# Research on the Vertical Vibration Characteristics of Hydraulic Screw Down System of Rolling Mill under Nonlinear Friction

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## Abstract

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## 1. Introduction

## 2. Principle of Hydraulic Screw Down System

## 3. Mathematical Model

#### 3.1. Nonlinear Flow Equation of Four-Way Valve

- ${C}_{\mathrm{d}}$: port flow coefficient of four-way valve;
- $W$: area grades of four-way valve;
- ${x}_{\mathrm{v}}$: spool displacement of four-way valve;
- ${p}_{\mathrm{L}}$: rod-less cavity pressure of hydraulic cylinder;
- $\rho $: oil density.

#### 3.2. Model of Servo Amplifier and Servo Valve

- $u$: output of vibration controller;
- ${K}_{\mathrm{p}1}$: gain of the servo amplifier;
- $i$: output current of servo amplifier;
- ${K}_{\mathrm{sv}}$: gain of servo valve.

#### 3.3. Flow Equation of Hydraulic Cylinder

- ${A}_{\mathrm{p}}$: cross-sectional area of rod-less cavity of cylinder;
- $x$: vibration displacement of cylinder piston rod;
- ${C}_{\mathrm{t}}$: leakage coefficient of hydraulic cylinder;
- $V$: rod-less cavity volume of cylinder;
- ${L}_{1}$: the initial stroke of the cylinder piston;
- ${\beta}_{\mathrm{e}}$: volume elastic modulus of oil.

#### 3.4. Nonlinear Friction

- $z$: average deformation of the sideburns;
- ${\sigma}_{0}$: stiffness coefficient of sideburns;
- ${\sigma}_{1}$: microscopic damping coefficient;
- ${\sigma}_{2}$: viscous damping coefficient;
- ${f}_{\mathrm{c}}$: coulomb force of friction;
- ${f}_{\mathrm{s}}$: maximum static friction force;
- ${v}_{\mathrm{s}}$: Stribeck speed.

- (1)
- When velocity $v=0$, $\frac{\mathrm{d}z}{\mathrm{d}t}=0$, the friction force is ${F}_{f}\left(0\right)={\sigma}_{0}z$ and is a constant. For the convenience of analysis, the constant is defined as ${f}_{\mathrm{s}}$, and ${\dot{F}}_{f}\left(0\right)=0$.
- (2)
- When the velocity $v\ne 0$, ${F}_{f}\left(v\right)$ can be expressed by Equation (6), and ${\dot{F}}_{f}\left(v\right)$ is continuously differentiable.

#### 3.5. Balance Equation of Cylinder Piston Force

- ${A}_{\mathrm{b}}$: the effective area of rod cavity of cylinder;
- ${m}_{\mathrm{t}}$: equivalent total mass of piston and load;
- ${c}_{1}$: viscous damping coefficient of piston;
- ${k}_{1}$: spring stiffness of load;
- ${F}_{f}\left(v\right)$: nonlinear friction;
- ${F}_{\mathrm{L}}$: external force;
- $v$: speed of hydraulic cylinder.

#### 3.6. System State Equation

#### 3.7. Vibration Controller

- ${K}_{\mathrm{P}}$: proportionality coefficient;
- ${K}_{\mathrm{I}}$: integral coefficient;
- ${K}_{\mathrm{D}}$: differential coefficient.

## 4. Numerical Simulation

#### 4.1. The Parameters Selection of Simulation Model

#### 4.2. Simulation Results and Analysis

#### 4.2.1. Influence of Different Equivalent Damping Coefficients on Nonlinear Dynamic Behavior

#### 4.2.2. Influence of Different Leakage Coefficients on Nonlinear Dynamic Behavior

#### 4.2.3. The Influences of Proportional Parameter of Vibration Controller on Nonlinear Dynamic Behavior

## 5. Conclusions

- (1)
- The friction force applied to the cylinder is nonlinear. The equivalent damping coefficient and internal leakage coefficient change with the working state and ambient temperature. The vibration model of the rolling mill is established by considering the influence of nonlinear friction and parameter uncertainty on the system characteristics.
- (2)
- When the vibration response of the hydraulic cylinder is analyzed by using a different damping coefficient, it is found that the vibration can be suppressed effectively by increasing the damping coefficient.
- (3)
- The hydraulic system will inevitably bring leakage problems with the aging of the sealing device. Through the analysis of the leakage coefficient, it is found that the vibration attenuation becomes slower and even periodic vibration appears with the increase of the leakage coefficient.
- (4)
- The controller is an effective mean to ensure accurate position control. Reasonable controller parameters can effectively suppress vibration, but fixed controller parameters will make negative effects on vibration suppression.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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

${A}_{\mathrm{p}}$ | $0.19635\text{}{\mathrm{m}}^{2}$ | ${K}_{\mathrm{p}1}$ | $0.0125\text{}\mathrm{A}/\mathrm{V}$ |

${c}_{1}$ | $2.25\times {10}^{6}\text{}\mathrm{N}\cdot \mathrm{s}/\mathrm{m}$ | ${k}_{1}$ | $2.5\times {10}^{9}\text{}\mathrm{N}/\mathrm{m}$ |

${m}_{\mathrm{t}}$ | $1200\text{}\mathrm{kg}$ | ${F}_{\mathrm{L}}$ | $2\times {10}^{6}\text{}\mathrm{N}$ |

${C}_{\mathrm{t}}$ | $5.0\times {10}^{-16}\text{}{\mathrm{m}}^{3}/\mathrm{s}\cdot \mathrm{P}\mathrm{a}$ | ${p}_{\mathrm{s}}$ | $1.8\times {10}^{7}\text{}\mathrm{P}\mathrm{a}$ |

${\beta}_{\mathrm{e}}$ | $7\times {10}^{8}\text{}\mathrm{P}\mathrm{a}$ | $\rho $ | $850\text{}{\mathrm{kg}/\mathrm{m}}^{3}$ |

${C}_{\mathrm{d}}$ | $0.62$ | $W$ | $0.025\text{}\mathrm{m}$ |

${K}_{\mathrm{sv}}$ | $0.01\text{}\mathrm{m}/\mathrm{A}$ | ${L}_{1}$ | $0.15\text{}\mathrm{m}$ |

$L$ | $0.26\text{}\mathrm{m}$ | ${A}_{\mathrm{b}}$ | $0.030159\text{}{\mathrm{m}}^{2}$ |

${p}_{\mathrm{b}}$ | $1.0\times {10}^{6}\text{}\mathrm{P}\mathrm{a}$ | ${p}_{\mathrm{t}}$ | $0.0\times {10}^{5}\text{}\mathrm{P}\mathrm{a}$ |

${\sigma}_{0}$ | $4.4178\times {10}^{9}\text{}\mathrm{N}/\mathrm{m}$ | ${\sigma}_{1}$ | $1\times {10}^{5}\text{}\mathrm{N}\cdot \mathrm{s}/\mathrm{m}$ |

${\sigma}_{2}$ | $1\times {10}^{5}\text{}\mathrm{N}\cdot \mathrm{s}/\mathrm{m}$ | ${f}_{\mathrm{c}}$ | $\mathrm{16,000}\text{}\mathrm{N}$ |

f_{s} | $\mathrm{22,000}\text{}\mathrm{N}$ | ${v}_{\mathrm{s}}$ | $0.1\text{}\mathrm{m}/\mathrm{s}$ |

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**MDPI and ACS Style**

Zhang, Y.; Jiang, W.; Zhu, Y.; Li, Z.
Research on the Vertical Vibration Characteristics of Hydraulic Screw Down System of Rolling Mill under Nonlinear Friction. *Processes* **2019**, *7*, 792.
https://doi.org/10.3390/pr7110792

**AMA Style**

Zhang Y, Jiang W, Zhu Y, Li Z.
Research on the Vertical Vibration Characteristics of Hydraulic Screw Down System of Rolling Mill under Nonlinear Friction. *Processes*. 2019; 7(11):792.
https://doi.org/10.3390/pr7110792

**Chicago/Turabian Style**

Zhang, Yongshun, Wanlu Jiang, Yong Zhu, and Zhenbao Li.
2019. "Research on the Vertical Vibration Characteristics of Hydraulic Screw Down System of Rolling Mill under Nonlinear Friction" *Processes* 7, no. 11: 792.
https://doi.org/10.3390/pr7110792