## 3 Theoretical Analyses
### Parameter Sensitivity Analyses
The peak of \(\omega_{0}=\omega_{1}\) mentioned above has entirely disappeared, which indicates that the vibration of the accessory structure has totally separated from that of the primary structure and becomes an independent FV. Moreover, the peak value of \(\omega_{0}=\omega_{3}\) marginally rises with a larger \(\lambda_{12}\). By the discrimination of the above three situations, the dynamic response with a various parameter has been adequately analyzed. Generally speaking, with the increase of \(\lambda_{12}\), AV decreases until it disappears. As \(\lambda_{01}\) increases, the first peak roughly matches \(\omega_{0}=\omega_{1}\) and the peak value significantly decreases with a rising \(\lambda_{12}\), and disappears when \(\lambda_{12}>5\). Moreover, the second peak initially corresponds to \(\omega_{0}=\omega_{2}\), but when the value of \(\lambda_{12}\) approaches that of \(\lambda_{13}\), the abscissa of this peak remains almost unchanged with the variation of \(\lambda_{12}\) and gradually approaches \(\omega_{0}=\omega_{3}\). As illustrated in Figure 12, the influence of the parameter \(\lambda_{13}\) on the dynamic response is investigated. Figure 12a shows the dynamic response of the accessory structure in the case of \(\lambda_{12}=0.5\). It is noteworthy that the peak value is remarkably increased with an increasing \(\lambda_{13}\) in the case of \(\omega_{0}=\omega_{1}\), denoting the dominance of AV; the other peak originally corresponds to \(\omega_{0}=\omega_{3}\) while \(\lambda_{13}<\lambda_{12}=0.5\), but becomes asymptotic to \(\omega_{0}=\omega_{2}\) as \(\lambda_{13}\) gets larger. Moreover, Figure 12b,c are detailed to obtain more specific analyses for the influence of \(\lambda_{13}\) on the dynamic response with different \(\lambda_{12}\). It is evident that similar conclusions can be drawn from Figure 12a,b. For the one-peak situation (Figure 12c), the peak of \(\omega_{0}=\omega_{3}\) abides by the same law as that in Figure 12a, before eventually becoming asymptotic to \(\omega_{0}=\omega_{2}\) when \(\lambda_{13}>\lambda_{12}\). The phenomenon above can be interpreted as the growth of AV with the increase of \(\lambda_{13}\). When \(\lambda_{13}\) is increased, which means that the natural frequency of the accessory structure in the independent FV system (as an independent structure) has been decreased. It can be inferred that the influence of AV will become salient. Figure 12: The dynamic response of the accessory structure with varying parameter \(\lambda_{13}\). (**a**) \(\lambda_{12}=0.5\); (**b**) \(\lambda_{12}=0.1\); (**c**) \(\lambda_{12}=10\). Combined with the analyses of \(\lambda_{12}\), it can be concluded that \(\lambda_{12}\) is a parameter determining the number of peaks, and \(\lambda_{13}\) is a parameter determining peak values. Both \(\lambda_{12}\) and \(\lambda_{13}\) determine the vibration type. AV is dominant when \(\lambda_{12}<\lambda_{13}\); as \(\lambda_{12}/\lambda_{13}\) gets larger, the vibration gradually becomes the independent FV. The dividing point is \(\lambda_{12}=\lambda_{13}\) from the curves. Based on the above analyses, the parameter \(\lambda_{12}\) is considered to be 0.5 in subsequent analyses to take the effects of both AV and independent FV into consideration, which can be seen from Figure 10.  
As illustrated in Figure 13, the influence of the damping ratio \(\xi_{1}\) on the dynamic response is investigated. It can be discovered that the increase of \(\xi_{1}\) suppresses the value of the first peak, but has the opposite effect on the value of the second peak. According to Figure 14, the value of the second peak decreases with the increase of the damping ratio \(\xi_{2}\), but the value of the first peak has not been influenced. Figure 15 illustrates the influence of the damping ratio \(\xi_{3}\) on the dynamic response, which is similar to that of \(\xi_{2}\), but the decrease influence of \(\xi_{3}\) on the second peak is greater.