Abstract: We consider a two coupled Duffing–van der Pol oscillators with a nonlinear coupling. The method of multiple scales is used to obtain a set of autonomous equations for the amplitudes and phases of the response of the system. The stability boundaries of fixed points of the approximate equations are obtained using Routh–Hurwitz criterion. The stability boundaries are drawn in various parameters space. The influence of nonlinear damping, detuning, nonlinearity and the coupling strength on the response dynamics is studied. Jump phenomenon is found to occur for a range of values of the parameters of the system. The results obtained from the approximate equations are verified by numerically solving the original system and good agreement is obtained.

Abstract: Two-to-one parametric resonance in transverse vibration of an axially accelerating viscoelastic string with geometric nonlinearity is investigated. The transport speed is assumed to be a constant mean speed with small harmonic variations. The nonlinear partial differential equation that governs transverse vibration of the string is derived from Newton's second law. The method of multiple scales is applied directly to the equation, and the solvability condition of eliminating secular terms is established. Closed-form solutions for the amplitude of the vibration and the existence conditions of nontrivial steady-state response in two-to-one parametric resonance are obtained. Some numerical examples showing effects of the mean transport speed, the amplitude and the frequency of speed variation are presented. Lyapunov's linearized stability theory is employed to analyze the stability of the trivial and nontrivial solutions for two-to-one parametric resonance. Some numerical examples highlighting the effects of the related parameters on the stability conditions are presented.

Abstract: The principal resonance of a van der Pol–Duffing oscillator to the combined excitation of a deterministic harmonic component and a random component has been investigated. By introducing a new expansion parameter e=e( e ,u 0 ) , the method of multiple scales is adapted for the strongly non-linear system. Then the method of multiple scales is used to determine the equations of modulation of response amplitude and phase. The behavior and the stability of steady-state response are studied by means of qualitative analysis. The effects of damping, detuning, bandwidth, and magnitudes of random excitations are analyzed. The theoretical analyses are verified by numerical results. Theoretical analyses and numerical simulations show that when the intensity of the random excitation increases, the non-trivial steady-state solution may change from a limit cycle to a diffused limit cycle. Under some conditions the system may have two steady-state solutions. Random jump may be observed under some conditions. The results obtained in the paper are adapted for a strongly non-linear oscillator that complement previous results in the literature for the weakly non-linear case.

Abstract: The present paper deals with the classical problem of linear sound propagation in tubes with isothermal walls. The perturbation technique of the method of multiple scales in combination with matched asymptotic expansions is applied to derive the first-order solutions and, in addition, the second-order solutions representing the correction due to boundary layer attenuation. The propagation length is assumed to be so large that in order to obtain asymptotic solutions which extend over the whole spatial range the first-order corrections to the classical attenuation rates of the different modes come into play as well. Starting with the case of the characteristic wavelength being large compared to the characteristic dimension of the duct, the analysis is then extended to the case where both of these quantities are of the same order of magnitude. Furthermore, the transmission line parameters and the transfer functions relating the sound pressures at the ends of the duct to the axial velocities are calculated.

Abstract: The steady-state transverse vibration of an axially moving string with geometric nonlinearity was investigated. The transport speed was assumed to be a constant mean speed with small harmonic variations. The nonlinear partial-differential equation that governs the transverse vibration of the string was derived by use of the Hamilton principle. The method of multiple scales was applied directly to the equation. The solvability condition of eliminating the secular terms was established. Closed form solutions for the amplitude and the existence conditions of nontrivial steady-state response of the two-to-one parametric resonance were obtained. Some numerical examples showing effects of the mean transport speed, the amplitude and the frequency of speed variation were presented. The Liapunov linearized stability theory was employed to derive the instability conditions of the trivial solution and the nontrivial solutions for the two-to-one parametric resonance. Some numerical examples highlighting influences of the related parameters on the instability conditions were presented.

Abstract: This paper studies the dynamics of the electrostatic transducers described by two nonlinearly coupled differential equations of motion submitted to two external periodic forces. The method of multiple scales is used to find solutions in the resonant and nonresonant cases. Chaotic behavior is found in terms of the amplitude of the second external force. A feedback controller is applied to drive the chaotic states of the system to an appropriately defined reference signal in spite of modelling errors, parametric variations and perturbing external forces. Computer simulations are provided to illustrate the operation of the designed control scheme.

Abstract: To investigate the principal resonance in transverse nonlinear parametric vibration of an axially accelerating viscoelastic string, the method of multiple scales is applied directly to the nonlinear partial differential equation that governs the transverse vibration of the string. To derive the governing equation, Newton's second law, Lagrangean strain, and Kelvin's model are respectively used to account the dynamical relation, geometric nonlinearity and the viscoelasticity of the string material. Based on the solvability condition of eliminating the secular terms, closed form solutions are obtained for the amplitude and the existence conditions of nontrivial steady-state response of the principal parametric resonance. The Lyapunov linearized stability theory is employed to analyze the stability of the trivial and nontrivial solutions in the principal parametric resonance. Some numerical examples are presented to show the effects of the mean transport speed, the amplitude and the frequency of speed variation.

Abstract: This paper presents an analysis of dynamic stability of an annular plate with a periodically varying spin rate subjected to a stationary in-plane edge load. The spin rate of the plate is characterised as the sum of a constant speed and a small, periodic perturbation. Due to this periodically varying spin rate, the plate may bring about parametric instability. In this work, the initial stress distributions caused by the periodically varying spin rate and the in-plane edge load are analyzed first. The finite element method is applied then to yield the discretized equations of motion. Finally, the method of multiple scales is adopted to determine the stability boundaries of the system. Numerical results show that combination resonances take place only between modes of the same nodal diameter if the stationary in-plane edge load is absent. However, there are additional combination resonances between modes of different nodal diameters if the stationary in-plane edge load is present.

Abstract: A new phenomenon for laminated composite plates undergoing dynamic post-buckling is presented. The buckled composite plate subjected to in-plane dynamic load not only gains more non-linearity by the quadratic term, which arises from static buckling, but also changes its dynamic behavior from pure parametric vibration to combined parametric and forcing traverse vibrations. The nonlinear behavior of a simply supported laminated composite plate undergoing dynamic post-buckling is investigated. The range of bifurcation parameter (forcing frequency) for chaotic motion is obtained, and the characteristics of the dynamic behavior of the laminated composite plate in post-buckling are unveiled. Hamilton's Principle, Galerkin's Method, and Lagrange's Equation are utilized to obtain the equations of motion with higher-order shear deformation. The differential equation has quadratic and cubic non-linearities as well as parametric and harmonic excitation terms. The method of multiple scales is used to determine the eq...

Abstract: The propagation of waves in a string for the case of timevarying length is a problem of one-dimensional wave propagation with moving boundary conditions. Confronted with the difficulty of finding an exact analytical method in order to solve such a problem, scientists and engineers have relied on numerical and approximate analytical methods. For simple cases, there are specific methods. However, more general mathematical methods are used for analytically solving such problems. These methods lead to the solutions of the functional or integro-differential equations. The method of separation of variables used by Tadashi Kotera leads to a complex solution. Such solutions are very cumbersome and complex. Recently, Y. M. Ram and J. Caldwell introduced a new method the method of distorted images. This method ! remains limited and does not take into account the energy of the system. However, these methods do not allow the deduction and the understanding of the physical phenomena. More recently, Stefan Kaczmarczyk and Wieslaw Ostachowicz used the method of multiple scales to study the longitudinal dynamics of hoisting cables and to predict the responses near the resonance region. Nicolas Gonzales studied the stability of the energy of the wave equation for periodic motion. The method used in this paper is an exact analytical method based on analytical functions and conformal transformations. It has been developed for solving a large class of systems with non-stationary dimensions. The main advantage of this method is that it gives the exact solution for all motions (parabolic, hyperbolic, etc.). The other advantages are the easy application, the exact formulation of the energy ratio, and the possibility of deducing the energy. By means of the method cited above, we have determined the exact solution of our problem, the modes, the energy ratio, the energy and discussed the results obtained. 2. SUMMARY OF L. GAFFOUR METHOD

Abstract: In Chapters 6, 7, 10 and 16 we had obtained exact solutions of the Schrodinger equation for specific potential energy variations. However, for most problems in quantum mechanics, it is extremely difficult to obtain exact solutions of the Schrodinger equation and one has to resort to approximate methods. The three most important approximate methods are (i) the perturbation method, (ii) the variational method and (iii) the JWKB method; the perturbation method will be discussed in this chapter. We may mention here that the variational method (which is discussed in Chapter 21) can give good estimate of the ground state energy by choosing an appropriate trial function; however, the method becomes quite cumbersome when one has to apply to higher excited states. On the other hand, the JWKB method (discussed in Chapter 17) gives an approximate but direct solution of the Schrodinger equation. The JWKB method is applicable when the potential energy variation is smoothly varying and when the Schrodinger equation is separable to a one-dimensional equation; if these conditions are satisfied, the method gives information about all the states of the problem.

Abstract: We investigated theoretically and experimentally the nonlinear responseof a clamped-clamped buckled beam to a subharmonic resonance of orderone-half of its first vibration mode. We used a multi-mode Galerkindiscretization to reduce the governing nonlinear partial-differentialequation in space and time into a set of nonlinearly coupledordinary-differential equations in time only. We solved the discretizedequations using the method of multiple scales to obtain a second-orderapproximate solution, including the modulation equations governing itsamplitude and phase, the effective nonlinearity, and the effectiveforcing. To investigate the large-amplitude dynamics, we numericallyintegrated the discretized equations using a shooting method to computeperiodic orbits and used Floquet theory to investigate their stabilityand bifurcations. We obtained interesting dynamics, such as phase-lockedand quasiperiodic motions, resulting from a Hopf bifurcation,snapthrough motions, and a sequence of period-doubling bifurcationsleading to chaos. Some of these nonlinear phenomena, such as Hopfbifurcation, cannot be predicted using a single-mode Galerkindiscretization. We carried out an experiment and obtained results ingood qualitative agreement with the theoretical results.