# Predicting the Dynamic Response of Dual-Rotor System Subject to Interval Parametric Uncertainties Based on the Non-Intrusive Metamodel

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model Description and Motion Equations

_{1}, J

_{d}

_{1}, J

_{p}

_{1}and m

_{2}, J

_{d}

_{2}, J

_{p}

_{2}are the mass, diameter moment of inertia and polar moment of inertia of the HP and LP rotors, respectively. There are mass imbalances on both of the rotors, denoted by ${e}_{1}$ and ${e}_{2}$. The angular rotating speeds of the LP and HP rotors are ${\omega}_{1}$ and ${\omega}_{2}$. The span of the system is L and the other locations are measured by their corresponding distances from the left end ${L}_{i},i=1,2,3,4$, as shown in Figure 1.

## 3. Non-Intrusive Interval Analysis of the System Based on Meta-Modeling

## 4. Results and Discussions

#### 4.1. Effect of Interval Mass Eccentricity

#### 4.2. Effect of Interval Bearing Stiffness

^{®}Core™ i7-8550U@1.8GHz. It should be noted that the actual speed interval calculated is 0–1400 rad/s and only a part of the results are presented in Figure 5. Moreover, the increment for two consecutive speed steps is small and the initial 300 periods of the vibration are skipped for every rotational speed to eliminate the transient effects. The above conditions will cause the calculation time in a single deterministic simulation to be relatively long. However, the difference of computation time between the two methods can still show their efficiency. For the steady-state dynamical response calculations, the average CPU time elapsed in the metamodel was 28.23 min, while it was 351.87 min in the scanning method. It is shown that the computational cost needed is significantly reduced in the metamodel. The above analyses verify the accuracy and efficiency of the developed interval method in the uncertain responses prediction of the dual-rotor system.

#### 4.3. Effect of Interval Geometric Length

_{1}to be uncertain as a result of different assemble conditions. The uncertain degree is chosen as 10%. Figure 7 presents the interval responses of the HP rotor under uncertain shaft length. We can find that the uncertainty has influences on the whole speed range though the physical parameter is related to the inner rotor. There are trivial peak shifts as well in both resonance peaks. However, the impacts of the uncertain length are weaker than the bearing stiffness which suggests that the dual-rotor is insensitive to the length. In the speed range right after the first peak, the bounds of the response and the deterministic curve overlapped with each other. This further proves the ability of the metamodel in the prediction of the interval response of the system evidenced by the fact that the deterministic curve is rigorously enclosed in the narrow range.

#### 4.4. Effect of Multi Interval Parameters

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**Dynamic response of the Lower Pressure (LP) rotor with uncertain imbalance on the LP rotor.

**Figure 4.**Dynamic response of the LP rotor with uncertain imbalance on the Higher Pressure (HP) rotor.

**Figure 8.**Dynamic responses of the dual-rotor with multi uncertain parameters: (

**a**) Interval responses of the HP rotor; (

**b**) Interval responses of the LP rotor.

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**MDPI and ACS Style**

Fu, C.; Feng, G.; Ma, J.; Lu, K.; Yang, Y.; Gu, F.
Predicting the Dynamic Response of Dual-Rotor System Subject to Interval Parametric Uncertainties Based on the Non-Intrusive Metamodel. *Mathematics* **2020**, *8*, 736.
https://doi.org/10.3390/math8050736

**AMA Style**

Fu C, Feng G, Ma J, Lu K, Yang Y, Gu F.
Predicting the Dynamic Response of Dual-Rotor System Subject to Interval Parametric Uncertainties Based on the Non-Intrusive Metamodel. *Mathematics*. 2020; 8(5):736.
https://doi.org/10.3390/math8050736

**Chicago/Turabian Style**

Fu, Chao, Guojin Feng, Jiaojiao Ma, Kuan Lu, Yongfeng Yang, and Fengshou Gu.
2020. "Predicting the Dynamic Response of Dual-Rotor System Subject to Interval Parametric Uncertainties Based on the Non-Intrusive Metamodel" *Mathematics* 8, no. 5: 736.
https://doi.org/10.3390/math8050736