## Inverse Methods for Continuous Balancing of Flexible Rolls

Tutkimustuotos ›

### Standard

**Inverse Methods for Continuous Balancing of Flexible Rolls.** / Jokinen, Kai.

Tutkimustuotos ›

### Harvard

*Inverse Methods for Continuous Balancing of Flexible Rolls*. Tampereen teknillinen yliopisto. Julkaisu, Vuosikerta. 944, Tampere University of Technology, Tampere.

### APA

*Inverse Methods for Continuous Balancing of Flexible Rolls*. (Tampereen teknillinen yliopisto. Julkaisu; Vuosikerta 944). Tampere: Tampere University of Technology.

### Vancouver

### Author

### Bibtex - Lataa

}

### RIS (suitable for import to EndNote) - Lataa

TY - BOOK

T1 - Inverse Methods for Continuous Balancing of Flexible Rolls

AU - Jokinen, Kai

N1 - Awarding institution:Tampere University of Technology

PY - 2010

Y1 - 2010

N2 - This thesis presents inverse methods to define imbalance distribution of rolls on the basis of whirling and bearing force measurements. Three different methods have been evaluated, of which the first is based on modal balancing of flexible rotors and is herein called “mode-based determination of distributed imbalance”. The other two methods are based on influence coefficient balancing. Accordingly, the second method uses well-chosen trial functions to define the imbalance distribution when forming an influence matrix; the third method expresses the imbalance distribution as the superposition of eigenmodes, and the orthogonality properties of these mode shapes are used to simplify the subsequent calculations. The goal of the present research is to improve the quality and efficiency of balancing paper machine rolls by the continuous balancing method wherein the balancing mass is distributed over the whole length of a roll tube. The proposed method is new and not known to be used in practice in roll balancing applications; rather, rolls are currently balanced with point masses on the basis of two or three-plane balancing principles. Rigid body balancing is accomplished by adding balancing masses to the roll ends, and whirling may be balanced by adding a third mass in the centre of the roll. The roll model used in calculations has significant effects on the accuracy of balancing results and the residual whirling of a balanced roll. Accordingly, a lot of effort has been dedicated towards the development of this model. The roll is modelled as a continuous system and mode shapes of a flexibly supported beam are used. It is shown that other boundary conditions, such as hinged or free supports, may also be used to achieve similar results. The advantage of using flexible mode shapes is that they enable the model to properly account for the shaft flexibility as well as the boundary conditions. The cross section of the roll is not constant. This circumstance affects not only the stiffness of the roll, but also the transverse inertia forces caused by rotation of the imbalanced roll. This work suggests taking this latter phenomenon into account by incorporating it into the mass matrix. So inertia forces of the roll endings are thereby added to the roll model, but it is assumed that these inertias do not modify the mode shapes of the roll. It is shown that all three proposed inverse methods lead to the same solution of the imbalance distribution if the roll whirling is known exactly. Since, in practice, there are always some measurement errors present, there is also a discernable limitation on the conformance of the imbalance distribution. This phenomenon is also studied, and a compromise solution with a limited number of trial functions is presented; this gives reasonable results without overreacting to observation errors.

AB - This thesis presents inverse methods to define imbalance distribution of rolls on the basis of whirling and bearing force measurements. Three different methods have been evaluated, of which the first is based on modal balancing of flexible rotors and is herein called “mode-based determination of distributed imbalance”. The other two methods are based on influence coefficient balancing. Accordingly, the second method uses well-chosen trial functions to define the imbalance distribution when forming an influence matrix; the third method expresses the imbalance distribution as the superposition of eigenmodes, and the orthogonality properties of these mode shapes are used to simplify the subsequent calculations. The goal of the present research is to improve the quality and efficiency of balancing paper machine rolls by the continuous balancing method wherein the balancing mass is distributed over the whole length of a roll tube. The proposed method is new and not known to be used in practice in roll balancing applications; rather, rolls are currently balanced with point masses on the basis of two or three-plane balancing principles. Rigid body balancing is accomplished by adding balancing masses to the roll ends, and whirling may be balanced by adding a third mass in the centre of the roll. The roll model used in calculations has significant effects on the accuracy of balancing results and the residual whirling of a balanced roll. Accordingly, a lot of effort has been dedicated towards the development of this model. The roll is modelled as a continuous system and mode shapes of a flexibly supported beam are used. It is shown that other boundary conditions, such as hinged or free supports, may also be used to achieve similar results. The advantage of using flexible mode shapes is that they enable the model to properly account for the shaft flexibility as well as the boundary conditions. The cross section of the roll is not constant. This circumstance affects not only the stiffness of the roll, but also the transverse inertia forces caused by rotation of the imbalanced roll. This work suggests taking this latter phenomenon into account by incorporating it into the mass matrix. So inertia forces of the roll endings are thereby added to the roll model, but it is assumed that these inertias do not modify the mode shapes of the roll. It is shown that all three proposed inverse methods lead to the same solution of the imbalance distribution if the roll whirling is known exactly. Since, in practice, there are always some measurement errors present, there is also a discernable limitation on the conformance of the imbalance distribution. This phenomenon is also studied, and a compromise solution with a limited number of trial functions is presented; this gives reasonable results without overreacting to observation errors.

M3 - Doctoral thesis

SN - 978-952-15-2501-8

T3 - Tampereen teknillinen yliopisto. Julkaisu

BT - Inverse Methods for Continuous Balancing of Flexible Rolls

PB - Tampere University of Technology

CY - Tampere

ER -