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 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 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 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 Effect of Nonlinear Dissipation on the Basin Boundaries of a Driven Two-Well Modified Rayleigh–Duffing Oscillator

2015 ◽  
Vol 25 (02) ◽  
pp. 1550024 ◽  
Author(s):  
C. H. Miwadinou ◽  
A. V. Monwanou ◽  
J. B. Chabi Orou
Keyword(s):  
Initial Conditions ◽  
Duffing Oscillator ◽  
Excitation Amplitude ◽  
Nonlinear Damping ◽  
Homoclinic Bifurcation ◽  
Nonlinear Dissipation ◽  
Transition To Chaos ◽  
Melnikov Criterion ◽  
Quadratic Nonlinearities ◽  
Basin Boundaries

This paper considers the effect of nonlinear dissipation on the basin boundaries of a driven two-well modified Rayleigh–Duffing oscillator where pure cubic, unpure cubic, pure quadratic and unpure quadratic nonlinearities are considered. By analyzing the potential, an analytic expression is found for the homoclinic orbit. The Melnikov criterion is used to examine a global homoclinic bifurcation and transition to chaos. Unpure quadratic parameter and parametric excitation amplitude effects are found on the critical Melnikov amplitude μ cr . Finally, the phase space of initial conditions is carefully examined in order to analyze the effect of the nonlinear damping, and particularly how the basin boundaries become fractalized.

A Comparison of the Effects of Nonlinear Damping on the Free Vibration of a Single-Degree-of-Freedom System

2012 ◽  
Vol 134 (2) ◽  
Author(s):  
Bin Tang ◽  
M. J. Brennan
Keyword(s):  
Free Vibration ◽  
Equations Of Motion ◽  
Nonlinear Damping ◽  
Degree Of Freedom ◽  
Damping Force ◽  
Single Degree Of Freedom ◽  
Sdof System ◽  
Single Degree ◽  
Dependent Term ◽  
Quadratic Damping

This article concerns the free vibration of a single-degree-of-freedom (SDOF) system with three types of nonlinear damping. One system considered is where the spring and the damper are connected to the mass so that they are orthogonal, and the vibration is in the direction of the spring. It is shown that, provided the displacement is small, this system behaves in a similar way to the conventional SDOF system with cubic damping, in which the spring and the damper are connected so they act in the same direction. For completeness, these systems are compared with a conventional SDOF system with quadratic damping. By transforming all the equations of motion of the systems so that the damping force is proportional to the product of a displacement dependent term and velocity, then all the systems can be directly compared. It is seen that the system with cubic damping is worse than that with quadratic damping for the attenuation of free vibration.

The DDA Method

2016 ◽  
pp. 90-102
Author(s):  
Katalin Bagi
Keyword(s):  
Numerical Integration ◽  
Displacement Field ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Discontinuous Deformation Analysis ◽  
Deformation Analysis ◽  
Short Overview ◽  
Abrupt Changes ◽  
Discontinuous Deformation ◽  
Iteration Technique

“DDA” stands for “Discontinuous Deformation Analysis”, suggesting that the displacement field of the analyzed domain shows abrupt changes on the element boundaries in the model. This chapter introduces the theoretical fundaments of DDA: mechanical characteristics of the elements together with the basic degrees of freedom, contact behavior, the equations of motion and their numerical integration with the help of Newmark's beta-method taking into account contact creation, loss and sliding with the help of an open-close iteration technique. Finally, a short overview on practical and scientific applications for masonry structures is given.

scholarly journals Treatment of Close Approaches in the Numerical Integration of the Gravitational Problem of N Bodies

1971 ◽  
Vol 10 ◽  
pp. 133-150 ◽  
Author(s):  
D. G. Bettis ◽  
V. Szebehely
Keyword(s):  
Numerical Integration ◽  
Body Problem ◽  
Considerable Increase ◽  
Equations Of Motion ◽  
Computer Time ◽  
Tidal Effects ◽  
Basic Ideas ◽  
Gravitational Problem ◽  
Considerable Detail ◽  
Numerical Approaches

AbstractOne of the main difficulties encountered in the numerical integration of the gravitational n-body problem is associated with close approaches. The singularities of the differential equations of motion result in losses of accuracy and in considerable increase in computer time when any of the distances between the participating bodies decreases below a certain value. This value is larger than the distance when tidal effects become important, consequently, numerical problems are encountered before the physical picture is changed. Elimination of these singularities by transformations is known as the process of regularization. This paper discusses such transformations and describes in considerable detail the numerical approaches to more accurate and faster integration. The basic ideas of smoothing and regularization are explained and applications are given.

scholarly journals Synchronisation phenomenon in three blades rotor driven by regular or chaotic oscillations

2018 ◽  
Vol 148 ◽  
pp. 06002
Author(s):  
Zofia Szmit ◽  
Jerzy Warmiński
Keyword(s):  
Time Series ◽  
Numerical Calculation ◽  
Duffing Oscillator ◽  
Equations Of Motion ◽  
Composite Beams ◽  
Chaotic Oscillations ◽  
Rotating System ◽  
Chaotic Motions ◽  
Structural Differences ◽  
Different Types

The goal of the paper is to analysed the influence of the different types of excitation on the synchronisation phenomenon in case of the rotating system composed of a rigid hub and three flexible composite beams. In the model is assumed that two blades, due to structural differences, are de-tuned. Numerical calculation are divided on two parts, firstly the rotating system is exited by a torque given by regular harmonic function, than in the second part the torque is produced by chaotic Duffing oscillator. The synchronisation phenomenon between the beams is analysed both either for regular or chaotic motions. Partial differential equations of motion are solved numerically and resonance curves, time series and Poincaré maps are presented for selected excitation torques.

A numerical investigation of the rolling-up of vortex sheets

1980 ◽  
Vol 373 (1752) ◽  
pp. 67-91 ◽  
Keyword(s):  
Numerical Integration ◽  
Numerical Investigation ◽  
Equations Of Motion ◽  
Additional Term ◽  
Vortex Sheet ◽  
Time Step ◽  
Vortex Sheets ◽  
Numerical Instabilities ◽  
Sinusoidal Disturbance ◽  
Induced Velocity

In this paper the development of a vortex sheet due to an initially sinusoidal disturbance is calculated. When determining the induced velocity in points of the vortex sheet, it can be represented by concentrated vortices but it is shown that it is analytically more correct to add an additional term that represents the effect of the immediate neighbourhood of the point considered. The equations of motion were integrated by a Runge-Kutta technique to exclude numerical instabilities. The time step was determined by the requirement that a quantity (Hamiltonian) that remains invariant as a result of the equations of motion, should not change more than a certain amount in the numerical integration of the equations of motion. One difficulty is that if a greater number of concentrated vortices are introduced to represent the vortex sheet, the effect of round-off errors becomes more important. The number of figures retained in the computations limits the number of concentrated vortices. Where the round-off errors have been kept sufficiently small, a process of rolling-up of vorticity clearly occurs. There is no point in pursuing the calculations much beyond this point, first because the representation of the vortex sheet by concentrated vortices becomes more and more inaccurate and secondly because viscosity will have the effect of transforming the rolled-up vortex sheet into a region of vorticity.

Sign in / Sign up

Export Citation Format

Share Document