Roll crown for the specific avoidance of quarter waves

A roll stand for producing flat rolled stock comprising:

• the roll stand has roll stand uprights;
• work rolls, or work rolls and backup rolls or work rolls, intermediate rolls and backup rolls are all mounted in the roll stand uprights;
• each of the rolls mounted in the roll stand uprights is rotatable about a respective rotational axis;
• where the work rolls or the work rolls and backup rolls are mounted in the roll stand uprights, the work rolls are supported in the uprights to be displaceable axially with respect to one another along the directions of their respective rotational axis, and where the work rolls, the intermediate rolls and the backup rolls are mounted in the roll stand uprights, the work rolls or the intermediate rolls are displaceable axially with respect to one another along the directions of their respective rotational axis;
• the work rolls or the intermediate rolls that are axially displaceable with respect to one another each has an effective barrel length (L);
• the work rolls or the intermediate rolls that are axially displaceable with respect to one another each has a curved contour, which extends over the entire effective barrel axial length (L);
• one of the two work rolls or of the two intermediate rolls that are axially displaceable with respect to one another has a first curved contour with a first radius (R1) at each location along the rotational axis thereof, the first radius (R1) corresponding to a combination of a base radius (R0) of the first curved contour and superposition of a first basis function (B1) and a first additional function (Z1);
• the other of the two work rolls or of the other two intermediate rolls that are axially displaceable with respect to one another has a second curved contour with a second radius (R2) at each location along the rotational axis thereof, the second radius (R2) corresponding to a combination of a base radius (R0) of the second curved contour and superposition of a second basis function (B2) and a second additional function (Z2);
• the basis functions (B1, B2) satisfy the relationships
• B
• ⁢
• ⁢
• 1
• =
• +
• A
• ·
• sin
• ⁡
• (
• 2
• ⁢
• ⁢
• φ
• L
• Ref
• ⁢
• (
• x
• +
• c
• )
• )
• -
• B
• ·
• x
• ⁢
• ⁢
• and
• B
• ⁢
• ⁢
• 2
• =
• -
• A
• ·
• sin
• ⁡
• (
• 2
• ⁢
• ⁢
• φ
• L
• Ref
• ⁢
• (
• x
• -
• c
• )
• )
• +
• B
• ·
• x
• or the relationships
• B1=+A′(x+c)5+A(x+c)3−B·x and
• B2=−A′(x−c)5−A(x−c)3+B·x
• and the additional functions (Z1, Z2) satisfy the relationships
• Z1=−α·Cx2−β·DX4 and
• Z2=−(2−α)·Cx2−(2−β)·Dx4
• or the relationships
• Z1=−α·C cos λx and
• Z2=−(2−α)·C cos λx
• where
• B1 and B2 are the first and second basis functions,
• Z1 and Z2 are the first and second additional functions,
• A and A′ are contour amplitudes,
• φ is a contour angle,
• LRef is a reference length,
• x is the location or the axial position, with respect to the center of the barrel,
• c is a contour displacement,
• B is a contour pitch,
• α and β are weighting factors,
• C and D are proportional factors and
• λ is a factor.