Materials with Complex Behaviour II pp 213-227

Part of the Advanced Structured Materials book series (STRUCTMAT, volume 16)

# Numerical Investigation of Chatter in Cold Rolling Mills

Chapter

## Abstract

In this work, considering a four high cold rolling mill and using a dynamic friction model, expressions for the variation of pressure in the roll bite have been developed. The effects of parameters used in the dynamic friction model on the variation of pressure and shear stress are investigated. The numerically obtained horizontal and vertical work roll deflections using the dynamic friction model have been compared with those obtained by the conventionally used constant friction model. The effects of rolling parameters like strip thickness; periodic back tension and strip velocity on the work roll deflections have been studied. This work will find applications in predicting the critical system parameters in cold rolling to avoid chatter.

### Keywords

Chatter Cold rolling mill Dynamic friction model Constant friction model Roll bite

### Nomenclature

M

Mass per unit length of the work roll (kg/m)

y

Vertical displacement of the work roll (m)

$$f^{s}$$

Reaction force from metal sheet (N/m)

$$D_{w}$$

Diameter of work roll (m)

$$D_{b}$$

Diameter of backup roll (m)

E

Young’s modulus of the material (GPa)

$${{\upmu}}$$

Poisson’s ratio of the material

$$f_{s}^{s}$$

$$f_{d}^{s}$$

Dynamic sheet force (N/m)

$$y_{s}$$

Work roll displacement due to the steady sheet force (m)

$$y_{d}$$

Work roll displacement due to the dynamic part of sheet force (m)

$$\dot{y}_{d}$$

Rate of change of dynamic roll gap displacement (m/s)

$$h_{c}$$

Gap between two work rolls (m)

$$h_{c0}$$

Gap between two work rolls at t = 0 (m)

$$\dot{h}_{c}$$

Rate of change of roll gap (m/s)

$$\omega_{n}$$

Natural frequency of the system not considering $$f_{d}^{s}$$ (Hz)

$$h_{1}$$

Strip thickness at entry (m)

$$h_{2}$$

Strip thickness at exit (m)

R

$$u_{1}$$

Strip velocity at entry (m/s)

$${{\uptau}}_{y}$$

Strip shear yield strength (MPa)

$$\sigma_{XX}$$

Normal stress in X-direction (MPa)

$$\sigma_{XY}$$

Normal stress in Y-direction (MPa)

$${{\uptau}}_{XY}$$

Shear stress (MPa)

m

Contact friction coefficient between the work roll and the strip

$${{\uptau}}_{\text{s}}$$

Shear stress at the surface of strip (MPa)

$$x_{1}$$

Distance measured from strip entry to the centerline of rolls (m)

$$x_{2}$$

Strip exit position (m)

$$x_{n}$$

Distance of neutral plane from the centerline of rolls (m)

$$m_{1}$$

Friction factor between $$x_{n}$$ and $$x_{1}$$ (considered positive)

$$m_{2}$$

Friction factor between $$x_{n}$$ and $$x_{2}$$ (considered negative)

p

Roll pressure (MPa)

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## Authors and Affiliations

1. 1.Mechanical Engineering DepartmentIndian Institute of Technology GuwahatiGuwahatiIndia
2. 2.Institute of Engineering and Computational MechanicsUniversity of StuttgartStuttgartGermany