FRACTIONAL DUFFING'S EQUATION AND GEOMETRICAL RESONANCE

We investigate the Fractional Duffing equation in the presence of nonharmonic external perturbations. We have applied the concept of Geometrical Resonance to this equation. We have obtained the conditions that should be satisfied by the external driving forces in order to produce high-amplitude periodic oscillations avoiding chaos. We also show that, for Duffing's equation with fractional damping, the perturbations that satisfy the Geometrical Resonance conditions are nonperiodic functions.

Ninety years of Duffing’s equation

In the paper the origin of the so named ?Duffing?s equation? is shown. The author?s generalization of the equation, her published papers dealing with Duffing?s equation and some of the solution methods are presented. Three characteristic approximate solution procedures based on the exact solution of the strong cubic Duffing?s equation are shown. Using the Jacobi elliptic functions the elliptic-Krylov-Bogolubov (EKB), the homotopy perturbation and the elliptic-Galerkin (EG) methods are described. The methods are compared. The advantages and the disadvantages of the methods are discussed.

Estimating Impulsive Loads in Duffing's Equation Using Two Methods

This work determines the time-varying impulsive loads, called inputs, in a nonlinear system using two novel input estimation inverse algorithms. Both algorithms use the extended Kalman filter with two different recursive estimators to determine impulsive loads. The extended Kalman filter generates the residual innovation sequences. The estimators use the residual innovation sequences to evaluate the magnitudes and, therefore, the onset time histories of the impulsive loads. Based on the two regression equations, a recursive least-squares estimator with a tunable fading factor is called a conventional input estimation with an adaptive weighting fading factor called an adaptive weighting input estimation. Both are used to estimate on-line inputs involving measurement noise and modeling errors. Numerical simulations of a nonlinear system, Duffing’s equation, demonstrate the accuracy of the proposed methods. Simulation results show that the proposed methods accurately estimate impulsive loads, and the AWIE approach has superior robust estimation capability than the CIE method in the nonlinear system.