# An Algebraic Approach for Identification of Rotordynamic Parameters in Bearings with Linearized Force Coefficients

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Mathematical Model of the Rotor-Bearing System

#### 2.2. Operation Velocity of the Rotor-Bearing System

- ${\dot{\varphi}}_{0}$ is the excitation frequency at the ramp beginning;
- $\ddot{\varphi}$ is the change ratio with respect to time of the excitation frequency;
- t is the time.

#### 2.3. Mathematical Model for Bearing Rotordynamic Parameters Identification

#### 2.3.1. Algebraic Identifier with Constant Operation Velocity

#### 2.3.2. Algebraic Identifier with Variable Operation Velocity

## 3. Results

#### 3.1. Algebraic Parameter Identification with Constant System Velocity

#### 3.2. Algebraic Parameter Identification with Variable System Velocity

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

## References

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**Figure 2.**Stiffness and damping parameters in bearings [40].

**Figure 3.**Rotor-bearing system scheme [40].

**Figure 5.**Identified stiffness parameters for bearing 1 at 600 rpm. (

**a**) ${k}_{xx}$, (

**b**) ${k}_{xz}$, (

**c**) ${k}_{zx}$, (

**d**) ${k}_{zz}$.

**Figure 6.**Identified damping parameters for bearing 1 at 600 rpm. (

**a**) ${c}_{xx}$, (

**b**) ${c}_{xz}$, (

**c**) ${c}_{zx}$, (

**d**) ${c}_{zz}$.

**Figure 7.**Identified stiffness parameters for bearing 2 at 600 rpm. (

**a**) ${k}_{xx}$, (

**b**) ${k}_{xz}$, (

**c**) ${k}_{zx}$, (

**d**) ${k}_{zz}$.

**Figure 8.**Identified damping parameters for bearing 2 at 600 rpm. (

**a**) ${c}_{xx}$, (

**b**) ${c}_{xz}$, (

**c**) ${c}_{zx}$, (

**d**) ${c}_{zz}$.

**Figure 10.**Identified stiffness parameters for bearing 1 with a linear ramp of excitation of $10rad/{s}^{2}$. (

**a**) ${k}_{xx}$, (

**b**) ${k}_{xz}$, (

**c**) ${k}_{zx}$, (

**d**) ${k}_{zz}$.

**Figure 11.**Identified damping parameters for bearing 1 with a linear ramp of excitation of $10rad/{s}^{2}$. (

**a**) ${c}_{xx}$, (

**b**) ${c}_{xz}$, (

**c**) ${c}_{zx}$, (

**d**) ${c}_{zz}$.

**Figure 12.**Identified stiffness parameters for bearing 2 with a linear ramp of excitation of $10rad/{s}^{2}$. (

**a**) ${k}_{xx}$, (

**b**) ${k}_{xz}$, (

**c**) ${k}_{zx}$, (

**d**) ${k}_{zz}$.

**Figure 13.**Identified damping parameters for bearing 2 with a linear ramp of excitation of $10rad/{s}^{2}$. (

**a**) ${c}_{xx}$, (

**b**) ${c}_{xz}$, (

**c**) ${c}_{zx}$, (

**d**) ${c}_{zz}$.

**Figure 14.**Identified stiffness parameters for bearing 1. (

**a**) ${k}_{xx}$, (

**b**) ${k}_{xz}$, (

**c**) ${k}_{zx}$, (

**d**) ${k}_{zz}$ at 50,000 rpm.

**Figure 15.**System vibratory response at node 4 with a linear ramp excitation with acceleration of $6000rad/{s}^{2}$.

**Figure 16.**Identified stiffness parameters for bearing 1 with a ramp excitation of $6000rad/{s}^{2}$. (

**a**) ${k}_{xx}$, (

**b**) ${k}_{xz}$, (

**c**) ${k}_{zx}$, (d) ${k}_{zz}$.

**Figure 17.**Identified damping parameters for bearing 1 with a ramp excitation of $6000rad/{s}^{2}$. (

**a**) ${c}_{xx}$, (

**b**) ${c}_{xz}$, (

**c**) ${c}_{zx}$, (

**d**) ${c}_{zz}$.

**Figure 18.**Identified stiffness parameters for bearing 2 with a ramp excitation of $6000rad/{s}^{2}$. (

**a**) ${k}_{xx}$, (

**b**) ${k}_{xz}$, (

**c**) ${k}_{zx}$, (d) ${k}_{zz}$.

**Figure 19.**Identified damping parameters for bearing 2 with a ramp excitation of $6000rad/{s}^{2}$. (

**a**) ${c}_{xx}$, (

**b**) ${c}_{xz}$, (

**c**) ${c}_{zx}$, (

**d**) ${c}_{zz}$.

Parameter | Value | Parameter | Value |
---|---|---|---|

Modulus of elasticity | $2\times {10}^{11}\mathrm{N}/{\mathrm{m}}^{2}$ | ${L}_{1}$ | $0.035\mathrm{m}$ |

Density | $7800\mathrm{kg}/{\mathrm{m}}^{3}$ | ${L}_{2}$ | $0.010\mathrm{m}$ |

Poisson ratio | $0.3$ | ${L}_{3}$ | $0.025\mathrm{m}$ |

${r}_{1}$ | $0.005\mathrm{m}$ | ${L}_{4}$ | $0.130\mathrm{m}$ |

${r}_{2}$ | $0.02\mathrm{m}$ | ${L}_{5}$ | $0.050\mathrm{m}$ |

${r}_{3}$ | $0.035\mathrm{m}$ | ${L}_{6}$ | $0.050\mathrm{m}$ |

Parameter | Value | Parameter | Value |
---|---|---|---|

${D}_{1}$ mass | $1.2\mathrm{kg}$ | ${D}_{2}$ mass | $1.0\mathrm{kg}$ |

${D}_{1}$ moment of inertia | $1.2\times {10}^{-3}\mathrm{kg}\xb7{\mathrm{m}}^{2}$ | ${D}_{2}$ moment of inertia | $1.0\times {10}^{-3}\mathrm{kg}\xb7{\mathrm{m}}^{2}$ |

${D}_{1}$ polar moment of inertia | $2.4\times {10}^{-3}\mathrm{kg}\xb7{\mathrm{m}}^{2}$ | ${D}_{2}$ polar moment of inertia | $2.0\times {10}^{-3}\mathrm{kg}\xb7{\mathrm{m}}^{2}$ |

${D}_{1}$ mass unbalance | $5\times {10}^{-7}\mathrm{kg}\xb7\mathrm{m}\angle 0rad$ | ${D}_{2}$ mass unbalance | $5\times {10}^{-7}\mathrm{kg}\xb7\mathrm{m}\angle \pi rad$ |

**Table 3.**Stiffness and damping bearing parameters [40].

Parameter | Bearing 1 (Node 4) | Bearing 2 (Node 8) |
---|---|---|

${k}_{xx}$ | $8\times {10}^{7}\mathrm{N}/\mathrm{m}$ | $5\times {10}^{7}\mathrm{N}/\mathrm{m}$ |

${k}_{xz}$ | $-1\times {10}^{7}\mathrm{N}/\mathrm{m}$ | $-2\times {10}^{7}\mathrm{N}/\mathrm{m}$ |

${k}_{zx}$ | $-6\times {10}^{7}\mathrm{N}/\mathrm{m}$ | $-4\times {10}^{7}\mathrm{N}/\mathrm{m}$ |

${k}_{zz}$ | $1\times {10}^{8}\mathrm{N}/\mathrm{m}$ | $7\times {10}^{7}\mathrm{N}/\mathrm{m}$ |

${c}_{xx}$ | $8\times {10}^{3}\mathrm{N}\xb7\mathrm{s}/\mathrm{m}$ | $6\times {10}^{3}\mathrm{N}\xb7\mathrm{s}/\mathrm{m}$ |

${c}_{xz}$ | $-3\times {10}^{3}\mathrm{N}\xb7\mathrm{s}/\mathrm{m}$ | $-1.5\times {10}^{3}\mathrm{N}\xb7\mathrm{s}/\mathrm{m}$ |

${c}_{zx}$ | $-3\times {10}^{3}\mathrm{N}\xb7\mathrm{s}/\mathrm{m}$ | $-1.5\times {10}^{3}\mathrm{N}\xb7\mathrm{s}/\mathrm{m}$ |

${c}_{zz}$ | $1.2\times {10}^{4}\mathrm{N}\xb7\mathrm{s}/\mathrm{m}$ | $8\times {10}^{3}\mathrm{N}\xb7\mathrm{s}/\mathrm{m}$ |

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**MDPI and ACS Style**

Mendoza-Larios, J.G.; Barredo, E.; Arias-Montiel, M.; Baltazar-Tadeo, L.A.; Landa-Damas, S.J.; Tapia-Herrera, R.; Colín-Ocampo, J.
An Algebraic Approach for Identification of Rotordynamic Parameters in Bearings with Linearized Force Coefficients. *Mathematics* **2021**, *9*, 2747.
https://doi.org/10.3390/math9212747

**AMA Style**

Mendoza-Larios JG, Barredo E, Arias-Montiel M, Baltazar-Tadeo LA, Landa-Damas SJ, Tapia-Herrera R, Colín-Ocampo J.
An Algebraic Approach for Identification of Rotordynamic Parameters in Bearings with Linearized Force Coefficients. *Mathematics*. 2021; 9(21):2747.
https://doi.org/10.3390/math9212747

**Chicago/Turabian Style**

Mendoza-Larios, José Gabriel, Eduardo Barredo, Manuel Arias-Montiel, Luis Alberto Baltazar-Tadeo, Saulo Jesús Landa-Damas, Ricardo Tapia-Herrera, and Jorge Colín-Ocampo.
2021. "An Algebraic Approach for Identification of Rotordynamic Parameters in Bearings with Linearized Force Coefficients" *Mathematics* 9, no. 21: 2747.
https://doi.org/10.3390/math9212747