scholarly journals The Method of Multiple Scales: Asymptotic Solutions and Normal Forms for Nonlinear Oscillatory Problems

1996 ◽  
Vol 21 (2) ◽  
pp. 245-252 ◽  
Author(s):  
NESTOR E. SANCHEZ
Keyword(s):  
Normal Forms ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Asymptotic Solutions

On Direct Methods for Constructing Nonlinear Normal Modes of Continuous Systems

1995 ◽  
Vol 1 (4) ◽  
pp. 389-430 ◽  
Author(s):  
Ali H. Nayfeh
Keyword(s):  
Normal Forms ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Direct Method ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Continuous Systems ◽  
Internal Resonances ◽  
Relief Valve ◽  
Nonlinear Normal

A direct method based on the method of normal forms is proposed for constructing the nonlinear normal modes of continuous systems. The proposed method is compared with the method of multiple scales and the methods of Shaw and Pierre and King and Vakakis by applying them to three conservative systems with cubic nonlinearities: (a) a hinged-hinged beam resting on a nonlinear elastic foundation, (b) a model of a relief valve (linear elastic spring attached to a nonlinear spring with a mass), and (c) a simply supported linear beam with nonlinear torsional springs at both ends. In the absence of internal resonance, the constructed nonlinear modes with all four methods are the same. The method of multiple scales seems to be the simplest and the least computationally demanding. The methods of multiple scales and normal forms are applicable to problems with and without internal resonances, whereas the present forms of the methods of Shaw and Pierre and King and Vakakis are not applicable to problems with internal resonances.

On the modulation of water waves on shear flows

1976 ◽  
Vol 347 (1651) ◽  
pp. 537-546 ◽  
Keyword(s):  
Water Waves ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Vries Equation ◽  
Harmonic Wave ◽  
Short Wave ◽  
Long Wave ◽  
Stokes Waves ◽  
The Stability ◽  
Wave Limit

The method of multiple scales is used to examine the slow modulation of a harmonic wave moving over the surface of a two dimensional channel. The flow is assumed inviscid and incompressible, but the basic flow takes the form of an arbitrary shear. The appropriate nonlinear Schrödinger equation is derived with coefficients that depend, in a complicated way, on the shear. It is shown that this equation agrees with previous work for the case of no shear; it also agrees in the long wave limit with the appropriate short wave limit of the Korteweg-de Vries equation, the shear being arbitrary. Finally, it is remarked that the stability of Stokes waves over any shear can be examined by using the results derived here.

scholarly journals Dynamic analysis of a parametrically excited golden Muga silk embedded pneumatic artificial muscle

2018 ◽  
Vol 211 ◽  
pp. 02008 ◽  
Author(s):  
Bhaben Kalita ◽  
S. K. Dwivedy
Keyword(s):  
Dynamic Analysis ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Quadratic Function ◽  
Artificial Muscle ◽  
Equation Of Motion ◽  
Time Response ◽  
Phase Portraits ◽  
Pneumatic Artificial Muscle ◽  
Pneumatic Muscle

In this work a novel pneumatic artificial muscle is fabricated using golden muga silk and silicon rubber. It is assumed that the muscle force is a quadratic function of pressure. Here a single degree of freedom system is considered where a mass is supported by a spring-damper-and pneumatically actuated muscle. While the spring-mass damper is a passive system, the addition of pneumatic muscle makes the system active. The dynamic analysis of this system is carried out by developing the equation of motion which contains multi-frequency excitations with both forced and parametric excitations. Using method of multiple scales the reduced equations are developed for simple and principal parametric resonance conditions. The time response obtained using method of multiple scales have been compared with those obtained by solving the original equation of motion numerically. Using both time response and phase portraits, variation of few systems parameters have been carried out. This work may find application in developing wearable device and robotic device for rehabilitation purpose.

On the asymptotic solution of the Orr–Sommerfeld equation by the method of multiple-scales

1968 ◽  
Vol 34 (1) ◽  
pp. 145-158 ◽  
Author(s):  
K. Kuen Tam
Keyword(s):  
Exact Solution ◽  
Velocity Profile ◽  
Asymptotic Solution ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Order Equation ◽  
Zeroth Order ◽  
Independent Variables ◽  
Sommerfeld Equation ◽  
Special Case

The method of multiple-scales is used to obtain the asymptotic solution of the Orr–Sommerfeld equation. For the special case of a linear velocity profile, the solution so obtained agrees well with an approximation of the exact solution which is known. For the general case, transformations on both the dependent and independent variables are introduced to obtain a zeroth-order equation which differs from the inner equation studied so far. On the ground of the favourable comparison for the special case, the asymptotic solution constructed is expected to be uniformly valid.

scholarly journals Stability and Hopf Bifurcation Analysis of an Epidemic Model by Using the Method of Multiple Scales

2016 ◽  
Vol 2016 ◽  
pp. 1-8 ◽  
Author(s):  
Wanyong Wang ◽  
Lijuan Chen
Keyword(s):  
Periodic Solution ◽  
Hopf Bifurcation ◽  
Time Delay ◽  
Epidemic Model ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Value Of Time ◽  
Nonlinear Incidence Rate ◽  
The Stability ◽  
Bifurcating Periodic Solution

A delayed epidemic model with nonlinear incidence rate which depends on the ratio of the numbers of susceptible and infectious individuals is considered. By analyzing the corresponding characteristic equations, the effects of time delay on the stability of the equilibria are studied. By choosing time delay as bifurcation parameter, the critical value of time delay at which a Hopf bifurcation occurs is obtained. In order to derive the normal form of the Hopf bifurcation, an extended method of multiple scales is developed and used. Then, the amplitude of bifurcating periodic solution and the conditions which determine the stability of the bifurcating periodic solution are obtained. The validity of analytical results is shown by their consistency with numerical simulations.

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