## 2 Materials and Methods  
### Modelling of the Flexible Spacrcraft  
Three-axis spacecrafts with flexible appendages have a complex model. It is difficult for us to obtain the relationship between vibrations and control torques. To stress the key point of this paper, here we provided a simplified spacecraft model: A flywheel actuated single-axis gas suspending rotary table (SGSRT) is applied. The diagram of the SGSRT is shown in Figure 1.  
As is shown in Figure 1, this system is mainly consists of a rigid hub and a flexible beam. A rigid beam is fixed on the opposite side of flexible beam. An eccentric mass is rotating around the tip of the rigid beam at a constant angular velocity. A reaction wheel is placed at the centre of the hub. Define \(OXYZ\) as the inertial frame and \(oxy\) as the frame fixed on the edge of rigid hub. \(\theta(t)\) is the rotation angle of the hub. \(y(x,t)\) is the displacement of place \(x\) in frame \(oxy\). \(\Omega\) and \(\Omega_{m}\) are the angular velocity of rigid hub and eccentric mass, respectively. The flexible beam is assumed to be an Euler-Bernoulli beam. The Newton-Euler method is used to build the mathematical model of the system. The dynamic model of the flexible system shown in Figure 1 is  
\[\left\{\begin{array}{l}\dot{\mathbf{J}}\dot{\mathbf{\Omega}}+\int\limits_{0}^{L}\mathbf{{r}_{b}}\times\mathbf{a_{b}}dx+\mathbf{r_{m}}\times\mathbf{a_{m}}\mathbf{m}_{m}=\mathbf{T}\\ E\frac{\partial^{4}y}{\partial x^{4}}+\sigma\Big{[}\frac{d^{2}y}{dt^{2}}+(R_{ hub}+x)\frac{d^{2}\theta}{dt^{2}}\Big{]}=0\end{array}\right. \tag{1}\]  
where \(\mathbf{J}\) is the rotational inertia matrix of the rigid hub. \(\mathbf{r_{b}}\) is the vector from the centre of the hub to an arbitrary mass point in the flexible beam. \(\mathbf{a_{b}}\) is the acceleration of mass points. \(\mathbf{r}_{m}\) is the vector from the centre of the hub to the eccentric mass. \(\mathbf{a_{m}}\) and \(\mathbf{m}_{m}\) are the acceleration and mass of the eccentric mass, respectively. \(\mathbf{T}\) is the actuator torque. \(R_{hub}\) is the radius of the rigid hub. \(EI\) is the flexural rigidity of the beam and \(\sigma\) is the surface density  
As the SGSRT is adopted in this study, Equation (1) can be simplified because it is a single-axis rotation system. The simplified model can be expressed as:  
\[\left\{\begin{array}{l}J\ddot{\theta}+\int\limits_{l}{(R_{hub}+x)\ddot{y}dm +W_{m}\Omega_{m}^{2}}=T\\ E\frac{\partial^{4}y}{\partial x^{4}}+\sigma\Big{[}\frac{d^{2}y}  
where \([D]\) is a matrix consisting of modes of the flexible beam, \(\eta\) is the modal coordinate vector. In consideration of the orthogonality of the mode. Equation (2) can be shown as:  
\[\left\{\begin{array}{l}\dot{j}\ddot{\theta}+\delta\ddot{\eta}+W_{m}\Omega_{m}^ {2}=T\\ \delta^{T}\ddot{\theta}+\ddot{\eta}+\dot{C}\dot{\eta}+K\eta=0\end{array}\right. \tag{4}\]  
where \(\delta\) is the coupling matrix of rotation and vibration. \(C\) and \(K\) are damping and stiffness matrices, respectively. It needs to be emphasized that the reaction wheel is assumed to be ideal equipment. That means it can generate ideal square torque. Furthermore, the influence on the system caused by eccentric mass is in sinusoidal law according to its expression. Hence, the two known-law kinds of torque will be adopted in the residual part.