# Validation of a Modal Work Approach for Forced Response Analysis of Bladed Disks

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Methods

#### 2.1. Unsteady Forcing Decomposition

#### 2.2. Modal Work Approach

#### 2.3. ANSYS Harmonic Analysis

## 3. Results and Comparisons

#### 3.1. Dummy Test Case

#### 3.1.1. Modal Analysis

#### 3.1.2. Forcing Function Distribution

#### 3.1.3. Harmonic Analysis with ANSYS

#### 3.1.4. Tremor-ANSYS Comparison

#### 3.2. LPT Bladed Rotor Test Case

#### 3.2.1. FEM Bladed Rotor Model

#### 3.2.2. Pre-Stressed Modal Analysis

#### 3.2.3. CFD Multi-Row Model

#### 3.2.4. Unsteady Multi-Row Analysis

#### 3.2.5. Aerodynamic Damping Evaluation

#### 3.2.6. Harmonic Analysis with ANSYS

#### 3.2.7. Tremor–ANSYS Comparison

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

BPF | Blade Passing Frequency |

CAD | Computer-Aided Design |

CFD | Computational Fluid Dynamic |

DFT | Discrete Fourier Transform |

EO | Engine Order |

FEM | Finite Element Method |

HCF | High Cycle Fatigue |

IBPA | Inter Blade Phase Angle |

IGV | Inlet Guide Vane |

LPT | Low-Pressure Turbine |

MSUP | Mode SUPerposition |

ND | Nodal Diameter |

PS | Pressure Side |

SS | Suction Side |

TREMOR | Tool for Response Evaluation by Modal wORk |

URANS | Unsteady Reynolds-Averaged Navier–Stokes |

ZZENF | Zig Zag shaped Excitation line in the Nodal diameter versus Frequency |

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**Figure 3.**First bending mode of the dummy test case ($\mathrm{ND}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\pm 4$).

**Figure 8.**Modal displacements for $\mathrm{ND}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\pm 7$ of the 1st bending family.

**Figure 9.**Unsteady lift coefficient for two different rotor profiles during 1/7 of the rotor revolution.

**Figure 11.**Unsteady forcing at $\mathrm{EO}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}7$ and $\mathrm{ND}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}-7$.

**Figure 12.**Unsteady forcing at $\mathrm{EO}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}126$ and $\mathrm{ND}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}+21$.

**Figure 14.**Check on forcing distribution mapping ($\mathrm{EO}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}7$).

**Figure 15.**Check on forcing distribution mapping ($\mathrm{EO}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}126$).

Maximum Amplitude of Total Displacement (m) | ||||
---|---|---|---|---|

ND = 0 | ND = +4 | ND = −4 | ND = +8 | |

Tremor | $9.36\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ | $8.95\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ | $8.60\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ | $8.12\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ |

ANSYS | $9.36\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ | $8.95\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ | $8.60\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ | $8.12\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ |

**Table 2.**Tremor–ANSYS comparison (1st family, out of resonance for $\mathrm{EO}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}4$, case 3).

Maximum Amplitude of Total Displacement (m) for EO = 4 | |||||
---|---|---|---|---|---|

ND = 0 | ND = +2 | ND = +4 | ND = +6 | ND = +8 | |

Tremor | $2.51\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ | $3.31\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ | $8.95\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ | $2.77\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ | $1.93\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ |

ANSYS | $2.51\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ | $3.31\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ | $2.78\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ | $1.94\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ |

Item | Unit | Disk | Blade |
---|---|---|---|

Material | Structural Steel | Aluminum | |

Young’s modulus | (GPa) | 200 | 72 |

Density | (kg/m^{3}) | 2810 | 7850 |

Poisson ratio | (-) | 0.3 | 0.33 |

Maximum Amplitude of Total Displacement (m) | ||
---|---|---|

IGV Wake (EO = 7) | Stator Wake (EO = 126) | |

Tremor | $4.77\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | $8.89\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{10}^{-9}$ |

ANSYS | $4.45\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | $9.40\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{10}^{-9}$ |

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**MDPI and ACS Style**

Pinelli, L.; Lori, F.; Marconcini, M.; Pacciani, R.; Arnone, A. Validation of a Modal Work Approach for Forced Response Analysis of Bladed Disks. *Appl. Sci.* **2021**, *11*, 5437.
https://doi.org/10.3390/app11125437

**AMA Style**

Pinelli L, Lori F, Marconcini M, Pacciani R, Arnone A. Validation of a Modal Work Approach for Forced Response Analysis of Bladed Disks. *Applied Sciences*. 2021; 11(12):5437.
https://doi.org/10.3390/app11125437

**Chicago/Turabian Style**

Pinelli, Lorenzo, Francesco Lori, Michele Marconcini, Roberto Pacciani, and Andrea Arnone. 2021. "Validation of a Modal Work Approach for Forced Response Analysis of Bladed Disks" *Applied Sciences* 11, no. 12: 5437.
https://doi.org/10.3390/app11125437