# A Comparison between Two Reduction Strategies for Shrouded Bladed Disks

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Background

#### 2.1. Cyclic Symmetry

**q**and

_{H}**q**and all the remaining inner dofs

_{L}**q**. More in detail,

_{I}**q**defines the nodes belonging to the High sector interface, whereas

_{H}**q**identifies the coordinates at the Low interface.

_{L}**q**and

_{H}**q**, where the phase angle φ is defined as:

_{L}**K**is the block-diagonal matrix, whose h

_{BD}^{th}block is the stiffness matrix corresponding to the h

^{th}harmonic index, the stiffness matrix of the whole bladed disk can be obtained [20] as

- −
**E**is an_{N}**N**×**N**Fourier matrix (**N**is the number of blocks, coincident with the number of nodal diameters h in this case), and ${\mathit{E}}_{\mathit{N}}^{\mathit{H}}$ is its conjugate transpose.- −
**I**is the identity matrix of dimension_{P}**P**,**P**being the dimension of each block of matrix**K**._{BD}- −
- ⊗ is the Kronecker product.

#### 2.2. Craig–Bampton Method

## 3. Reduction Methods for Bladed Disks

#### 3.1. Direct Method

**NC**and

**L**indicate the partition of constraint and fixed-interface modes of the single sector with cyclic symmetry boundary conditions, corresponding to ${\mathit{q}}_{\mathit{N}\mathit{C}}$ and ${\mathit{q}}_{\mathit{L}}$, respectively.

#### 3.2. Two-Step Method

#### 3.3. Remarks about the Two Approaches

- The direct method requires one reduction (${\mathit{R}}_{\mathit{D}}$ matrix) for each harmonic index h, while the two-step method requires a first reduction (${\mathit{R}}_{\mathbf{1}}$ matrix), which is performed only once, followed by a second reduction (${\mathit{R}}_{\mathbf{2}}$ matrix) for each harmonic index.
- The direct method requires a lower number of constraint modes $\mathit{\Psi}$ to be computed with respect to the two-step method, where the number of static analyses necessary to generate the constraint modes ${\mathit{\Psi}}_{\mathbf{1}}$ is proportional to the number of contact dofs and to the number of inter-sector interface dofs.
- In the direct method, the linear modal analyses (one per each harmonic index h, necessary to compute the fixed-interface modes $\mathit{\varphi}$, are computed on the full model of the fundamental sector. In the two-step method, only one modal analysis of the full model of the fundamental sector is performed to obtain ${\mathit{\varphi}}_{\mathit{1}}$, and the modal analyses performed in the second reduction operate on an already reduced model.
- Fixed-interface modes $\mathit{\varphi}$ computed with the direct method respect the cyclic symmetry properties of the bladed disk, while fixed-interface modes ${\mathit{\varphi}}_{\mathbf{1}}$ assume fixed inter-sector interfaces.

## 4. Numerical Results

#### 4.1. First Test Cases for Accuracy and Efficiency Analysis

_{1}of fixed-interface modes during the 1st step (Z

_{1}= 60, 120 and 180). For each set, three subsets are generated with an increasing number Z

_{2}of slave modes (Z

_{2}= 20, 40 and 60) in the 2nd step.

_{2}slave modes in the 2nd step. In this way, ROMs with the same size are compared. Of course, for the two-step method, the overall calculation time, necessary to perform both steps, is given for a fair comparison.

_{1}= 60). Results are summarized in Figure 5 and Figure 6.

_{2}is increased from 40 to 60. We interpret this result as a lack of accuracy of the ROM generated in the 1st step because of the fixed-interface boundary conditions applied to the sector interface nodes ${\mathit{q}}_{\mathit{H}}$ and ${\mathit{q}}_{\mathit{L}}$.

_{1}= 120 and Z

_{1}= 180, respectively. In Figure 7, the effect of the number of modes of the 1st reduction (Z

_{1}) is shown for the first 10 natural frequencies associated with h = 1.

_{2}= 60) as the final one generated with the direct method (Figure 4). It is observed that increasing the number of modes retained in the 1st step significantly affects the accuracy of the final ROM, specifically for modes #9 and #10. This behavior agrees with the interpretation given about the results in Figure 5 because of the (fixed) boundary conditions applied to the disk interfaces, and the reduction basis used in the 1st step must include a number of modes (Z

_{1}) significantly higher (up to three times) than the number of modes retained in the 2nd step (Z

_{2}), to accurately represent the dynamics of the bladed disk.

_{1}= 180 modes at the 1st step and Z

_{1}= 60 modes in the 2nd step. In this way, the ROMs will have the same final size and, according to the previous analysis, comparable accuracy.

#### 4.2. Additional Test Case for Efficiency Analysis.

- the calculation of the constraint modes in the 1st step is the most time-consuming operation in the two-step process,
- the computation of the fixed-interface modes is the most time-consuming step of the direct method.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 4.**Test Case A (linear elements)—accuracy of the direct method for h = 1 (Up) and h = 13 (Low).

**Figure 8.**Particular of the Test Case C mesh (

**a**) at the blade root and (

**b**) at the blade tip (blade-shroud contact area, right).

Test Case | A | B | ||
---|---|---|---|---|

Blades | 27 | 81 | ||

Sector | ||||

Element Type | Linear | Quadratic | Linear | Quadratic |

Elements | 16,561 | 16,561 | 5627 | 5627 |

Nodes | 6052 | 30,920 | 2454 | 11,672 |

Direct Method | Two-Step Method | ||
---|---|---|---|

1st Set of ROMs | 2nd Set of ROMs | 3rd Set of ROMs | |

Z | (Z_{1}; Z_{2}) | (Z_{1}; Z_{2}) | (Z_{1}; Z_{2}) |

20 | 60; 20 | 120; 20 | 180; 20 |

40 | 60; 40 | 120; 40 | 180; 40 |

60 | 60; 60 | 120; 60 | 180; 60 |

Full | Direct Method | Two-Step Method | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

(Hz) | Z_{1} = 60 | Z_{1} = 120 | Z_{1} = 180 | |||||||||

Z = 20 | Z = 40 | Z = 60 | Z_{2} = 20 | Z_{2} = 40 | Z_{2} = 60 | Z_{2} = 20 | Z_{2} = 40 | Z_{2} = 60 | Z_{2} = 20 | Z_{2} = 40 | Z_{2} = 60 | |

200.18 | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% |

488.70 | 0.01% | 0.01% | 0.00% | 0.01% | 0.01% | 0.00% | 0.01% | 0.01% | 0.00% | 0.01% | 0.01% | 0.00% |

1044.80 | 0.02% | 0.00% | 0.00% | 0.02% | 0.00% | 0.00% | 0.02% | 0.00% | 0.00% | 0.02% | 0.00% | 0.00% |

1211.40 | 0.07% | 0.01% | 0.01% | 0.07% | 0.01% | 0.01% | 0.07% | 0.01% | 0.01% | 0.07% | 0.01% | 0.01% |

1943.90 | 0.02% | 0.00% | 0.00% | 0.03% | 0.01% | 0.01% | 0.03% | 0.01% | 0.00% | 0.03% | 0.00% | 0.00% |

3439.40 | 0.27% | 0.01% | 0.00% | 0.27% | 0.01% | 0.00% | 0.27% | 0.01% | 0.00% | 0.27% | 0.01% | 0.00% |

3895.00 | 1.51% | 0.08% | 0.01% | 1.51% | 0.08% | 0.01% | 1.51% | 0.08% | 0.01% | 1.51% | 0.08% | 0.01% |

6389.70 | 0.05% | 0.03% | 0.00% | 0.05% | 0.03% | 0.00% | 0.05% | 0.03% | 0.00% | 0.05% | 0.03% | 0.00% |

6915.70 | 0.37% | 0.03% | 0.01% | 0.41% | 0.07% | 0.04% | 0.38% | 0.04% | 0.01% | 0.37% | 0.04% | 0.01% |

7775.40 | 0.12% | 0.01% | 0.00% | 0.46% | 0.35% | 0.34% | 0.16% | 0.04% | 0.03% | 0.13% | 0.02% | 0.01% |

Test Case | Node # | Direct (s) | Two-Step (s) | Two-Step/Direct |
---|---|---|---|---|

B-Lin | 2454 | 328 | 641 | 1.95 |

B-Quad | 6052 | 779 | 1169 | 1.50 |

A-Lin | 11,672 | 322 | 336 | 1.04 |

A-Quad | 30,920 | 1120 | 620 | 0.55 |

Test Case | Node # | Direct (s) | Two-Step (s) | Two-Step/Direct |
---|---|---|---|---|

B-Lin | 2454 | 328 | 641 | 1.95 |

B-Quad | 6052 | 779 | 1169 | 1.50 |

A-Lin | 11,672 | 322 | 336 | 1.04 |

C-Lin | 13,885 | 646 | 110 | 0.17 |

A-Quad | 30,920 | 1120 | 620 | 0.55 |

C-Quad | 89,228 | 2244 | 332 | 0.15 |

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**MDPI and ACS Style**

Sommariva, A.; Zucca, S.
A Comparison between Two Reduction Strategies for Shrouded Bladed Disks. *Appl. Sci.* **2018**, *8*, 1736.
https://doi.org/10.3390/app8101736

**AMA Style**

Sommariva A, Zucca S.
A Comparison between Two Reduction Strategies for Shrouded Bladed Disks. *Applied Sciences*. 2018; 8(10):1736.
https://doi.org/10.3390/app8101736

**Chicago/Turabian Style**

Sommariva, Alessandro, and Stefano Zucca.
2018. "A Comparison between Two Reduction Strategies for Shrouded Bladed Disks" *Applied Sciences* 8, no. 10: 1736.
https://doi.org/10.3390/app8101736