August 19, 2019

# Automatic detuning of steam turbine rotor blades’ eigenfrequencies away from critical areas

*Industry: Turbomachinery | Product: pSeven | Company: Ural Turbine Works (UTW)*

## Objective and Methodology

One of the most important issues to ensure operational safety is the tuning of the rotor blades’ eigenfrequencies under operating conditions away from possible resonances with the rotation frequencies and pulsations of the steam flow. Within the described project, this task is solved using an automated module for determining the eigenfrequencies of the blades.

Automated module for blades eigenfrequencies detuning presented in this study is a part of dedicated automated solution for 3D simulation-based turbine design. Place of these simulation and optimization tools in design chain is shown on Fig.1. The solution was developed by DATADVANCE and deployed at the Ural Turbine Works simulation department.

All automation modules have common user interface based on pSeven platform. Due to web-based nature of the platform, the solution can be deployed on the server and accessed via web-browser. The user can switch between simulations, enter input parameters, choose simulation statements presets and browse the results. Modules for mechanical, thermal and flow simulations are supplemented with effective optimization tools of pSeven in order to enable automated simulation-driven design. All modules are connected to database and can retrieve initial data from preliminary design study or boundary conditions from other modules.

*Fig.1 Part of the design chain covered by automated simulation modules *

We introduce the **Adaptive Design** technique of the Adaptive Design of Experiments (ADoE) approach in pSeven, which allows creating uniform or non-uniform sample that satisfy given constraints. This approach works well for heavy simulation models and allows reducing significantly the number of points in the final sample that violate the constraints. The module accepts the given geometry of the blades and a set eigenmodes for analysis. Variable parameters are geometry characteristics of the blade fillets, shroud ring etc. As a result, we obtain a uniform set of parameters describing the blades away from the resonance.

## Challenge

Tuning the rotor blades’ eigenfrequencies away from resonance can be considered as a constraints satisfaction optimization problem (CSP). The solution of the CSP is the single configuration of parameters that satisfy the constraints. However, in real life applications the problem usually includes the technological constraints, which are hard to formalize, and therefore single (the first obtained) optimal solution may be not suitable. So, it would be more useful to look for multiple different solutions in the feasible domain.

## Solution

**1. Simulation model and problem statement**

Optimization study is based on the automated module for determining the eigenfrequencies of the blades. One of the simplest load cases is the centrifugal load without additional forces. In order to evaluate eigenfrequencies we have to consider full assembly of the blade, disk and shroud ring. In this case, we focus on T-edged blade and dovetail-shaped shroud connection. However, the module allows automated assembling of similar models for several types of connectors. Final geometry model is shown on Fig.2.

*Fig.2 Full assembly of the blade and parameters for the study *

Meshing and simulation are carried out in Ansys Mechanical and wrapped with automation scripts, including the empirical rules for mesh setup. Total mesh size is about 30000 cells. Simulation model has cyclic symmetry, and we consider only centrifugal load for a given rotation rate. All bodies have contacts with tension.

The simulation results are the eigenmodes and eigenfrequencies of the blade. In order to setup optimization problem, the user needs to specify the one or several modes for consideration. In this case, we will focus only on first flexural mode.

The variables for the optimization study are fillets radii and shroud ring thickness (see Fig.2), however other parameters can be included if needed. Parameters boundaries are presented in Table 1.

**Table 1. Variable parameters and bounds**

Parameter |
Description |
Lower bound |
Upper bound |

Rp, mm | Fillet radius (low, inside) | 5 | 10 |

Rs, mm | Fillet radius (low, outside) | 5 | 10 |

Rb, mm | Fillet radius (up) | 5 | 10 |

H, mm | Shroud ring rim | 3 | 15 |

Constraint for blade eigenfrequencies values is constructed as function of allowed and restricted frequency ranges. These ranges come from design methodology and in fact are located near the multiples of the basic rotation rate.

This constraint function divides feasible and unfeasible blades geometries and should automatically fit the given resonance regions. We used the piecewise-quadratic function, which is negative for restricted frequencies and positive elsewhere.

Fig.3 represent this constraint function for particular case. One may note that the first flexural mode of the blade hits the restricted frequencies range. This is a typical use case to apply automated eigenfrequency detuning.

*Fig.3 Restricted frequencies ranges and constraint function*

Problem statement may also include constraints for maximum stress in different parts of the blade. In this case, we set upper limits for the stress in the lock and in the shroud rim equal to 320 MPa. Final set of constraints is presented in Table 2.

**Table 2. Problem constraints and their values**

Constraint |
Description |
Bounds |

c | Artificial function (distance from resonance) | >0 |

stressT, MPa | Max stress in shroud rim | <320 MPa |

stress, MPa | FMax stress in disk lock | <320 MPa |

**2. Adaptive DoE method and results **

As mentioned above, tuning the rotor blades’ eigenfrequencies away from resonance can be considered as a constraints satisfaction optimization problem (CSP). However, the solution of the CSP is the single configuration of parameters that satisfy the constraints. For the subsequent design process, it is useful to investigate all the possible configurations leading to the desired result. That is why we implemented adaptive DoE approach.

Adaptive DoE methodology is based on the concept of SBO (Surrogate-Based Optimization). Starting as simple DoE strategy, it iteratively refines the internal knowledge of constraint functions as new points arrive from the simulation model. This method allows to save significant amount of simulation runs while obtaining uniform feasible sample of given size. It has proved its efficiency in case of heavy simulations models when the total number of evaluations is crucial. It can be also used in sample-based mode for offline iterative sample collection.

Following the general idea of pSeven algorithmic core, this algorithm is adaptive and requires the minimum number of settings (number of designs and search intensity).

## Results

In presented case, we compared the number of feasible designs found by aDoE to the number of feasible designs in uniform DoE. Adaptive DoE found 21 feasible geometries from 80 simulation runs in total, while uniform DoE sample contained only 9 feasible points. Moreover, this advantage will only grow with extension of total simulation budget.

The feasible designs are presented on Fig.4 in parallel coordinates.

*Fig.4 Feasible designs and ranges in parallel coordinates*

Note that the feasible designs set is almost uniform and contains various geometries. It allows performing trade-off studies and select the best design. It can be especially useful when the problem includes the technological constraints, which are hard to formalize and therefore single optimal solution may be not suitable.

## Conclusion

The automated module for rotor blades’ eigenfrequencies detuning is introduced as a part of custom solution for simulation automation. Optimization problem statement was formulated for detuning routine and then solved using a novel effective adaptive DoE algorithm. Based on SBO methodology, it allows obtaining a variety of blades geometries away from resonance areas within limited evaluations budget.

Problem statement can be easily extended for additional parameters and constraints. Together with full automation of the simulation for wide range of geometries and load cases, it makes the presented automated module a useful tool for simulation-based steam turbine design.

This study was presented at the NAFEMS World Congress 2019. Presentation slides can be found here.

*By Anton Saratov, Head of Application Engineering Department, DATADVANCE, and Mikhail Stepanov, Dmitri Kshesinskii, Ural Turbine Works, Russia. *