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In this paper the optimal parameters of the fractional-order Voigt type dynamic vibration absorber (DVA) are analytically studied for two cases, named as H∞ and H2 optimization criteria. At first the approximately analytical solution is obtained by the averaging method when the primary system is subjected to harmonic excitation. Then the optimal fractional coefficient and order are obtained based on H∞ optimization criterion, which is designed to minimize the maximum amplitude magnification factor of the primary system. Based on H2 optimization criterion, the optimal fractional parameters are obtained to reduce the total vibration energy of the primary system over the whole-frequency range. The comparisons of the approximate solutions with the numerical ones in the two cases are fulfilled, and the results verify that the approximately analytical solutions are correct and satisfactorily precise. At last the control performance of the fractional-order Voigt type DVA is compared with the classical integer-order counterpart, and it could be concluded that the fractional-order DVA has superiority in vibration engineering, and fractional-order element could replace the traditional damper and spring simultaneously in some cases.

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.

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 toH∞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.

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 linear quadratic optimal state regulator is designed based on optimal control theory, which is applied to get the optimal control force of the active fractional-order suspension system. A stiffness-damping system is added to the passive fractional-order suspension system. Based on the criteria, i.e. the force arising from the accessional stiffness-damping system should be as close as possible to the optimal control force of the active fractional-order suspension system, the parameters of the optimized passive fractional-order suspension system are obtained by least square algorithm. An Oustaloup filter algorithm is adopted to simulate the fractional-order derivatives. Then, the simulation models of the three kinds of fractional-order suspension systems are developed respectively. The simulation results indicate that the active and optimized passive fractional-order suspension systems both reduce the value of vehicle body vertical acceleration and improve the ride comfort compared with the passive fractional-order suspension system, whenever the vehicle is running on a sinusoidal surface or random surface.

In this paper, the computation schemes for periodic solutions of the forced fractional-order Mathieu-Duffing equation are derived based on incremental harmonic balance (IHB) method. The general forms of periodic solutions are founded by the IHB method, which could be useful to obtain the periodic solutions with higher precision. The comparisons of the approximate analytical solutions by the IHB method and numerical integration are fulfilled, and the results certify the correctness and higher precision of the solutions by the IHB method. The dynamical analysis of strongly nonlinear fractional-order Mathieu-Duffing equation is investigated by the IHB method. Then, the effects of the excitation frequency, fractional order, fractional coefficient, and nonlinear stiffness coefficient on the complex dynamical behaviors are analyzed. At last, the detailed results are summarized and the conclusions are made, which present some useful information to analyze and/or control the dynamical response of this kind of system.

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.

In this paper a single degree-of-freedom semi-active oscillator with time delay is researched. By averaging method, the first-order approximately analytical solution is obtained, and the stability condition is also established based on the Lyapunov theory. The analytical results show that the amplitude and the stability condition of the steady-state solution are all periodic functions of time delay, with the same period as the excitation one. Moreover, another simple case, namely the semi-active oscillator without time delay, is also investigated based on the first-order approximately analytical solution, and the result shows that the steady-state solution in this case is unconditionally stable. The comparisons of the analytical solution and the numerical one are fulfilled, and the results verify the correctness and satisfactory precision of the first-order approximately analytical solution. At last, the selection or design of an appropriate time delay to improve control performance through the first-order approximately analytical solution is studied.

In this paper the primary resonance of Duffing oscillator with fractional-order derivative is researched by the averaging method. At first the approximately analytical solution and the amplitude–frequency equation are obtained. Additionally, the effect of the fractional-order derivative on the system dynamics is analyzed, and it is found that the fractional-order derivative could affect not only the viscous damping, but also the linear stiffness, which is characterized by the equivalent damping coefficient and the equivalent stiffness coefficient. This conclusion is remarkably different from the existing research results about nonlinear system with fractional-order derivative. Moreover, the comparisons of the amplitude–frequency curves 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. At last, the effects of the two parameters of the fractional-order derivative, i.e. the fractional coefficient and the fractional order, on the amplitude–frequency curves are investigated, which are different from the traditional integer-order Duffing oscillator.