Showing papers on "Multiple-scale analysis published in 1994"

Dissolution or growth of soluble spherical oscillating bubbles

Abstract: A new theoretical formulation is presented for mass transport across the dynamic interface associated with a spherical bubble undergoing volume oscillations. As a consequence of the changing internal pressure of the bubble that accompanies volume oscillations, the concentration of the dissolved gas in the liquid at the interface undergoes large-amplitude oscillations. The convection-diffusion equations governing transport of dissolved gas in the liquid are written in Lagrangian coordinates to account for the moving domain. The Henry's law boundary condition is split into a constant and an oscillating part, yielding the smooth and the oscillatory problems respectively. The solution of the oscillatory problem is valid everywhere in the liquid but differs from zero only in a thin layer of the liquid in the neighbourhood of the bubble surface. The solution to the smooth problem is also valid everywhere in the liquid; it evolves via convection-enhanced diffusion on a slow timescale controlled by the Peclet number, assumed to be large. Both the oscillatory and smooth problems are treated by singular perturbation methods: the oscillatory problem by boundary-layer analysis, and the smooth problem by the method of multiple scales in time. Using this new formulation, expressions are developed for the concentration field outside a bubble undergoing arbitrary nonlinear periodic volume oscillations. In addition, the rate of growth or dissolution of the bubble is determined and compared with available experimental results. Finally, a new technique is described for computing periodically driven nonlinear bubble oscillations that depend on one or more physical parameters. This work extends a large body of previous work on rectified diffusion that has been restricted to the assumptions of infinitesimal bubble oscillations or of threshold conditions, or both. The new formulation represents the first self-consistent, analytical treatment of the depletion layer that accompanies nonlinear oscillating bubbles that grow via rectified diffusion.

On Nonlinear Modes of Continuous Systems

Abstract: We use several methods to study the nonlinear modes of one-dimensional continuous systems with cubic inertia and geometric nonlinearities. Invariant manifold and perturbation methods applied to the discretized system and the method of multiple scales applied to the partial-differential equation and boundary conditions are discussed and their equivalence is demonstrated. The method of multiple scales is then applied directly to the partial-differential equation and boundary conditions governing several nonlinear beam problems.

Nonlinear vibrations of a beam-spring-mass system

Abstract: The nonlinear response of a simply supported beam with an attached spring-mass system to a primary resonance is investigated, taking into account the effects of beam midplane stretching and damping. The spring-mass system has also a cubic nonlinearity. The response is found by using two different perturbation approaches. In the first approach, the method of multiple scales is applied directly to the nonlinear partial differential equations and boundary conditions. In the second approach, the Lagrangian is averaged over the fast time scale, and then the equations governing the modulation of the amplitude and phase are obtained as the Euler-Lagrange equations of the averaged Lagrangian. It is shown that the frequency-response and force-response curves depend on the midplane stretching and the parameters of the spring-mass system. The relative importance of these effects depends on the parameters and location of the spring-mass system.

Parametric resonances at subcritical speeds in discs with rotating frictional loads

Abstract: This paper is concerned with the parametric resonances in a stationary classical annular disc when excited by a rotating mass-spring-damper system together with a frictional follower load. An analysis by the method of multiple scales is performed to reveal the existence of instabilities associated with subcritical parametric resonances, and other instabilities of the backward waves in modes with nodal diameters. The latter are shown to be driven by friction and not to be dependent upon the rotational speed. A state-space analysis, with truncated modes, is used to investigate the effect of varying the friction, stiffness, mass and damping prameters in a series of simulated problems. The results obtained from the state-space eigenvalue method tend to support the conclusions of the multiple scales analysis.

A theoretical and experimental investigation of a three-degree-of-freedom structure

Abstract: The response of a structure to a simple-harmonic excitation is investigated theoretically and experimentally. The structure consists of two light-weight beams arranged in a T-shape turned on its side. Relatively heavy and concentrated weights are placed at the upper and lower free ends and at the point where the two beams are joined. The base of the ‘T’ is clamped to the head of a shaker. Because the masses of the concentrated weights are much larger than the masses of the beams, the first three natural frequencies are far below the fourth; consequently, for relatively low frequencies of the excitation, the structure has, for all practical purposes, only three degrees of freedom. The lengths and weights are chosen so that the third natural frequency is approximately equal to the sum of the two lower natural frequencies, an arrangement that produces an autoparametric (also called an internal) resonance. A linear analysis is performed to predict the natural frequencies and to aid in the design of the experiment; the predictions and observations are in close agreement. Then a nonlinear analysis of the response to a prescribed transverse motion at the base of the ‘T’ is performed. The method of multiple scales is used to obtain six first-order differential equations describing the modulations of the amplitudes and phases of the three interacting modes when the frequency of the excitation is near the third natural frequency. Some of the predicted phenomena include periodic, two-period quasiperiodic, and phase-locked (also called synchronized) motions; coexistence of multiple stable motions and the attendant jumps; and saturation. All the predictions are confirmed in the experiments, and some phenomena that are not yet explained by theory are observed.

Damping of parametrically excited single-degree-of-freedom systems

Abstract: A single-degree-of-freedom parametrically excited system coupled with a Lanchester damper, a mass-dashpot device, is studied. The two equations governing the total system are solved using the method of multiple scales for the case of principal parametric resonance. Steady-state solutions are obtained and the effect of the various system parameters examined. The stability analysis for the steady-state solution is also carried out. Results show that this damper can limit the maximum response of the main system and delay the onset of the force threshold necessary to trigger a non-trivial stable response.

A ray method of solving problems connected with a shock interaction

Abstract: Problems connected with a shock interaction of a thin elastic cylindrical bar and an elastic sphere with an Uflyand-Mindlin plate of a finite size are considered. In enumerated problems, an impact process is accompanied by material local deformations and propagation of wave surfaces of a strong or weak discontinuity in elastic bodies coming in contact. The local deformations are taken into account in terms of the Hertz's contact theory, but the dynamic deformations behind the fronts of incident and reflected waves are determined by means of truncated power series with variable coefficients (what is known as ray series). These two types of deformations are mated on the contact region boundary. As the truncated ray series, as a rule, are not uniformly applicable in the whole region of the wave solution existence, then for their improvement the method of “forward-area” regularization is used which is based on a combination of the recurrent equations of the ray method with the method of multiple scales. The method of power series (ray method) allows one to find the main characteristics of the impact theory in an analytical form.

Nonlinear analysis of the forced response of a beam with three mode interaction

Abstract: An analysis is presented for the primary resonance of a clamped-hinged beam, which occurs when the frequency of excitation is near one of the natural frequencies,ωn Three mode interaction (ω2 ≈ 3ω1 and ω3 ≈ ω1 + 2ω2) is considered and its influence on the response is studied The case of two mode interaction (ω2 ≈ 3ω1) is also considered to compare it with the case of three mode interaction The straight beam experiencing mid-plane stretching is governed by a nonlinear partial differential equation By using Galerkin's method the governing equation is reduced to a system of nonautonomous ordinary differential equations The method of multiple scales is applied to solve the system Steady-state responses and their stability are examined Results of numerical investigations show that there exists no significant difference between both modal interactions' influences on the responses

Nonlinear electrohydrodynamic stability of a fluid layer : effect of a tangential electric field

Abstract: The weakly nonlinear electrohydrodynamic stability of fluid layer sandwiched between two semi-infinite fluids is investigated. The nonlinear theory of perturbation is applied for symmetric and anti-symmetric modes. The method of multiple scales is used to expand the various perturbation quantities to yield the linear and successive nonlinear partial differential equations of the various orders. The solutions of these equations are obtained. The application of the boundary conditions leads to two nonlinear Schrodinger equations. It is found that the presence of the tangential field plays a stabilizing role and can be used to suppress the instability of the system at a given wavenumber which is unstable linear stability. Numerical calculations show a global stability for certain wavenumbers. A local instability is also observed in the graphs. The field plays a dual role. It is observed that the change of the layer thickness redistributes the stable areas.

Dynamic response of spinning blades subjected to gyroscopic motion

Abstract: This paper presents an investigation into the vibration and stability of a blade spinning with respect to a nonfixed axis. Due to the motion of the spin axis, parametric instability of the blade may occur in certain situations. In this work, the discretized equations of motion are first formulated by the finite element technique. Then the system equations are transformed, by a special modal analysis procedure, into independent sets of first-order simultaneous differential equations. Each set of differential equations is solved analytically by the method of multiple scales if the precessional speed of the spin axis is assumed to be small compared to the spin rate of the blade, yielding the system response and the expressions for the boundaries of the unstable regions. Finally, the effects of system parameters on the changes in these boundaries are studied numerically.

Suppression of flutter by parametric excitations

Abstract: 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 perioddoubling bifurcations that culminates in chaos. The complex motions are characterized by using phase planes, power spectra, Lyapunov exponents, and dimensions.

Pulse Propagation in Nonlinear Optical Fibers using Phase-Sensitive Amplifiers

Abstract: : A mathematical model for pulse propagation in a nonlinear fiber optic communications line is presented where linear loss in the fiber is balanced by a chain of periodically spaced, Phase Sensitive Amplifiers (PSAs). A multiple scale analysis is employed to average over the strong, rapidly varying and periodic perturbations to the governing nonlinear Schroedinger equation (NLS). The analysis indicates that the averaged evolution is governed by a fourth order nonlinear diffusion equation which evolves on a length scale much greater than that of the typical soliton period. In a particular limit, stable steady state hyperbolic secant solutions of the averaged equation are analytically found to exist provided a minimum amount of over amplification is supplied. Further, arbitrary initial conditions within a wide stability region exponentially decay onto the steady state. Outside of this analytic regime, extensive numerical simulations indicate that soliton-like steady states exist and act as exponential attractors for a wide region of parameter space. These simulations also show that the averaged evolution is quite accurate in modeling the full NLS with loss and phase sensitive gain. The bifurcation structure of the fourth order equation is explored. A subcritical bifurcation from the trivial solution is found to occur for a specific over amplification value. Further, a limit point, or fold, is also found which connects the stable branch of solutions with the unstable branch from the subcritical bifurcation. The bifurcation structure can be further explored in parameter space with the use of AUTO which is capable of tracking steady state solutions for a wide range of parameters. For larger amplifier spacings, a small dispersive radiation field is generated from the periodic forcing of the loss and gain. The NLS with variably-spaced PSAs is then considered in an effort

Nonlinear stability of Halley cometosheath with transverse plasma motion

Abstract: Weakly nonlinear MHD stability of the Halley cometosheath determined by the balance between the outward ion-neutral drag force and the inward Lorentz force is investigated including the transverse plasma motion as observed in the flanks with the help of the method of multiple scales. The eigenvalues and the eigenfunctions are obtained for the linear problem and the time evolution of the amplitude is obtained using the solvability condition for the solution of the second order problem. The diamagnetic cavity boundary and the adjacent layer of about 100 km thickness is found unstable for the travelling waves of certain wave numbers. Halley ionopause has been observed to have strong ripples with a wavelength of several hundred kilometers. It is found that nonlinear effects have stabilizing effect.

Long-Disturbance Stability of Nonlinear Wave Regimes in a Falling Liquid Film

Abstract: So called Kuramoto-Sivashinsky equation is considered. The nonlinear stability of periodic steady-state travelling solutions with respect to long length disturbances is investigated. The method of multiple scales is used It is shown that if their wave numbers are near the neutral wave number, then the nonlinear evolution of the initially infmitesimal waves are described by equation that is similar to the Ginzburg-Landau equation. But as far as stability is concerned there is essential difference between them. It follows from this equation that all of these waves are unstable to the sideband disturbances, while the same solutions of Ginzburg­ Landau equation are stable to ones. If their wave numbers are far away from the neutral wave number, another nonlinear evolution equation is obtained that allow us to determine the conditions of the double periodic regime arising.

Analysis of the limit cycle of a generalised van der pol equation by a time transformation method

Abstract: The properties of the limit cycle of a generalised van der Pol equation of the form u + u = e (1—u2n)u, where e is small and n is any positive integer, are investigated by applying a time transformation perturbation method due to Burton. It is found that as n increases the amplitude of the limit cycle oscillation decreases and its period increases. The time transformation solution is compared with the solution derived using the method of multiple scales and with a numerical solution. It is found that, to first order in e, the time transformation solution for the limit cycle agrees better with the numerical solution than the multiple scales solution. Both perturbation solutions give the same result for the period of the limit cycle to second order in e. The accuracy of the time transformation solution decreases as n increases.