The Method of Multiple Scales for Ordinary Differential Equations

1996 ◽  
pp. 267-409 ◽  
Author(s):  
J. Kevorkian ◽  
J. D. Cole
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Multiple Scales ◽  
Method Of Multiple Scales

scholarly journals Nonlinear Response of Cantilever Beams to Combination and Subcombination Resonances

1998 ◽  
Vol 5 (5-6) ◽  
pp. 277-288 ◽  
Author(s):  
Ali H. Nayfeh ◽  
Haider N. Arafat
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Virtual Work ◽  
Equations Of Motion ◽  
Nonlinear Ordinary Differential Equations ◽  
Response Curves ◽  
First Order ◽  
Aspect Ratios

The nonlinear planar response of cantilever metallic beams to combination parametric and external subcombination resonances is investigated, taking into account the effects of cubic geometric and inertia nonlinearities. The beams considered here are assumed to have large length-to-width aspect ratios and thin rectangular cross sections. Hence, the effects of shear deformations and rotatory inertia are neglected. For the case of combination parametric resonance, a two-mode Galerkin discretization along with Hamilton’s extended principle is used to obtain two second-order nonlinear ordinary-differential equations of motion and associated boundary conditions. Then, the method of multiple scales is applied to obtain a set of four first-order nonlinear ordinary-differential equations governing the modulation of the amplitudes and phases of the two excited modes. For the case of subcombination resonance, the method of multiple scales is applied directly to the Lagrangian and virtual-work term. Then using Hamilton’s extended principle, we obtain a set of four first-order nonlinear ordinary-differential equations governing the amplitudes and phases of the two excited modes. In both cases, the modulation equations are used to generate frequency- and force-response curves. We found that the trivial solution exhibits a jump as it undergoes a subcritical pitchfork bifurcation. Similarly, the nontrivial solutions also exhibit jumps as they undergo saddle-node bifurcations.

Nonlinear Vibrations of a Thin Plate Under Simultaneous Internal and External Resonances

2010 ◽  
Vol 132 (5) ◽  
Author(s):  
Usama H. Hegazy
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Thin Plate ◽  
Nonlinear Response ◽  
Phase Plane ◽  
Multiple Scales ◽  
Approximate Solutions ◽  
State Response ◽  
The Stability ◽  
Parametric And External Excitations

The dynamic behavior of a rectangular thin plate under parametric and external excitations is investigated. The motion of the thin plate is modeled by coupled second-order nonlinear ordinary differential equations. Their approximate solutions are sought by applying the method of multiple scales. A reduced system of four first-order ordinary differential equations is determined to describe the time variation of the amplitudes and phases of the vibration in the horizontal and vertical directions. The steady-state response and the stability of the solutions for various parameters are studied numerically, using the frequency-response function and the phase-plane methods. It is also shown that the system parameters have different effects on the nonlinear response of the thin plate. Moreover, the chaotic motion of the thin plate is found by numerical simulation.

Study on Free Oscillations of Systems with Even Nonlinearities by Means of the Method of Multiple Scales

2007 ◽  
Vol 10-12 ◽  
pp. 193-197
Author(s):  
J.M. Wen ◽  
Z.C. Cao
Keyword(s):  
Differential Equations ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Nonlinear Differential Equations ◽  
Approximate Solutions ◽  
Free Oscillations ◽  
Analytical Technique ◽  
Spring System ◽  
Mass Spring ◽  
Good Agreement

An analytical technique, namely the method of multiple scales, is applied to solve the differential equations of free oscillations with even nonlinearities in a mass-spring system. Unlike other perturbation methods, the method of multiple scales is effective in determining the transient response as well as determining the approximation to the frequency of the nonlinear system. In this paper, the periodic solutions of the even nonlinear differential equations have been obtained by using the method of multiple scales. Compared with the numerical examples, the approximate solutions are in good agreement with exact solutions. The numerical and analytical solutions have clearly shown that there exists the so-called drift phenomenon in the free oscillations of systems with even nonlinearities without any external excitation.

Characterization of a Tunable MEMS RF Filter

2006 ◽  
Author(s):  
Bashar K. Hammad ◽  
Elihab M. Abdel-Rahman ◽  
Ali H. Nayfeh
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Input Signal ◽  
Multiple Scales ◽  
Distributed Parameter ◽  
Filter Performance ◽  
Rf Filter ◽  
Wide Range ◽  
Global Modes

We present a reduced-order analytical model to describe the response of a tunable MEMS RF filter to an input signal whose frequency is in the neighborhood of the passband. It extends our earlier model by allowing for the application of independent DC voltages in addition to an AC input signal. The model is obtained by discretizing the distributed-parameter system using a Galerkin procedure. It consists of two second-order nonlinearly coupled ordinary-differential equations. Using the method of multiple scales, we determine four first-order nonlinear ordinary-differential equations describing the amplitudes and phases of the modes. We found that mismatch between the natural frequencies of the resonators modifies the global modes significantly, leading to localization of the response in either the input or the output beam. We found that the filter can be tuned to operate linearly for a wide range of VAC by choosing a DC voltage that makes the effective nonlinearities vanish. Amplifying the input signal VAC to improve the filter performance creates multi-valued responses beyond a threshold in the case of non-zero effective nonlinearities.

scholarly journals Supercritical Nonlinear Vibration of a Fluid-Conveying Pipe Subjected to a Strong External Excitation

2016 ◽  
Vol 2016 ◽  
pp. 1-21 ◽  
Author(s):  
Yan-Lei Zhang ◽  
Hui-Rong Feng ◽  
Li-Qun Chen
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Nonlinear Vibration ◽  
Internal Resonance ◽  
Multiple Scales ◽  
Excitation Amplitude ◽  
Transverse Displacement ◽  
Gyroscopic System ◽  
Response Curves ◽  
Equilibrium Configurations

Nonlinear vibration of a fluid-conveying pipe subjected to a transverse external harmonic excitation is investigated in the case with two-to-one internal resonance. The excitation amplitude is in the same magnitude of the transverse displacement. The fluid in the pipes flows in the speed larger than the critical speed so that the straight configuration becomes an unstable equilibrium and two curved configurations bifurcate as stable equilibriums. The motion measured from each of curved equilibrium configurations is governed by a nonlinear integro-partial-differential equation with variable coefficients. The Galerkin method is employed to discretize the governing equation into a gyroscopic system consisting of a set of coupled nonlinear ordinary differential equations. The method of multiple scales is applied to analyze approximately the gyroscopic system. A set of first-order ordinary differential equations governing the modulations of the amplitude and the phase are derived via the method. In the supercritical regime, the subharmonic, superharmonic, and combination resonances are examined in the presence of the 2 : 1 internal resonance. The steady-state responses and their stabilities are determined. The various jump phenomena in the amplitude-frequency response curves are demonstrated. The effects of the viscosity, the excitation amplitude, the nonlinearity, and the flow speed are observed. The analytical results are supported by the numerical integration.

The Method of Multiple Scales in Solving Nonlinear Dynamic Differential Equations of Gear Systems

2013 ◽  
Vol 274 ◽  
pp. 324-327
Author(s):  
J.F. Nie ◽  
M.L. Zheng ◽  
G.B. Yu ◽  
J.M. Wen ◽  
B. Dai
Keyword(s):  
Differential Equations ◽  
Analytical Solutions ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Nonlinear Differential Equations ◽  
Transmission Error ◽  
Gear System ◽  
Approximate Analytical Solutions ◽  
Response Equation ◽  
Gear Systems

To obtain exact analytical solutions of differential equations of gear system dynamics due to the difficulty of solving complicated differential equations. Only the approximate analytical solutions can be determined. The method of multiple scales is one of the most powerful, popular perturbation methods. The dynamic model which describes the torsional vibration behaviors of gear system has been introduced accurately in this paper. The differential equation of gear system nonlinear dynamics exhibiting combined nonlinearity influence such as time-varying stiffness, tooth backlash and dynamic transmission error (DTE) has been proposed. The theory of multiple scales method has been presented in solving nonlinear differential equations of gear systems and the frequency response equation has been obtained. The fact that the approximate analytical solution by using the method of multiple scales is in good agreement with the exact solutions by numerically integrating differential equations has proved that the method of multiple scales is one of the most frequently used methods in solving differential equations, especially for large and complicated differential equations.

Dynamic Behavior of an AMB/Supported Rotor Subject to Parametric Excitation

2006 ◽  
Vol 128 (5) ◽  
pp. 646-652 ◽  
Author(s):  
Y. A. Amer ◽  
M. H. Eissa ◽  
U. H. Hegazy ◽  
A. S. Sabbah
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Multiple Scales ◽  
Initial Conditions ◽  
Dynamical Behavior ◽  
Approximate Solutions ◽  
Sensitive Dependence ◽  
Response Function Method ◽  
The Stability ◽  
Gyroscopic Effects

The dynamical behavior of a parametrically excited simple rigid disk-rotor supported by active magnetic bearings (AMB) is investigated, without gyroscopic effects. The principal parametric resonance case is considered and studied. The motion of the rotor is modeled by a coupled second-order nonlinear ordinary differential equations. Their approximate solutions are sought applying the method of multiple scales. A reduced system of four first-order ordinary differential equations are determined to describe the time variation of the amplitudes and phases of the vibration in the horizontal and vertical directions. The steady-state response and the stability of the solutions for various parameters are studied numerically, using the frequency response function method. The numerical results show that the system behavior includes multiple solutions, jump phenomenon, and sensitive dependence on initial conditions. It is also shown that the system parameters have different effects on the nonlinear response of the rotor. Results are compared to previously published work.

Ordinary differential equations

2018 ◽  
Author(s):  
Joseph F. Boudreau ◽  
Eric S. Swanson
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Software Design ◽  
Multiple Scales ◽  
Numerical Algorithms ◽  
Object Oriented ◽  
Step Size ◽  
Derivative System ◽  
Adaptive Step ◽  
Classical And Quantum Mechanics

This chapter surveys the ordinary differential equations (ODEs) that occur in classical and quantum mechanics, and describes both numerical algorithms and appropriate software design for solving them. Systems of ordinary differential equations, together with a few constants of integration, can in most cases be regarded as a means of defining a function (the “solution”). In this chapter, we develop an object-oriented architecture that applies integrators of the Runge-Kutta family to create these functions. Together with an automatic derivative system for generating partial derivatives from functions of one or more variables, the differential equation solver becomes a powerful tool for solving a variety of few-body problems in classical Hamiltonian systems. This chapter presents a blend of numerical algorithms, physics, and computing techniques. The phenomenon of energy drift is discussed and used to motivate symplectic solvers. Techniques such as adaptive step size and possible problems with stability and multiple scales are also discussed.

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