Second-Order Perturbation Analysis of Low-Amplitude Traveling Waves in a Periodic Nonlinear Chain

2013 ◽  
Author(s):  
Smruti R. Panigrahi ◽  
Brian F. Feeny ◽  
Alejandro R. Diaz
Keyword(s):  
Traveling Waves ◽  
Perturbation Analysis ◽  
Multiple Scales ◽  
Periodic Structures ◽  
Second Order ◽  
Quadratic Nonlinearity ◽  
Wave Number ◽  
Nonlinear Chain ◽  
Multiple Scales Analysis ◽  
Low Amplitude

Traveling waves in one-dimensional nonlinear periodic structures are investigated for low-amplitude oscillations using perturbation analysis. We use second-order multiple scales analysis to capture the effects of quadratic nonlinearity. Comparisons with the linear and cubical nonlinear cases are presented in the dispersion relationship, group velocity and phase velocity and their dependence on wave number and amplitude of oscillation. Quadratic nonlinearity is shown to have a significant effect on the behavior.

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.

Higher-Order Multiple Scales Analysis of Weakly Nonlinear Lattices With Implications for Directional Stability

2018 ◽  
Author(s):  
Matthew D. Fronk ◽  
Michael J. Leamy
Keyword(s):  
Acoustic Waves ◽  
Local Stability ◽  
Multiple Scales ◽  
Periodic Structures ◽  
Plane Waves ◽  
Weakly Nonlinear ◽  
Nonlinear Springs ◽  
Nonlinear Lattices ◽  
Multiple Scales Analysis ◽  
The Stability

Recent focus has been given to nonlinear periodic structures for their ability to filter, guide, and block elastic and acoustic waves as a function of their amplitude. In particular, two-dimensional (2-D) nonlinear structures possess amplitude-dependent directional bandgaps. However, little attention has been given to the stability of plane waves along different directions in these structures. This study analyzes a 2-D monoatomic shear lattice composed of discrete masses, linear springs, quadratic and cubic nonlinear springs, and linear viscous dampers. A local stability analysis informed by perturbation results retained through the second order suggests that different directions become unstable at different amplitudes in an otherwise symmetrical lattice. Simulations of the lattice’s equation of motion subjected to both line and point forcing are consistent with the local stability results: waves with large amplitudes have spectral growth that differs appreciably at different angles. The results of this analysis could have implications for encryption strategies and damage detection.

scholarly journals Perturbations of nonlinear autonomous oscillators

1994 ◽  
Vol 35 (4) ◽  
pp. 445-468 ◽  
Author(s):  
P. B. Chapman
Keyword(s):  
General Theory ◽  
Fourier Coefficients ◽  
Multiple Scales ◽  
Second Order ◽  
Multiple Scales Method ◽  
Suitable Parameter ◽  
Non Linear

AbstractA general theory is given for autonomous perturbations of non-linear autonomous second order oscillators. It is found using a multiple scales method. A central part of it requires computation of Fourier coefficients for representation of the underlying oscillations, and these coefficients are found as convergent expansions in a suitable parameter.

Wave Propagation in Two-Dimensional Nonlinear Periodic Lattices

2009 ◽  
Author(s):  
Raj K. Narisetti ◽  
Massimo Ruzzene ◽  
Michael J. Leamy
Keyword(s):  
Wave Propagation ◽  
Linear Models ◽  
Periodic Structures ◽  
Harmonic Wave ◽  
Excitation Amplitude ◽  
Pass Band ◽  
Perturbation Approach ◽  
Nonlinear Stiffness ◽  
Two Dimensional ◽  
Low Amplitude

This paper investigates wave propagation in two-dimensional nonlinear periodic structures subject to point harmonic forcing. The infinite lattice is modeled as a springmass system consisting of linear and cubic-nonlinear stiffness. The effects of nonlinearity on harmonic wave propagation are analytically predicted using a novel perturbation approach. Response is characterized by group velocity contours (derived from phase-constant contours) functionally dependent on excitation amplitude and the nonlinear stiffness coefficients. Within the pass band there is a frequency band termed the “caustic band” where the response is characterized by the appearance of low amplitude regions or “dead zones.” For a two-dimensional lattice having asymmetric nonlinearity, it is shown that these caustic bands are dependent on the excitation amplitude, unlike in corresponding linear models. The analytical predictions obtained are verified via comparisons to responses generated using a time-domain simulation of a finite two-dimensional nonlinear lattice. Lastly, the study demonstrates amplitude-dependent wave beaming in two-dimensional nonlinear periodic structures.

Bang-Bang Versus Linear Control of a Second-Order Rate-Type Servomotor

1960 ◽  
Vol 82 (1) ◽  
pp. 66-72 ◽  
Author(s):  
Philip F. Meyfarth
Keyword(s):  
Limit Cycle ◽  
Function Approximation ◽  
Second Order ◽  
Describing Function ◽  
Response Characteristics ◽  
Order Rate ◽  
Maximum Value ◽  
Bang Bang Control ◽  
Low Amplitude ◽  
Rate Type

A simple nonlinear scheme for controlling a second-order rate-type servomotor is described. This scheme is referred to as “bang-bang” control since the input to the servomotor is made to “bang” from its maximum value in one direction to its maximum value in the other direction depending only on the sign of an error signal. This bang-bang system oscillates in a continuous high-frequency, low-amplitude limit cycle. The nature of this limit cycle is studied by the describing function approximation and by an exact method. The step and frequency response characteristics of the bang-bang system are discussed and compared with the characteristics of a simple linear system. It is shown that many aspects of the behavior of the bang-bang system can be predicted from rather simple considerations.

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.

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