Model modification and feature study of Duffing oscillator

With the development of chaos theory, Duffing oscillator has been extensively studied in many fields, especially in electronic signal processing. As a nonlinear oscillator, Duffing oscillator is more complicated in terms of equations or circuit analysis. In order to facilitate the analysis of its characteristics, the study analyzes the circuit from the perspective of vibrational science and energetics. The classic Holmes-Duffing model is first modified to make it more popular and concise, and then the model feasibility is confirmed by a series of rigorous derivations. According to experiments, the influence of driving force amplitude, frequency, and initial value on the system is finally explained by the basic theories of physics. Through this work, people can understand the mechanisms and characteristics of Duffing oscillator more intuitively and comprehensively. It provides a new idea for the study of Duffing oscillators and more.

An analytical approach to obtain exact solutions of some space-time conformable fractional differential equations

In this paper, the sine-Gordon expansion method is used to obtain analytical solutions of the conformable space-time generalized reaction Duffing model and conformable space-time Eckhaus equation with the aid of symbolic computation. These equations can be reduced into ordinary differential equations (ODEs) using a suitable wave transformation with a predicted polynomial-type solution.

A Modified Matlock–Duffing Model for Two-Dimensional Ice-Induced Vibrations of Offshore Structures With Geometric Nonlinearities

Oil and gas exploration and production have been expanding in Arctic waters. However, numerical models for predicting the ice-induced vibrations (IIV) of offshore structures are still lacking in the literature. This study aims to develop a mathematical reduced-order model for predicting the two-dimensional IIV of offshore structures with geometric coupling and nonlinearities. A cylindrical structure subject to a moving uniform ice sheet is analyzed using the well-known Matlock model, which, in the present study, is extended and modified to account for a new empirical nonlinear stress–strain rate relationship determining the maximum compressive stress (MCS) of the ice. The model is further developed through the incorporation of ice temperature, brine content, air volume, grain size, ice thickness, and ice wedge angle effects on the ice compressive strength. These allow the effect of multiple ice properties on the ice–structure interaction to be investigated. A further advancement is the inclusion of an equation allowing the length of failed ice at a point of failure to vary with time. A mixture of existing equations and newly proposed empirical relationships is used. Structural geometric nonlinearities are incorporated into the numerical model through the use of Duffing oscillators, a technique previously proposed in vortex-induced vibration studies. The model is validated against results from the literature and provides new insights into IIV responses including the quasi-static, randomlike chaotic, and locked-in motions, depending on the ice velocity and system nonlinearities. This numerical Matlock–Duffing model shows a potential to be used in future IIV analysis of Arctic cylindrical structures, particularly fixed offshore structures, such as lighthouses, gravity bases, and wind turbine monopiles.

Exact solutions of some fractional differential equations arising in mathematical biology

In the last decades Exp-function method has been used for solving fractional differential equations. In this paper, we obtain exact solutions of fractional generalized reaction Duffing model and nonlinear fractional diffusion–reaction equation. The fractional derivatives are described in the modified Riemann–Liouville sense. The fractional complex transform has been suggested to convert fractional-order differential equations with modified Riemann–Liouville derivatives into integer-order differential equations, and the reduced equations can be solved by symbolic computation.

Fractional sub-equation method for the fractional generalized reaction Duffing model and nonlinear fractional Sharma-Tasso-Olver equation

In this paper the fractional sub-equation method is used to construct exact solutions of the fractional generalized reaction Duffing model and nonlinear fractional Sharma-Tasso-Olver equation.The fractional derivative is described in the Jumarie’s modified Riemann-Liouville sense. Two illustrative examples are given, showing the accuracy and convenience of the method.