# Shock Vibration Control of SDOF Systems with Tubular Linear Eddy Current Dampers
## 4 Transient Response Analysis
### Influence of Shock Impulse Characteristics and ECD Mechanical Parameters  
In this section, various peak responses are mainly analyzed instead of the whole-time histories of the SDOF system with various ECDs when subjected to different forms of impulsive excitation. Maximum structural deformation is typically a crucial design criterion in the design of dampers for shock vibration control, and the transient response analysis is useful for the performance evaluation of ECDs. The intended maximum displacement of the SDOF system with the ECD is, therefore, set to be approximately 15% of the intended maximum displacement of the case without a damper. In this section, different values of the dimensionless critical relative velocity \(\gamma\) have been considered in the peak response analysis, and the dimensionless maximum damping force \(\beta\) is an independent design variable obtained through multiple iterations to reach the intended maximum displacement. For achieving the target maximum displacement, other responses are discussed, including the maximum damper force \(f_{0}\), the time to maximum displacement \(t_{0}\), and the time required for decaying to one-third of the intended maximum displacement \(t_{1/3}\). The last item is approximately 5% of the maximum displacement of the system without a damper, and the reason to discuss it is that sometimes it is the threshold value to use the devices repeatedly in some cases. Meanwhile, as mentioned before, shock excitation is determined by its maximum value, duration, and shape. Figure 3 depicts the three types of shock excitation used in this section including the rectangular shock, initial peak triangular shock, and half-cycle  
Figure 5: Time\(-\)histories for displacement, velocity, acceleration, and energy of the SDOF system subjected to half-cycle sine shock: (**a**) Without damper; (**b**) With an ECD. sine shock. The duration of each shock \(t_{d}\) is set to be 10 milliseconds so that the ratio of the shock duration to the period of the structure \(t_{d}/T_{n}=0.2<0.25\), which satisfies the definition of an impulse. For the peak acceleration of each shock, it is adjusted so that the impulse is the same for each shock. Given that the peak force for each shock is \(p_{0}\), the impulses for the three forms of shock excitation are \((2/\pi)p_{0}t_{d}\), \(p_{0}t_{d}/2\), and \(p_{0}t_{d}\). Similarly, the corresponding peak acceleration is adjusted as shown in Table 2.  
However, when the critical relative velocity and the maximum damping force are dimensionless, the peak acceleration values of the three forms of loadings are considered the same at 30 g to reflect the same impulse. Figure 6 shows the required dimensionless damping force of the ECD and responses of the SDOF system with an ECD with a varying dimensionless critical relative velocity subjected to half-cycle sine, initial peak triangular, and the rectangular shock when reaching the target maximum displacement. From Figure 6a, it can be seen that there is a minimal dimensionless maximum damping force for all shock loadings, and the system with the ECD subjected to rectangular shock always requires the minimum value. In general, the larger maximum damping force of the ECD represents a larger dimension and a higher cost, and therefore, the optimal value of the dimensionless maximum damping force is significant. Similarly, a minimal time of the target maximum displacement can also be noted in Figure 6c, and the values are always the largest and the smallest for rectangular and initial peak triangular shock, respectively. This implies that the ECD can be designed for optimal control when reaching the target maximum displacement.