On the stability of periodic motions of an unbalanced rigid rotor on lubricated journal bearings

A theoretical investigation is carried out on the orbital motions of a symmetrical, unbalanced, rigid rotor subjected to a constant vertical load and supported on two lubricated journal bearings. In order to determine the fluid film forces, the short bearing theory is adopted.A method is illustrated that makes it possible to determine the analytical equation of the orbit as an approximated solution of the system of non-linear differential equations of motion of the journal axis. A procedure is also described for evaluating the stability of the solution found. Diagrams of the curves delimiting, in the working plane of the rotor -m, the areas of stability of the various periodic solutions determined are provided.Finally, the results obtained are compared and combined with those provided by a direct integration of the motion equation made using the Runge-Kutta method.Nomenclature C radial clearance - D = 2R bearing diameter - E mass unbalance cecentricity - Fx, Fy fluid film force components - fi = Fi/W dimensionless fluid film force components - L bearing length - M one half rotor mass - % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbnL2yY9% 2CVzgDGmvyUnhitvMCPzgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqe% fqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYlf9irVeeu0d% Xdh9vqqj-hEeeu0xXdbba9frFf0-OqFfea0dXdd9vqaq-JfrVkFHe9% pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaaca% qabeaadaabauaaaOqaaiaad2gacqGH9aqpcaWGnbGaam4qaiabeM8a% 3naaCaaaleqabaGaaGOmaaaakiaac+cacqaHdpWCcaWGxbaaaa!471F!\[m = MC\omega ^2 /\sigma W\] dimensionless one half rotor mass - R bearing radius - T = 2 synchronous orbit period - t time - W load per bearing - X, Y, Z coordinates - x = X/C; y = y/C; z = Z/L dimensionless coordinates - oil dynamic viscosity - = E/C dimensionless mass unbalance eccentricity - = (RL/W)/(R/C) ² (L/D) ² modified Sommerfeld number - = t dimensionless time = periodic orbit frequency - = 2/ frequency ratio - journal angular velocity - (·) dimensionless time derivative     

Static and dynamic behaviour of a rigid rotor on journal bearings

This experimental investigation concerns the static and dynamic behaviour of a rigid balanced rotor on journal bearings. Running conditions were chosen in order to obtain relatively high eccentricity ratios near the instability threshold speeds.The results, compared also with the theoretical predictions of some authors, have substantially shown the importance of examining the static behaviour near the instability threshold speed to allow a correct interpretation of some aspects of the dynamic behaviour.In order to clarify the discrepancies between the observed and the theoretically predicted dynamic behaviour, some further tests have been carried out.

---

Sommario E' stato esaminato sperimentalmente sia il comportamento statico che quello dinamico di un rotore rigido e bilanciato sostenuto da cuscinetti cilindrici lubrificati.Le condizioni di funzionamento sono state scelte in modo tale che, in prossimità della soglia di instabilità, il valore del rapporto di eccentricità risultasse relativamente grande.I risultati, confrontati anche con le indicazioni teoriche di altri autori, hanno sostanzialmente evidenziato l'importanza dell'esame del comportamento statico in prossimità della soglia di instabilità al fine di interpretare correttamente alcuni aspetti del fenomeno dell'instabilità.Sono state inoltre effettuate una serie di ulteriori indagini con lo scopo di chiarire le discordanze fra il comportamento dinamico osservato e quello previsto dalla teoria.

     

Steady-state behaviour of flexible rotordynamic systems with oil journal bearings

This paper deals with the long term behaviour of flexible rotor systems, which are supported by nonlinear bearings. A system consisting of a rotor and a shaft which is supported by one oil journal bearing is investigated numerically. The shaft is modelled using finite elements and reduced using a component mode synthesis method. The bearings are modelled using the finite-length bearing theory. Branches of periodic solutions are calculated for three models of the system with an unbalance at the rotor. Also self-excited oscillations are calculated for the three models if no mass unbalance is present. The results show that a mass unbalance can stabilize rotor oscillations.     

A numerical method on estimation of stable regions of rotor systems supported on lubricated bearings

ＩｎｔｒｏｄｕｃｔｉｏｎＷｈｅｎｃｏｎｔｒｏｌｌｉｎｇｔｈｅｄｙｎａｍｉｃｓｔａｂｉｌｉｔｙｏｆｌａｒｇｅｒｏｔａｔｉｎｇｍａｃｈｉｎｅｒｙ ,ｎｏｔｏｎｌｙｔｈｅｐｒｏｂｌｅｍｗｈｅｔｈｅｒｅｑｕｉｌｉｂｒｉｕｍｓｔａｔｅｏｆｔｈｅｓｙｓｔｅｍｉｓｓｔａｂｌｅｍｕｓｔｂｅｓｏｌｖｅｄ ,ｂｕｔａｌｓｏｔｈｅｒｅｇｉｏｎｏｆａｓｙｍｐｔｏｔｉｃｓｔａｂｉｌｉｔｙｎｅｅｄｔｏｂｅｄｅｔｅｒｍｉｎｅｄ .Ｗｈｅｎｔ→∞ ,ｓｏｌｕｔｉｏｎｓｕｎｄｅｒｉｎｉｔｉａｌｃｏｎｄｉｔｉｏｎｓｗｉｔｈｉｎｓｕｃｈｒｅ…     

Operator‐splitting method for the analysis of cavitation in liquid‐lubricated herringbone grooved journal bearings

This paper presents an operator‐splitting method (OSM) for the solution of the universal Reynolds equation. Jakobsson–Floberg–Olsson (JFO) pressure conditions are used to study cavitation in liquid‐lubricated journal bearings. The shear flow component of the oil film is first solved by a modified upwind finite difference method. The solution of the pressure gradient flow component is computed by the Galerkin finite element method. Present OSM solutions for slider bearings are in good agreement with available analytical and experimental results. OSM is then applied to herringbone grooved journal bearings. The film pressure, cavitation areas, load capacity and attitude angle are obtained with JFO pressure conditions. The calculated load capacities are in agreement with available experimental data. However, a detailed comparison of the present results with those predicted using Reynolds pressure conditions shows some differences. The numerical results showed that the load capacity and the critical mass of the journal (linear stability indicator) are higher and the attitude angle is lower than those predicted by Reynolds pressure conditions for cases of high eccentricities. Copyright © 2004 John Wiley & Sons, Ltd.     

Performance of hydrodynamic journal bearing under the combined influence of textured surface and journal misalignment: A numerical survey

A wisely chosen geometry of micro textures with the favorable relative motion of lubricated surfaces in contacts can enhance tribological characteristics. In this paper, a computational investigation related to the combined influence of bearing surface texturing and journal misalignment on the performances of hydrodynamic journal bearings is reported. To this end, a numerical analysis is performed to test three texture shapes: square “SQ”, cylindrical “CY”, and triangular “TR”, and shaft misalignment variation in angle and degree. The Reynolds equation of a thin viscous film is solved using a finite differences scheme and a mass conservation algorithm (JFO boundary conditions), taking into account the presence of textures on both full film and cavitation regions. Preliminary results are compared with benchmark data and are consistent with a positive enhancement in misaligned bearing performances (load carrying capacity and friction). The results suggest that the micro-step bearing mechanism is a key parameter, where the micro-pressure recovery action present in dimples located at the second angular part of the bearing (from 180° to 360°) can compensate for the loss on performances caused by shaft misalignment, while the micro-pressure drop effect at the full film region causes poor performances. Considering the right arrangement of textures on the contact surface, their contours geometries can have a significant impact on the performance of misaligned journal bearings, particularly at high eccentricity ratios, high misalignment degrees and when the misalignment angle α approaches to $0\mathrm{°}$ or $180\mathrm{°}$.     

Spectral element analysis of herringbone‐grooved journal bearings with groove–ridge discontinuity

Many researchers have investigated the performance of herringbone‐grooved journal bearings (HGJBs). However, few have yet mentioned the issue of film thickness discontinuities in HGJBs with a finite number of grooves. Most studies have involved the application of a finite difference method for discretization. The present work utilizes the spectral element method to calculate the pressure distribution and dynamic coefficients of HGJBs, in which the thickness of the fluid film changes abruptly in the groove–ridge region. Conservation of mass is adopted to solve the problem. Additionally, the present method can be adopted for grooves with curvy geometry. The numerical results were compared with the analytical solution for a one‐dimensional slider bearing and an HGJB. It also shows that for the case of HGJB, the numerical result by the present method is more accurate than the numerical results found in the literature (Trans. ASME J. Tribol. 2000; 122 : 103–109, Int. J. Numer. Meth. Heat Fluid Flow 2002; 12 : 518–540). Furthermore, employing the present method with the Elrod algorithm can improve the accuracy of deriving loads of HGJBs when cavitation occurs. In addition, the result displays the efficiency of the present method by observing the CPU time. Therefore, the approach can be employed to compute the critical mass of a HGJB. The influence of changing groove angle, groove depth, groove width, and the eccentricity on the critical mass are discussed. Observing the variations in critical mass shows that when the eccentricity is small, a larger groove angle, a lower groove depth, and smaller groove width correspond to a higher critical mass of the HGJB. Copyright © 2010 John Wiley & Sons, Ltd.     

表面涂敷聚四氟乙烯固体润滑薄膜的混合式陶瓷球轴承与全钢轴承性能对比分析

以受轴向载荷的三点接触球轴承为研究对象，采用Heathcote轴承滚动滑动理论，计算了轴承球与套圈之间的表面滑动，并根据固体表面受力变形，计算了固体薄膜润滑情况下轴承球与滚道间的牵引力，对比分析了混合式陶瓷球轴承和全钢轴承在高速下产生的热量、表面剪应力、旋滚比以及接触角等重要轴承性能参数．结果表明：在PTFE固体薄膜润滑下，陶瓷球轴承在40 000r／min、3000N载荷下所产生的热量仅为钢轴承的62．6％，旋滚比为钢轴承的18％．试验结果对特殊工况下陶瓷球轴承的设计和应用具有参考价值．