## 3 Results and Discussion  
### Analysis and Design of Optimal Compensate Criterion  
For many studies about vibration suppression of flexible spacecraft, their simplified mathematical model can be expressed as a 2-order vibration system, for example, like the model used in [7]. This means the 2-order vibration system can be used to illustrate the vibration characters of flexible spacecraft. In this section, the typical second order vibration system will be used for analysis of the effectiveness of disturbance compensation and to confirm the optimal compensate criterion. Sinusoidal disturbance and square compensation are adopted based on practical engineering. It should be noted that the frequencies of two kinds of torques are assumed to be the same. Assume there are two periodic exciting forces, \(F_{1}\) and \(F_{2}\), acting on the system. Then the system shown in Equation (5) can be expressed as:  
\[m\ddot{x}+c\dot{x}+kx=F_{1}+F_{2} \tag{6}\]  
where  
\[F_{1}=\left\{ \begin{array}{ll}B,&\frac{2j\pi}{\omega_{f}}\leq t<\frac{(2j+1) \pi}{\omega_{f}}\\ -B,&\frac{(2j+1)\pi}{\omega_{f}}\leq t<\frac{(2j+1)\pi}{\omega_{f}}\end{array}  \right.,\;B\geq 0,\;j=0,1,2,\ldots\]  
\[F_{2}=-A\sin\left(\omega_{f}t\right),\;A\geq 0\]  
\(\omega_{f}\) is the natural frequency of both \(F_{1}\) and \(F_{2}\). \(B\) and \(A\) are the amplitudes of the square force and sinusoidal force, respectively. Periodic square force \(F_{1}\) can be expressed by Fourier series as:  
\[F_{1}=\frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left(a_{n}\cos n\omega_{f}t+b_{n} \sin n\omega_{f}t\right) \tag{7}\]  
where the coefficients are:  
\[a_{0}=0,\;a_{n}=0,\;b_{n}=\left\{ \begin{array}{ll}\frac{4B}{n\pi^{2}},&n=1, 3,5,\ldots\\ 0&n=2,4,6,\ldots\end{array} \right.\]In this series, the fundamental harmonic has the maximal amplitude. Then, the right side of the equal sign in Equation (6) can be expressed as:  
\[F_{1}+F_{2}=\left(\frac{4B}{\pi}-A\right)\sin\omega_{f}t+\frac{4B}{\pi}\left( \frac{1}{3}\sin 3\omega_{f}t+\frac{1}{5}\sin 5\omega_{f}t+\cdots\right) \tag{8}\]  
Here, we introduce an auxiliary parameter called amplitude ratio, which can be expressed as \(\beta=\frac{B}{A}\). Substitute Equation (8) into Equation (6), and replace \(B\) by \(\beta\), we have:  
\[m\ddot{x}+c\dot{x}+kx=\sum_{i=1}^{\infty}P_{i}\sin\left((2i-1)\omega_{f}t\right) \tag{9}\]  
where  
\[P_{i}= \begin{cases}\frac{(4\beta-\pi)A}{\pi},&i=1\\ \frac{4 AB}{(2i-1)\pi},&i=2,3,4,\ldots\end{cases} \]  
Solving Equation (9), we obtain  
\[x(t)=\sum_{i=1}^{\infty}\left\{ \begin{array}{ll}e^{-\zeta\omega_{g}t}\left( x_{0}\cos\omega_{d}t+\frac{\dot{x}_{0}+\zeta\omega_{n}x_{0}}{\omega_{d}}\sin \omega_{d}t\right)\\ +Q_{i}e^{-\zeta\omega_{n}t}\left[\sin\varphi_{i}\cos\omega_{d}t+\frac{\omega _{n}}{\omega_{d}}(\zeta\sin\varphi_{i}-\lambda(2i-1)\cos\varphi_{i})\sin \omega_{d}t\right]\\ +Q_{i}\sin\left((2i-1)\omega_{f}t-\varphi_{i}\right)\end{array} \right\} \tag{10}\]  
where \(x_{0}=x(0)\), \(\dot{x}_{0}=\dot{x}(0)\) are the initial conditions. \(\lambda=\frac{\omega_{f}}{\omega_{n}}\) is the frequency ratio. \[Q_{i}=\frac{P_{i}/k}{\sqrt{\left(1-\left(2i-1\right)^{2}\lambda^{2}\right)^{2 }+\left((4i-2)\zeta\lambda\right)^{2}}},\ \varphi_{i}=\arctan\frac{(4i-2)\zeta\lambda}{1-(2i-1)^{2}\lambda^{2}}\]  
For zero initial conditions system, the solution shown in Equation (10) can be expressed as:  
\[x(t)=\sum_{i=1}^{\infty}\left\{ \begin{array}{ll}Q_{i}e^{-\zeta\omega_{n}t} \left[\sin\varphi_{i}\cos\omega_{d}t+\frac{\omega_{n}}{\omega_{d}}(\zeta\sin \varphi_{i}-\lambda(2i-1)\cos\varphi_{i})\sin\omega_{d}t\right]\\ +Q_{i}\sin\left((2i-1)\omega_{f}t-\varphi_{i}\right)\end{array} \right\} \tag{11}\]  
Equation (11) is the final displacement formula. The first line of Equation (11) will be decreased because of damping, and only the second line will remain when system becomes stable. We can find the displacement is strongly associated with the frequency and amplitude ratio, and the fundamental harmonic is vital in the compensated force design. Numerical simulations were provided to provide the laws.