## 3 Dynamic Response for an Accessory Structure Equipped on the Primary Structure
### Theoretical Model
The steady-state dynamic response for the system can be expressed as:  
\[x_{1}=X_{1}e^{i\omega_{0}t} \tag{5}\]  
\[x_{2}=X_{2}e^{i\omega_{0}t} \tag{6}\]  
Substituting Equations (3)-(6) into Equations (1) and (2), the complex coefficients \(X_{1}\) and \(X_{2}\) can be obtained  
\[X_{1}=\frac{\big{(}-\omega_{0}^{2}m_{2}+i\omega_{0}c_{2}+k_{2}\big{)}F_{1}+ \big{(}i\omega_{0}c_{2}+k_{2}\big{)}F_{2}}{\big{(}-\omega_{0}^{2}m_{2}\big{)} \big{(}i\omega_{0}c_{2}+k_{2}\big{)}+\big{(}-\omega_{0}^{2}m_{2}+i\omega_{0}c _{2}+k_{2}\big{)}\big{(}-\omega_{0}^{2}m_{1}+i\omega_{0}c_{1}+k_{1}\big{)}} \tag{7}\]  
\[X_{2}=\frac{\big{(}i\omega_{0}c_{2}+k_{2}\big{)}F_{1}+\big{(}-\omega_{0}^{2}m _{1}+i\omega_{0}c_{1}+k_{1}+i\omega_{0}c_{2}+k_{2}\big{)}F_{2}}{\big{(}-\omega _{0}^{2}m_{2}\big{)}\big{(}i\omega_{0}c_{2}+k_{2}\big{)}+\big{(}-\omega_{0}^{2 }m_{2}+i\omega_{0}c_{2}+k_{2}\big{)}\big{(}-\omega_{0}^{2}m_{1}+i\omega_{0}c _{1}+k_{1}\big{)}} \tag{8}\]  
If the subsidiary structure is not subjected to the external load (i.e., the surface orifice gate is not directly affected by the fluctuating pressure induced by flood discharge), Equation (8) can be rewritten as  
\[m_{1}\ddot{x}_{1}+c_{2}\Big{(}\dot{x}_{1}-\dot{x}_{2}\Big{)}+k_{2}(x_{1}-x_{2 })+c_{1}\dot{x}_{1}+k_{1}x_{1}=P_{1}(t)+P_{2}(t) \tag{1}\]0  
It is noted that the dynamic response of the subsidiary structure should be calculated by Equation (8) when the surface outlet radial gate is partially opened, and the outflow water directly impacts the bottom of the gate. Moreover, the displacement of the accessory structure should be calculated by Equation (9) when the surface outlet radial gate is closed or largely opened and no interaction between outflow water and radial gate is generated. As the phase information is included in the above expressions of \(X_{2}\), the moduli of \(X_{2}\) with and without considering the external excitation \(P_{2}(t)\) are deduced and given as follows to facilitate following analysis:  
\[m_{1}\ddot{x}_{1}+c_{2}\Big{(}\dot{x}_{1}-\dot{x}_{2}\Big{)}+k_{2}(x_{1}-x_{2 })+c_{1}\dot{x}_{1}+k_{1}x_{1}=P_{1}(t)+P_{2}(t) \tag{1}\]1  
\[m_{1}\ddot{x}_{1}+c_{2}\Big{(}\dot{x}_{1}-\dot{x}_{2}\Big{)}+k_{2}(x_{1}-x_{2 })+c_{1}\dot{x}_{1}+k_{1}x_{1}=P_{1}(t)+P_{2}(t) \tag{1}\]2  
where  
\[m_{1}\ddot{x}_{1}+c_{2}\Big{(}\dot{x}_{1}-\dot{x}_{2}\Big{)}+k_{2}(x_{1}-x_{2 })+c_{1}\dot{x}_{1}+k_{1}x_{1}=P_{1}(t)+P_{2}(t) \tag{1}\]3  
\[m_{1}\ddot{x}_{1}+c_{2}\Big{(}\dot{x}_{1}-\dot{x}_{2}\Big{)}+k_{2}(x_{1}-x_{2 })+c_{1}\dot{x}_{1}+k_{1}x_{1}=P_{1}(t)+P_{2}(t) \tag{1}\]4  
The parameters in Equations (12) to (16) can be expressed as  
\[m_{1}\ddot{x}_{1}+c_{2}\Big{(}\dot{x}_{1}-\dot{x}_{2}\Big{)}+k_{2}(x_{1}-x_{2 })+c_{1}\dot{x}_{1}+k_{1}x_{1}=P_{1}(t)+P_{2}(t) \tag{1}\]5  
\[m_{1}\ddot{x}_{1}+c_{2}\Big{(}\dot{x}_{1}-\dot{x}_{2}\Big{)}+k_{2}(x_{1}-x_{2 })+c_{1}\dot{x}_{1}+k_{1}x_{1}=P_{1}(t)+P_{2}(t) \tag{1}\]6  
\[m_{1}\ddot{x}_{1}+c_{2}\Big{(}\dot{x}_{1}-\dot{x}_{2}\Big{)}+k_{2}(x_{1}-x_{2 })+c_{1}\dot{x}_{1}+k_{1}x_{1}=P_{1}(t)+P_{2}(t) \tag{1}\]7  
\[m_{1}\ddot{x}_{1}+c_{2}\Big{(}\dot{x}_{1}-\dot{x}_{2}\Big{)}+k_{2}(x_{1}-x_{2 })+c_{1}\dot{x}_{1}+k_{1}x_{1}=P_{1}(t)+P_{2}(t) \tag{1}\]8  
\[m_{1}\ddot{x}_{1}+c_{2}\Big{(}\dot{x}_{1}-\dot{x}_{2}\Big{)}+k_{2}(x_{1}-x_{2 })+c_{1}\dot{x}_{1}+k_{1}x_{1}=P_{1}(t)+P_{2}(t) \tag{1}\]9  
\[m_{2}\ddot{x}_{2}+c_{2}\Big{(}\dot{x}_{2}-\dot{x}_{1}\Big{)}+k_{2}(x_{2}-x_{1} )=P_{2}(t) \tag{2}\]0  
\[m_{2}\ddot{x}_{2}+c_{2}\Big{(}\dot{x}_{2}-\dot{x}_{1}\Big{)}+k_{2}(x_{2}-x_{1} )=P_{2}(t) \tag{2}\]1  
It is worth noting that the aforementioned theoretical model is developed from the passive control theory. Conventional theory analysis of structures equipped with DVAs mainly focuses on the dynamic response of primary structures rather than the accessory structures (i.e., the DVA), and the damping of primary structures as well as the external force on accessory structures are generally ignored in the existing literature.