# Forces and coefficient of friction during cylindrical three roller bending

Technical Paper

## Abstract

Three roller bending process is one of the processes of forming cylindrical shells, for manufacturing drums, pressure vessels, windmill towers, etc. During the bending process, forces on the rollers must not exceed their bending capacity. Along with operational and material parameters, the forces exerted during bending and power consumption are affected by the coefficient of friction at the roller-plate interface. It is difficult to determine the coefficient of friction practically during three roller bending process. An attempt is made to determine the forces and friction coefficient at each roller–plate interface, through derived mathematical model, using experimental results. The variation of coefficient of friction under the higher loading conditions was also determined as a case study. With the present mathematical formulation along with the coefficient of friction, the forces at each roller–plate interface could be determined more precisely. The determination of forces and friction coefficient helps in deciding operational and design parameters of the bending machine, respectively.

## Keywords

Three roller bending Coefficient of friction Bending forces Bending power Cylindrical bending Rolling friction Friction

## Nomenclatures

a

Distance between the axis of two bottom rollers

ae, ax

Perpendicular distance of the vertical forces on the bottom roller at the entry and the exit side, respectively, from the top roller contact point

Ftangential

Tangential force acting at the interface of the two bodies

Fnormal

Normal force acting at the point of contact of the two bodies

Ht, He, Hx

Horizontal force on the top roller the bottom roller at the entry and the bottom roller on the exit side, respectively

It, Ib

Moment of inertia on the top roller and the bottom rollers, respectively

J

Normal force at the contact between the plate and the bottom roller at the entry side

K

Strength coefficient

Me, Mx

Bending moment of the resultant force at the bottom roller on the entry and the exit side, respectively, about the top roller contact point

Mexternal

External bending moment based on applied forces

Minternal

Internal bending moment based on the geometry of material its properties

Nt, Nb

Revolution per minute of the top roller and bottom rollers, respectively

n

Strain hardening exponent

P

Normal force at the top roller contact point

pt, pe, px

Power consumed during bending at the top roller the bottom roller at the entry and the bottom roller on the exit side, respectively

pe-shaft, px-shaft

Power consumed in rotating just the shaft of the bottom roller at the entry and the bottom roller on the exit side, respectively

ppl

Power consumed only in bending and driving plate across the rollers. It does not include the power consumed by motors and shaft of the rollers

pn(av)

Average of pe-shaft and px-shaft Q Normal force at the contact between the plate and the bottom roller at the exit side

RP, RJ, RQ

Resultant force at the top roller the bottom roller at the entry and the bottom roller on the exit side, respectively

R

Radius of the circle tangent all the three rollers

rt, rb

Radius of the top roller and the bottom rollers, respectively

t

Thickness of the plate

U

Displacement of the top roller

ue, ux

Perpendicular distance of the horizontal forces at the bottom roller at the entry and the exit side, respectively, from the top roller contact point

Vt, Ve, Vx

Vertical force on the top roller the bottom roller at the entry and the bottom roller on the exit side, respectively

W

The weight of the plate

xe, xx

The horizontal distance of the contact points of the bottom roller at the entry and the exit side, respectively, from the reference point

ye, yx

The vertical distance of the contact points of the bottom roller at the entry and the exit side, respectively, from the reference point

Λ

Angular acceleration of the top and the bottom rollers

μ

Friction coefficient (CF)

μt, μe, μx

CF at the roller-plate interface for the top roller, the bottom roller at the entry and the bottom roller on the exit side, respectively

θt, θe, θx

Contact angle at the top roller, the bottom roller at the entry, and the bottom roller on the exit side, respectively

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