## 2 Integrated Modeling
### Vibration Source Subsystem  
In previous integrated modeling research, the models of the vibration source subsystem mainly applied two forms: (1) power spectral density and (2) harmonic superposition of fixed frequency and amplitude. The first modeling method is to multiply the power spectral density function of the flywheel vibration source obtained from field measurements or theoretical analysis by the transfer function from the vibration source to the relevant position of interest, so as to obtain the frequency domain distribution function and ultimately obtain the micro  
Figure 1: Framework block diagram of an integrated model. vibration response information through integrators. However, it is easy to cause the loss of narrow-band characteristics under small damping during the process of integrators. The second modeling method ignores the disturbing characteristics of the flywheel, that is, under an actual control subsystem, it has different rotation speeds during each imaging, resulting in different harmonic characteristics. [PERSON] has proven that micro-vibration causes the blurring of satellite imaging mainly due to the harmonic characteristics of the flywheel micro-vibration disturbance source [26]. So, this paper establishes a micro-vibration cosine harmonic superposition vibration source model, which not only satisfies the simulation of the overall model in the time-domain but also considers the fact that the flywheel has different micro-vibration disturbance characteristics due to the different flywheel rotation speeds. The flywheel micro-vibration disturbance model based on experience is composed of discrete cosine harmonics, and its amplitude is proportional to the square of the flywheel rotation speed. The expression of disturbance force or disturbance torque \(X_{j}(t)\) is:  
\[X_{j}(t)=\sum_{i=1}^{n}C_{ij}n^{2}\cos(h_{ij}t+p_{j}) \tag{1}\]  
where \(C_{ij}\) is the harmonic amplitude coefficient, \(n\) is the flywheel rotation speed, \(p_{j}\) is the initial phase, and \(h_{ij}\) is the corresponding harmonic factor. Perform Fourier transform on Equation (1) and the ignore negative frequency interval, and get:  
\[X_{j}(\omega)=\sum_{i=1}^{n}\pi C_{ij}n^{2}[\cos(p_{j})+j\sin(p_{j})]\delta( \omega-h_{ij}n). \tag{2}\]  
It can be concluded from Equation (2) that in a Campbell diagram with the flywheel rotation speed as the X axis, frequency as the Y axis, and amplitude as the Z axis, the peaks are at \(\omega=h_{ij}*n\), and the peaks values are proportional to the square of the flywheel rotation speed. Typical disturbance peaks appear on a series of harmonic lines whose frequency is proportional to the flywheel rotation speed. The satellite has three flywheels installed in three directions. A single flywheel includes the disturbance force (X, Y, Z-direction) and the disturbance moment (X, Y-direction). The disturbance moment in the Z direction is very small, so it is not considered. Table 1 shows the number of typical harmonics for each disturbance force/torque. As an example, a Campbell diagram of the disturbance torque (X-direction of flywheel) for one flywheel (Y-direction of satellite) and highlighted four harmonics are given in Figure 2, and Table 2 shows the relevant parameters of these four harmonics. When the flywheel rotation speed is \(n\), the time-domain micro-vibration disturbance of one flywheel on the j-direction can be obtained by Equation (3):  
\[T_{d}^{j}(t)=\sum_{i=1}^{n}A_{ij}(t)\cos[n(t)\cdot h_{ij}\cdot\tau_{t}(t)+p_{ ij}(t)] \tag{3}\]  
where \(A_{ij}\) and \(p_{ij}\) are respectively the amplitude and the initial phase of each order harmonic obtained by interpolating the harmonic characteristics in test data, and \(\tau_{t}\) is the simulation time.