There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Assessing CO2 emissions in China's iron and steel industry: A dynamic vector autoregression model

Analytical study of a quasi-zero stiffness coupling using a torsion magnetic spring with negative stiffness

Large amplitude vibration of fractionally damped viscoelastic CNTs/fiber/polymer multiscale composite beams

In this paper, the primary resonance of Duffing oscillator with two kinds of fractional-order derivatives is investigated analytically. Based on the averaging method, the approximately analytical solution and the amplitude–frequency equation are obtained. The effects of the two kinds of fractional-order derivatives on the system dynamics are analyzed, and it is found that these two kinds of fractional-order derivatives could affect not only the linear viscous damping, but also the linear stiffness, which could be characterized by the equivalent damping coefficient and the equivalent stiffness coefficient. The different effects are analyzed based on these two deduced equivalent parameters, when the two fractional orders are limited in the typical intervals, i.e. p1∈[0 1] and p2∈[1 2]. Moreover, the comparisons of the amplitude–frequency curves obtained by the approximately analytical solution and the numerical integration are fulfilled, and the results certify the correctness and satisfactory precision of the approximately analytical solution. Especially, the effects of the parameters in the second kind of fractional-order derivative are studied when the coefficient of the first kind of fractional-order derivative is zero or not. At last, two special cases for the coefficient of the second kind of fractional-order derivative are analyzed, which could make engineers obtain satisfactory vibration control performance and keep the frequency characteristic almost unchanged. These results are very useful in vibration control engineering.

Primary resonance of Duffing oscillator with two kinds of fractional-order derivatives

Primary resonance of Duffing oscillator with fractional-order derivative

In this paper, the incremental harmonic balance (IHB) method is extended to analyze the dynamical properties of fractional-order nonlinear oscillator, and the general forms of the periodic solutions for the oscillator are founded based on IHB method, which is useful to obtain the solutions with higher precision. The fractional-order Duffing oscillator is selected at first to test the feasibility of IHB method for this kind of system. The comparisons between the numerical results with the approximate analytical solutions by IHB method and averaging method verify the correctness and higher precision of IHB method. Then the nonlinear dynamics of fractional-order Mathieu–Duffing oscillator is researched by IHB method, where the coupling effects of the two kinds of typical nonlinearities are especially analyzed. Moreover, the influences of the system parameters in the fractional-order derivative on the amplitude–frequency curves are also researched by IHB method. At last, the detailed results are summarized and the conclusions are made, which presents some useful information to analyze and/or control the dynamical response of this kind of system.

Dynamical analysis of fractional-order nonlinear oscillator by incremental harmonic balance method

A new type of dynamic vibration absorber (DVA) with negative stiffness is studied in detail. At first, the analytical solution of the system is obtained based on the established differential motion equation. Three fixed points are found in the amplitude-frequency curves of the primary system. The design formulae for the optimum tuning ratio and optimum stiffness ratio of DVA are obtained by adjusting the three fixed points to the same height according to the fixed-point theory. Then, the optimum damping ratio is formulated by minimizing the maximum value of the amplitude-frequency curves according to optimization principle. According to the characteristics of negative stiffness element, the optimum negative stiffness ratio is also established and it could still keep the system stable. In the end, the comparison between the analytical and the numerical solutions verifies the correctness of the analytical solution. The comparisons with three other traditional DVAs under the harmonic and random excitations show that the presented DVA performs better in vibration absorption. This result could provide theoretical basis for optimum parameters design of similar DVAs.

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Parameters Optimization for a Kind of Dynamic Vibration Absorber with Negative Stiffness

In this paper the optimal control and parameters design of fractional-order vehicle suspension system are researched, where the system is described by fractional-order differential equation The li

https://journals.sagepub.com/doi/pdf/10.1177/0263092317717166

Xing Haijun

Papers

Primary resonance of Duffing oscillator with fractional-order derivative

Primary resonance of Duffing oscillator with two kinds of fractional-order derivatives

Dynamical analysis of fractional-order nonlinear oscillator by incremental harmonic balance method

Parameters Optimization for a Kind of Dynamic Vibration Absorber with Negative Stiffness

Optimal control and parameters design for the fractional-order vehicle suspension system: