Do you know why the axial roll is cone shape?

The three main rolls in the ring mill are the main roll, the mandrel and the axial roll. This means that these three rolls wear quickly and that these rolls should be replaced frequently. And among these three rolls, axial roll is the most expensive roll. Because the axial roll is cone shape, it is difficult to machine. 
Why is the axial roll not a cylindrical shape that is easy to machine, but a cone shape that is difficult to machine? I have to explain the linear velocity of the axial roll to tell the reason. I explained the linear velocity of the main roll before 
YouTube(What is the linear velocity)
In the ring mill, the main roll and axial roll rotate the ring.  The linear velocities of the ring by the main roll(V1) and by the axial roll(V2) should be same.  This ensures that the ring remains in the center of the ring mill. When the linear velocity(V2) of the axial roll is faster or slower than the linear velocity(V1) of the main roll, the center of the ring will be offset from the center of the machine. If the center of the ring does not match the center of the equipment, the ring is not round but it could be a oval.  Moreover there could be various defects in the shape of the ring.




The linear velocity Vm on the main roll surface is Vm = Rm * Nm (Rm: main roll radius, Nm: main roll rotation speed(rpm)). The linear velocity of the ring at the contact point with the main roll is same as the linear velocity of the main roll(Vm) assuming there is no slip between the main roll and the ring. 
The linear velocity of the ring(Vr) is proportional to the radius of the ring(Rr).
Vr = Rr * Nr
(Vr: linear velocity of the ring, Rr: radius of the ring, and Nr: rotational speed of the ring(rpm)).
The linear velocity of the ring is zero at the center of the ring and is maximum(V) at the outer surface of the ring.  The linear velocity at the outer surface of the ring is not the same as the linear velocity at the inner surface. The linear velocity of the ring is not a single value and changes linearly from the inner surface of the ring to the outer surface. The total amount of the linear velocity of the ring is the area of ​​the linear velocity distributed from the inner surface to the outer surface.



The linear velocity is constant at the outer surface of the ring in contact with the main roll.
* The linear velocity at the outer surface of the ring is constant.

But the linear velocity at the top and bottom surface of the ring is not constant.  Since the radius of the ring at each point on the top and bottom of the ring is different, the linear velocity at each point is different.
* The linear velocity at the top and bottom surface of the ring is not constant but varies linearly.

The shape of the axial roll is a cone shape because the linear velocity at the top and bottom face of the ring in contact with the axial roll is different according to the radius of the ring.  Because the axial roll is a cone shape, the radius of rotation on the surface of the axial roll varies linearly.  That means the linear velocity of the axial roll varies linearly like the linear velocity of the ring's top and bottom surface in contact with the axial roll.
* The linear velocity of the axial roll varies linearly because the axial roll is cone shape.

In the case of the main roll, the linear velocity on the main roll surface is constant because the radius of rotation is constant. Therefore, the linear velocity at the outer diameter surface of the ring in contact with the main roll is also uniform.  However, the linear velocity at the top and bottom surfaces of the ring in contact with the axial roll varies linearly with the radius of the ring.  Because the axial roll is a cone shape, the radius of rotation at each position of the axial roll surface changes, so that the linear velocity also changes linearly.  The linear velocity at the vertex of the cone, which is the center of rotation of the axial roll, is 0, and the linear velocity of the axial roll at the outer surface of the ring where the TR(Tracer Roll) touches is the maximum.  When the vertex of the axial roll and the center of the ring coincide, the linear velocity distribution of the ring exactly coincides with the linear velocity distribution of the axial roll.
* The linear velocity of the axial roll matches the linear velocity of the ring because the axial roll is cone shape.

If the axial roll is a cylindrical shape rather than a cone shape, the linear velocity of the ring on the top and bottom of the ring in contact with the axial roll is not linearly distributed but constant.  Therefore, the linear velocity of a ring that varies linearly with the radius of the ring can not match the linear velocity of the axial roll.
* If the axial roll is cylindrical shape, the linear velocity of the axial roll can not match the linear velocity of the ring.


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