation of motion in which the geometrical nonlinear effect is taken into account. Averaged equation is derived by applying the method of multiple scales and the bifurcation diagram is theoretically described. Then, it is clarified that the amplitude of the cantilever-probe can be reduced by increasing the nonlinear feedback gain. By using our own making AFM, we demonstrate the nonlinear dynamics of a “van der Pol” type self-excited cantilever. Steady state amplitude of self-excited oscillation is reduced to 8 nm.

Abstract: A thin liquid layer of a non-Newtonian film falling down an inclined plane that is subjected to non-uniform heating has been considered. The temperature of the inclined plane is assumed to be linearly distributed and the case when the temperature gradient is positive or negative is investigated. The film flow is influenced by gravity, mean surface-tension and thermocapillary force acting along the free surface. The coupling of thermocapillary instability and surface-wave instabilities is studied for two-dimensional disturbances. A non-linear evolution equation is derived by applying the long-wave theory and the equation governs the evolution of a power-law film flowing down an inclined plane. The linear stability analysis shows that the film flow system is stable when the plate temperature is decreasing in the downstream direction while it is less stable for increasing temperature along the plate. Weakly non-linear stability analysis using the method of multiple scales has been investigated and this leads to a secular equation of the Ginzburg-Landau type. The analysis shows that both supercritical stability and subcritical instability are possible for the film flow system. The results indicate the existence of finite-amplitude waves and the threshold amplitude and non-linear speed of these waves are influenced by thermocapillarity. The results for the dilatant as well as pseudoplastic fluids are obtained and it is observed that the result for the Newtonian model agrees with the available literature report. The influence of non-uniform heating of the film flow system on the stability of the system is compared with the stability of the corresponding uniformly heated film flow system.Copyright © 2007 by ASME

Abstract: The primary resonance of a Duffing oscillator with two distinct time delays in state feedback under narrow-band random excitation is investigated in detail by using the method of multiple scales. First, the equations of modulation of response amplitude and phase are determined. Then, the expressions of the first-order and the second-order steady-state moments and their stable regions are obtained by introducing the equivalent detuning frequency and the equivalent damping ratio. For the case of two distinct time delays, the appropriate choices of the combinations of the feedback gains and the difference between two time delays are discussed from the viewpoint of vibration control. Finally, the theoretical analyses are well verified through numerical simulations.

Abstract: We present a reduced-order model and closed-form expressions describing the response of a micromechanical filter made up of two clamped-clamped microbeam capacitive resonators coupled by a weak microbeam. The model accounts for geometrical and electrical nonlinearities as well as the coupling between them. It is obtained by discretizing the distributed-parameter system using the Galerkin procedure. The basis functions are the linear undamped global mode shapes of the unactuated filter. Closed-form expressions for these mode shapes and the coressponding natural frequencies are obtained by formulating a boundary-value problem (BVP) that is composed of five equations and twenty boundary conditions. This problem is transformed into solving a system of twenty linear homogeneous algebraic equations for twenty constants and the natural frequencies. We predict the deflection and the voltage at which the static pull-in occurs by solving another boundary-value problem (BVP). We also solve an eigenvalue problem (EVP) to determine the two natural frequencies delineating the bandwidth of the actuated filter. Using the method of multiple scales, we determine four first-order nonlinear ODEs describing the amplitudes and phases of the modes. We found a good agreement between the results obtained using our model and the published experimental results. We found that the filter can be tuned to operate linearly for a wide range of input signal strengths by choosing a DC voltage that makes the effective nonlinearities vanish.Copyright © 2007 by ASME

Abstract: This study applies the method of multiple scales to obtain periodic solutions of a two-pulley belt system with clearance-type nonlinearity. The purpose is to explain the published numerical results and clarify how design parameters affect the system dynamics. The validity of the perturbation method for such strong nonlinearity is evaluated. The closed-form frequency response relation is determined at the first order, and an implicit expression is obtained for the second order approximation. The preload applied to the accessory determines the softening level of the nonlinearity. Larger preload leads to less disengagement and less softening. For a considerable range of practical parameter values, the analytical solutions well approximate the numerical results from harmonic balance.Copyright © 2007 by ASME

Abstract: We study the nonlinear response of a system with cubic and quartic nonlinearities to an amplitude-modulated excitation whose carrier frequency is much higher than the natural frequency of the system. The method of multiple scales is used to derive a second-order equation governing the components of the response of the system near its natural frequency. In this equation, the amplitude of the excitation, which is a function of time, appears as parametric excitation. The result of the perturbation analysis was used to study the response of the system to constant amplitude and harmonically modulated excitations. For harmonically modulated excitations, various resonances occur and these were shown to correspond to external combination resonances. Two coupled first-order ordinary differential equations describe the evolution of the amplitude and phase. The evolution equations are used to determine the steady-state motion and representative frequency-response curves are presented for each resonance. The stability of the steady-state solution is investigated. The response amplitude has a multivalued curve and symmetric single value curve for the variation of the amplitude with the detuning parameter σ and the coefficient of external excitation f respectively. The response amplitude is not affected and has the same magnitude for decreasing values of the damping factor μ. The response amplitude loses stability for increasing values of the coefficient of cubic term α.

Abstract: The aim of this work is to study periodic oscillations and bifurcations in cellular nonlinear networks composed by oscillatory cells and connected through arbitrary couplings. In order to characterize each oscillator by using amplitude and phase variables, a method based on a generalized version of the describing function technique is proposed. Furthermore, by exploiting the method of multiple scales a set of ordinary differential equations governing the amplitude and phase dynamics is derived. The results also permit to study accurately weakly connected oscillatory networks. Finally, the method is compared to a spectral technique, based on the harmonic balance approach, by considering a chain of Chua's circuits.