# Numerical Study on Vibration Response of Compressor Stator Blade Considering Contact Friction of Holding Ring

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Methods

#### 2.1. Frictional Contact Model

_{0}—critical displacement amplitude of the relative motion/μm; u—displacement of the relative motion/μm; μ—dynamic friction coefficient and N—normal forces of contact surfaces/N.

_{0}is the ratio of the critical friction μN and tangential stiffness K

_{d}on the contact surface, which can be described by the following formula.

_{1}is the radius of the first contact pair/m; R

_{2}is the radius of the second contact pair over m; ν

_{1}is the first contact to Poisson ratio; ν

_{2}is the second contact to Poisson ratio; E

_{1}is the elastic modulus of the first contact pair/Pa and E

_{2}is the elastic modulus of the second contact pair/Pa.

_{d}of the contact pair can be calculated by the following formula.

^{−2}and ν is the Poisson ratio of the object.

#### 2.2. Vibration Response Analysis Method

_{eq}and equivalent damping coefficient C

_{eq}are defined on the model contact surface according to the geometric position of the element. By averaging the equivalent stiffness and damping coefficients under a certain normal load N, tangential stiffness and damping coefficients of each element matrix are obtained as shown in Figure 2.

**K**and damping matrix

**C**of the spring damping element are 12 × 12 matrices. Suppose x, y and z are the directions on the node, then the stiffness matrix

**K**and damping matrix

**C**are:

_{n}is the normal contact stiffness between contact surfaces/N·m

^{−1}and n is the number of spring-damping units between contact surfaces.

#### 2.3. Airflow Load Analysis

^{−3}; U, u, v, w—velocity and its coordinate components/m·s

^{−1}; t—time/s; T—temperature/K; E, F, G—convectional momentum flux/kg·s

^{−2}·m

^{−1}; E

_{v}, F

_{v}, G

_{v}—viscosity momentum flux/kg·s

^{−2}·m

^{−1}; C

_{p}—specific heat/J·kg

^{−1}·K

^{−1}and λ—heat transfer coefficient/ W·m

^{−2}·K

^{−1}.

## 3. Computational Model

#### 3.1. Finite Element Model of Compressor Stator Blade

^{3}. Some of the parameters used here are commonly used parameter ranges in engineering. Aerodynamic excitation force was applied to the blade surface. In ANSYS, solid185 elements were used in combination to simplify the overall structure into a finite element model of stator blade nonlinear vibration, in which a few solid186 elements were used in some excessive areas.

#### 3.2. Stator Blade Load Analysis Model

^{5}. Figure 8 shows a grid diagram of the fluid domain during CFD calculation. The boundary conditions were calculated using total temperature and pressure at the entrance and mass flow at the exit. Corresponding to a working condition point of the compressor model, the total inlet pressure was 75,000 Pa. The total temperature of import was 300 K. The outlet mass flow rate was 72 kg/s.

## 4. Results and Discussion

#### 4.1. Nonlinear Vibration Results of Different Models

#### 4.2. Vibration Response under a Particular Exciting Force

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 9.**Response normalization amplitude curves under different normal loads with 40 N exciting force. (

**a**) Circumferential amplitude curve. (

**b**) Axial amplitude curve.

**Figure 10.**The equivalent stiffness coefficients of different normal loads at the resonant frequency of 40 N excitation force.

**Figure 11.**Response normalization amplitude curves under different normal loads with 4 N exciting force. (

**a**) Circumferential amplitude curve. (

**b**) Axial amplitude curve.

**Figure 12.**The equivalent stiffness coefficients of different normal loads at the resonant frequency of 4 N excitation force.

**Figure 13.**Response normalization amplitude curves under different exciting force with 30 N normal load. (

**a**) Circumferential amplitude curve. (

**b**) Axial amplitude curve.

**Figure 14.**The equivalent stiffness coefficients of different exciting force at the resonant frequency of 30 N normal load.

**Figure 15.**Compressor stator blade airflow excitation force. (

**a**) Circumferential force. (

**b**) Axial force.

**Figure 17.**Vibration response curve of stator blade under this condition. (

**a**) Circumferential amplitude curve. (

**b**) Axial amplitude curve.

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**MDPI and ACS Style**

Ma, J.; Liu, Z.; Zhang, D.; Xie, Y.
Numerical Study on Vibration Response of Compressor Stator Blade Considering Contact Friction of Holding Ring. *Appl. Sci.* **2023**, *13*, 6380.
https://doi.org/10.3390/app13116380

**AMA Style**

Ma J, Liu Z, Zhang D, Xie Y.
Numerical Study on Vibration Response of Compressor Stator Blade Considering Contact Friction of Holding Ring. *Applied Sciences*. 2023; 13(11):6380.
https://doi.org/10.3390/app13116380

**Chicago/Turabian Style**

Ma, Jiaobin, Zhufeng Liu, Di Zhang, and Yonghui Xie.
2023. "Numerical Study on Vibration Response of Compressor Stator Blade Considering Contact Friction of Holding Ring" *Applied Sciences* 13, no. 11: 6380.
https://doi.org/10.3390/app13116380