# Roll Eccentricity Signal Detection and Its Engineering Application Based on SFFT-IAA
## 2 Description of Roll Eccentricity Problems  
Currently, the roller system of the mainstream single-stand reversible cold rolling mill consists of six rollers of different diameters. In the rolling process of plates and strips, the rollers are pressed against each other under the rolling force generated by the hydraulic cylinder and are rotated at a high speed driven by the motor drive system. Then, the AGC system changes the size of the roller gaps by adjusting the position of the hydraulic cylinder to obtain the thickness of the exporting strip steel. Figure 1 illustrates the working principle of the six-roller reversible cold rolling mill. According to the spring equation for rolling mills, the exit thickness of the strip steel is [19]:  
\[h=S_{0}+\frac{P}{M} \tag{1}\]  
where \(h\) is the exit thickness of the strip steel, \(S_{0}\) is the unloaded roller gap, \(P\) is the rolling force, and \(M\) is the stiffness of the rolling mill. It can be deduced from Equation (1) that the exit thickness fluctuation \(\Delta h\) after thickness control is:  
\[\Delta h=\frac{\Delta P}{M_{\text{C}}} \tag{2}\]  
\[M_{\text{C}}=\frac{\partial P}{\partial h}=M(1-C) \tag{3}\]  
where \(M_{\text{C}}\) is the effective stiffness of the rolling mill, \(\partial P\) is the fluctuation of the rolling force, \(\partial h\) is the thickness deviation of the rolled piece, and \(C\) is the compensation coefficient at the position of the roller gap. \[\Delta P=(\Delta H-\Delta h)Q \tag{4}\]  
where \(\Delta H\) is the entry thickness of the rolled piece, and \(Q\) is the plastic stiffness. If the roller eccentricity amount \(e\) is considered, Equation (2) can be rewritten as:  
\[\Delta h=e+\frac{(\Delta H-\Delta h)Q}{M_{\text{C}}} \tag{5}\]  
Then, we have:  
\[\Delta h=\frac{M_{\text{C}}}{M_{\text{C}}+Q}e^{+}\frac{Q}{M_{\text{C}}+Q}\Delta H \tag{6}\]  
When the common thickness gauge control method is used in the thickness control system, the value of \(C\) ranges from 0 to 1, the value of \(M_{\text{C}}\) ranges from \(M\) to \(\infty\), and the roll eccentricity \(e\) will be nearly or even completely reflected in the thickness deviation of the rolled piece, which cannot be eliminated. Figure 2 illustrates the impacts of roll eccentricity. Figure 1: Schematic diagram of rollers in a rolling mill. As shown in Figure 2, when eccentricity \(e\) exists, the actual thickness of the strip decreases by \(\Delta h_{e}\) and the rolling force increases by \(\Delta P_{e}\). In this case, the control system considers that the thickness of the strip increases. Therefore, under the action of the controller, the system operates in the direction of reducing the thickness of the strip, which leads to a low thickness accuracy and deteriorated control quality. In actual situations, when roll eccentricity exists, the roller will vibrate at a high frequency during high-speed rotations, resulting in periodic oscillations in the practical roll gap of the rolling mill, rolling force, and thickness of the exit strip steel. Roll eccentricity can be divided into two main types: the rotational eccentricity generated when the center of rotation of the roller does not coincide with its geometric center and the elliptical eccentricity caused by the irregular shape of the roller itself. In engineering practice, both types of roll eccentricity may exist. According to the causes of roll eccentricity and the working mechanism of rolling mills, the unified model of roll eccentricity can be expressed as the superposition of a series of cosine periodic waves obtained by integer multiples of the roller rotation. Thus, the frequency of an eccentricity component induced by the rotational eccentricity is the same as the rotational frequency of the roller, and the frequency of the eccentricity component of the roller caused by ellipticity is two times the rotation period [20].