# Modeling and Test Results of an Innovative Gyroscope Wave Energy Converter
## 3 Theoretical Analysis  
Assuming that the gyroscope rotor rotates about the \(Z\)-axis at an angular velocity \(\dot{\varphi}\), the floating platform and the flywheel both rotate around the \(X\)-axis at an angular velocity \(\dot{\delta}\) by the motion of the wave. Then, the flywheel coordinate system is used as the moving  
Figure 1: Reference frames. Figure 2: Principle of GWEC. coordinate system (the unit vector of the moving coordinate system is \(\overset{\rightarrow}{i}\overset{\rightarrow}{,j}\overset{\rightarrow}{,k}\) respectively), the angular velocity of the gyroscope rotor relative to the inertial space is:  
\[\overset{\rightarrow}{\omega}=\dot{\delta}\cos\varepsilon\cdot\overset{ \rightarrow}{i}+\dot{\varepsilon}\cdot\overset{\rightarrow}{j}+(\dot{\delta} \sin\varepsilon+\dot{\varphi})\cdot\overset{\rightarrow}{k} \tag{1}\]  
where \(\varepsilon\) is the gyroscope precession angle. Then, the angular moment of the rotor can be obtained (where \(J_{x},J_{y},J_{z}\) are the moments of inertia in x, y, z directions). \[H=J_{x}\dot{\delta}\cos\varepsilon\cdot\overset{\rightarrow}{i}+J_{y}\dot{ \varepsilon}\cdot\overset{\rightarrow}{j}+J_{z}(\dot{\delta}\sin\varepsilon+ \dot{\varphi})\cdot\overset{\rightarrow}{k} \tag{2}\]  
The change of the precession angle is the key to wave energy generation. According to the expression of angular momentum, the equation of motion of the y1 is written:  
\[J_{y}\ddot{\varepsilon}+(J_{x}-J_{z})\dot{\delta}^{2}\sin\varepsilon\cos \varepsilon-J_{z}\dot{\varphi}\dot{\delta}\cos\varepsilon=T_{y} \tag{3}\]  
where \(T_{y}\) is the control torque in the direction of the precession axis, the PTO can be considered as a spring damping system, and Equation (3) can be rewritten as:  
\[J_{y}\ddot{\varepsilon}+(J_{x}-J_{z})\dot{\delta}^{2}\sin\varepsilon\cos \varepsilon-J_{z}\dot{\varphi}\dot{\delta}\cos\varepsilon=-c_{y}\dot{ \varepsilon}-k_{y}\varepsilon \tag{4}\]  
where \(c_{y}\) is the damping coefficient and \(k_{y}\) is the spring coefficient. When \(\varepsilon\) is small, \(\sin\varepsilon\) is approximately equal to 0. It can be noticed in Equation (4) that the second term of the above formula is two orders of magnitude smaller than the others, so it can be ignored to reduce the amount of calculation. Then, Equation (4) can be rewritten as follows:  
\[J_{y}\ddot{\varepsilon}+c_{y}\dot{\varepsilon}+k_{y}\varepsilon=J_{z}\dot{ \varphi}\dot{\delta}\cos\varepsilon \tag{5}\]  
In order to obtain the main behavioral characteristics and dynamic performance of the gyroscope device, Equation (5) has been linearized around the position \(\varepsilon\) approximately equal to 0. \[J_{y}\ddot{\varepsilon}+c_{y}\dot{\varepsilon}+k_{y}\varepsilon=J_{z}\dot{ \varphi}\dot{\delta} \tag{6}\]  
It can be observed that Equation (6) is a typical second-order system, which can be rewritten as:  
\[\ddot{\varepsilon}+2\zeta\omega_{n}\dot{\varepsilon}+\omega_{n}^{2} \varepsilon=J_{z}\dot{\varphi}\dot{\delta}/J_{y} \tag{7}\]  
where equivalent damping ratio \(\zeta=\frac{c_{y}}{2\sqrt{J_{y}k_{y}}}\), free oscillation frequency \(\omega_{n}=\sqrt{\frac{k_{y}}{J_{y}}}\). When a certain input \(\dot{\delta}\) is given to the system, we can observe the output as:  
\[(-J_{y}\omega^{2}\sin(\omega t)+c_{y}\omega\cos(\omega t)+k_{y}\sin(\omega t) )\varepsilon_{0}=J_{z}\dot{\varphi}\dot{\delta} \tag{8}\]  
where \(\omega\) is the frequency of the wave, \(\varepsilon=\varepsilon_{0}\sin(\omega t)\) by assuming the gyroscope precession change periodically. Equation (9) can be obtained using \(k_{y}=J_{y}\omega_{n}^{2}\). Figure 2:0  
In this paper, the energy \(P\) extracted from the damper is considered to be available for power generation, then:  
Figure 2:1It can be observed from Equation (10) that the maximum power acquisition is when the system resonates.