scholarly journals Equivalent Mathematical Representation of Second-Order Damped, Driven Nonlinear Oscillators

2013 ◽  
Vol 2013 ◽  
pp. 1-11
Author(s):  
Alex Elías-Zúñiga ◽  
Oscar Martínez-Romero
Keyword(s):  
Wavelet Transform ◽  
Equations Of Motion ◽  
Nonlinear Oscillators ◽  
Duffing Equation ◽  
Phase Portraits ◽  
Mathematical Representation ◽  
Continuous Wavelet ◽  
Oscillatory Systems ◽  
Pendulum Equation ◽  
Nonlinear Equations Of Motion

The aim of this paper focuses on applying a nonlinearization method to transform forced, damped nonlinear equations of motion of oscillatory systems into the well-known forced, damped Duffing equation. The accuracy obtained from the derived equivalent equations of motion is evaluated by studying the amplitude-time, the phase portraits, and the continuous wavelet transform diagrams of the cubic-quintic Duffing equation, the generalized pendulum equation, the power-form elastic term oscillator, the Duffing equation with linear and cubic damped terms, and the pendulum equation with a cubic damped term.

scholarly journals Investigation of the Equivalent Representation Form of Strongly Damped Nonlinear Oscillators by a Nonlinear Transformation Approach

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Alex Elías-Zúñiga ◽  
Oscar Martínez-Romero
Keyword(s):  
Equations Of Motion ◽  
Nonlinear Oscillators ◽  
Transformation Method ◽  
Nonlinear Transformation ◽  
Equivalent Representation ◽  
Oscillatory Systems ◽  
Representation Form ◽  
Engineering Problems ◽  
Linear And Nonlinear ◽  
Strongly Damped

We use a nonlinear transformation method to develop equivalent equations of motion of nonlinear homogeneous oscillatory systems with linear and nonlinear odd damping terms. We illustrate the applicability of our approach by using the equations of motion that arise in many engineering problems and compare their amplitude-time curves with those obtained by the numerical integration solutions of the original equations of motion.

Analysis of the Nonlinear Dynamics of the Timoshenko Flexible Beams Using Wavelets

2011 ◽  
Vol 7 (1) ◽  
Author(s):  
J. Awrejcewicz ◽  
A. V. Krysko ◽  
V. Soldatov ◽  
V. A. Krysko
Keyword(s):  
Nonlinear Dynamics ◽  
Wavelet Transform ◽  
Continuous Wavelet Transform ◽  
Chaotic Dynamics ◽  
Equations Of Motion ◽  
Stability Loss ◽  
Continuous Wavelet ◽  
Flexible Beams ◽  
Advantages And Disadvantages ◽  
Timoshenko Type

Regular and chaotic dynamics of the flexible Timoshenko-type beams is studied using both the standard Fourier (FFT) and the continuous wavelet transform methods. The governing equations of motion for geometrically nonlinear Timoshenko-type beams are reduced to a system of ODEs using both finite element method (FEM) and finite difference method (FDM) to ensure the reliability of numerical results. Scenarios of transition from regular to chaotic vibrations and beam dynamical stability loss are analyzed. Advantages and disadvantages of various wavelet functions are discussed. Application of continuous wavelet transform to the investigation of transitional and chaotic phenomena in nonlinear dynamics is illustrated and discussed.

scholarly journals Continuous Wavelet Transform of Schwartz Distributions in DL2′(Rn), n ≥ 1

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1106
Author(s):  
Jagdish N. Pandey
Keyword(s):  
Wavelet Transform ◽  
Function Space ◽  
Continuous Wavelet Transform ◽  
Inversion Formula ◽  
Continuous Wavelet ◽  
The Real ◽  
Schwartz Distributions ◽  
Distributional Sense ◽  
Wavelet Inversion

We define a testing function space DL2(Rn) consisting of a class of C∞ functions defined on Rn, n≥1 whose every derivtive is L2(Rn) integrable and equip it with a topology generated by a separating collection of seminorms {γk}|k|=0∞ on DL2(Rn), where |k|=0,1,2,… and γk(ϕ)=∥ϕ(k)∥2,ϕ∈DL2(Rn). We then extend the continuous wavelet transform to distributions in DL2′(Rn), n≥1 and derive the corresponding wavelet inversion formula interpreting convergence in the weak distributional sense. The kernel of our wavelet transform is defined by an element ψ(x) of DL2(Rn)∩DL1(Rn), n≥1 which, when integrated along each of the real axes X1,X2,…Xn vanishes, but none of its moments ∫Rnxmψ(x)dx is zero; here xm=x1m1x2m2⋯xnmn, dx=dx1dx2⋯dxn and m=(m1,m2,…mn) and each of m1,m2,…mn is ≥1. The set of such wavelets will be denoted by DM(Rn).

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