Effect of minimum friction coefficient on vibration stability in cold rolling mill

doi:10.1016/j.triboint.2021.106958

Tribology International

Volume 159, July 2021, 106958

https://doi.org/10.1016/j.triboint.2021.106958 Get rights and content

Highlights

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A mathematical model is proposed to predict the lower limit friction coefficient.
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The model is proposed by monotonicity analysis based on mixed lubrication theory.
•
The model is validated to be accurate by previous literature and actual rolling test.
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The optimization of the rolling condition is designed to increase mill vibration rolling stability.

Abstract

Mill chatter is a major restriction in the high-speed rolling of high-strength and thin strips. To maintain mill stability, the friction coefficient must be kept within a certain range. Unfortunately, there is no model to calculate the lower limit of the friction. In this paper, a new model is developed to capture the minimum friction coefficient. This model is proposed based on mixed lubrication theory and monotonicity analysis in the friction curve. The verification results showed that the established model could accurately predict the minimum friction and the friction balance in tandem rolling mills. The model was utilized to design the optimal rolling conditions to expand the stability range of the friction and improve mill vibration stability.

Graphical abstract

Introduction

Tandem cold rolling mills are key pieces of equipment in the iron and steel-making industry. Taking a tin-plate steel-strip production line as an example, Fig. 1(a) shows the basic rolling process of a universal crown control mill (UCM). The production line contains five tandem stands, and all strips are rolled in the five-pass process stage. With the increased demand for high dimensional accuracy in production and high-efficiency rolling, the development of rolling process stability control technology requires special attention. However, the third-octave mode chatter often occurs in cold rolling of high-strength and thin strips [1].

Yun et al. [2] reviewed existing chatter models and reported that the friction factor model has a close relationship with mill chatter. Tlusty et al. [3] explored the effect of the friction coefficient on the amplitudes of vibrations using a simplified chatter model. However, the quantitative relationship between the friction coefficient and the limiting rolling speed has not been accurately described. Hu et al. [4,5] proposed a dynamic rolling process model and improved its suitability for chatter studies. Zhao et al. [6,7] proposed a third-octave chatter model based on Hu's model and investigated the effects of the rolling speed and the friction coefficient on mill stability. However, in Hu and Zhao's research, a constant friction coefficient was adopted. Zeng et al. [8] analyzed a Hopf bifurcation in the cold rolling process using a nonlinear friction model and improved the vibration stability by designing a linear and nonlinear controller. The main problem in Zeng's publications is that the Sim friction model cannot accurately describe the friction in the roll gap during cold rolling. Using a combination of the viscosity and friction, Tan et al. [9] proposed a dynamic friction model and found that the prediction results were more accurate with the dynamic friction model. The usage of a non-constant friction model is necessary to improve the rolling mill chatter model. The non-constant friction model can be applied to the chatter model if the lubrication regime belongs to a specific lubrication state under specific rolling conditions.

In cold rolling, a lubricant recirculation supply system is widely used to reduce the rolling load and improve the surface quality of the strip. Sun et al. [10] reported that the formation of lubricating film before rolling had a great influence on rolling conditions such as rolling force, rolling power, reduction and friction coefficient. Wilson et al. [11] proposed that the lubrication state could be described as a boundary regime, mixed lubrication regime, and hydrodynamic lubrication regime. Azushima et al. [12] studied the influence of rolling speed on friction coefficient, and considered that the lubrication state can be divided into boundary lubrication, mixed lubrication and plastohydrodynamic lubrication, as shown in Fig. 1(b). In low-speed rolling, the lubrication state is boundary lubrication [13]. With the increase in the speed, the lubrication state changes to mixed lubrication, and full-film hydrodynamic lubrication will occur when the rolling speed is extremely high [14].

To reveal the nonlinear friction characteristics in the roll gap, mixed and hydrodynamic lubrication theories have attracted extensive attention from scholars, and outstanding results have been achieved. Le et al. [15,16] proposed a semi-empirical friction model to predict the friction coefficient and investigated the effect of the film thickness and the speed on the friction factor. Lu et al. [17] developed a mixed film lubrication model in the inlet zone of the roll gap, and the effects of the speed and the tension on the film thickness were studied. Saniei et al. [18] presented a mixed lubrication model to analyze the pressure distribution in the roll gap using the Reynolds equation. Liu et al. [19] found that the prediction result of rolling force was more accurate by using mixed lubrication friction model. However, these lubrication theories were not used for the analysis of mill chatter stability in the above studies. Fujita et al. [20]reported that the lubrication state had a close relationship with the friction coefficient and chatter. Heidari et al. [21] developed a nonlinear friction and mill chatter model based on mixed lubrication theory and found that the mixed lubrication friction model was more precise for mill chatter predictions. Wang et al. [22] investigated the influence of the speed roll roughness and the roll radius on mill vibration stability using mixed lubrication friction theory. Jeng et al. [23] investigated the effects of rolling conditions of cold rolling on the surface quality of aluminum sheets by a mixed lubrication model. Cao et al. [24] reported that the rolling speed, tension, and viscosity had a significant effect on third-octave mode mill chatter. However, in these reports, the mathematical quantitative relationship between the friction and the vibration stability was still unclear.

Through rolling experiments, Kimura et al. [25] found that the friction coefficient in a single stand must remain in a certain range to avoid chatter, as shown in Fig. 1(c). Mehrabi et al. [26] verified the existence of the upper limit of the friction coefficient in Kimura's study by establishing a finite element cold rolling mill chatter model. Lu et al. [27] proposed a nonlinear mathematical mill vibration stability criterion and analyzed the effect of the upper friction limit on the critical rolling speed quantitatively. Heidari et al. [28] found that when the friction coefficient was small, the mill vibration stability decreased as the friction coefficient decreased, which proved the existence of the lower limit of the friction coefficient. Fujita et al. [29] reported that the friction coefficient balance between the No. 4 and 5 stands must be controlled within a certain range of 0.005–0.007, i.e., $μ_{4} - μ_{5} = 0.005 \sim 0.007$ . Gao et al. [30] proposed a structure-process-control coupled mill chatter model and investigated the lower critical rolling speed caused by the minimum friction coefficient by applying the Routh stability criterion. However, the specific mechanical mechanism that causes the lower limit of the friction has not been successfully explained.

Despite the great achievements made in the past decades, it is still not easy to quantitatively describe the relationship between friction and vibration stability. Moreover, a method to obtain the lower limit of the friction coefficient is still unknown. Hence, the main motivation and contributions of this paper are to propose a mathematical model to capture the minimum limit of the friction coefficient and optimize the tension, roll roughness, and viscosity to increase the vibration stability in tandem cold rolling mills.

This paper is organized as follows. First, mixed lubrication theory is introduced to obtain the friction coefficient in the high-speed rolling stage. Second, an attempt is made to propose a mathematical lower friction coefficient model based on the analysis of the monotonicity in the friction coefficient curve. Third, the proposed mathematical model is evaluated by comparing the findings in previous literature and actual industrial rolling tests. Finally, the rolling conditions are optimized to increase mill vibration stability.

Section snippets

Interfacial mixed lubrication theory

The inlet microstructures of the surfaces between the roll and strip are illustrated as Fig. 2. The hydrodynamic compressive pressure of the lubricant can be determined by the Reynolds equation [31]: $\bar{v} \frac{\partial h_{t}}{\partial x} - \frac{\partial h_{t}}{\partial x} + \frac{\partial}{\partial x} (\frac{h_{t}^{3}}{12 η} \frac{\partial p}{\partial x}) = 0$ $\frac{d p}{d x} = 6 η (v_{r} + v_{i n}) \frac{h - h_{a}}{h^{3}}$ where $\bar{v} = \frac{v_{r} + v_{i n}}{2}$ .

Considering the pressure and thermal effect on oil viscosity, the oil viscosity is calculated by the Roelands model [32]: $η (p, T) = η_{0} \exp {(\ln η_{0} + 9.67) [{(1 + \frac{p}{1.96 \times 10^{8}})}^{z} \times {(\frac{T_{0} - 138}{T - 138})}^{s_{0}} - 1]}$ where $z = \frac{θ}{5.1 \times 10^{- 9} (\ln η_{0} + 9.67)}$ . In this study, $z \approx 0.6$ ,

Derivation of mathematical minimum friction model

In the cold rolling, researchers believed that [12,29,30,40] the friction coefficient tends to increase in the in high speed full-film hydrodynamic lubrication stage. This means that there is a minimum value of the friction coefficient with the change of the rolling conditions. The minimum friction can be obtained by monotonicity analysis, which is given as follows: $\frac{\partial μ_{M i x e d}}{\partial (v η / p)} = 0$

The influence of the unit oil film pressure on the emulsion viscosity is shown in Fig. 4. The oil viscosity

Results and discussions

Fujita et al. [29] determined that there was a certain range of the friction coefficient balance between stands No. 4 and 5 through rolling experiments. To evaluate the performance of the proposed mathematical model, experiments were implemented to validate the prediction accuracy of the proposed mathematical model using the raw data in Fujita's research. Based on Eq. (41), the calculated minimum friction coefficients of stands No. 4 and 5 were 0.0134, and 0.0083, respectively. The calculated

Conclusions

The main purpose of this study was to propose and validate a novel mathematical model to predict the minimum limit of the friction coefficient in a high-speed cold rolling mill. The key results are as follows:

(1)
A mathematical prediction model for the lower limit of the friction coefficient was proposed by deriving the friction curve in the hydrodynamic lubrication regime. The pressure peak and the sharp change in the direction of friction stress near the neutral point are the main reasons for the

CRediT authorship contribution statement

Xing Lu: Methodology, Data curation, Visualization, Investigation, Writing - original draft. Jie Sun: Conceptualization, Methodology, Supervision, Validation, Writing - review & editing. Zhen Wei: Data curation. Guangtao Li: Data curation. Dianhua Zhang: Conceptualization, Supervision, Validation, Writing - review & editing.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

Funding: This work was supported by the National Natural Science Foundation of China [grant numbers 51774084, 51634002, 52074085 ], the National Key R&D Program of China [grant number 2017YFB0304100], the Fundamental Research Funds for the Central Universities [grant numbers N2004010], and the LiaoNing Revitalization Talents Program XLYC1907065.

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Effect of minimum friction coefficient on vibration stability in cold rolling mill

Highlights

Abstract

Graphical abstract

Introduction

Section snippets

Interfacial mixed lubrication theory

Derivation of mathematical minimum friction model

Results and discussions

Conclusions

CRediT authorship contribution statement

Declaration of competing interest

Acknowledgments

CIRP Ann - Manuf Technol

Int J Mach Tool Manufact

Int J Mach Tool Manufact

J Mater Process Technol

J Manuf Process

J Mater Process Technol

J Mater Process Technol

J Mater Process Technol

Tribol Int

Tribol Int

J Mater Process Technol

J Mater Process Technol

Mech Syst Signal Process

Tribol Int

Int J Mech Sci

Experimental and numerical investigation of the vibration characteristics in a hot plate rolling mill based on multibody dynamics

ISIJ Int

Review of chatter studies in cold rolling

Int J Mach Tool Manufact

Stability analysis of chatter in tandem rolling mills—Part 1: single- and multi-stand negative damping effect

J Manuf Sci E-T ASME

Stability analysis of chatter in tandem rolling mills—Part 2: the regenerative effect

J Manuf Sci E-T ASME

Hopf bifurcation control for rolling mill multiple-modal-coupling vibration under nonlinear friction

J Vib Acoust

Mechanics of sheet metal forming