# Optimization Procedure and Toolchain for Roll Dynamic Geometry

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Optimization Procedure

#### 2.2. Simulator

#### 2.3. Target Function

#### 2.4. Optimizer

#### 2.5. Case Example

## 3. Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Väänänen, P. Turning of Flexible Rotor by High Precision Circularity Profile Measurement and Active Chatter Compensation. Licensiate Thesis, Teknillinen Korkeakoulu, Espoo, Finland, 1993. [Google Scholar]
- Kotamäki, M.J. In-Situ Measurement and Compensation Control in External Grinding of Large Cylinders, Acta Polytechnica Scandinavica; Finnish Academy of Technology: Helsinki, Finland, 1996. [Google Scholar]
- Nyberg, T.R. Dynamic Macro Topography of Large Slowly Rotating Cylinders, Acta Polytechnica Scandinavica; Finnish Academy of Technology: Helsinki, Finland, 1993. [Google Scholar]
- Kuosmanen, P. Predictive 3D Roll Grinding Method for Reducing Paper Quality Variations in Coating Machines; Helsinki University of Technology Publications in Machine Design; 2/2004; Helsinki University of Technology: Espoo, Finland, 2004. [Google Scholar]
- Juhanko, J. Dynamic Geometry of a Rotating Paper Machine Roll. Ph.D. Thesis, Aalto University, Espoo, Finland, 2011. [Google Scholar]
- Tiainen, T. Multi-Probe Roundness Measurement of Large Rotors. Ph.D. Thesis, Aalto University, Espoo, Finland, 2020. [Google Scholar]
- Widmaier, T. Optimisation of the Roll Geometry for Production Conditions; Telojen Geometrian Optimointi Tuotanto-Olosuhteita Varten. Ph.D. Thesis, Aalto University, Espoo, Finland, 2012. [Google Scholar]
- Julkunen, T. Paperikoneiden on the Dynamic Balancing of Steel Tube Rolls of Paper Machines. Ph.D. Thesis, Helsinki University of Technology, Espoo, Finland, 1974. [Google Scholar]
- Savolainen, M.T. Paperikoneen Development of the Measuring and Balancing System for the Paper Machine Rolls. Master’s Thesis, Helsinki University of Technology, Espoo, Finland, 1996. [Google Scholar]
- ISO 12181-1:2011; Geometrical Product Specifications (GPS). Roundness. Part 1: Vocabulary and Parameters of Roundness. International Organization for Standardization: Geneva, Switzerland, 2011.
- Arora, J.S. Computational design optimization: A review and future directions. Struct. Saf.
**1990**, 7, 131–148. [Google Scholar] [CrossRef] - Daxini, S.; Prajapati, J. Parametric shape optimization techniques based on Meshless methods: A review. Struct. Multidiscip. Optim.
**2017**, 56, 1197–1214. [Google Scholar] [CrossRef] - ISO 12181-2:2011; Geometrical Product Specifications (Gps). Roundness. Part 2: Specification Operators. International Organization for Standardization: Geneva, Switzerland, 2011.
- Powell, M.J.D. An efficient method for finding the minimum of a function of several variables without calculating derivatives. Comput. J.
**1964**, 7, 155. [Google Scholar] [CrossRef] - ISO 12180-1:2011; Geometrical Product Specifications (GPS)—Cylindricity—Part 1: Vocabulary and Parameters of Cylindrical Form. International Organization for Standardization: Geneva, Switzerland, 2011.
- Braibant, V.; Fleury, C. Shape optimal design using b-splines. Comput. Methods Appl. Mech. Eng.
**1984**, 44, 247–267. [Google Scholar] [CrossRef] - Keskinen, E. Continuous balancing method for long flexible rotors. In Proceedings of the IMAC-XX: A Conference on Structural Dynamics, Los Angeles, CA, USA, 4–7 February 2002; pp. 511–515. [Google Scholar]

**Figure 1.**Optimization procedure and software components. From the initial geometry parameters (i.e., a set number of control points), the simulator generates a CAD model, meshes and deforms it. The target function evaluator utilizes mesh libraries to estimate the target function value from the deformed mesh provided by the simulator. In each iteration of the optimization, the optimizer selects new geometry parameters for the simulator.

**Figure 2.**Schematic picture of the target function. The target function is defined as the mean RMS roundness error of equally spaced outer cross-sections of the roll.

**Figure 3.**Used parametric model of roll geometry with control points shown in purple (

**a**) and boundary conditions applied in the simulation with fixed displacements of nodes at the end cross-sections (

**b**).

**Figure 4.**Progress of the optimization. For each iteration, Powell’s conjugate direction method was used to obtain new control point values for estimating the target function.

**Figure 5.**(

**a**) Undeformed initial geometry (stationary). (

**b**) Deformed initial geometry (rotating). (

**c**) Wall thickness variation of initial geometry.

**Figure 6.**(

**a**) Undeformed optimized geometry (stationary). (

**b**) Deformed optimized geometry (rotating) with minimized cross-section roundness errors. (

**c**) Wall thickness variation of optimized geometry.

**Figure 7.**Visualization of von Mises stresses in the rotating initial geometry (

**a**) and rotating optimized geometry (

**b**).

**Figure 8.**Roundness profiles from 10 evenly spaced cross-sections along the length axis of the roll for (

**a**) undeformed initial geometry (stationary) (

**b**) deformed initial geometry (rotating) (

**c**) undeformed optimized geometry (stationary) and (

**d**) deformed optimized geometry (stationary).

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

Density | 7800 $\frac{\mathrm{kg}}{{\mathrm{m}}^{3}}$ |

Young’s modulus | 215 $\mathrm{GPa}$ |

Poisson’s ratio | 0.3 |

Rotating speed | 1200 $\mathrm{rad}/\mathrm{s}$ |

Initial control point radius | 1.3 $\mathrm{m}$ |

Upper boundary for control point radius | 1.3 $\mathrm{m}$ |

Lower boundary for control point radius | 1.05 $\mathrm{m}$ |

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**MDPI and ACS Style**

Tiainen, T.; Laine, S.; Viitala, R.
Optimization Procedure and Toolchain for Roll Dynamic Geometry. *Machines* **2022**, *10*, 498.
https://doi.org/10.3390/machines10070498

**AMA Style**

Tiainen T, Laine S, Viitala R.
Optimization Procedure and Toolchain for Roll Dynamic Geometry. *Machines*. 2022; 10(7):498.
https://doi.org/10.3390/machines10070498

**Chicago/Turabian Style**

Tiainen, Tuomas, Sampo Laine, and Raine Viitala.
2022. "Optimization Procedure and Toolchain for Roll Dynamic Geometry" *Machines* 10, no. 7: 498.
https://doi.org/10.3390/machines10070498