^{1}

^{2}

^{2}

^{1}

^{2}

Nonlinear dynamic rolling forces in the vertical and horizontal directions are, respectively, established, considering the impact of vertical and horizontal directions vibration of rolls. Then a vertical-horizontal coupling nonlinear vibration dynamic model of rolling mill rolls is proposed, based on the interactions between this dynamic rolling force and mill structure. The amplitude-frequency equations of the main resonance and inner resonance are carried out by using multiple-scale method. The characteristics of amplitude frequency under nonlinear stiffness, damping, and amplitude of the disturbance are obtained by adopting the actual parameters of 1780 rolling mills. Finally, the bifurcation behavior of the system is studied, and it is found that many dynamic behaviors such as period, period-3 motion, and chaos exist in rolling mill, and this behavior could be restrained effectively by choosing proper system parameters.

The vibration of rolling mill often occurs in rolling process. The occurrence of vibration not only affects the quality of rolling products, but also leads to breakdown of the rolling equipment. In order to understand the vibration behaviors of mills, a number of models have been developed during the past few decades [

In this paper, nonlinear dynamic rolling forces in the vertical and horizontal directions are proposed, respectively. Then a vertical-horizontal coupling vibration dynamic model of rolling mill rolls is constructed based on the interactions between this dynamic nonlinear rolling force and mill structure. Then the amplitude-frequency characteristics of the main resonance and inner resonance are analyzed under the nonlinear stiffness, damping, and the amplitude of the disturbance stiffness. Finally, the conditions of different dynamical motions are obtained by analyzing bifurcation behavior of the system, which could provide theoretical base for understanding of vibration mechanism of mill.

Considering the influence of vertical and horizontal vibration of roll, the dynamic deformation process of strip is shown in Figure

The dynamic deformation process of strip.

As shown in Figure

Considering the elastic flattening of the rolls, the roll gap is treated as a parabolic curve [

Assuming that the thickness

Then the velocity at entry position along horizontal direction can be derived form (

Considering the influence of roll horizontal vibration, the equivalent velocity at the entry position will be composed of two parts: the entrance velocity

From Figure

Under vibration conditions, the bulk of metal in deformation zone is not constant, so the equation of constant mass throughput is no longer suitable for vibration conditions. Then a new principal of metal flow per second will be proposed, and the dynamic flow at any arbitrary position

Where the volume in deformation zone range from

Then the speed

When the horizontal component of surface speed of roll is equal to (

According to slab method [

The pressure diagram of strip.

Based on force balance theory in deformation zone by Von Karman [

By taking account of the assumption of homogeneous deformation [

Integrating (

The unit rolling force by (

The rolling force in horizontal and vertical direction can be obtained as follows:

In (

Set

Based on the assumption that the mass of working rolls are much smaller than that of backup rolls, the mass of the working rolls may be neglected [

The vertical-horizontal coupling dynamic model of rolling mill rolls.

The dynamic equation in Figure

Assuming that the structure of rolling mill and vibration are symmetrical in relation to the rolled strip [

Under steady conditions, the external disturbance force

Substituting (

Substitute (

Equation (

Equation (

Assuming that the external disturbance

By using multiple scales method, one has

Set (

Substituting (

Set the solution of (

Substituting (

In the case of main resonance, set

The polar coordinate form of

By substituting (

Assuming that

Substituting (

When the system has a periodic motion, (

Taking the 1780 rolling mills of Chengde Steel Co as an example, the parameters of this mill are listed as follows:

Parameters of rolling force can be listed as follows: ^{3}, ^{3}, ^{3}, and ^{3}.

Figures

Main resonance amplitude-frequency curve of different nonlinear stiffness

In Figure

In Figure

Main resonance amplitude frequency curve of different structural damping

Main resonance amplitude frequency curve of different nonlinear stiffness

Figure

In Figure

Curve of main resonance amplitude frequency with different disturb amplitude

Figures

Curve of inner resonance amplitude frequency in horizontal direction with disturb amplitude

When

When

Curve of inner resonance amplitude frequency in vertical direction with disturb amplitude

When

When

In Figure

In Figure

According to (

Bifurcation characteristics of coupling system with disturb amplitude

The phase diagrams and Poincare maps are shown in Figures

Periodic motion when

Phase diagram

Poincare map

Figure

Period-3 motion when

Phase diagram

Poincare map

Chaotic motion when

Phase diagram

Poincare map

The nonlinear rolling force model of rolling mill in the vertical and horizontal directions is built. On this basis, the dynamic model of nonlinear vertical-horizontal coupling vibration model of rolling mill is proposed, considering the influence of mill structure.

By means of multiple-scale method, the amplitude-frequency equations of main resonance and inner resonance of coupling system of rolling mill rolls are carried out. The simulation adopting the actual parameters of rolling mill is analyzed. It is found that the amplitude of vibration increases with an increase of stiffness and external disturb; but the maximum value of the main resonance will decrease as the increase of structure damp; when changing nonlinear stiffness, jump phenomenon will arise both in main resonance and in inner resonance, so choosing proper parameter will restrain resonance vibration of rolling mill.

The bifurcation characteristics of vertical-horizontal coupling system of rolling mill roll are studied, and it is found that the system has different motions such as period motion, period-3 motion, and chaos, and choosing proper parameters may change the motion state of rolling mill.

Arbitrary distance from the centerline of the rolls

Distance of the exit plane from the centerline of the rolls

Distance of the entry plane from the centerline of the rolls

Distance of the neutral plane from the centerline of the rolls

Variation of the horizontal displacement of rolls

Rate of change of the horizontal position of roll bite

Rate of change of roll horizontal displacement

Arbitrary distance from the asymmetry line of the rolls

Roll vertical displacement

Strip velocity at exit

Strip velocity at entry

Roll velocity

Strip horizontal velocity at any arbitraty position from the centerline of the rolls

The equivalent horizontal velocity at entry

Variation of the strip thickness at exit

Strip thickness at entry

Strip thickness at any arbitrary distance from the centerline of the rolls

Forward tensile stress at exit

Backward tensile stress at entry

Horizontal tensile stress at any arbitrary distance from the centerline of the rolls

Shear stress

Interface pressure

Friction factor

Shear yield strength

The rolling force in horizontal direction

Rolling force in vertical direction

Roll radius

Volume flow in deformation zone range from

The rate of volume flow change in deformation zone range from

Equivalent stiffness between upper rolls and upper supporting posts

Equivalent stiffness between upper rolls and upper beam

Equivalent stiffness between lower rolls and lower supporting posts

Equivalent stiffness between lower rolls and lower supporting posts

Equivalent damping between upper rolls and upper supporting posts

Equivalent damping between upper rolls and upper beam

Equivalent damping between lower rolls and lower supporting posts

Equivalent damping between lower rolls and lower supporting posts

Equivalent mass of upper rolls

Equivalent mass of lower rolls

External disturbance of upper rolls

External disturbance of lower rolls.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This research is supported by National Natural Science Foundation of China (Grant no. 51105324), Natural Science Foundation of Hebei Province of China (Grant no. E2014501006), and Hebei Province Science and Technology Support Program (Grant no. 13211907D).