1 2 0 CARTA SEXTA Y ÚLTIMA

Back Tension = 134*103N

Front Tension = 191*103N

This set of data yields the cluster angles and force components (see fig.3»l6) below, which may be compared

with those given in section 3*3 (following equation (3«4l)).

3 0° _{F1 = PT} ®2 = 40.3° F2 = 0.535 PT e3 = 23.3° F3 S 0.219 PT *4 = 59.7° F4 = 0.391 PT e_ _{= 78.3° F5} ss O .336 PT CD _<J\ = 40.8° _{F6 = 0.266 PT} e7 s 4.4° F7 = 0.191 PT CD _{CO = 22.2° F8 3 0.423 PT}

(the cluster is symmetrical under these conditions so the right-half values are identical).

The rolling load P^, is calculated as 3°23*10^N under these conditions, with a deformed workroll radius of

-3

30.9*10 ^m, This represents a fairly light loading for this mill.

The gain matrix corresponding to this standard set of data, and evaluated at eight points across the strip is given in Appendix 3 (section A3.8)0 Every element in the matrix has been treated with a simple scalar multiplier (the same for each element) to give these

values. This was done in the light of early plant tests, to give values in closer agreement with reality than

the untreated model. The multiplier is 0.0054. One possible reason for this requirement is the simplistic approach to the calculation of the actual , strip shape

(equation (3.113)) in which Young*s modulus is used as a multiplier. It is very probable that, due to the

plastic nature of the rolling process, a value of gradient on the portion of the stress-strain curve above the yield point should be used rather than Young's modulus (which is, of course, the gradient below the yield point). For the material in question, this upper portion of the curve flattens out very significantly, and the factor given above is quixe feasible. All the matrices to be dis­ cussed have been processed in this way to allow direct comparison.

The matrix of A5 .8 may be compared with that given in A p •9 which was derived by Gunawardene for similar

conditions (Ref.16, section 6.6, p.151 ). It can be seen that the two models are in good basic agreement, although the present model's computer execution time is only a small fraction of that of Gunawardene8s model. Further­ more, it can be seen that the matrix produced by the present model exhibits the absolute symmetry which is expected under the conditions for which it was run

(i.e. g^j s g ^ ^ (9 j)^* whereas numerical errors in Gunawardene*s model have disrupted this to some extent

Another feature of the matrix is that, as expected, vertical columns sum to zero (within rounding errors);

since shape is displayed with respect to mean the

average value across the strip (down the column) must be zero. The horizontal rows should also sum to zero (if each As-U-Roll is moved by the same amount , 3. pure gauge

change will result - not a shape change), but this is not actually the case , small errors being present. The

discrepancy is due to numerical errors, but is generally less than the errors in the matrix of A3.9- One unusual feature common to both models is that As-U-Roll number 2 appears to have a greater effect upon the portion of strip nearer to As-U-Roll number 1 than does As-U-Roll number 1 itself (i.ec This is at first sight, incorrect, and has not been conclusively observed on the plant, but a tentative explanation is possible (this applies to the present model, i.e. to the matrix given in A3 . 8 ) and is now proffered.

Figures 3*4l(a) to 3*4l(d) give the workroll deflection graphs produced by the model during calculation of the

matrix of A3 .8 . Graph (a) is the deflection due to motion of As-U-Roll 1 only, graph (b) is for As-U-Roll 2 only, etc. As-U-Rolls 8 ,7$6 and 3 simply produced mirror images of the graphs for As-U-Rolls 1,2,3 and 4 re­

spectively. Consider graphs (c) and (d), and notice that the deflection profiles are extremely similar both in form and magnitude, being simply shifted laterally to

coincide with the appropriate As-U-Roll position. This is due to the fact that for both these As-U-Rolls there is

o ~

r~

m o 3.'31 D.F:?. 3. cr>2 1.23

D ] STRNCF. RLONG ROi

Fig. 3-41 Contd

plenty of* strip to either side of the As-U-Roll location, and the strip edges have no effect therefore. Consider now graph (b). Here, the edge of the strip becomes

significant. It is placed at 0.045m on the horizontal scale, whilst As-U-Roll 2 is at 0.275m. The downward motion of. the As-U-Roll rack will therefore cause a certain amount of force on that part of the workroll which is unsupported to the left of the strip edge. This is treated as a

contilever and is expected to deflect much more than when strip is present (as will be seen by comparing the maxi­ mum deflection of graph (b) with graphs (c) or (d)).

Graph (b) exhibits two distinct portions to the deflection. The portion over the strip (from about O.l^m to 0.92m)

exhibits similar behaviour to the right-hand portions of graphs (c) and (d), tending to "bottom out" at

approximately -2.8*10 -m. The cantilevered portion, however, deflects more easily and causes thinning of the strip edge as it runs into the supported portion thus building up the entire graph. Now, in graph (a),

As-U-Roll 1 is virtually coincident with the strip edge. Therefore, its influence on the supported portion of the strip is not as great as that of As-U-Rolls 2,3 and 4. The "supported" portion of the deflection curve, such as it is, appears to "bottom out" therefore at say -1.5*10“^m. However, the As-U-Roll (l) is not actually over the un­ supported portion of the workroll (as would be the case

that due to As-U-Roll 1, hence the entries in the gain matrix. It is stressed that the Author does not have

great confidence in this "explanation” , and clearly more work is possible in this area.