# An Evaluation of the Design Parameters of a Variable Bearing Profile Considering Journal Perturbation in Rotor–Bearing Systems

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## Abstract

**:**

## 1. Introduction

## 2. Equations Related to the Hydrodynamic Lubrication Regime

- ▪
- The pressure value: $p=0$ at position $\theta =0$
- ▪
- The pressure gradient and the pressure value: $\frac{\partial p}{\partial \theta}=p=0$ at position $\theta ={\theta}_{c}$.

## 3. Three-Dimensional Misalignment Model

## 4. Design of the Bearing Profile

- ▪
- Curved type:

- ▪
- Linear type:

## 5. Dynamic Characteristics of the Rotor–Bearing System

## 6. Numerical Solution

## 7. Results and Discussions

#### 7.1. Effect of Mesh Density and Validation of the Current Model

#### 7.2. Effect of the 3D Misalignment on the Characteristics of the System

#### 7.3. Modification Effect on the Characteristics of Misaligned Journal Bearing

#### 7.4. Effect of Modification on the Journal Trajectory

## 8. Conclusions and Remarks

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Simplified drawing of an ideal case (without Mis.). (

**a**) Side view and (

**b**) Three-dimensional representation.

**Figure 4.**The considered system ([28] edited).

**Figure 6.**Effect of mesh density on the dimensionless critical speed and dimensionless maximum pressure.

**Figure 7.**Three-dimensional Misalignment effect for a wide range of Length/Diameter ratio on the (

**a**) Dimensionless maximum pressure and (

**b**) dimensionless minimum film thickness; ε

_{r}= 0.6, Δ

_{v}= Δ

_{h}=0.56.

**Figure 8.**Three-dimensional misalignment effect for a wide range of Length/Diameter ratio on the (

**a**) Dimensionless equivalent stiffness coefficient and (

**b**) dimensionless critical speed; ${\epsilon}_{r}=0.6,{\Delta}_{v}={\Delta}_{h}=0.56$.

**Figure 9.**Effect of 3D misalignment for a wide range of Length/Diameter ratios on the four dimensionless stiffness coefficients; ${\epsilon}_{r}=0.6,{\Delta}_{v}={\Delta}_{h}=0.56$.

**Figure 10.**Effect of 3D misalignment for a wide range of Length/Diameter ratios on the four dimensionless damping coefficients; ${\epsilon}_{r}=0.6,{\Delta}_{v}={\Delta}_{h}=0.56$.

**Figure 11.**Consequence of changing the bearing design using two forms of modification on (

**a**) the dimensionless maximum pressure and (

**b**) the dimensionless minimum film thickness: ${\rm Y}=\Gamma =0.25$.

**Figure 12.**Three-dimensional dimensionless pressure distributions when $L/D=1.0$ (left) and $L/D=2.0$ (right). (

**a**) Ideal case, (

**b**) 3D misalignment (${\mathsf{\Delta}}_{v}={\mathsf{\Delta}}_{h}=0.56)$ and (

**c**) 3D misalignment (${\mathsf{\Delta}}_{v}={\mathsf{\Delta}}_{h}=0.56$) and curved modification $\left({\rm Y}=\Gamma =0.25\right)$.

**Figure 13.**Three-dimensional dimensionless film thickness when $L/D=1.0$. (

**a**) Ideal case, (

**b**) 3D misalignment (${\mathsf{\Delta}}_{v}={\mathsf{\Delta}}_{h}=0.56)$, (

**c**) 3D misalignment (${\mathsf{\Delta}}_{v}={\mathsf{\Delta}}_{h}=0.56$) and linear modification $\left({\rm Y}=\Gamma =0.25\right)$ and (

**d**) 3D misalignment (${\mathsf{\Delta}}_{v}={\mathsf{\Delta}}_{h}=0.56$) and curved modification $\left({\rm Y}=\Gamma =0.25\right)$.

**Figure 14.**Effect of changing the geometry of the bearing using two forms of modification on (

**a**) The dimensionless equivalent stiffness and (

**b**) the dimensionless critical speed; ${\rm Y}=\Gamma =0.25$.

**Figure 15.**The journal center trajectories for the unmodified system, due to position perturbation when $L/D=1.0$ at (

**a**) half the critical speed, (

**b**) critical speed, and (

**c**) greater than ${\Omega}_{crit}$.

**Figure 16.**The journal center trajectories due to position perturbation when $L/D=1.0$ (

**a**) Modified and unmodified bearings at $0.5{\Omega}_{c}$ of unmodified bearing, (

**b**) linear (left) and curved (right) modification at ${\Omega}_{c}$ of unmodified bearing, and (

**c**) linear (left) and curved (right) modification at greater than ${\Omega}_{c}$ of unmodified bearing.

**Figure 17.**Comparison between the variation of the eccentricity ratio with time for the unmodified and modified bearings.

**Table 1.**The comparison between the results of the current work and the result of Reference [17].

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## Share and Cite

**MDPI and ACS Style**

Hamzah, A.A.; Abbas, A.F.; Mohammed, M.N.; Aljibori, H.S.S.; Jamali, H.U.; Abdullah, O.I.
An Evaluation of the Design Parameters of a Variable Bearing Profile Considering Journal Perturbation in Rotor–Bearing Systems. *Designs* **2023**, *7*, 116.
https://doi.org/10.3390/designs7050116

**AMA Style**

Hamzah AA, Abbas AF, Mohammed MN, Aljibori HSS, Jamali HU, Abdullah OI.
An Evaluation of the Design Parameters of a Variable Bearing Profile Considering Journal Perturbation in Rotor–Bearing Systems. *Designs*. 2023; 7(5):116.
https://doi.org/10.3390/designs7050116

**Chicago/Turabian Style**

Hamzah, Adawiya Ali, Abbas Fadhil Abbas, M. N. Mohammed, H. S. S. Aljibori, Hazim U. Jamali, and Oday I. Abdullah.
2023. "An Evaluation of the Design Parameters of a Variable Bearing Profile Considering Journal Perturbation in Rotor–Bearing Systems" *Designs* 7, no. 5: 116.
https://doi.org/10.3390/designs7050116