scholarly journals Second-order perturbation analysis of low-amplitude traveling waves in a periodic chain with quadratic and cubic nonlinearity

Wave Motion ◽  
2017 ◽  
Vol 69 ◽  
pp. 1-15 ◽  
Author(s):  
Smruti R. Panigrahi ◽  
Brian F. Feeny ◽  
Alejandro R. Diaz
Keyword(s):  
Traveling Waves ◽  
Perturbation Analysis ◽  
Second Order ◽  
Order Perturbation ◽  
Cubic Nonlinearity ◽  
Low Amplitude ◽  
Periodic Chain

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 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 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.

Investigation of the Spin Hamiltonian Parameters of Yb3+ in CaWO4 Crystal

2004 ◽  
Vol 59 (12) ◽  
pp. 943-946 ◽  
Author(s):  
Hui-Ning Dong ◽  
Shao-Yi Wu
Keyword(s):  
Hyperfine Structure ◽  
Local Structure ◽  
Reasonable Agreement ◽  
Second Order ◽  
Order Perturbation ◽  
Superposition Model ◽  
Spin Hamiltonian ◽  
Structure Constants ◽  
Hamiltonian Parameters ◽  
G Factors

In this paper, the spin Hamiltonian parameters g factors g∥ and g⊥ of Yb3+ and hyperfine structure constants A∥ and A⊥ of 171Yb3+ and 173Yb3+ in CaWO4 crystal are calculated from the two-order perturbation formulae. In these formulae, the contributions of the covalence effects, the admixture between J =7/2 and J =5/2 states as well as the second-order perturbation are included. The needed crystal parameters are obtained from the superposition model and the local structure of the studied system. The calculated results are in reasonable agreement with the observed values. The results are discussed.

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