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.

scholarly journals A Transformation Method for Solving Conservative Nonlinear Two-Degree-of-Freedom Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Alex Elías-Zúñiga ◽  
Daniel Olvera Trejo ◽  
Inés Ferrer Real ◽  
Oscar Martínez-Romero
Keyword(s):  
Numerical Integration ◽  
Nonlinear Equations ◽  
Equations Of Motion ◽  
Nonlinear Oscillators ◽  
Transformation Method ◽  
Degree Of Freedom ◽  
Nonlinear Transformation ◽  
Equivalent Representation ◽  
Representation Form ◽  
Two Degree Of Freedom

A nonlinear transformation approach based on a cubication method is developed to obtain the equivalent representation form of conservative two-degree-of-freedom nonlinear oscillators. It is shown that this procedure leads to equivalent nonlinear equations that describe well the numerical integration solutions of the original equations of motion.

scholarly journals Energy Method to Obtain Approximate Solutions of Strongly Nonlinear Oscillators

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Alex Elías-Zúñiga ◽  
Oscar Martínez-Romero
Keyword(s):  
Potential Energy ◽  
Analytical Solutions ◽  
Energy Method ◽  
Equations Of Motion ◽  
Nonlinear Oscillators ◽  
Approximate Solutions ◽  
Equivalent Representation ◽  
Strongly Nonlinear ◽  
Representation Form ◽  
Strongly Nonlinear Oscillators

We introduce a nonlinearization procedure that replaces the system potential energy by an equivalent representation form that is used to derive analytical solutions of strongly nonlinear conservative oscillators. We illustrate the applicability of this method by finding the approximate solutions of two strongly nonlinear oscillators and show that this procedure provides solutions that follow well the numerical integration solutions of the corresponding equations of motion.

scholarly journals Equivalent Representation Form of Oscillators with Elastic and Damping Nonlinear Terms

2013 ◽  
Vol 2013 ◽  
pp. 1-11
Author(s):  
Alex Elías-Zúñiga ◽  
Daniel Olvera ◽  
Inés Ferrer Real ◽  
Oscar Martínez-Romero
Keyword(s):  
Numerical Integration ◽  
Structural Engineering ◽  
Duffing Oscillator ◽  
Equations Of Motion ◽  
Fractional Power ◽  
Polymeric Materials ◽  
Nonlinear Damping ◽  
Equivalent Representation ◽  
Representation Form ◽  
Dynamics Response

In this work we consider the nonlinear equivalent representation form of oscillators that exhibit nonlinearities in both the elastic and the damping terms. The nonlinear damping effects are considered to be described by fractional power velocity terms which provide better predictions of the dissipative effects observed in some physical systems. It is shown that their effects on the system dynamics response are equivalent to a shift in the coefficient of the linear damping term of a Duffing oscillator. Then, its numerical integration predictions, based on its equivalent representation form given by the well-known forced, damped Duffing equation, are compared to the numerical integration values of its original equations of motion. The applicability of the proposed procedure is evaluated by studying the dynamics response of four nonlinear oscillators that arise in some engineering applications such as nanoresonators, microresonators, human wrist movements, structural engineering design, and chain dynamics of polymeric materials at high extensibility, among others.

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 A METHOD OF CHAOS CONTROL BASED ON THE LINEAR AND NONLINEAR TRANSFORMATION OF SYSTEM VARIABLES

2000 ◽  
Vol 49 (5) ◽  
pp. 849
Author(s):  
LUO XIAO-SHU ◽  
LIU MU-REN ◽  
FANG JING-QING ◽  
KONG LIN-JIANG ◽  
TANG GUO-NING
Keyword(s):  
Chaos Control ◽  
Nonlinear Transformation ◽  
Linear And Nonlinear ◽  
System Variables

Linear and Nonlinear Dynamics of Cantilevered Cylinders in Axial Flow

2014 ◽  
Author(s):  
Ahmad Jamal ◽  
Michael P. Païdoussis ◽  
Luc G. Mongeau
Keyword(s):  
Equations Of Motion ◽  
Axial Flow ◽  
Experimental System ◽  
Vital Role ◽  
Viscous Force ◽  
Flexible Cylinder ◽  
Force Coefficients ◽  
Precise Calculation ◽  
Linear And Nonlinear ◽  
Nonlinear Equations Of Motion

Understanding and prediction of the dynamics of slender flexible cylinders in axial flow is of interest for the design and safe operation of heat exchangers and nuclear reactors, specifically that of heat exchanger tubes, nuclear fuel elements, control rods, and monitoring tubes. In such fluid-structure interaction problems, the fluid forces acting on the flexible structure play a vital role in defining its dynamics. Therefore, a precise calculation of the coefficients associated to these forces, such as the longitudinal and normal viscous force coefficients, and base drag coefficient in the equation of motion is imperative. The present work is aimed at (i) calculating these force coefficients for a cantilevered slender flexible cylinder, fitted with an ogival end-piece, in axial flow and (ii) conducting experiments on the same system. In the calculation of these force coefficients, the parameters of the experimental system are used, so that the theoretically predicted dynamics would be representative of the actual physical system. These calculated force coefficients are then incorporated in the linear and nonlinear equations of motion and the predicted dynamics are compared with those of the experiments. The comparison shows good agreement between the theoretical and experimental results.

Application of Fractional Calculus for Dynamic Problems of Solid Mechanics: Novel Trends and Recent Results

2009 ◽  
Vol 63 (1) ◽  
Author(s):  
Yuriy A. Rossikhin ◽  
Marina V. Shitikova
Keyword(s):  
Fractional Calculus ◽  
Degrees Of Freedom ◽  
Solid Mechanics ◽  
State Of The Art ◽  
Dynamic Problems ◽  
Engineering Problems ◽  
Multilayered Systems ◽  
Linear And Nonlinear ◽  
Linear And Nonlinear Systems

The present state-of-the-art article is devoted to the analysis of new trends and recent results carried out during the last 10years in the field of fractional calculus application to dynamic problems of solid mechanics. This review involves the papers dealing with study of dynamic behavior of linear and nonlinear 1DOF systems, systems with two and more DOFs, as well as linear and nonlinear systems with an infinite number of degrees of freedom: vibrations of rods, beams, plates, shells, suspension combined systems, and multilayered systems. Impact response of viscoelastic rods and plates is considered as well. The results obtained in the field are critically estimated in the light of the present view of the place and role of the fractional calculus in engineering problems and practice. This articles reviews 337 papers and involves 27 figures.

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