# Dynamic Torque Limitation Principle in the Main Line of a Mill Stand: Explanation and Rationale for Use

^{*}

## Abstract

**:**

## 1. Introduction

^{2}within a batch or 50 N/mm

^{2}within a plate [11].

^{plus}) system [12]. The spindle bearing is located near the middle of the shaft. Each spindle is individually counterbalanced with a hydraulic cylinder, levers, and coupling rods vertically and (if using CVC

^{plus}) horizontally [13,14].

- -
- optimizing the functions to control the electric drive;
- -
- reducing the rigidity of mechanical gears;
- -
- monitoring the elastic torque by special software, etc.

## 2. Statement of Problem

^{2}. The firmware of the stand controller uses an algorithm based on this chart.

- (1)
- Impact torque amplitude at the moment of capture (Figure 3a, door 2) is nearly double the configured rolling torque (up to 5.8 MN·m vs. 3 MN·m).
- (2)
- Torsional oscillations overlaying a sine wave caused by shaft rotation feature a high amplitude, especially in case of the bottom spindle (Figure 3b).
- (3)
- Engine and spindle torques change identically with slight differences in the maximum torque and the time attain it. As mechanical and electrical systems are inertial, the maximum shaft torque (Figure 3a, door 2) is attained earlier than that of the engine (door 1). The difference is 0.1–0.15 s.
- (4)

## 3. Analysis of Elastic Torque Components when Choosing Play

_{0}at constant engine torque M

_{1}. When the acceleration begins and there is no mechanical connection between inertial masses (the second mass is motionless; ω

_{2}= 0), the engine evenly accelerates to the initial speed:

_{12}, we obtained the following differential equation:

_{av}= ε

_{0}, i.e., the average acceleration after an elastic impact is the same as prior thereto.

_{12}, natural frequency of elastic oscillations ω

_{12}, initial engine speed at the moment of play choice ω

_{init}, and average elastic torque M

_{12c}determined by the load torque. It ought to be mentioned that the first two parameters are basically constants, as an electric drive does not affect them in any way. The study of how the two remaining parameters affect the dynamic amplification factor is of interest. It ought to be mentioned that the initial speed at the moment of play choice is a function of two other parameters—initial acceleration of the electric drive and angular play in the gear. See the structural layout of a model of a dual-mass system in Figure 4b. See the equipment parameters for Mill 5000 required to plot this layout in the Table 1.

_{12}torque change graph based on Equation (4), temporary dependencies of the separate components calculated using Equations (5) and (6), and an M

_{c}static load torque graph. They correspond to the following conditions: ε

_{av}= ε

_{0}= 1.6 rad/s

^{2}, M

_{c}= 1,910,000 N∙m (nominal engine torque), gear play 10° (overestimated value for qualitative evaluation of processes). We can see that component M

_{12δ}caused by an elastic impact has a significantly lower amplitude than component M

_{12y}caused by elastic oscillations in the gear. Amplitude ratio M

_{12δmax}/M

_{12ymax}= 0.21; M

_{12δ}torque amplitude features a non-linear dependence on the play and increases insignificantly if the play becomes wider.

_{12δmax}/M

_{12ymax}ratio varies from 0.067 to 0.21. Therefore, the second component has an amplitude that is 4.7–15 times higher than that of the first component. It ought to be mentioned that this conclusion covers the initial moment of time immediately after capture, because in a real system dissipative forces result in unequal dampening of oscillations. This issue requires further studies using an electric drive model, which is why it is not considered here.

_{12δ}may be somewhat reduced. For instance, if ε

_{0}= 0.16 rad/s

^{2}and maximum play is 10°, the M

_{12δmax}/M

_{12ymax}ratio = 0.067, i.e., 3 times lower than when accelerations are the same.

_{12c}; it basically determines the first maximum of the dynamic torque in the shaft. This torque may be divided into a constant component and a variable component ${M}_{12c}={J}_{2}\cdot {\epsilon}_{av}+{M}_{c}$. It may be compensated by negative dynamic torque, i.e., by means of negative acceleration after capture.

_{12}and moments of inertia of the dual-mass system. Initial speed ${\omega}_{\mathrm{init}}=\sqrt{2\cdot {\epsilon}_{0}\cdot \delta}$ is determined by angle δ. That is why when the play is chosen and δ = 0, this component equals zero.

- -
- ensure play closure by means of pre-acceleration to minimize the second component;
- -
- decelerate the electric drive after capture with a set negative acceleration.

_{2}are not considered as we cannot impact them, then the negative acceleration must be (−ε

_{av}), where ${\epsilon}_{av}=\frac{{M}_{1}-{M}_{12}}{{J}_{1}+{J}_{2}}$ is the average acceleration after the play has been chosen. That means that the deceleration rate must be equal to the pre-capture acceleration.

_{12c}in the range of ±50% of nominal values is provided below. The graph depicting dependence of the dynamic amplification factor on the average torque and the play is given in Figure 6a. It was plotted using Equation (7) at initial acceleration ε

_{0}= 1.6 rad/s

^{2}, changes of torque of load in the range of 1–2 MN·m and angular play of 0.017–0.17 rad.

## 4. Dynamic Torque Limitation Principle

- -
- caused by an impact at angular play closure;
- -
- caused by elastic properties of the mechanical gear;
- -
- caused by settings of the automatic electric drive speed control system (ACS).

- (1)
- Before the shock loading, the system is pre-accelerated with the minimum initial acceleration to choose play regardless of its value. This helps to compensate for component M
_{12δ}caused by an impact (lower component of the elastic torque). - (2)
- In order to compensate for component M
_{12y}, after the shock loading, the speed is reduced using set negative acceleration (deceleration rate).

- -
- metal is captured when the electric drive is accelerated;
- -
- the speed at the time of capture must be equal to the value required to compensate for the dynamic speed control error;
- -
- when to start accelerating the drive is determined by the distance between the “head” of a slab and the stand calculated as follows:$$S=({V}_{t}+0.5{V}_{ff})\cdot ({t}_{2}-{t}_{1})$$
- -
- the speed is reduced after capture according to the linear law with set negative acceleration (according to simulation results, the optimal acceleration for the electric drive of rolling mill stand 5000 rolls varies from −2.5 to −3.5 rad/s
^{2}).

_{ff}/(t

_{2}− t

_{1}), known values of threading speed V

_{t}and forward flow speed V

_{ff}(final pre-acceleration rate). This is used to calculate the distance from the “head” of a slab to the stand that is required for pre-acceleration to start.

## 5. Discussion of the Results

## 6. Conclusions

- (1)
- Caused by an elastic impact at angular play closure.
- (2)
- Caused by elastic properties of the shaft line.
- (3)
- Caused by settings of the automatic electric drive speed control system.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**Functional chart of the electric drive speed control system (contoured blocks are the ones that provide pre-acceleration).

**Figure 3.**Waveforms of engine torques (

**a**) and elastic torques in the spindle shafts of the top and bottom rolls, horizontal stand, Mill 5000 (

**a**,

**b**).

**Figure 4.**Kinematic configuration of a dual-mass system with elastic coupling and play (

**a**) and the structural layout of the model (

**b**): J

_{1}= J

_{E}—first mass (engine) moment of inertia; J

_{2}—given second mass moment of inertia (work and backup rolls); c

_{12}—stiffness of the elastic coupling; M

_{12}—elastic torque; W(p)—transfer function of the mechanical part.

**Figure 5.**Torque component change graphs (disregarding dissipative forces): 1—elastic torque M

_{12}in the shaft; 2—component M

_{12δ}caused by elastic impact; 3—component M

_{12y}caused by elastic oscillations; 4—static load torque M

_{12c}.

**Figure 6.**Dependencies of the dynamic amplification factor on the average torque and gear play at constant angular acceleration (

**a**), and on angular acceleration and play at the nominal torque of load (

**b**).

Parameter | Designation | Value |
---|---|---|

Stiffness of the elastic coupling | c_{12} | 5,934,842 N·m/rad |

Moment of inertia of the first moving mass (engine) | J_{1} | 125,000 kg·m^{2} |

Moment of inertia of the second moving mass | J_{2} | 114,571 kg·m^{2} |

Natural frequency of elastic oscillations | ω_{12} | 9.96 rad/s |

Initial acceleration of the electric drive | ε_{0} | 1–3 rad/s^{2} |

Gear play | δ | 1–10° |

Average elastic torque | M_{12c} | 1.9 MN·m |

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**MDPI and ACS Style**

Gasiyarov, V.R.; Khramshin, V.R.; Voronin, S.S.; Lisovskaya, T.A.; Gasiyarova, O.A. Dynamic Torque Limitation Principle in the Main Line of a Mill Stand: Explanation and Rationale for Use. *Machines* **2019**, *7*, 76.
https://doi.org/10.3390/machines7040076

**AMA Style**

Gasiyarov VR, Khramshin VR, Voronin SS, Lisovskaya TA, Gasiyarova OA. Dynamic Torque Limitation Principle in the Main Line of a Mill Stand: Explanation and Rationale for Use. *Machines*. 2019; 7(4):76.
https://doi.org/10.3390/machines7040076

**Chicago/Turabian Style**

Gasiyarov, V.R., V.R. Khramshin, S.S. Voronin, T.A. Lisovskaya, and O.A. Gasiyarova. 2019. "Dynamic Torque Limitation Principle in the Main Line of a Mill Stand: Explanation and Rationale for Use" *Machines* 7, no. 4: 76.
https://doi.org/10.3390/machines7040076