Stability of the damped cubic-quintic duffing oscillator

Project Abstract

The stability analysis of the damped cubic-quintic Duffing oscillator is investigated in this research. The system's dynamics are governed by a nonlinear differential equation that includes both cubic and quintic terms, in addition to damping. The study focuses on understanding the conditions under which the oscillator exhibits stable behavior and the influence of the damping parameter on its stability. To analyze the stability of the system, linear stability analysis and numerical simulations are employed. The linear stability analysis involves examining the equilibrium points of the system and determining their stability properties based on the eigenvalues of the linearized system matrix. By analyzing the stability of the equilibrium points, insights into the overall behavior of the oscillator can be gained. Numerical simulations are conducted to validate the results obtained from the linear stability analysis. These simulations involve solving the nonlinear differential equation numerically for various initial conditions and parameter values. The simulations provide a more comprehensive understanding of the system's dynamics, including the presence of periodic, quasi-periodic, or chaotic behavior. The influence of the damping parameter on the stability of the damped cubic-quintic Duffing oscillator is a key focus of this research. It is observed that varying the damping parameter leads to changes in the system's stability characteristics. Specifically, the critical damping value is identified, beyond which the system transitions from stable to unstable behavior. The relationship between the damping parameter and the onset of instability is analyzed in detail. Moreover, the effect of the cubic and quintic nonlinear terms on the stability of the system is investigated. These nonlinear terms introduce additional complexities to the system's dynamics, affecting its stability properties. By analyzing the interplay between the nonlinear terms and the damping parameter, a deeper understanding of the system's behavior is achieved. Overall, this research contributes to the understanding of the stability of the damped cubic-quintic Duffing oscillator and provides insights into the role of damping and nonlinearities in shaping the system's dynamics. The findings have implications for various fields, including nonlinear dynamics, mechanical vibrations, and control systems, where similar oscillator models are encountered.

Project Overview

<p> </p><p><strong>1.0 &nbsp; &nbsp; INTRODUCTION</strong></p><p><strong>&nbsp;</strong></p><p><strong>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;</strong>Most real life problems are nonlinear in nature, this has made the study of nonlinear systems which are very complex an important area of study and research. The Duffing oscillator with viscous damping is one of such important nonlinear system which can be generally described by the following equation of motion:</p><p>&nbsp;</p><p>&nbsp;</p><p>(1)</p><p>where</p><p>.</p><p>&nbsp;</p><p>For the un-damped system, the total potential energy is given by:</p><p>, then we have a Hamiltonian system where H is the Hamiltonian. In this case, one can equate H to a constant and integrate for the displacement .</p><p>Where we have positive damping ( , we have the important property, .</p><p>If we set &nbsp;in (1), we obtain,</p><p>&nbsp;</p><p>(2)</p><p>where,</p><p>The system (2), describes the motion of the viscously damped cubic Duffing oscillator which can be used to model conservative double well oscillators which can occur in magneto-elastic mechanical systems [20]. A good and illustrating example of such system was described in [30]. The cubic Duffing equation can as well be used to model the nonlinear spring-mass system (hardening and softening) [2], [9], as well as the motion of a classical particle in a double well potential [20], [25]. The system (2) with forcing was proposed by Correig in [6] as a model of microseism time series and have been used in [22] to model the prediction of earthquake occurrence.</p><p>Generally, the viscously damped and forced cubic-quintic Duffing oscillator with random noise which can be obtained by setting , in (1) is given by the equation</p><p>(3)</p><p>where, ,</p><p>is the damping co-efficient.</p><p>is the proper or resonant frequency.</p><p>are the co-efficient of nonlinearity.</p><p>is the random noise.</p><p>Equation (3) with &nbsp;was used in [8] to model the transverse oscillations of a nonlinear beam.</p><p>We can as well write (3) as a system in the form,</p><p>&nbsp;</p><p>(4)</p><p>where,</p><p>&nbsp;</p><p>is a tri-stable potential or a triple well potential.</p><p>Setting &nbsp;in equation (4), then we get</p><p>&nbsp;</p><p>(5)</p><p>where &nbsp;implies as well from (4),</p><p>. &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; (6)</p><p>The stability matrix of the system (4) is given by,</p><p>(7)</p><p>where the eigenvalues must satisfy the equation det</p><p>(8)</p><p>&nbsp;</p><p><strong>1.1 PURPOSE OF STUDY</strong></p><p>This study is aimed at first, observing the effect of the added quintic nonlinear term to the damped cubic Duffing oscillator, secondly, to show that we can only obtain centres and saddles for any arbitrary set of parameters in the case of an un-damped cubic-quintic Duffing oscillator, thirdly, to observe the effect of positive/negative damping coefficient to the Duffing oscillator and finally to observe the effect strong nonlinearity to the damped Duffing oscillator.</p><p>&nbsp;</p><p><strong>1.2 &nbsp; DEFINITION OF ESSENTIAL TERMS</strong></p><p><strong>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<u>Positive Damping</u></strong></p><p>This is understood as the basic damping in which case, the damping acts opposite the direction of the velocity of a given system thereby bringing the system to equilibrium at a given time. This is very feasible in all physical systems.</p><p><strong><u>Negative Damping</u></strong></p><p>In this case we mean to represent a situation where the damping acts in the same direction as the velocity. This results in oscillations with increasing or growing amplitude. Physically, this is very possible, many physical systems behave this way, as was noted in [34] where catastrophic failures recorded for a large, long stroke and high speed extrusion press was attributed to negative damping. In [35], it was also noted that systems such as the laser driven pendulum studied in [36] and systems close to thermodynamic equilibrium exhibit such damping, [37] investigated the asymptotic behavior of the solutions for nonlinear wave equations of Kirchhoff type with positive/negative damping, it was also remarked in [38] that negative damping is responsible for LCOs (limit cycle oscillations) arising in nonlinear aero-elastic systems. Motivated by the fact that the drop in steady-state wind turbine rotor thrust with wind speed above rated would lead to negative damping of the barge-pitch mode and contribute to the large system-pitch motions, [39] investigated the influence of control on the pitch damping of a floating wind turbine and interesting results were obtained. Though practical measures are taken to prevent/reduce negative damping in most physical systems, its occurrence persist after a given period of time. Systems like the nose wheel shimmy of an airplane exhibits negative damping while the airplane is in motion as it eventually comes in contact with an external object (say a pebble on the runway ). We must note that it is usually very hard to construct simple systems that exhibit negative damping. Obviously negative damping is no abstract or mathematical idealization as it is frequently obtainable and observable in most physical systems.</p> <br><p></p>