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(1) Suppose that B is the average width of strip, h is the average thickness, T is the total tension measured by the tension meter. Then, the average tension stress of strip is BhT (2) In order to quickly calculate the strip shape and display its curve, suppose B=n•bi, hi≈h (i=1, 2, …, n). A simplified formula of tension stress σi can be expressed as NNNnNniniiiniiii 1 11 1 (3) where N means the average radius force of all channels. By Eqs. (2) and (3), the residual stress (stress deviation) ∆σi is obtained as NN T i (4) The calculated value T of total tension is obtained by sin211 niiiiniiNhb T (5) According to Hook’s law, when the metal has elastic deformation, the stress ∆σi is proportional to the elastic deformation ,/ lli that is iiE ll 21 (6) where E is elastic modulus, is Poisson ratio. Internationally, usually the basic unit shape I is [3] 5 510 10 lllllI (7) where l′ and l are the lengths of the longest element and the shortest element, respectively, and ∆l is the length difference between them. According to Eqs. (6) and (7), the measured shape S can be obtained as 52510)()1(10)()( EylylyS i ii (8) After the actual shape S(y) subtracting the goal shape Sa(y), the online shape deviation ∆S is obtained as ∆S(yi)=S(yi)−Sa(yi) (9) Passing ∆S to the shape control computer, the real time shape status can be displayed and the strip shape can be controlled by the shape control means of mill at the same time.
4 Integral shape detecting roller 4.1 Features of integral shape detecting roller The core component of shape detecting system is the detecting roller, which is mainly composed of the main roller body, the elastic block and the Magneto- elastic sensor. The strength and accuracy of signal have direct impact on the precision of the shape detecting system. As shown in Fig. 5, in the assembling process of integral detecting roller, firstly the sensors are fixed in the elastic block which applied pre-pressure to the sensor, and then the blocks are embedded in the symmetrical rectangular slots of the main roller body in turn. The differential circuits are composed of two radial symmetrical sensors for eliminating signal drifting which may be caused by temperature, humidity, vibration, magnetic field and other factors. So, the integral shape detecting roller can significantly improve the shape detecting precision and the anti-interference ability. The structure of integral shape detecting roller has excellent performance of easy installation and good maintenance, effectively avoiding scratching strip surface. The assembled detecting roller is shown in Fig. 6.
4.2 Internal hole size optimization of elastic block In the shape detecting process, the elastic block is the most critical component for ensuring steady work of the detecting roller. As shown in Fig. 5, the sensitivity of sensor has a great relationship with the radial position A and the width B of internal hole in the elastic block. The reciprocal pressure difference of sensor is set as the objective function to optimize A and B by ANSYS [19]. Considering the precisions of machining and assembly, finally A and B are determined as 163 and 50 mm, respectively. The optimized structure of detecting roller not only has good mechanical transfer properties, but also has the maximum Mises stress in the safe range. Figure 7 shows a good linear relationship between the radial force of detecting roller and the detecting force of sensor.