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A coupled vibration model of hot rolling mill rolls under multiple nonlinear effects is established by considering the nonlinear spring force produced by the hydraulic cylinder, the nonlinear friction between the work rolls, the dynamic variation of rolling force, and the effect of external excitation as well as according to the structural constraints of a four-high hot rolling mill in the vertical and horizontal directions. The amplitude-frequency response equation of rolling mill rolls is determined by using a multiple-scale approximation method. Furthermore, use of actual data for simulation indicates that the internal resonance is the main cause of coupling vibration of the rolling mill rolls. In addition, changes in the movement displacement of the hydraulic cylinder and the coupling parameters strongly affect the coupling system of the rolling mill rolls. Finally, the study of the dynamic bifurcation characteristics of the rolling mill rolls indicates that, with varying external excitation amplitude, the vibration of rolls alternates between periodic motion, period-doubling motion, and chaotic motion in both vertical and horizontal directions. This is one of the reasons for the appearance of periodic light and dark stripes on the strip surface. Furthermore, the range of the external excitation amplitude (_{0}) at which the rolling mill roll system vibrates violently, that is, 5.68_{0} < 5.84_{0} > 6.12

Hot rolling is an important process in the manufacturing and processing of steel strips. It is a high-temperature and large-reduction process, which imparts large fluidity in the workpiece. In addition, with the application of various new processing technologies, rolling mill vibration is an inevitable problem, which potentially loosens and wears out the mechanical parts, shortens the rolling mill life, and causes vibration of steel strips, thereby reducing the overall product quality and accuracy [

Most studies on rolling mill vibration focus on a single vibration system, such as vertical, horizontal, or torsional vibration. For instance, Sun et al. considered the interaction between strip tension and thickness and analyzed the effect of multiple-source external disturbance on rolling mill vibration; they established a vertical vibration model of the rolling mill and further improved the control accuracy of tension and thickness of the hot rolling mill through optimization design [

In this study, we consider the effects of the nonlinear spring force of the hydraulic cylinder of a rolling mill, the nonlinear friction force between rolls, the nonlinear rolling force, and the structural constraints of the rolling mill and establish a coupling vibration model of a four-high hot rolling mill. We investigate the effects of tuning parameters, nonlinear stiffness, and coupling parameters on the amplitude-frequency characteristics of the vibration system. The simulation results show that a process of energy exchange occurs between the vertical and horizontal directions of the rolling mill vibration system. When the vibration parameters change, a jumping phenomenon occurs in both directions, causing instability of the coupling vibration system of the rolling mill. A change in the nonlinear spring force coefficient of the hydraulic cylinder considerably affects the vibration state of the rolling mill rolls, and changes in the coupling parameters lead to the complex nonlinear phenomenon of the vibration system. Thus, the coupling vibration model of a hot rolling mill established in this paper is confirmed to be effective. Finally, the bifurcation and chaos behavior of the coupled vibration system of the rolling mill under dynamic external excitation are studied. It is found that different periodic motions exist and the vibration alternates among different forms, which is one of the reasons for the appearance of periodic light and dark stripes on the strip surface. The results of this study can provide certain theoretical reference and technical support for reducing or restraining rolling mill vibration.

The mechanical structure of a hot rolling mill can be simply illustrated as shown in Figure

Structure of 1780 hot rolling mill.

Furthermore, because the diameter and rotation speed of the backup rolls and work rolls differ, when the rolling speed changes, the friction force on the strip changes, and the rolling force of the roll system is not in the vertical direction. In other words, during tension rolling, the impact load of the hot rolling mill is extremely large when it bites into the strip, and the bearing bracket will impact the mill housing, which will degrade the stability of the rolling mill system, and the rolling force of the rolling mill rolls does not act in the vertical direction. Therefore, to improve the accuracy of a vibration model of the hot rolling mill, the nonlinear vibration factors of the rolling mill rolls in the horizontal direction must be considered. Consequently, by considering the effect of the nonlinear force of the rolling mill in the vertical and horizontal directions and combining it with rolling mill vibration, we can better explore the chatter mechanism and vibration behavior of a hot rolling mill.

A hydraulic cylinder has many advantages in practical application and is widely used in the hydraulic screwdown device of a rolling mill. In this study, we consider a double-acting single-piston servo hydraulic cylinder in a hot rolling mill as an example for the analysis; its structure diagram is presented in Figure

Structure diagram of a double-acting single-piston servo hydraulic cylinder.

As Figure _{1} and _{2} are the effective areas of piston rodless cavity and rod cavity of the hydraulic cylinder, respectively. _{L1} and _{L2} are the volumes of oil in the pipeline between the valve and the rodless cavity and between the valve and the rod cavity, respectively. _{0} is the initial position of the piston,

During the working of the hydraulic cylinder, the piston can be regarded as a rigid body. The change in the piston displacement changes the pressure and oil volume in the two oil cavities, changing the oil stiffness accordingly. Therefore, _{e} is the elastic modulus of the oil volume and _{1} and _{2} are the volumes of the rodless cavity and rod cavity, respectively.

Because _{L1} and _{L2} are very small compared with the volumes of the two cavities and may be ignored, the spring stiffness of the hydraulic cylinder can be simplified as

Therefore, the spring stiffness of the hydraulic cylinder can be regarded as a function of vibration displacement, and the change law between them is shown in Figure

Curve of the spring stiffness with respect to the vibration displacement of the hydraulic cylinder.

By applying Taylor expansion to equation (

Due to the symmetry of spring potential energy

From the derivation of equation (

Equation (

The calculation accuracy of rolling force greatly affects the distribution of friction force and strip quality in the deformation zone. Many expressions for rolling force under different working conditions have been proposed by analyzing, simplifying, and regressing the characteristics of nonlinear coupling vibration of the rolling mill and dynamic components of rolling force based on actual tests [

Dynamic rolling process in the deformation zone.

As Figure _{b} and _{f} are the tension at the entrance and exit of the rolling, respectively. _{0} and _{1} are the thickness at the entrance and exit of the workpiece in steady state, respectively. _{2} is the thickness at the exit of the workpiece in dynamic rolling, _{2} = _{1} + _{μ} is the friction force on the work rolls:_{p} is the influence coefficient of the stress state, and _{c} is the horizontal projection length of the roll contact arc in the deformation area:_{0} − _{1} −

Considering that the friction coefficient between roll gaps varies with the fluctuation of rolling speed during rolling, Robert’s formula for friction coefficient is adopted [_{1} and _{2} are the friction characteristic coefficients, whose values are determined by specific friction model parameters; generally, _{1} = 0.51 and _{2} = 0.001. _{0} − _{1}; hence, equation (_{0} is the friction coefficient of the roll gap in the steady state and Δ

In steady-state rolling, the Taylor expansion of equation (

Here

Curve of rolling force affected by the coupling of vertical vibration displacement and horizontal vibration velocity.

According to the structure diagram of the four-high hot rolling mill shown in Figure

Nonlinear coupled vibration model of hot rolling mill rolls.

As Figure _{0} cos_{0} is the external excitation amplitude, and _{1} and _{1} are the equivalent stiffness and damping between the upper roll system and the frame and the archway column, respectively, and _{2} and _{2} are the equivalent stiffness and damping between the upper roll system and the steel strip, respectively.

The friction on the roll is small in the vertical direction and can be neglected. Therefore, from equations (_{μ} (

According to the Lagrange theory, the coupled vibration dynamic equation of the hot rolling mill rolls can be obtained as follows:

The multiscale method has strong problem-solving ability and good computing ability, and considering that it is widely used in solving coupled vibration system problems, we use the multiscale method to solve the dynamic response of the system [_{0} = _{1} = _{n} (_{n} = _{n}. The solution for equation (

Substituting equations (

The solution of equation (

Considering the internal resonance of the system, _{2} + _{1} = _{2} + _{1} are set, where _{1} are the tuning parameters; _{2} and represent the value range between them. To avoid the duration term,

We set _{1} = _{2} − _{1} − _{1}_{1} and _{2} = _{1} − _{2}. Considering the occurrence of a periodic movement in the coupling system of the hot rolling mill rolls, _{1} and _{2}, the amplitude-frequency response equation of the coupling vibration system can be written as

Table

Structural and technological parameters of the 1780 hot rolling mill.

Parameters | Value |
---|---|

_{e} (GPa) | 1.6 |

_{1} (m^{2}) | 0.6361 |

_{2} (m^{2}) | 0.3243 |

0.110 | |

_{0} (m) | 0.064 |

1.5 | |

0.42 | |

_{0} (m·s^{−1}) | 2.5 |

_{b} (MPa) | 5.5 |

_{f} (MPa) | 3.8 |

_{0} (m) | 0.0141 |

_{1} (m) | 0.0082 |

_{0} (°C) | 996 |

1.44 × 10^{5} | |

_{1} (N·s·m^{−1}) | 5.20 × 10^{3} |

_{1} (N·m^{−1}) | 7.31 × 10^{9} |

_{2} (N·s·m^{−1}) | 8.85 × 10^{5} |

_{2} (N·m^{−1}) | 2.08 × 10^{10} |

_{0} (MN) | 0.55 |

Ε | 0.01 |

Figure ^{−3}) and _{1} = −3.3. An energy exchange process is observed between the vertical and the horizontal direction, which is a special phenomenon that occurs in response to internal resonance. When tuning parameter _{1}≈_{2} and _{2}, which imply that the external excitation frequency is close to the natural frequency in both directions and that resonance occurs. In addition, Figure

Amplitude-frequency curve of coupling vibration of the hot rolling mill roll system.

Figure ^{−3}), there are multiple solutions, the amplitude increases, and the resonance region expands. This indicates that the change in the piston position of the hydraulic cylinder greatly affects the vibration state of the rolling mill rolls. In actual rolling, hydraulic oil with good temperature characteristics and large bulk modulus of elasticity should be used as much as possible. Moreover, seals with good performance should be selected to prevent an increase in the level of impurities due to the long-term use of oil. However, the change in nonlinear stiffness is caused by the change in viscosity, leading to abnormal vibration.

Amplitude-frequency curve of the rolling mill rolls as a function of parameter

Figure _{1}, which is composed of horizontal vibration speed _{1} < −3, there are multiple solutions and the vibration amplitude increases gradually, causing severe vibration of rolling mill rolls. The vibration form of the rolling mill in this range of coupling parameter is complex and variable. As _{1} increases, the vibration state of the rolling mill gradually stabilizes. This further indicates that the study of the coupling vibration of the hot rolling mill rolls is meaningful and would provide some guiding significance to explain the vibration phenomenon of rolling mills.

Amplitude-frequency curve of the rolling mill rolls as a function of coupling term parameter _{1}.

In this study, using the bifurcation and chaos theory, we determine the critical value and parameter range from the bifurcation solution of the coupling vibration system of the hot rolling mill rolls, so as to restrain and avoid rolling mill vibration.

Figure _{0}.

Bifurcation diagram of the vertical vibration under varying external excitation amplitude.

Phase path of the vertical vibration under varying external excitation amplitude: (a) _{0} = 5.54_{0} = 5.60_{0} = 5.90_{0} = 6.05_{0} = 6.55

Poincare section of vertical vibration under varying external excitation amplitude: (a) _{0} = 5.54_{0} = 5.60_{0} = 5.90_{0} = 6.05_{0} = 6.55

Figure _{0} < 5.54_{0}, the steady-state vibration becomes period-2 motion, the corresponding phase path is a closed curve formed after two circles (Figure _{0}, the vibration displacement will also increase. As shown in Figure _{0} < 5.84_{0} continues to increase, there is a period-doubling bifurcation, that is, the vibration becomes period-6 motion (Figures _{0} > 6.12

Figure _{0} are shown in Figures

Bifurcation diagram of horizontal vibration of the hot rolling mill rolls under varying external excitation amplitude.

Phase paths of horizontal vibration of the rolling mill rolls under varying external excitation amplitude: (a) _{0} = 5.31_{0} = 5.95_{0} = 6.20

Poincare section of horizontal vibration of the rolling mill rolls under varying external excitation amplitude, (a) _{0} = 5.31_{0} = 5.95_{0} = 6.20

When _{0} < 5.31_{0} gradually increases, the horizontal vibration system directly enters chaotic motion after period-2 motion, and when _{0} > 5.95_{0} > 6.20

Figures _{0}, namely, 5.68e5 N < _{0} < 5.84_{0} > 6.12

In this study, a coupled vibration model of hot rolling mill rolls under the effect of multiple nonlinear forces was established. Below is the summary of the analysis and steps involved in establishing the model.

By analyzing the actual structure and working principle of a double-acting single-piston servo hydraulic cylinder, the nonlinear spring force produced by it was obtained. Considering the velocity fluctuation at the entry of the strip workpiece in the horizontal direction and the changes in the vibration displacement in the vertical direction, the nonlinear friction force between the rolls was obtained. Finally, by considering the dynamic variation of the rolling force, the coupling vibration model of a four-high hot rolling mill under the effect of multiple nonlinearity was established.

Based on the amplitude-frequency response equation, and by using the actual rolling parameters of the 1780 four-high hot rolling mill, the main reason for the severe resonance of the rolling mill was found to be the occurrence of the internal resonance when the external excitation frequency is close to the derived frequency in the vertical and horizontal directions. This results in the instability of the system and occurrence of the jump phenomenon. Further, changes in the movement displacement of the hydraulic cylinder and the coupling term parameters considerably contribute to the changes in the amplitude and resonance range of the coupling vibration system of the hot rolling mill rolls.

From the study of the bifurcation characteristics of the coupled vibration system of the hot rolling mill under varying external excitation amplitude, it was found that period and period-doubling motions exist in both vertical and horizontal directions, and the vibration alternates between different forms. Therefore, periodic light and dark stripes appear on the strip. The results indicate that the abnormal vibration of rolling mill rolls can be mitigated if the external excitation amplitude is maintained below the critical value.

The structural and technological data of rolling mill used to support the findings of this study are included within the article.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This research was supported by National Natural Science Foundation of China (Grant no. 61973262) and Hebei Province Natural Science Funds for the Joint Research of Iron and Steel (Grant no. E2019203146).