Eventos Anais de eventos
COBEM 2023
27th International Congress of Mechanical Engineering
Analysis of vibration dynamics with nonlinear damping by means of the Krylov-Bogoliubov method.
Submission Author:
Robert Batista Neves , AM , Brazil
Co-Authors:
Robert Batista Neves, Gustavo Cunha da Silva Neto, Danilo de Santana Chui, Bruna Santiago , Matheus Aleme
Presenter: Robert Batista Neves
doi://10.26678/ABCM.COBEM2023.COB2023-1740
Abstract
Predicting the effects of oscillatory phenomena in mechanical systems is important during mechanical design. This is mostly due to the nonlinear dynamic behavior of structures. One of the main causes of collapse in vibrating mechanical systems is the so-called fluttering, mathematically formulated by limit cycles, which linear models are unable to describe. The Van der Pol (VdP) equation is an example of a model that satisfactorily describes vibrations with nonlinear damping. However, it is known that this type of equation does not have an analytical solution, because of that the proper analyses is more difficult. In this sense, the Krylov-Bogoliubov (K-B) method presents an approximate solution for certain differential equations, in which VdP can be included, that contains dynamic information pertinent to the model, including indicating the characteristic of limit cycles, unlike the solutions obtained by numerical integration. Therefore, this project seeks to analyze the dynamics of nonlinear vibrations modeled by the Van der Pol equation using the K-B method, confronting the solutions with those obtained numerically. As an alternative to analyze the behavior of the VdP, it was initially sought a tool in which it was not necessary to know its solution. In this case, the system was linearized, as of the conditions required by the Hartman-Grobman theorem, which guarantees the validity of linearization of dynamical systems with respect to hyperbolic points. For a better analysis of particular solutions, the fourth-order Runge-Kutta (R-K) method was used to approximate the time response curves. Finally, the K-B method and the refined K-B method were used to find expressions for the VdP unknown function. Comparing each of the approaches, it was noted that they all agreed with each other, according to the scope of themselves. The regimes analyzed by the linearized system could be observed in the numerical solutions, however, for values farther from the equilibrium point, the presence of limit cycles is noticeable. Moreover, the K-B method presented considerable similarity with the other approaches, expressed by low values of mean square error in relation to the solutions by R-K.
Keywords
nonlinear vibrations, Krylov-Bogoliubov method, Van der Pol oscillator, limit cycle oscillation, Differential equations
