# Shake Table Testing of Standard Cold-Formed Steel Storage Rack
## 4 Damping of the System  
Another important dynamic characteristic is the structure's damping, which could be a significant source of energy dissipation. The damping of rack structures attributes to the dissipated energy of the rack components due to yielding and deformations as well as the friction in connections and the sliding of the pallets. However, it is hardly possible to identify or mathematically describe each of these energy dissipation mechanisms in an actual building [46]. Limited investigations on the damping of rack structures have been reported so far from previous shake table tests. Previous studies emphasized that the effect of the amplitude of motion on the damping ratio is significant, and the structural damping is affected by the looseness of the connections. However, a damping ratio for the seismic design of rack structures has not yet been proposed with a high level of confidence. However, proposing a precise damping ratio is beyond the scope of this study and indeed needs more in situ testing of loaded and unloaded racks. The damping characteristic of structures and buildings is mathematically known as the damping ratio (\(\xi\)), which is the normalized damping coefficient of the structure with respect to its critical damping. The fundamental period of vibration, T, and its corresponding damping ratio are the main parameters to predict the behaviour of buildings and structures under given dynamic loads. One of the few practical ways to determine the fundamental period of a structure and the damping ratio related to that period is by obtaining the frequency-response curve experimentally. This task can be carried out by performing sinusoidal sweep testing (a forced vibration test) using sinusoidal (harmonic) loading over a range of frequencies. The concept is to excite the structure with harmonic loading such that at certain periods (periods of vibration, T\({}_{i}\)) the structure experiences resonance. Damping ratios corresponding to each frequency of vibration can be determined by using the half power band method, as displayed in Figure 18.  
Then, the damping ratio can be calculated by the equation below:  
\[\xi(\%)=\frac{f_{b}-f_{a}}{2f_{n}} \tag{1}\]  
where \(f_{n}\) is the forcing frequency at resonance and \(f_{a}\) and \(f_{b}\) are the calculated frequencies at a response amplitude of \(\
icefrac{{1}}{{\sqrt{2}}}\) of the peak response amplitude [47]. As mentioned earlier, after every step of the shake table tests, the structure was subjected to a sine sweep wave with an acceleration amplitude of \"0.02g\" in order to evaluate the damping ratio and vibration period elongations due to cumulative damage in the system. Table 5 shows the damping ratios as well as the fundamental vibration period of the system calculated from the sine sweep tests conducted after each of the test steps for both non-scaled and scaled time-domain conditions. Damping ratios were then calculated by applying the half-power band method on the sine sweep test results. The results of this table show an increase in the damping ratios after each increment of the shake table  
Figure 17: Failure at beam to upright connection. test with increasing seismic intensity. The corresponding increase in the amount of lateral movement of the frame is basically the result of the interactions of the hooks in the beam end connectors and upright slots. As the displacement amplitude in the scaled tests in the time domain was smaller than in the non-scaled tests, lower damping ratios were obtained for the former condition. Figure 19 shows the frequency-response curves of the sine sweep test results used to obtain the damping ratio as well as the fundamental period of vibration. For the non-scaled tests, the period was visibly changed due to the apparent damage of the system in dissipative zones (e.g., connections), and the system lost stiffness. The increase in the damping ratio due to enhancing the intensity of the earthquake can also be seen in the graphs by focusing on the width of the curves that were broadened.