## 3 Feature Analysis of Vibration Information in GIS
### Vibration Information Analysis  
In the three-phase common box GIS, it is assumed that the three phases A, B, and C are arranged in an equilateral triangle, and the top view and the cross-sectional view are as shown in Figure 3. The wire distance is a, the length is l, and the three-phase current is symmetrical power frequency, \(f=50\) Hz. There are currents in the A, B, and C three-phase conductors, and interactive electromotive forces are bound to occur. Assuming \(i_{A}=I_{m}\sin(2\pi\beta t)\), \(i_{B}=I_{m}\sin(2\pi\beta t-2\pi/3)\) and \(i_{C}=I_{m}\sin(2\pi\beta t-4\pi/3)\), \(F_{AB}\) and \(F_{AC}\) are the electromotive forces generated by A-phase conductor under the action of B-phase current and C-phase current, respectively. \(F_{ABx}\) and \(F_{ABy}\) are the components of \(F_{AB}\) on \(x\) and \(y\)-axis, respectively. \(F_{ACx}\) and \(F_{ACy}\) are the components of \(F_{AC}\) on x and y-axis respectively. The sum of \(F_{ABx}\) and \(F_{ACx}\) and the sum of \(F_{ABy}\) and \(F_{ACy}\) are \(F_{Ax}\) and \(F_{Ay}\), respectively. The resultant force of \(F_{Ax}\) and \(F_{Ay}\) is the total electric power \(F_{A}\) of A-phase conductor. Thus, the electric power of the A-phase conductor is calculated as follows:  
\[ \begin{array}{l}F_{Ax}=F_{ABx}+F_{ACx}\\ \qquad=\frac{\mu\omega}{2\pi a}l_{m}^{2}(\sin(2\pi ft)\sin(2\pi ft-2\pi/3)-\sin (2\pi ft)\sin(2\pi ft-4\pi/3))\sin(\pi/6)=K\sin(4\pi ft)\end{array}  \tag{1}\]  
\[ \begin{array}{l}F_{Ay}=F_{ABy}+F_{ACy}\\ \qquad=\frac{\mu\omega}{2\pi a}l_{m}^{2}(\sin(2\pi ft)\sin(2\pi ft-2\pi/3)+ \sin(2\pi ft)\sin(2\pi ft-4\pi/3))\cos(\pi/6)=K(1-\cos(4\pi ft))\end{array}  \tag{2}\]  
\[ \begin{array}{l}\left\{\begin{array}{l}\mathbf{F}_{A}=\sqrt{F_{Ax}^{2}+F_ {Ay}^{2}}\alpha\\ \alpha=\arctan(F_{Ay}/F_{Ax})\end{array}\right.\end{array}  \tag{3}\]  
\[ \begin{array}{l}\left\{\begin{array}{l}\left|\mathbf{F}_{A}\right|=\frac {\sqrt{3}}{2}|K\sin(2\pi ft)|\\ \alpha=\pi(2f)t\pm k\pi,k=0,1,2,\ldots\end{array}\right.\end{array}  \tag{4}\]  
where \(\mu_{0}\) is vacuum permeability and \(K=(-\sqrt{3}\mu_{0}\,I^{2}{}_{m}l)/(8\pi a)\). It can be seen from (4) that the frequency of the amplitude change of the A-phase conductor is 2f (the period of the absolute value of the sine function is twice the original function), and the frequency of the angle change is also 2f (the tangent function period is \(\pi\)). Therefore, the frequency of the A-phase conductor electric power is 100 Hz, and the electric power of the B and C phases are also 100 Hz [20]. Combined with Figure 3 and the electrodynamic analysis of the three-phase conductor, it can be seen that the excitation of the GIS shell vibration is derived from the frequency 100 Hz electrodynamic action, that is, the GIS vibration information is expressed as an output response under 100 Hz forced vibration. Based on the periodicity of the vibration signal, the spectral description based on Fourier  
Figure 3: Mechanical analysis of three-phase GIS: (**a**) top view; (**b**) cross-section drawn. transformation becomes an effective means of GIS vibration feature extraction. According to the principle of Fourier transformation, the time domain and frequency domain information of three operating conditions are compared under the 1000 A current as shown in Figure 4.  
It can be concluded that:  
(1) In addition to the excitation spectrum of 100 Hz, the GIS vibration signal also has stability information of other frequency points. Under the premise that the GIS system is a damping system, the vibration response characteristics indicate that the GIS system is not a linear damping system. (2) According to the principle of non-linear vibration [29; 30], under the action of simple harmonic vibration the frequency response of the nonlinear system contains a super-harmonic signal of integer multiples of the excitation frequency and a sub-harmonic signal of integers.