## 3 Dynamic Response for an Accessory Structure Equipped on the Primary Structure
### Parameter Sensitivity Analysis of the Theoretical Model without Considering Excitation \(P_{2}\)  
According to the code for the seismic design of buildings, the damping ratio of large-scale concrete structures is generally about 0.05, and that of accessory structures which are made of steel and lower than 50 m is considered equal to 0.04. Moreover, the mass ratio \(m_{12}\) is considered equal to 50,000, which indicates that the mass of the accessory structure is far less than that of the primary structure. Based on the aforementioned parameter values, Figure 10 gives the frequency response for the subsidiary structure of the two-degrees-of-freedom system shown in Figure 9 with varying parameter \(\omega_{12}\). It is noted that there are two peaks in the amplitude coefficient curves in Figure 10 when the parameter \(\omega_{12}\
eq 1\), and there is only one peak when parameter \(\omega_{12}=1\). Therefore, it is understandable that the two peaks coincide with each other when the natural frequency of the primary structure equals that of accessory structure. As illustrated in Figure 10, the peak corresponding to the lower external excitation frequency is always higher than the rest peak. Generally speaking, the vibration frequency of the accessory structure is usually higher than that of the large-scale primary structure and the value of \(\omega_{12}\) is considered to be 0.1 in the following analysis. Therefore, it is considered that the dynamic response of the subsidiary structure in the case of \(\omega_{0}=\omega_{1}\) is greater than that in the case of \(\omega_{0}=\omega_{2}\). The possible reason for this is that the dynamic response of the subsidiary structure is significantly amplified by the resonance of the primary structure when \(\omega_{0}=\omega_{1}\) and the amplification effect is not so obvious when \(\omega_{0}=\omega_{2}\), because the external force does not act directly on the accessory structure.  
As shown in Figure 11, the influence of the damping ratio \(\xi_{2}\) on the dynamic response of the accessory structure was investigated. It is noted that the peak value decreases with increasing damping ratio \(\xi_{2}\) in the case of \(\omega_{0}=\omega_{2}\) (i.e., \(\omega_{01}=10\)), and the amplitude coefficient slightly increases with increasing \(\xi_{2}\) at the end of the curves. Moreover, the varying parameter \(\xi_{2}\) has no influence on the first peak of the curves, which indicates that the maximum possible value for the amplification factor is not reduced.  
As illustrated in Figure 12, the influence of the damping ratio \(\xi_{1}\) on the accessory structure's dynamic response is investigated. The first peak value significantly decreases with increase of the damping ratio \(\xi_{1}\) in the case of \(\omega_{0}=\omega_{1}\) (i.e., \(\omega_{01}=1\)), and the second peak value slightly decreases with increasing \(\xi_{1}\) when \(\omega_{0}=\omega_{2}\) (i.e., \(\omega_{01}=10\)). Therefore, the peak dynamic response of the accessory structure will be effectively reduced when the primary structure's damping ratio increases and the values of the parameters \(\omega_{12}\), \(\xi_{2}\) and \(m_{12}\) remain unchanged.  
Figure 11: Frequency response for the subsidiary structure with varying parameter \(\xi_{2}\). (**a**) the entire graph; (**b**) partial enlarged graph.  
Figure 10: Frequency response of the subsidiary structure with varying parameter \(\omega_{12}\).  
According to the analysis, the mass ratio \(m_{12}\) has no influence on the accessory structure's dynamic response under the condition of \(\omega_{12}=0.1\), \(\xi_{1}=0.05\) and \(\xi_{2}=0.04\). To save space, the figure for the frequency response of the subsidiary structure with varying parameter \(m_{12}\) is not given.