## 2 Mathematical Modeling  
The TRID system proposed before mainly consists of a torsional spring element and rotary inertia. Compared with the TRID system, the ARID system consists of an active driver and a rotary inertia. The simplified analytical models of the suspended structure with the ARID/TRID system are developed in order to investigate the dynamic characteristics of the ARID system, which are shown in Figure 1. The model consists of two degrees of freedom used as generalized coordinators, which are the structural pendular angle \(\theta\) and relative rotation angle of the rotary inertia with respect to the ground base \(\phi\). The suspended structure has the swing length \(l\), suspended particle mass \(m\)and damping coefficient \(c\). The control system rotary inertia mass is \(m_{a}\) and the rotary inertia is \(J_{a}\). The acceleration at the suspended structure lifting point is \(a_{x0}(t)\). The rotational stiffness coefficient of the ARID system is \(k_{a}\). The damping coefficient is \(c_{a}\). The output torque of the ARID system is \(M_{a}(t)\). The motion equations [44] of the structure with the ARID system model are deduced based on the Lagrangian difference method as  
\[(m+m_{a})l^{2}\ddot{\partial}+c\dot{\partial}+(m+m_{a})gl\sin\theta=-(m+m_{a}) l\cos\theta a_{x0}(t)+c_{a}\Big{(}\dot{\varnothing}-\dot{\theta}\Big{)}+k_{a}( \varnothing-\theta)-M_{a}(t) \tag{1}\]  
\[J_{a}\ddot{\varnothing}+c_{a}\Big{(}\dot{\varnothing}-\dot{\theta}\Big{)}+k_ {a}(\varnothing-\theta)=M_{a}(t) \tag{2}\]  
where \(J_{a}=m_{a}r^{2}\), \(c=2\Big{[}m+m_{a}l^{2}\Big{]}\omega_{m}\xi\), \(\omega_{m}=\sqrt{g/l}\), \(c_{a}=2m_{a}r^{2}\omega_{a}\xi_{a}\), and \(\omega_{a}=\sqrt{k_{a}/l_{a}}\) are the required parameters. \(r\) is the radius of the mass inertia, \(\xi\) is the structural damping ratio, and \(\xi_{a}\) is the ARID damping ratio. Generally, the damping force and torsion resistance of the ARID system are very small compared with the control forces, which can be neglected (\(c_{a}\) and \(k_{a}\) are 0) [44].  
Equation (2) gives the damping force, torsion resistance, and output active control force of the ARID system. The motion equations of the system are linearized. Furthermore, the classical Linear Quadratic Regulator (LQR) algorithm is used in the system and the gain matrices \(\mathbf{A}\) can be obtained [44]. The state vector is \(Z(t)=\Big{(}\theta(t),\phi(t),\dot{\theta}(t),\dot{\phi}(t)\Big{)}^{T}\), and the active torque \(M_{a}(t)\) is given as  
\[M_{a}(t)=-\mathbf{A}\;Z(t). \tag{3}\]  
The rotational stiffness coefficient of the TRID system is \(k_{t}\). The damping coefficient is \(c_{t}\). Compared with the ARID system, the TRID system cannot output the active control torque. However, the TRID system is a kind of passive control system, and it needs frequency modulation according to the structure. The motion equations of the structure with the TRID system model are deduced based on the Lagrangian difference method as  
\[(m+m_{a})l^{2}\ddot{\partial}+c\dot{\partial}+(m+m_{a})gl\sin\theta=-(m+m_{a} )l\cos\theta a_{x0}(t)+c_{t}\Big{(}\dot{\varnothing}-\dot{\theta}\Big{)}+k_{ t}(\varnothing-\theta) \tag{4}\]  
\[J_{a}\ddot{\varnothing}+c_{t}\Big{(}\dot{\varnothing}-\dot{\theta}\Big{)}+k_{ t}(\varnothing-\theta)=0 \tag{5}\]  
where \(c_{t}=2m_{a}r^{2}\omega_{t}\xi_{t}\), \(k_{t}=\omega_{t}{}^{2}J_{a}\), and \(\xi_{t}\) is the TRID damping ratio. For the TRID system, the key parameters \(c_{t}\) and \(k_{t}\) of the torsional spring have a great influence on the control effectiveness.  
Figure 1: The simplified analytical model of the suspended structure with the control system: (**a**) The structure with the Active Rotary Inertia Driver (ARID) system; (**b**) the structure with the Tuned Rotary Inertia Damper (TRID) system.