Dynamic vibration absorber (DVA) with large auxiliary mass has better control performance, but it is also more bulky. Therefore, the mass ratio (the ratio of auxiliary mass of DVA to mass of controlled object) is usually limited to make the DVA easy to install and suitable for engineering practice. In this paper a grounded type DVA with lever component is proposed, which aims to increase the effective mass and reduce unnecessary mass to improve control performance of the DVA. Firstly, the motion differential equations of the DVA are established and solved. Secondly, the optimum parameters are obtained based on H∞ and H2 optimization criterion. Then, the performances of the grounded type DVA equipped with and without the lever are investigated. Finally, the control performance of the DVA is compared with other three typical DVAs under H∞ and H2 criterion. In this type DVA there are no global optimum parameters, and larger frequency ratio will get better control performance. If the amplification ratio (the ratio of lever power arm to lever resistance arm) is greater than 1, the introduced lever will contribute to control performance of the DVA. Its control performance is better than those of other three typical DVAs. The use of the lever can increase the effective mass of the DVA, thereby improving the control performance of the DVA. The DVA can achieve good performance at small mass ratio by adjusting amplification ratio, which may provide theoretical basis for the design of new kinds of DVAs.

A linear single degree-of-freedom (SDOF) oscillator with fractional-order PID controller of acceleration feedback is investigated by the averaging method, and the approximately analytical solution is obtained. Moreover, the numerical solution of the system is obtained by the step-down order method and the power series method progressively. The effects of the parameters in fractional-order PID controller on the dynamical properties are characterized by some new equivalent parameters. The proportional component of fractional-order PID controller is characterized in the form of equivalent mass. The integral component of fractional-order PID controller is denoted in the form of the equivalent linear damping and equivalent mass. The differential component of fractional-order PID controller is denoted in the form of the equivalent linear negative damping and equivalent mass. Those equivalent parameters could distinctly illustrate the effects of the parameters in fractional PID controller on the dynamical response. A comparison between the analytical solution with the numerical results is made, and their satisfactory agreement verifies the correctness of the approximately analytical results. The effects of the parameters in fractional-order PID controller on control performance are further analyzed by some performance parameters of the time response. Finally, the robustness of the fractional-order PID controller based on acceleration feedback is demonstrated through the control of a SDOF quarter vehicle suspension model.

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The complicated dynamic behavior of a fractional-order Duffing system with slow variable parameter excitation is investigated. The stability and bifurcation behavior of the fast subsystem are analyzed by using the dynamic theory of fractional-order systems. The pitchfork bifurcation, Hopf bifurcation and limit cycle bifurcation are discussed in detail, and it was found that Hopf bifurcation only happens while the fractional order is bigger than 1. On the other hand, the influence of the amplitude of parametric excitation on cluster oscillation models is discussed. The results show that amplitude regulates cluster oscillation models with different bifurcation types. The point–point cluster oscillation only relates to pitchfork bifurcation. The point–cycle cluster oscillation includes pitchfork bifurcation and Hopf bifurcation. The point–cycle–cycle cluster oscillation involves three kinds of bifurcation, i.e., the pitchfork bifurcation, Hopf bifurcation and limit cycle bifurcation. The larger the amplitude, the more bifurcation types are involved. The research results of cluster oscillation and its generation mechanism will provide valuable theoretical basis for mechanical manufacturing and engineering practice.