## 1 Introduction  
Smart materials and vibration control technology can suppress vibration of flight vehicles and allow them to operate beyond the traditional flutter boundary, improve ride comfort, and minimize vibration fatigue damage [1]. Due to the controllable and varied physical properties of smart materials, they are also used for living infrastructures, mechanical structures, and seismic vibration controls for decades [2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35]. Smart materials and structures that possess electromechanical characteristics have been widely used for active vibration control or semi-active vibration control, such as piezoelectric damper [4; 5; 6; 7; 8], eddy current damper [9], magnetostrictive spring [10], magneto-rheological fluid (MRF) damper [11; 12; 13], electro-rheological fluid (ERF) damper [14; 15; 16; 17], shape memory alloy (SMA) [18; 19; 20; 21], and electromagnetic and piezoelectric shunt damper [22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35]. Smart materials have one or more properties changed by the external stimuli, such as temperature, stress, electric field or magnetic field. Active vibration control uses piezoelectric ceramics to generate the force or displacement based on the external applied voltage [4; 5; 6; 7; 8; 9]. To suppress the structure's vibration, ERF and MRF dampers are widely used as semi-active element in suspension systems [11; 12; 13; 14; 15; 16; 17]. ER and MR fluids are transformed from the liquid state into the solid state in milliseconds by applying an electric or a magnetic field. Shape memory alloys can change its material properties based on temperature [18; 19; 20; 21]. Since [PERSON] proposed the original dynamic vibration absorber (DVA) in a decade, DVAs are successfully applied in the vibration control of civil structures, naval architectures, and aerospace [36; 37]. The DVA consist of a stiffness element, a damping component, and an oscillator. The working principle of the DVA is to transfer the vibration energy to the oscillators by a specified design for a certain frequency range. Because the effective working frequency span of DVAs is narrow only for a certain designed frequency range, [PERSON] et al. considered the optimal tuning of the absorber parameters including damping to improve the DVA's design [38]. Furthermore, the optimal determinations of parameters were investigated for the absorber with nonlinear damping and different loading characteristics in the following studies [39]. Among the above studies, the absorbers' parameters are constant and time invariant; therefore, the vibration absorber is passive. To improve the performance of the passive vibration absorbers, many researchers investigated semi-active and active designs [40; 41]. The semi-active absorbers have the capability of tuning their working resonant frequencies by adjusting the stiffness and/or damping of the absorbers. One of the adjustments can be realized either through the active change of geometry or through the material properties of the constituents. Thus, they are called actively tuned vibration absorbers (ATVAs) [42]. The other types of ATVA are made of the smart materials whose physical properties can be controlled by the external input. Among all ATVA designs, the ones involved with the use of smart materials, such as piezoelectric materials [4; 5; 6; 7; 8; 9], magnetorheological elastomer (MRE) [12], electrorheological fluid (ERF) [16; 17], and shape memory alloy and shape memory polymer [19], are frequently seen. ATVAs have advantages of being more stable in system performance and simpler in controller design. Since [PERSON] discovered the electrorheological effect [43], the reversible and controllable material properties make ERF attractive for many applications, such as vibration dampers, shock absorbers, clutches, and valves [44; 45; 46; 47; 48; 49].