Slow Passage through Resonance in Mathieu's Equation

2003 ◽  
Vol 9 (6) ◽  
pp. 685-707 ◽  
Author(s):  
Leslie Ng ◽  
Richard Rand ◽  
Michael O'Neil
Keyword(s):  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Initial Conditions ◽  
Analytic Approximation ◽  
Slow Flow ◽  
Wkb Method ◽  
Flow Equations ◽  
Mathieu’S Equation ◽  
Flow Approximation ◽  
Slow Passage

We investigate slow passage through the 2:1 resonance tongue in Mathieu's equation. Using numerical integration, we find that amplification or de-amplification can occur. The amount of amplification (or de-amplification) depends on the speed of travel through the tongue and the initial conditions. We use the method of multiple scales to obtain a slow flow approximation. The Wentzel-Kramers-Brillouin (WKB) method is then applied to the slow flow equations to obtain an analytic approximation.

Slow Passage Through Resonance in Mathieu’s Equation

2002 ◽  
Author(s):  
Leslie Ng ◽  
Richard Rand ◽  
Michael O’Neil
Keyword(s):  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Initial Conditions ◽  
Analytic Approximation ◽  
Slow Flow ◽  
Wkb Method ◽  
Flow Equations ◽  
Mathieu’S Equation ◽  
Flow Approximation ◽  
Slow Passage

We investigate slow passage through the 2:1 resonance tongue in Mathieu’s equation. Using numerical integration, we find that amplification or de-amplification can occur. The amount of amplification (or de-amplification) depends on the speed travelling through the tongue and the initial conditions. We use the method of multiple scales to obtain a slow flow approximation. The WKB method is then applied to the slow flow equations to get an analytic approximation.

Slow flow solutions and stability analysis of single machine to infinite bus power systems

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
M. G. Suresh Kumar ◽  
C. A. Babu
Keyword(s):  
Power Systems ◽  
Power System ◽  
Single Machine ◽  
Multiple Scales ◽  
Operating Conditions ◽  
Phase Portraits ◽  
Slow Flow ◽  
Approximate Analytical Expression ◽  
Flow Equations ◽  
Load Angle

Abstract Nonlinearity is a major constraint in analysing and controlling power systems. The behaviour of the nonlinear systems will vary drastically with changing operating conditions. Hence a detailed study of the response of the power system with nonlinearities is necessary especially at frequencies closer to natural resonant frequencies of machines where the system may jump into the chaos. This paper attempt such a study of a single machine to infinite bus power system by modelling it as a Duffing equation with softening spring. Using the method of multiple scales, an approximate analytical expression which describes the variation of load angle is derived. The phase portraits generated from the slow flow equations, closer to the jump, display two stable equilibria (centers) and an unstable fixed point (saddle). From the analysis, it is observed that even for a combination of parameters for which the system exhibits jump resonance, the system will remain stable if the variation of load angle is within a bounded region.

Harmonically Excited Delay Equation for Machine Tool Vibrations

2018 ◽  
Author(s):  
János Lelkes ◽  
Tamás Kalmár-Nagy
Keyword(s):  
Multiple Scales ◽  
Phase Portraits ◽  
Slow Flow ◽  
Flow Equations ◽  
Vibration Model ◽  
External Sources ◽  
Tool Vibrations ◽  
Local And Global Bifurcations ◽  
Direct Numerical Integration ◽  
Delay Differential

A machining tool can be subject to different kinds of excitations. The forcing may have external sources (such as rotating imbalance or misalignment of the workpiece) or it can arise from the cutting process itself (e.g. chip formation). We investigate the classical tool vibration model which is a delay-differential equation with a quadratic and cubic nonlinearity and periodic forcing. The method of multiple scales was used to derive the slow-flow equations. The resonance curves of the system are similar to those for the Duffing-equation, having a hardening characteristic. Stability analysis for the fixed points of the slow-flow equations was performed. Local and global bifurcations were studied and illustrated with phase portraits and direct numerical integration of the original equation. Subcritical Hopf, saddle-node and heteroclinic bifurcations were found.

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.

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