Second Order Perturbation Analysis of a Forced Nonlinear Mathieu Equation

2012 ◽  
Author(s):  
Venkatanarayanan Ramakrishnan ◽  
Brian F. Feeny
Keyword(s):  
Perturbation Analysis ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Mathieu Equation ◽  
Second Order ◽  
Order Perturbation ◽  
Cubic Nonlinearity ◽  
First Order ◽  
Order Expansion ◽  
Direct Forcing

The present study deals with the response of a forced nonlinear Mathieu equation. The equation considered has parametric excitation at the same frequency as direct forcing and also has cubic nonlinearity and damping. A second-order perturbation analysis using the method of multiple scales unfolds numerous resonance cases and system behavior that were not uncovered using first-order expansions. All resonance cases are analyzed. We numerically plot the frequency response of the system. The existence of a superharmonic resonance at one third the natural frequency was uncovered analytically for linear system. (This had been seen previously in numerical simulations but was not captured in the first-order expansion.) The effect of different parameters on the response of the system previously investigated are revisited.

Second-Order Perturbation Analysis of In-Plane Blade-Hub Dynamics of Horizontal-Axis Wind Turbines

2018 ◽  
Author(s):  
Ayse Sapmaz ◽  
Brian F. Feeny
Keyword(s):  
Perturbation Analysis ◽  
Multiple Scales ◽  
Perturbation Expansion ◽  
Second Order ◽  
Horizontal Axis ◽  
Dynamic Responses ◽  
Order Perturbation ◽  
Resonance Case ◽  
Horizontal Axis Wind Turbine ◽  
Rotor Angle

This paper deals with a second-order perturbation analysis of the in-plane dynamic responses of both tuned and mistuned three-blade-hub horizontal-axis wind-turbine equations. The blades are under effect of gravitational and cyclic aerodynamics forces and centrifugal forces. Although the blades and hub equations are coupled, they can be decoupled by changing the independent variable from time to rotor angle and by using a small parameter approximation. A second-order method of multiple scales is applied in the rotor-angle domain to analyze in-plane blade-hub dynamics. A superharmonic resonance case at one third the natural frequency was revealed. This resonance case was not captured by a first-order perturbation expansion. The relationship between response amplitude and frequency is studied. The effect of blade mistuning on the coupled blade-hub dynamics are taken into account.

Nonlinear Dynamics of a Travelling Beam Subjected to Multi-Frequency Parametric Excitation

2014 ◽  
Vol 592-594 ◽  
pp. 2076-2080 ◽  
Author(s):  
Bamadev Sahoo ◽  
L.N. Panda ◽  
Goutam Pohit
Keyword(s):  
Parametric Excitation ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Equation Of Motion ◽  
Cubic Nonlinearity ◽  
First Order ◽  
Reduced Equations ◽  
Stable Periodic ◽  
Bifurcation And Stability ◽  
First Order Differential Equations

This paper deals with two frequency parametric excitation in presence of internal resonance. The cubic nonlinearity is inserted into the equation of motion by considering the mid-line stretching of the beam. The perturbation method of multiple scales is applied directly to the governing nonlinear fourth order integro-partial differential equation of motion. This leads to a set of first order differential equations known as the reduced equations or normalized reduced equations, which are utilized to determine the additional instability zones, appeared in the trivial state stability plot, the bifurcation and stability of fixed-points, periodic, quasi-periodic, mixed mode and chaotic responses. The transition of system behaviour from stable periodic to unstable chaotic occurs through intermittency route

Nonlinear electrohydrodynamic Kelvin–Helmholtz instability conditions of an interface between two fluids under the effect of a normal periodic electric field

1990 ◽  
Vol 68 (6) ◽  
pp. 479-494 ◽  
Author(s):  
E. F. El Shehawey ◽  
N. R. Abd El Gawaad
Keyword(s):  
Differential Equation ◽  
Electric Field ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Mathieu Equation ◽  
Second Order ◽  
Surface Charges ◽  
Helmholtz Instability ◽  
Order Problem ◽  
Two Fluids

The electrohydrodynamic Kelvin–Helmholtz instability conditions of an interface separating two dielectric streaming fluids, stressed by a normal electric field in absence of surface charges on the interface, are studied. We use the method of multiple scales to solve nonlinear equations. In the first-order problem we obtained Mathieu's differential equation. For the second-order problem, we obtain the nonhomogeneous Mathieu equation and we use the method of multiple scales to obtain a sequence of equations. In the third-order problem, we obtain the second-order differential equation of periodic coefficients; we also obtain a formula for surface elevation, and we determine the instability conditions.

A Perturbation Analysis of the Wavemaking of a Ship, with an Interpretation of Guilloton’s Method

1975 ◽  
Vol 19 (03) ◽  
pp. 140-148
Author(s):  
F. Noblesse
Keyword(s):  
Approximate Solution ◽  
Free Surface ◽  
Perturbation Analysis ◽  
Order Approximation ◽  
Second Order ◽  
Regular Perturbation ◽  
First Order ◽  
Perturbation Problem ◽  
Sinkage And Trim ◽  
Basic Ideas

A thin-ship perturbation analysis, suggested by Guilloton's basic ideas, is presented. The analysis may be regarded as an application of Lighthill's method of strained coordinates to a regular perturbation problem. An inconsistent second-order approximation in which the Laplace equation is satisfied to first order, and the boundary conditions both at the free surface and on the ship hull are satisfied to second order, is derived. When sinkage and trim, incorporated into the present analysis, are ignored, this approximate solution is shown to be essentially equivalent to the method of Guilloton.

Reduced Order Model for MEMS Cantilever Resonators Under Soft AC Voltage Near Natural Frequency

2012 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Israel Martinez
Keyword(s):  
Natural Frequency ◽  
Nonlinear Response ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Electrostatic Force ◽  
Reduced Order Model ◽  
Order Model ◽  
Reduced Order ◽  
First Order ◽  
Electrostatically Actuated

The nonlinear response of an electrostatically actuated cantilever beam microresonator is investigated. The AC voltage is of frequency near resonator’s natural frequency. A first order fringe correction of the electrostatic force and viscous damping are included in the model. The dynamics of the resonator is investigated using the Reduced Order Model (ROM) method, based on Galerkin procedure. Steady-state motions are found. Numerical results for the uniform microresonator are compared with those obtained via the Method of Multiple Scales (MMS).

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