# A New Geometrical Design to Overcome the Asymmetric Pressure Problem and the Resulting Response of Rotor-Bearing System to Unbalance Excitation

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## Abstract

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## 1. Introduction

## 2. Governing Equations and the Mathematical Model

## 3. The Dynamic Characteristics of the Finite Length Bearing

## 4. Unbalance Excitation and Linear Stability Analysis

## 5. Numerical Solution

## 6. Results and Discussions

_{min}) and the dimensionless critical speed (CS) are plotted against the number of nodes. It can be seen that increasing the number of nodes $\left(N\times M\right)$ beyond 16,471 has a trivial effect on both results despite the narrow range of the y-axes of H

_{min}and CS. Therefore, this number is adopted in this analysis. Furthermore, the current model is also verified using the results presented by reference [22], as shown in Table 1. A relatively low and high value of the Summerfield numbers is used in the comparison shown in this table. It can be seen that very good agreements are found for the values of ${K}_{XX}$ and ${K}_{YY}$ for the two Summer field numbers.

_{min}and P

_{max}values for this case are 0.4 and 0.645, respectively. The presence of the misalignment increases P

_{max}by 63.8% to 1.057 and reduces H

_{min}by 85.7% to 0.0572, representing a significant change in the designed values of the system. In addition, the film thickness distribution is no longer prismatic, and the pressure distribution is not symmetric about the middle axis of the bearing. Such a level of minim film thickness reduces the bearing life due to the expected high friction and wear ranges. In order to overcome this problem, the load-carrying capacity of the bearing should be reduced significantly to increase the value of the minimum film thickness. However, this is not an ideal solution as the bearing is usually designed for a specified supported load. Therefore, any other solution to this problem keeps the same designed load-carrying capacity without sacrificing acceptable levels of the minimum film thickness represents an important outcome. The values of the geometrical design parameters of the bearing profile can play a significant role in this direction. Improving the geometry of the bearing by removing material from the inner surface of the bearing to compensate for the reduction in the film thickness related to the misalignment has significantly positive effects on film thickness and pressure values. Figure 5c illustrates these effects where the dimensionless minimum film thickness becomes 0.244, and the maximum pressure reduces to 0.724. This represents several times improvement in the value of H

_{min}and a decrease of 31.5% in P

_{max}. The distribution shapes have also improved, particularly the pressure distribution shape, where the pressure spikes are reduced and shifted away from the bearing edges due to the improvement in the film thickness levels at the zones where the misalignment has the most negative effect.

_{max}and elevating the level of the H

_{min}for the same designed load-carrying capacity. Therefore, the investigation proceeded to examine the effect of the geometrical parameters on the main dynamic characteristics and the response of the rotor-bearing system to an unbalanced excitation. Figure 6 shows the variation of the equivalent stiffness coefficient for the modified and unmodified bearing geometry with a wide range of 3D misalignment parameters. Acceding this misalignment parameter range is insufficient for practical considerations as the minimum film thickness becomes extremely thin. It can be seen that for the ideal case (without misalignment) where $\Delta {v}_{max}=\Delta {h}_{max}=0$, the value of ${K}_{eq}$ is 1.347. As the misalignment parameters increase, the values of ${K}_{eq}$ also, increase with a maximum value of 3.161 when $\Delta {v}_{max}=\Delta {h}_{max}=0.59$. This increase means, in other words, an elevation in the level of the critical speed (which will be explained later) of the rotor bearing system, but large drawbacks accompany it in terms of H

_{min}and P

_{max,}as illustrated in the previous figure. Therefore, it cannot be considered an enhancement of the system’s performance. On the other hand, the geometrical modification of the profile reduces the ${K}_{eq}$ in comparison with the misalignment case. The reduction in ${K}_{eq}$ at the extreme case when $\Delta {v}_{max}=\Delta {h}_{max}=0.59$ is 32.64% (${K}_{eq}=2.129)$. Despite this reduction in ${K}_{eq}$ as a result of introducing the geometrical change in the bearing profile, the equivalent stiffness values for the whole considered range of misalignment parameters remain higher than the stiffness of the ideal case where $\Delta {v}_{max}=\Delta {h}_{max}=0$. The improvement in ${K}_{eq}$ for the cases of $\Delta {v}_{max}=\Delta {h}_{max}=0.59$ is 58.1% in comparison with the ideal case. This outcome represents a significant improvement in the system performance due to the geometrical design changes, the previously mentioned reduction in P

_{max}, and the significant increase in H

_{min}. The modification parameters for the results presented in this figure are $A=B=0.25$. The effect of the design parameter $B$ on the equivalent stiffness is shown in Figure 7, and the optimum value of the other design parameter (parameter $A$) was found to be 0.25. It can be seen that increasing the length of modification along the axial direction of the bearing (parameter $B$) reduces the values of ${K}_{eq}$. However, the equivalent stiffness coefficient is not continued to change significantly when $B\ge 0.25$. This value of $B\left(0.25\right)$ also gives the best outcome in terms of Pmax and Hmin, which are therefore adopted for the main analysis in this work.

_{min}and reducing the levels of P

_{max}resulting from the 3D misalignment case and also resulting in higher critical speed in comparison with the ideally aligned bearing system. Furthermore, analyzing the system response under an unbalance excitation showed that the modification does not cause higher transient amplitude for the rotation around the steady state position at the high operating speed of rotation. This important outcome can be further advanced to consider the 3D misalignment in analyzing the response to unbalance excitation, which is an extremely complex case as the model requires a solution in space for the equations of motions. The authors intend to address this important issue in the near future work as both the misalignment and the unbalance excitation are common machinery problems.

## 7. Conclusions and Remarks

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Schematic representation of the journal bearing. (

**a**) Ideal case (without misalignment), (

**b**) Representation of the 3D deviation, and (

**c**) Variable profile.

**Figure 2.**The rotor-bearing system ([37] edited).

**Figure 5.**3D dimensionless film thickness (

**left**) and 3D dimensionless pressure distribution (

**right**). (

**a**) Perfect case, (

**b**) Misaligned and (

**c**) Misaligned and Modified.

**Figure 6.**Effect of geometrical modification on the equivalent stiffness coefficient for a range of 3D misalignment parameters.

**Figure 7.**Effect of geometrical parameter (B) along the bearing width on the equivalent stiffness coefficient for the bearings under misalignment $\left(\Delta {v}_{max}=\Delta {h}_{max}=0.59\right)$.

**Figure 8.**Trajectories of the journal center under different rotational speeds and unbalanced excitation. (

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

**b**) At 0.75 of the critical speed, and (

**c**) At the critical speed. The dimensionless critical speed is 2.6789 (6540.93 rpm).

**Figure 9.**Journal trajectory when the rotational speed is 3000 rpm (1.0 $\mathsf{\Gamma}$) for different values of geometrical bearing profile parameters. (

**a**) without mod, (

**b**) $A=B=0.1$, (

**c**) $A=B=0.2$, (

**d**) $A=B=0.25$ and (

**e**) $A=B=0.3$.

**Figure 10.**Journal trajectory when the rotational speed is 3000 rpm (1.0 $\mathsf{\Gamma}$) when $A=0.25$ and using different values of $B$. (

**a**) without mod, (

**b**) $B=0.1$, (

**c**)$B=0.2$, (

**d**) $B=0.25$(

**e**) $B=0.3$ and (

**f**) $B=0.4$.

**Figure 11.**Journal trajectories for the unmodified and modified bearing profile relative to the steady state position (in terms of ${X}^{\prime}$ and ${Y}^{\prime}$ coordinates) when the rotational speed is 3000 rpm (1.0 $\mathsf{\Gamma})$ and $A=B=0.25$.

**Figure 12.**Journal trajectories for the unmodified and modified bearing profile relative to the steady state position (in terms of ${X}^{\prime}$ and ${Y}^{\prime}$ coordinates) when the rotational speed is equal to the critical speed of the modified system, 2.8636 (6991.917 rpm) and $A=B=0.25$.

**Figure 13.**Variation of ${X}^{\prime}$ and ${Y}^{\prime}$ with the time (T = $\mathsf{\Omega}t)$ in rad for the modified and modified bearing. (

**a**) variation of ${X}^{\prime}$ and (

**b**) variation of ${Y}^{\prime}$.

**Table 1.**Comparison with the results of Ref. [22].

SF No. | K_{yy} | K_{xx} | ||
---|---|---|---|---|

Ref. [22] | Results of the Presented Model | Ref. [22] | Results of the Presented Model | |

0.319 | 2.10 | 1.99 | 3.35 | 3.34 |

1.220 | 2.30 | 2.21 | 1.62 | 1.69 |

Case | As It Is | $\mathbf{Modified}\left(\mathit{A}=\mathit{B}=0.25\right)$ |
---|---|---|

Ideal | 2.6789 | 2.8636 |

Mis. $\Delta {v}_{max}=\Delta {h}_{max}=0.5$ | 3.4366 | 3.2539 |

Mis. $\Delta {v}_{max}=\Delta {h}_{max}=0.59$ | 4.0347 | 3.4121 |

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**MDPI and ACS Style**

Jamali, H.U.; Aljibori, H.S.S.; Al-Tamimi, A.N.J.; Abdullah, O.I.; Senatore, A.; Mohammed, M.N.
A New Geometrical Design to Overcome the Asymmetric Pressure Problem and the Resulting Response of Rotor-Bearing System to Unbalance Excitation. *Axioms* **2023**, *12*, 812.
https://doi.org/10.3390/axioms12090812

**AMA Style**

Jamali HU, Aljibori HSS, Al-Tamimi ANJ, Abdullah OI, Senatore A, Mohammed MN.
A New Geometrical Design to Overcome the Asymmetric Pressure Problem and the Resulting Response of Rotor-Bearing System to Unbalance Excitation. *Axioms*. 2023; 12(9):812.
https://doi.org/10.3390/axioms12090812

**Chicago/Turabian Style**

Jamali, Hazim U., H. S. S. Aljibori, Adnan Naji Jameel Al-Tamimi, Oday I. Abdullah, Adolfo Senatore, and M. N. Mohammed.
2023. "A New Geometrical Design to Overcome the Asymmetric Pressure Problem and the Resulting Response of Rotor-Bearing System to Unbalance Excitation" *Axioms* 12, no. 9: 812.
https://doi.org/10.3390/axioms12090812