# Shock Vibration Control of SDOF Systems with Tubular Linear Eddy Current Dampers
## 1 Introduction  
Structural vibration control methods and devices have been well-developed to suppress vibration [1; 2; 3]. Unlike force loadings with a relatively long duration, impulse loadings are a special form of loading that can be applied to a structure in a very short period. In general, when the ratio of the duration of loading to the natural period \(td/T0<0.25\), the loading can be defined as an impulse [4; 5]. In reality, impulsive loadings can be generated from many sources, including free-fall impacts, collisions, explosions, short-duration pulse-type seismic excitations, aircraft landing, braking loads, and many more. Although the duration of impulse loadings is relatively short, the peak forces exerted on the structure are usually quite high. As a result, the loadings can generate high responses and damage the structure consequently. This means that the design of the structure or the corresponding measures to prevent shock is crucial. In order to alleviate high responses caused by shockpulses, a wide range of control devices have been proposed and analyzed. Based on the stability maximization criterion [6], various types of dynamic vibration absorbers have been designed to decay transient responses [7; 8; 9]. Moreover, some nonlinear devices [10] and semi-active [11] or active control systems [12; 13] were also proposed for shock vibration control. Compared to the complicated vibration control system, it is simpler and more direct to exert resistance forces on the structure by dampers to reduce structural transient responses caused by impulsive loadings. Hence, it is necessary to calculate and analyze structural transient responses under shock excitation. Responses of the structure without dampers or with just linear dampers can be easily calculated when subjected to simple loading forms, such as sinusoidal loading and rectangular loading [5; 14]. Moreover, the responses of the structure with nonlinear fluid viscous dampers (FVDs) under impulsive loadings have been analyzed [15; 16]. Recently, the eddy current damper (ECD) has gained growing interest in the field of structural vibration suppression [17; 18] and auxiliary braking systems [19]. The eddy current damper mainly consists of a stator and a relatively movable part (i.e., permanent magnets and the conductor), and the damping force is actually the Lorentz force generated from the conductor by cutting the magnetic lines of flux. If the conductor is replaced by the closed conductive coil, the ECDs will transfrom into electromagnetic dampers [20]. In the ECD, the permanent magnets (PMs) and the conductor are contactless, which implies that wear and fatigue can be avoided to result in less maintenance and a longer lifetime. In contrast to the conventional FVD, another merit of the ECD is that it does not need a working fluid, and thus, the ECD does not suffer from fluid-related issues, including leaks, degradation, and high temperature. In eddy current tuned mass dampers (ECTMDs), the ECD can function as a planar damping component [17; 18; 21], or it can directly act on the structure in the forms of a rotary type [22; 23] and linear type [24]. When the relative velocity between the PMs and the conductor is not high, the damping force of the ECD is nearly linear with the change in velocity [21]. This usually happens in the ECD without speed amplification devices such as the EC-TMD, or when the external excitations are gentle. However, if the relative velocity is amplified by the ball-screw mechanism [25] and rack and gear [26], or when external excitations are intense as mentioned above, the relative velocity can be quite high and the ECD shows strong nonlinearity in the force-velocity relationship due to the induced magnetic field resisting original magnetic field [27].