Introduction.

The complete description of the mill gain calculation is described in this chapter. Sections 4.2 to 4.7 describe the calculation of mill constants. Sections 4.8 to 4.11 describe the roll force model, roll pressure and deflection calculation, thickness and stress profile

calculation. The final two sections describe the complete model and the gain matrix calculation.

The mill width is divided into 67 sections or multiples of 67 sections. This odd number 67 is chosen to match the back-up roll dimensions to its segments. To be more precise, for one segment of the back-up roll (see fig. 4.1), the ratio between the portion in contact with the second intermediate roll b to non contact area (& - b) is an integer if the mill width is divided into 67

sections. That is lengths b and (-6 - b) can be divided into an integer number of sections. The width of each section is given by

\

4-. 2. Strip width adjustment.

The width of the strip w has also to he_s adjusted so that the strip width will have an integer number of sections. This is done in * the following manner. The strip is placed in the mill so that the centre of the strip width lies on the vertical line passing through the mill centre. If the edge of the strip lies inside a section (see fig. 4*.2) the distance between the edge of section and the edge of the strip is calculated. This is denoted by Aw ._s

where (r + l) is the integer number of the section in which the strip edge lies inside.

k.J. Strip dimensions.

either rectangular or parabolic the model considers only these two types of profile. If the strip centre line thickness h and the amount of strip camber h are_{m c}

specified then the strip profile can be obtained as shown If Aw > -_S 5

_c*

ws = [(N - r - l) - (r + l)J dx = (N - 2r - 2)dx (^.2)

If Aw _{s &} then

a 3)

Since most of the input gauge profiles are

below.

An equation for a parabolic profile as shown in fig. can be written as (variables defined in

fig. 4.3a)

y(x) = h (^.*0

As shown in fig. *K3b the thickness at any point distance x from the left hand end of the mill can be written as

h(x) = hw + 2y(x)

(*•5)

If the strip is rectangular then this profile can be obtained by putting hQ = 0 in eq. (*K 6).

The input thickness profile is given by

V x> " - 2V tr_m - D 2

(4.7)

The output thickness h9(x) can be calculated if the

reduction is known. This can he done by specifying the centreline output strip thickness. Let this be h^. Assuming that there is constant reduction across strip width, h^(x)

is given by h2 w =

lm

The mean output thickness is given by

therefore the deviation of output gauge from mean is given

BjL. (A. 8) assumes constant reduction which implies that the output strip has perfect shape. This is only an

initialization process and the deviation Ah^(x) given by eq. (A. 10) will be updated at later stages, since the work roll profile will be deformed when forces are applied. A. k. Back-up roll profile.

distance the back-up roll axis is deflected. The back-up roll profile calculation for a given As-U-Roll movement is

( M )

When the As-U-Rolls are moved by a certain

described here. When the As-U-Roll is moved vertically

upwards or downwards the mechanics are designed in such a way that the centre c^ of disc B rotates about a fixed point c^ (see section 2.7.3) as shown in fig. 4.4a. If c

is the distance between c^ and c^, the centre c^

describes a circle of radius c with centre at c^. This means that, since disc B is solid, any point on the disc describes a circular arc with radius c. Let z be the vertical movement made by the As-U-Roll. Since the disc B is geared to racks, any point on the circumference also experiences a net movement of z. This point also rotates about c^ and therefore the angle of rotation 0 about c^ as shown in fig. A. 4b is given by

where R is the distance between c^ to the racks. Therefore the net vertical distance y travelled by c^ or any other point on the circumference is given by

On either side of each segment there are two such discs which can be moved independently. The profile of the back-up roll between two racks are calculated assuming a linear relationship. Fig. 4*. 5^ shows the profile when the first rack is moved by a distance z^ with zero movement

(4.11)

y = c-sin0 = o-sin( ). (4.12)

in all other racks. The profile y^(x^) can be calculated from

where Z is the distance between two racks. Fig. 5b shows the profile when only the first and second racks are moved by distances z^ and ’ z^. In this case the profiles yi(xi) and corresponding to first and second segments are given by

The profile for the case when alternative racks are moved by the same amount (say z^) is shown in fig.4.5c. Fig.*h51 shows the profile when all racks are racks are moved by the same amount then obviously the whole of the back-up roll will be moved vertically. The same result can be achieved by moving the two screwdowns situated at both ends of the back-up roll, by the same amount.

(4.14) and

(4.15)

moved by different amounts (say z^, z^, Zg). When all