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Fractional-order derivatives provide a powerful tool for the characterization of mechanical properties of viscoelastic materials. Fractional oscillators are mechanical models of viscoelastically damped structures, the viscoelastic damping of which is described by fractional-order constitutive equations. This paper proposes sliding mode control for a two-degree-of-freedom fractional Zener oscillator. Firstly, a virtual fractional oscillator is generated by means of a state transformation. Then, the total mechanical energy in the virtual oscillator is determined as the sum of the kinetic energy, the potential energy, and the fractional energy. Furthermore, sliding mode control for the fractional Zener oscillator is designed, in which the Lyapunov function is defined by the total mechanical energy. Finally, numerical simulations are conducted to validate the effectiveness of the proposed controllers.

Hereditary effects of exponentially damped oscillators with past histories are considered in this paper. Nonviscously damped oscillators involve hereditary damping forces which depend on time-histories of vibrating motions via convolution integrals over exponentially decaying functions. As a result, this kind of oscillators are said to have memory. In this work, initialization for nonviscously damped oscillators is firstly proposed. Unlike the classical viscously damped ones, information of the past history of response velocity is necessary to fully determine the dynamic behaviors of nonviscously damped oscillators. Then, initialization response of exponentially damped oscillators is obtained to characterize the hereditary effects on the dynamic response. At last, stability of initialization response is proved and the hereditary effects are shown to gradually recede with increasing of time.

This paper presents mechanical energy and equivalent viscous damping for a single-degree-of-freedom fractional Zener oscillator. Differential equation of motion is derived in terms of fractional Zener constitutive equation of viscoelastic materials. A virtual fractional oscillator is generated via a state transformation. Then, based on the diffusive model for fractional integrators, the stored energy in fractional derivatives with orders lying in (0, 1) and (2, 3) is determined. Thus, the total mechanical energy in the virtual oscillator is determined. Finally, fractional derivatives are split into three parts: the equivalent viscous damping, equivalent stiffness, and equivalent mass. In this way, the fractional differential equation is simplified into an integer-order differential equation, which is much more convenient to handle in engineering.

Vibration control of two-degree-of-freedom (2DOF) exponentially damped oscillators is considered. Firstly, the equations of motion for 2DOF exponentially damped oscillators are established by using the nonviscous damping models that depend on the past history of motion. Secondly, the equations of motion for 2DOF exponentially damped oscillators are transformed into state equations with six dimensions by using a set of internal variables. Thirdly, Sliding control design is proposed. Furthermore, this control method can also be used in the multi-degree-of-freedom (MDOF) exponentially damped oscillators. Finally, numerical simulations are provided to illustrate the above control designs.

The optimal control problems of exponentially damped system are considered in this paper. Exponentially damped system involves convolution integrals over exponentially decaying functions, which is used as damping models in viscoelastically damped structures. The dynamic constraint is in the form of a differential equation that includes integer derivatives and the “integral” term. The performance index considered is a function of both the state and the control variables. The traditional state-space approach is extended to the optimal control problems of exponentially damped system, A direct numerical technique is used to solve the resulting equations. Numerical simulations are provided to illustrate the above control design.