## 3 Results and Discussion
### Simulations for Flexible Spacecraft  
The former subsection demonstrated that the square compensation force can be used to mitigate the vibration caused by sinusoidal disturbance. To demonstrate the effectiveness of compensation in flexible spacecrafts, numerical simulations and experiments are implemented. In this subsection, the flexible spacecraft model in Equation (4) is adopted, and simulation results are presented. The system is actuated by a sinusoidal disturbance and a square compensate torque produced by a periodic rotated eccentric mass and a reaction wheel, respectively. Residual vibrations with different combinations of frequency and amplitude ratios will be given to show the effectiveness. Here, the torque \(T\) can be expressed as:  
\[T=T_{1}+T_{2} \tag{12}\]  
where \(T_{1}\) and \(T_{2}\) have the same expression as \(F_{1}\) and \(F_{2}\) in Equation (4), respectively. \(T_{1}\) is given by the ideal reaction wheel. \(T_{2}\) is from the periodic eccentric mass. \(T_{1}\) and \(T_{2}\) can be expressed as:  
\[T_{1}= \begin{cases}U,&\frac{2i\pi}{\omega_{m}}\leq t<\frac{(2i+1)\pi}{\omega_ {m}}\\ -U,&\frac{(2i+1)\pi}{\omega_{m}}\leq t<\frac{\pi(i+1)\pi}{\omega_{m}}\\ &T_{2}=-W_{m}\Omega_{m}^{2}\end{cases} ,\ U\geq 0,\ j=0,1,2,\ldots\]  
Equation (4) can be further expressed as:  
\[\left\{ \begin{array}{c}\int\ddot{\theta}+\delta\ddot{\eta}=\sum\limits_{i=1 }^{\infty}P_{i}\sin((2i-1)\omega_{m}t)\\ \delta\mathbf{T}\ddot{\theta}+\ddot{\eta}+\mathbf{C}\dot{\eta}+\mathbf{K}\eta =0\end{array} \right. \tag{13}\]  
where  
\[P_{i}= \begin{cases}\frac{(4\beta-\pi)}{\pi}\Omega_{m}^{2}P_{m}\sigma_{m}m_{m},&i=1\\ \frac{4\Omega_{m}^{2}P_{m}\sigma_{m}m_{m}\delta}{(2i-1)\pi},&i=2,3,4,\ldots\\ \beta=\frac{U}{\Omega_{m}^{2}P_{m}\sigma_{m}m_{m}}\end{cases} \]  
Substitute the first part of Equation (13) into the second formula of Equation (13), we will have:  
\[\left(\mathbf{I}_{3}-\frac{\delta^{T}\delta}{J}\right)\ddot{\eta}+\mathbf{C} \dot{\eta}+\mathbf{K}\eta=-\frac{\delta^{T}}{J}\sum\limits_{i=1}^{\infty}P_{ i}\sin((2i-1)\omega_{m}t) \tag{14}\]  
Modal solutions are complex to be predicted analytically, not to mention the multi-dimension modal. Thus, we will directly present the simulation results. Parts of parameters are measured values of experimental equipment used in our study. Others are calculated in a mathematical way. The moment of inertia of the rigid hub is \(J=15\) kg\(\cdot\)m\({}^{2}\). The damping ratios of the vibration are \(\zeta_{1}=\zeta_{2}=\zeta_{3}=0.05\), the natural frequencies are \(\omega_{n1}=2.11\) rad\(/\)s, \(\omega_{n2}=13.28\) rad\(/\)s, \(\omega_{n3}=37.15\) rad\(/\)s. The coupling matrix is \(\delta=\left[ \begin{array}{ccc}2.65&0.54&0.21\end{array} \right]\) kg\(\cdot\)m\({}^{2}\). In simulations, the mass of rotating eccentric mass is \(m_{m}=0.4\) kg, and its eccentric radius is \(\sigma_{m}=0.08\) m. The states of system are initialised to be zero. The maximum of sinusoidal disturbance will vary in a certain range depending on each frequency ratio. Square compensation torques were assumed to be ideally outputted, which means it can perfectly fit the sinusoidal disturbance. The frequency ratio range is set from 0.3 to 3.2, and the amplitude ratio range is set from 0.3 to 1. Those settings are based on the results in the previous subsection. Amplitudes of flexible beam tip residual vibration under different combinations of frequency and amplitude ratios are calculated by 'Simulation' in 'Matlab' software. Simulation results are illustrated in Figures 6 and 7.  
when the amplitude ratio is near \(\beta=0.7\), the displacement of the flexible tip is minimal. This law is particularly obvious when the frequency ratio exceeds 0.5. Those results just proved the optimal criterion. Variation amplitudes are so weak that the trend is not so clear when the frequency ratio is smaller than 0.5. Figure 7a,b are the amplitude history of the flexible beam tip acceleration. The tip acceleration is a real effect when the amplitude ratio is located near \(\beta=0.7\).