# Parameter Optimization for Ship Antiroll Gyros
## 2 Dynamical Model of Ship's Antiroll Gyro
### Joint Dynamical Equation of Ship and Antiroll Gyro  
As displayed in Figure 2, the antiroll gyro was mounted on the ship deck. The antiroll gyro consisted of a precession axis, frame, rotor, and rotor spindle. \(O\xi\eta\zeta\) is the absolute coordinate system, and \(\phi\) is the roll angle of the hull around \(O\zeta\). _Oxyz_ is the relative coordinate system, and \(Oz\) is the rotating axis of the rotor. \(Oy\) is the rotating axis of the outer frame, and \(\beta\) is the precession angle of the gyro around \(Oy\).  
Figure 1: The stress of the ship.  
The mass of the gyro's outer frame was neglected, and the influence of the ship's movement on other degrees of freedom on the gyro was not considered. The rotor was an axisymmetric rigid body whose moment of inertia around \(Oz\) was \(I_{z}\), the rotor speed was constant at \(\omega_{0}\), and the momentum moment constant of \(Ox\) and \(Oy\) was \(J\). According to the Euler equation of the motion of rigid bodies [23], the motion equation of the antiroll gyro relative to \(Oxyz\) was as follows:  
\[\left\{\begin{array}{l}M_{x}=\dot{f}\ddot{\phi}\cos\beta+h_{0}\dot{\beta}\\ M_{y}=\dot{f}\ddot{\phi}+\dot{f}\dot{\phi}^{2}\sin\beta\cos\beta-h_{0}\dot{\phi }\cos\beta\\ M_{z}=I\dot{\omega}_{0}=0\end{array}\right. \tag{9}\]  
where \(M_{x},M_{y}\), and \(M_{z}\) are the components of the resultant external torque on \(Oxyz\), \(\dot{\beta}\) is the precession angular velocity, \(\ddot{\beta}\) is the precession angular acceleration, and \(h_{0}\)\(=\) I\({}_{z}\omega_{0}\) is the momentum moment constant of the gyro. If \(Oy\) was considered as the input axis, then input torque \(M_{y}\) caused the gyro to precess, and then a torque \(M_{x}\) output on \(Ox\) occurred. The output torque \(M_{x}\) from the above motion equation was projected into \(O\xi\eta\zeta\). Given the stability of the high-speed spinning gyro, the angular velocity \(\dot{\phi}\) was considerably smaller than \(\omega_{0}\), and its second derivative was ignored. Equation (9) could then be simplified as follows:  
\[\left\{\begin{array}{l}M_{\xi}=h_{0}\dot{\beta}\cos\beta\\ M_{\zeta}=h_{0}\dot{\beta}\sin\beta\\ M_{y}=\dot{f}\ddot{\beta}-h_{0}\dot{\phi}\cos\beta\end{array}\right. \tag{10}\]  
The damping device was added in the precession direction of the antiroll gyro, and the precession of the gyro was restricted appropriately depending on the characteristics of damping to improve the antirolling effect. The total damping torque \(M_{y}\) could be expressed as \(M_{y}=C\dot{\beta}\), and \(C\) was defined as the gyro's damping coefficient in units of Ns/m, which was the ratio of the damper's damping force installed in the gyro to the movement speed of the damper's piston rod.  
Furthermore, the motion equation of the ship and the mathematical model of the antiroll gyro could be simultaneously established to obtain the ship antiroll gyro's motion equation:  
\[\left\{\begin{array}{l}\ddot{\phi}=-c_{1}\dot{\phi}-c_{3}\dot{\phi}^{3}-k_{1 }\phi-k_{3}\phi^{3}-k_{5}\phi^{5}-h_{0}\dot{\beta}\cos(\beta)/\left(I_{\phi \phi}+J_{\phi\phi}\right)+Dh\alpha(t)/\left(I_{\phi\phi}+J_{\phi\phi}\right)\\ \dot{\beta}=h_{0}\dot{\phi}\cos(\beta)/J-C\dot{\beta}/J\end{array}\right. \tag{11}\]