# Shock Vibration Control of SDOF Systems with Tubular Linear Eddy Current Dampers
## 4 Transient Response Analysis
### Comparison of the ECD and FVD in Shock Vibration Control  
In addition to the influence of the mechanical parameters of the ECD on structural responses, performance evaluation and comparison of ECDs with FVDs in shock vibration control are also necessary, since the latter is the most widely used energy dispassion device. The force-velocity relationship of the FVD can be expressed as (14), in which \(c_{a}\) and \(\alpha\) are the damper constant and the damper exponent, respectively. However, different from the steady-state response of the structure under constant harmonic excitation [38], the concept of equal energy dissipation for the structural response during impulsive loading is pointless, so it is hard to directly compare the performance of the FVD and ECD for shock vibration control. As a result, by introducing two target responses, the maximum displacement and the maximum damper force (which are significant in structural responses), the mechanical parameters of the FVD (the damper constant and the damper exponent) and ECD (the maximum damping force and the critical relative velocity) can be determined. In this section, the target maximum displacement is set to be the same as that in Section 4.3. Given that the damper exponent of the FVD \(\alpha\) usually ranges from 0.35 to 1.00 for vibration control [39], the target maximum damper force of the ECD is taken to be the same as that of the FVD when the damper exponent is 0.5 and when the maximum displacement of the SDOF system with the FVD reaches the target maximum displacement. Because of the same maximum displacement and damper force, after determining the mechanical parameters of the ECD and the FVD, other responses, including the time of maximum displacement and the time required for decaying to one-third of the target maximum displacement, will be discussed. \[F_{FVD}=c_{a}sgn\left(\dot{u}\right)\left|\dot{u}\right|^{\alpha} \tag{14}\]  
Figure 7 shows the comparison of the two time indexes and two energy indexes when the system is equipped with the FVD and the ECD, respectively. It can be observed from Figure 7a,b that at the same maximum displacement and damper force, the SDOF system with the ECD requires a slightly longer time to reach the target maximum but much less time to decay to one-third of the target maximum displacement. This means that the ECD is more advantageous in some design cases where the structure requires to return to the initial state. From an energy perspective, Figure 7c,d illustrates that compared to the FVD, the ECD slightly reduces the energy input of the system and also dissipates less energy. This is particularly true when the system is subjected to an initial peak triangular shock. The displacement time history of the SDOF systems with an ECD and an FVD subjected to half-cycle sine shock is shown in Figure 8a, and the differences in the two time indexes between the ECD and the FVD can be clearly seen. From Figure 8b, it can be observed that when the velocity is less than 0.27 m/s, the ECD provides the lower damper force, although it provides the higher damping force when the velocity is higher than 0.27 m/s. Moreover, the shock vibration control performance of the ECD and the FVD under seismic excitation is also considered in this study. Figure 9 shows the acceleration time histories of the EW El Centro wave, and Figure 10 shows the displacement time histories of the SDOF system in three conditions, i.e., without damper, with an FVD, and with an ECD. In this case, the damper constant and the damper exponent of the FVD are 12 kN\(\cdot\)s/m and 0.7, respectively. The parameters of the ECD are determined by the design method in [38], with \(v_{cr}=0.786v_{max}=0.29\) m/s and \(F_{max}=c_{a}\left|v_{max}\right|^{\alpha}\), where \(v_{max}\) is the maximum velocity of this SDOF system with a FVD under seismic excitation. Considering that the power spectral densities of the ground motions are larger between 1-2 Hz, the natural period of the structure is adjusted to 10 s to meet the definition of impulsive loading.