sin(0 o Br)

'7 ■ . L ( » ; - h v ) ?- _{5 7} ... (3-36)

and Fc = F7sin(e7 + e4)

...(3>37)

sin(05 - ^ )

(The same equations will yield F _ and F _ as above)._{/K ^>K} Now consider roll J. It is known that 0 g =

from previous discussion, therefore sufficient information is available to obtain Fg and Fgp by horizontal and

vertical resolution as:

F si nO^ - 0 „) + F sin(0£ + ^ p ) Ffi = -2--- 2---- 3---- 2*--- S---- , (3 .38)

sin20g

and F sin0o - F_T)sin0or)

f6R = F6 + -J _{s m P g} ^ ... ... (3.39)

The remaining unknown angles 9 g and ®g^ can now be found. At roll B, resolving perpendicular to Fg gives

and solution in the direction of Fg gives

Fg = FgeosOg - © g ) + F^cos (

8-

F6E> F7E’ & 6E and i*1 the above equations. The reader may, if he so wishes, verify that the overall effect of reactions from the mill housing balances the rolling load i . e,

f5c°s 5 + F g C O S g + FgRcos gR + F 5 Rc o s 5 R . Pt

The various equations in this section therefore completely specify the geometry and overall load dis­ tribution pattern in the upper half of the mill cluster as shown in figure 3»l6. A. set of results is given below for screwdown rack at 8 operator's display units, left

and right hand side eccentrics at 5 and 7 units respectively and mean As-U-Roll rack position at +1 unit.

e, 0 F,₁ -Pm_T CD _to 3 37.5° _{e2E = 38.3° F2 =} 0.639P_t F2R = 0.628Pt = 22.4° _{®3R = 21.7° F3 = 0.24lPT F 3R =} 0.228Pt CD =

59-5°

position for the screwdown rack position, setting 0^=zero (no As-U-Rolls in lower half) and applying the analysis from equation

Although this section has defined the overall pattern of load distribution, it is of course necessary to examine the way in which the load varies across the mill for shape control purpose. This is considered in a later section.

3.4 Rolling Load and Roll Flattening Calculations In order to quantify the forces discussed in the previous section it is now necessary to obtain knowledge of the rolling load (P^). On a four-high mill, this can be had from load cells placed between the mill screws and backup roll chocks. In the Sendzimir mill it would be difficult (not to mention extremely costly) to obtain a direct measurement of rolling load, and only an indirect indication is available. This takes the form of indication of the differential pressure in the hydraulic screwdown cylinderso From knowledge of the cylinder dimensions, this can be converted to rack-pull in some convenient units (e.g. MN). The mill manufacturers then provide a rule-of-thumb conversion from rack-pull to rolling load. Accurate measurement of rolling load is therefore un­ available.

The measurement of rolling load is not however necessary for operation of the on-line control scheme, but only for use in the static mill model (which is run off-line). The value of rolling load used in the static

P,p = f (w, h_^ ,h^ 3T_^ , , k E

$

)

(where the various symbols are defined at the beginning of the chapter).

The drawback is that since no accurate measure of mill rolling load is available,it is not easy to assess the accuracy of the value of thus calculated. For this reason, steps have been taken to allow the use of a rolling load model which is well tested, but which has often been rejected in the past, for models requiring rapid execution, on the grounds of computational

difficulties.

The ’'yardstick’1 as it were, by which the accuracy of cold rolling models has traditionally been judged, is the work of Orowan (22), although this work itself is not suitable for efficient computer mechanization. The work of Orowan set this standard of accuracy by removing arbitrary simplifications imposed by previous models. Due to the need for more rapidly evaluated models,

various workers have simplified the theory by judicious re-incorporation of some of the simplifying factors, made possible by the understanding of Orowan's work. Typical examples are given in (23) to (27), but the most widely accepted of these from an accuracy point of view is

probably the theory of Bland and Ford (23). Unfortunately Bland and Ford's model involves iterative solution of

implicit simultaneous equations; which renders it, at first sight, unsuitable for use in models (such as the present static model - see later sections) requiring

several rolling load evaluations. To overcome this difficultys Bryant and Osborn (26) have proposed an

explicit solution by introducing further simplifications and Carlton, Edwards and Thomas (28) have subsequently extended this work. Despite the simplifications, the model of Bryant and Osborn compares acceptably with the more accurate models under certain conditions, and has

been used by other workers in the area under discussion(l4), The author has removed some of the objections to

the use of Bland and Fords 1 model (in a mill off-line static model) by the use of a fast, but little known, algorithm for solution of the equations, (Note that for applications requiring on-line calculation of rolling load, such as mill scheduling and automatic set up systems, this method would probably still not be fast

enough under stringent timing constraints). The algorithm involves the use of a modification to the secant method, which can have a greatly beneficial effect upon the

solution time under certain conditions - in the case of this static model, convergence to within 0.3% is achieved after typically only two iterations through the process outlined below.

Bland and Ford's model assumes that the arc of contact remains circular during rolling, as depicted in figure 3.17. The deformed roll radius is given by

P (N/m)