Showing papers on "Multiple-scale analysis published in 1996"

Multiple Scale and Singular Perturbation Methods

Abstract: 1. Introduction.- 1.1. Order Symbols, Uniformity.- 1.2. Asymptotic Expansion of a Given Function.- 1.3. Regular Expansions for Ordinary and Partial Differential Equations.- References.- 2. Limit Process Expansions for Ordinary Differential Equations.- 2.1. The Linear Oscillator.- 2.2. Linear Singular Perturbation Problems with Variable Coefficients.- 2.3. Model Nonlinear Example for Singular Perturbations.- 2.4. Singular Boundary Problems.- 2.5. Higher-Order Example: Beam String.- References.- 3. Limit Process Expansions for Partial Differential Equations.- 3.1. Limit Process Expansions for Second-Order Partial Differential Equations.- 3.2. Boundary-Layer Theory in Viscous, Incompressible Flow.- 3.3. Singular Boundary Problems.- References.- 4. The Method of Multiple Scales for Ordinary Differential Equations.- 4.1. Method of Strained Coordinates for Periodic Solutions.- 4.2. Two Scale Expansions for the Weakly Nonlinear Autonomous Oscillator.- 4.3. Multiple-Scale Expansions for General Weakly Nonlinear Oscillators.- 4.4. Two-Scale Expansions for Strictly Nonlinear Oscillators.- 4.5. Multiple-Scale Expansions for Systems of First-Order Equations in Standard Form.- References.- 5. Near-Identity Averaging Transformations: Transient and Sustained Resonance.- 5.1. General Systems in Standard Form: Nonresonant Solutions.- 5.2. Hamiltonian System in Standard Form Nonresonant Solutions.- 5.3. Order Reduction and Global Adiabatic Invariants for Solutions in Resonance.- 5.4. Prescribed Frequency Variations, Transient Resonance.- 5.5. Frequencies that Depend on the Actions, Transient or Sustained Resonance.- References.- 6. Multiple-Scale Expansions for Partial Differential Equations.- 6.1. Nearly Periodic Waves.- 6.2. Weakly Nonlinear Conservation Laws.- 6.3. Multiple-Scale Homogenization.- References.

Multiple-Scale Analysis of the Quantum Anharmonic Oscillator

Abstract: Conventional weak-coupling perturbation theory suffers from problems that arise from the resonant coupling of successive orders in the perturbation series. Multiple-scale perturbation theory avoids such problems by implicitly performing an infinite reordering and resummation of the conventional perturbation series. Multiple-scale analysis provides a good description of the classical anharmonic oscillator. Here, it is extended to study the Heisenberg operator equations of motion for the quantum anharmonic oscillator. The analysis yields a system of nonlinear operator differential equations, which is solved exactly. The solution provides an operator mass renormalization of the theory.

Nonlinear piezoelectric vibration absorbers

Abstract: The effect nonlinear electrical shunts on the performance of piezoelectric vibration absorbers for linear systems is presented. The equations of motion are derived in a format suitable for perturbation analysis. An electrical shunt containing cubic and quadratic elements is coupled to the structure via the piezoelectric effect. Nonlinearities are introduced as a combination of the square and/or cube of the charge flowing in the linear inductive - resistive (LR) shunt. Turning the shunt near a structural mode causes mechanical energy to be transformed to electric energy and dissipated by the resistive element in the shunt in a manner analogous to a damped vibration absorber. Analysis is carried out using the method of multiple scales. Simulation results are also presented.

Nonlinear stability of Kelvin-Helmholtz waves in magnetic fluids stressed by a time-dependent acceleration and a tangential magnetic field

Abstract: The nonlinear stability of surface waves propagating between two superposed streaming magnetic fluids is investigated. The fluids are stressed by a constant tangential magnetic field and a vertical periodic acceleration. The solution employs the method of multiple scales. Owing to the periodicity, resonant cases appear. Two parametrically nonlinear Schrodinger equations are derived for the resonant cases to describe the elevation of weakly nonlinear capillary waves. The standard nonlinear Schrodinger equation is satisfied for the non-resonant case. Necessary and sufficient conditions for stability are obtained. A formula for the surface elevation is obtained in each case. It is found that the magnetic field, the velocities and the frequency of the applied periodic force play dual roles in the resonant region. Investigation of the stability criterion by nonlinear perturbation shows that an increase in the acceleration frequency has a stabilizing effect. The stabilizing role of the frequency is due to the destabilizing effect of the amplitude of the periodic acceleration.

Mechanical-Wave Filtering in a Periodically Corrugated Elastic Plate

Abstract: The method of multiple scales is employed to analyze the interaction of SH modes in an elastic plate having periodically corrugated outerfaces. Two types of resonant conditions leading to two-mode as well as four-mode interactions are considered. The results of the analysis are utilized to develop ultrasonic mechanical wave filters operating on frequency bands centered at the resonant frequency. The stop-band filter frequency response is presented in terms of the power reflection coefficient. The characteristics of reflection of these filters are enhanced by imposing amplitude taper on the periodic corrugations.

The Method of Multiple Scales for Ordinary Differential Equations

Abstract: Various physical problems are characterized by the presence of a small disturbance which, because it is active over a long time, has a non-negligible cumulative effect. For example, the effect of a small damping force over many periods of oscillation is to produce a decay in the amplitude of an oscillator. A more interesting example having the same physical and mathematical features is that of the motion of a satellite around Earth. Here the dominant force is a spherically symmetric gravitational field. If this were the only force acting on the satellite, the motion would be periodic (for sufficiently low energies). The presence of a thin atmosphere, a slightly nonspherical Earth, a small moon, a distant sun, and so on, all produce small but cumulative effects which, after a sufficient number of orbits, drastically alter the nature of the motion.

Evolution equations and stability results for finite-depth Wilton ripples

Abstract: The method of multiple scales, in both space and time, is used to derive a set of five coupled non-linear partial differential equations. These are the evolution equations which describe, up to cubic order, the motion of a small amplitude finite depth capillary-gravity wavetrain caused by the interaction of the fundamental mode and its second harmonic. Solutions to these equations are obtained which correspond physically to sinusoidal wavetrains. The stability of these wavetrains to small perturbations, which may be at any angle to the wave, is examined. Regions of stability are identified.

Nonlinear dynamics of crane operation at sea

Abstract: The transfer operation of a crane ship becomes dangerous when sea state 3 is reached. The sea state may directly or indirectly cause large motions of the container being hoisted by creating instabilities of the crane load's motion. A simplified model is studied and used to illustrate how the instabilities could arise due to the combination of a one-to-one internal resonance and a primary (additive) resonance or a parametric (multiplicative) resonance. The method of multiple scales is used to derive four ordinarydifferential equations describing the amplitudes and phases of the two modes. The resulting two sets of modulation equations are used to study the equilibrium and dynamic solutions and their stability. The response could be a single-mode solution or a twomode solution. A combination of a shooting technique and Floquet theory is used to calculate limit cycles and ascertain their stability. The numerical results indicate the existence of a sequence of perioddoubling bifurcations that culminates in chaos, multiple attractors, intermittency of type I, and cyclicfold bifurcations. The excitation parameters that lead to complex motions, including chaos, are identified.

Nonlinear piezoelectric vibration absorbers

Abstract: The effect of quadratic nonlinear electrical shunts on the performance of piezoelectric vibration absorbers is presented. The equations of motion are derived in a format suitable for perturbation analysis. An electrical shunt containing cubic and quadratic elements is coupled to the structure via the piezoelectric effect. Nonlinearities are introduced as a combination of the square and/or cube of the charge flowing in the linear inductive-resistive (LR) shunt. Tuning the shunt near a structural mode causes mechanical energy to be transformed to electrical energy and dissipated by the resistive element in the shunt in a manner analogous to a damped vibration absorber. Analysis is carried out using the method of multiple scales. Simulation results are also presented.

The nonlinear response and passive vibration isolation of rigid bodies

Abstract: The nonlinear response of an automotive engine on mounts is investigated. The engine and mount system is represented by a planar three degree of freedom system consisting of a rigid body connected to ground through flexible supports. The nonlinear response of this three degree of freedom system is established using the method of multiple scales. A representative frequency response of the system is presented using parameter values associated with an in-line four cylinder engine running at hot idle. This frequency response shows that when 1:1 and 2:1 internal resonances exist between the linear natural frequencies of the system, multiple steady state solutions can exist. This nonlinear system response occurs due to the system geometry and therefore nonlinear force-deflection characteristics of the flexible supports are not required. Additionally, the isolation characteristics of the system are established based on the system response. The efficient transfer of energy from one mode to another is exploited to enhance the isolation performance of the system.

Primary resonance of piezoelectric resonators

Abstract: It is found that the nonlinear behavior of piezoelectric resonators is more complicated than that predicted by previous theories. Actually the resonators work as parametrically excited vibration systems. Mode-coupled equations are obtained by means of the boundary integration. The method of multiple scales is used to solve the equations. Finally the ordinary differential equations that govern the evolution of the amplitude and phase of the fundamental and the harmonics, respectively, are derived. The nonlinear behavior of the resonator in the vicinity of primary resonance is thoroughly studied. Experiments are conducted for thickness-vibration LiNbO/sub 3/ resonators. For comparison, the behavior at the parametrically resonant regime is also discussed.

Effects of Bearing and Shaft Asymmetries on the Resonant Oscillations of a Rotor-Dynamic System

Abstract: Parametric steady-state vibrations of an asymmetric rotor while passing through primary resonance and the associated stability behavior are analyzed. The undamped case is considered and the equations of motion are rewritten in a form suitable for applying the method of multiple scales. Sensitivity to the bearing as well as shaft asymmetries of the oscillations due to unbalance excitation is evaluated. Expressions for amplitude and frequency modulation functions are obtained and are specialized to yield the steady-state solutions near primary resonance. Frequency-amplitude relationships that result from combined parametric and mass unbalance excitations are derived. Stability regions in the parameter space are obtained based on the time evolution of the amplitude and phase of the steady-state motions. The effects of bearing asymmetry on the amplitude and phase of the resonant oscillations are brought out. The sensitivity of vibrational and stability characteristics to various rotor-dynamic system parameters is illustrated through a numerical investigation.