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This quantity of differential stress is a measure of strip shape, and is displayed by the ASEA Stressometer shapemeters on the Sendzimir mill in question.

Substituting (3*11^)i (3»H5) and (3*116) into (3.117) gives A ^n = Es • JH hN 1 - H HN Q=1 i H S. Q _o (Nm““ ) . • . .(3.118) for N=19...9 JH

Thus, shape is positive where the strip is tighter than the mean, and negative where it is slacker. For strip to have "perfect” shape, this internal stress dis­ tribution equation should give = 0 for all N. Non­ zero values of lead to the internal stresses in the strip forming "latent" (bad) shape. If these stresses grow large enough to overcome the section modulus of

the material, then the strip will visibly buckle, forming "manifest" (bad) shape (see section ;lv, chapter 1.).

The input gauge profile of the strip (hN in equation (3 *118)) is known either from an estimate of the

characteristics of the incoming strip (on the first pass) or by reading the output gauge profile stored at a number of points during the previous pass. The output gauge

profile (H^j) is calculated as follows, using the knowledge of the mean output gauge (from the plant instrumentation)

The values of given by equation (3 . H 8 ) must be given at poinxs which coincide with the centres c>f the rotors on the ASEA Stressometer, if any simple model/plant comparison is to be made. Let the shapemeter have JD

rotors of width L^(m). Note that is an odd number (31 for the present Z mill) so as to place a rotor centre at the strip centre. Fig.3.38 shows the strip passing over the segmented shapemeter roll. The number of shape­ meter rotors covered by the strip is given by

Ls .

Tp-

R

which will probably not be an integer at this stage, but must be made such. Since is an odd integer, and fig. 3.38 is symmetrical about its vertical centre-line, then the number of shapemeter rotor centres covered by the strip (Jjj) must also be odd. An integerised version of

(3.119) is obtained by truncating its fractional part:- L,

fractional part set to zero.

This is then tested to ascertain whether it is even or odd (e.g. by dividing by 2 and testing for a remainder). Consideration of fig. 3.38 shows that if i^ is even, then Jjj = ijj+1; whereas if i^ is odd it needs no

The value of L in fig. 3-38 is then found hy­_{ps °}

Ls “

lr

^

L _PS = Cm;

o

The distance from the LH end of the mill at which *fcll

the N value of gauge must be calculated to coincide -with a shapemeter rotor centre, can then be found as

x*T s _N Ltt_U + L + (N-ljL^ (m) for Nsls...,J„ . • .(3.121)_{ps R a H}

Now the WR deflection is known at M^. points along the roll (from section 3-9) 9 also measured from the front of the mill, but it is most improbable that the values of x^ given by (3.121) will correspond precisely with

values from the set of M^. values of x^ used in section 3*9 to find y^. . To find WR def lection at points corresponding

M

to (3.121) therefore, a curve could be fitted to the values of y^. previously calculated for the WR (section

XM

3.9J and the Jw values of y read off it. Fitting a XN

single high-order curve and interpolating in this way, is prone to numerical inaccuracies however, and"the method employed instead (see below) is more accurate, although somewhat laborious to set down on paper. (The computer mechanization is quite simple of course).

For each x^ value given by (3.121), a search is made through the values (at which WR deflection is known) until the nearest value of x-w to x„T is found. The x.,_{M N M} values on either side of this value (i.e. x^ ^ and

are also taken: so that three values of* x., are con-_{’ M} sidered, with the value of* falling within the range

2

of the three. A quadratic of the form y= aK +bx+c is then fitted to the three points. We therefore have

Wx a-vr X,, _{N M-l} + b,, X,, n + C , T_{N M-l N} M-l r _VT = a,T x,, + b,T x,. + c,T_{N M N M N} XM = aXT x,, n + b.T x,. _ + c,, y^ N M+l N M+l N

bN =

yw

”yw

2 2

yw - yw

2

2

2 2

2

2

The WR deflection corresponding to the point x^ measured from the front of the mill (i.e. corresponding to a shapemeter rotor centre) is then given by

(Note that if a value of should fall so close to one end of the WR that the nearest x., value is the last on the_M

give a smooth deflection profile, the fitting of a quad- radtic to any three consecutive points will introduce negligible errors.

When rolling strip in a four-high rolling mill, the conditions around the roll-bite are such that if a bending profile is forced onto the upper WR, then the lower WR will always adopt the inverse profile. The strip will therefore always look symmetrical about its horizontal axis. It is thought however, that this condition will not apply in a Sendzimir mill, since the lower WR is not free to deflect to the same extent. Therefore, if the in­ coming strip has the gauge profile of fig. 3»39(i) and good shape, to maintain the good shape whilst imparting a per-unit reduction the roll-bite must adopt in the limit the profile shown in fig.3-39(ii) (neglecting elastic recovery of gauge and the lower WR camber). As has been mentioned previously, although fig. 3«39(ii)

looks extreme, it is grossly exaggerated as c^ is some four or five orders of magnitude less than L^. Due to this relatively minute lateral bending of the strip, it is expected that it will elastically recover after rolling roll, then the end three values of x^ are used).

Since the number of forces taken to act on the WR (JT_) and the number of points at which deflection of the_Wr WR was calculated were chosen to be large enough to

F'