ACOUSTIC ANALYSIS OF HYDRODYNAMIC AND ELASTO-HYDRODYNAMIC OIL LUBRICATED JOURNAL BEARINGS*

BOUAZIZ Slim, FAKHFAKH Tahar

Dynamic of Mechanical Systems Research Unit (UDSM), University of Sfax, Sfax, Tunisia,

E-mail: slim.bouaziz1@gmail.com

HADDAR Mohamed

Mechanics Modelling and Production Research Unit (U2MP) and National School of Engineers of Sfax (ENIS), BP.1173, 3038, University of Sfax, Sfax, Tunisia

ACOUSTIC ANALYSIS OF HYDRODYNAMIC AND ELASTO-HYDRODYNAMIC OIL LUBRICATED JOURNAL BEARINGS*

BOUAZIZ Slim, FAKHFAKH Tahar

Dynamic of Mechanical Systems Research Unit (UDSM), University of Sfax, Sfax, Tunisia,

E-mail: slim.bouaziz1@gmail.com

HADDAR Mohamed

Mechanics Modelling and Production Research Unit (U2MP) and National School of Engineers of Sfax (ENIS), BP.1173, 3038, University of Sfax, Sfax, Tunisia

The purpose of the present paper is to investigate the effect of elastic deformation of bearing liner on the acoustic behavior of oil lubricated journal bearings. Analysis is performed for hydrodynamic(HD) and elasto-hydrodynamic (EHD) lubrications. Dynamic behavior and acoustical properties are investigated through an analysis of pressure fluctuation calculated from the Reynolds equation governing the flow in the clearance space of the journal bearing. This is solved numerically using the finite difference method with the successive over relaxation technique. In elasto-hydrodynamic lubrication, the finite element method with in iteration scheme is adopted to solve both Reynolds equation and the three-dimensional elasticity equation representing the displacement field in the bearing shell. The results show that the sound pressure level of the bearing is markedly influenced by the flexibility of the bearing liner, the viscosity of lubricant and the load applied to journal. HD analysis shows that the journal centre’s orbit, from a disturbed position, converges to the static equilibrium position faster than EHD lubrication. The results of the present paper could aid in the design of low-noise rotor-bearing systems supported by oil lubricated journal bearings.

vibration amplitude, pressure level, lubricated bearing, sound pressure

Introduction

Rotating machines are important assets in most of the industries. Bearings of rotating machinery are complex which needs accurate and reliable prediction of its dynamic characteristics and acoustic properties. Bearings, used for supporting the rotating part of machinery, are one of the crucial elements by which the safe operation of the machinery is ensured. In recent years, with continuing demands for increased performance, many rotating industrial machines are now being designed for operation at high speed, a trend which has resulted in increased mechanical vibration and acoustic problems. Many researchers have studied the vibration characteristics of bearings[1,2], but there is relatively little information regarding their acoustic properties. Therefore, bearing acoustic properties should be determined in order to solve bearingassociated problems and develop quieter systems. From this point of view, Rho and Kim[3]investigated the acoustical properties of hydrodynamic journal bearings through frequency analysis of oil pressure fluctuation through nonlinear transient analysis. Furthermore, Miettinen and Andersson[4]focus on the Acoustic Emission (AE) measurement method for monitoring the lubrication situation in grease lubricated rolling bearing. The aim of their investigation was to clarify how the contaminants in the grease influence the AE of the rolling bearing. Mirhadizadeh and Mba[5]presented an experimental test aimed at understanding the influence of speed and load on generation of AE in a hydrodynamic bearing. The research presented in this work showed that the power losses associated with hydrodynamic bearing has a direct influence on the generation of AE. Rho et al.[6]studied the effects of design parameters on the noise produced byrotor-bearing systems supported by oil lubricated journal bearings. They also presented the effects of radial clearance and the width of the bearing, lubricant viscosity, and mass eccentricity of the rotor on the noise of the bearing. Ban et al.[7]proposed a numerical investigation to determinate the sound characteristics of roller bearings operating under radial load. For the sake of simplifying the analysis, they assumed that the roller bearings are infinitely long, a noise source due to pressure fluctuation of oil film is taken as a line noise source, and acoustic energy losses in the bearing are neglected. To obtain sound characteristics of the bearing, the rolling contact load and the sound pressure level distributions were calculated for various operating conditions. Moreover, the noise caused by oil pressure fluctuation in hydrodynamic journal bearings, using a transmission theory of plane waves, was studied by Rho et al.[8]. Gobert et al.[9]studied the interaction between a boundary-layer flow and an elastic plate using direct numerical simulation and taking into account the full coupling between the fluid flow and the flexible wall. The sound radiation levels were shown to be increased in the presence of flexible walls with however significant differences in the radiated pressure levels for different coupling assumptions. Recently, Bannwar et al.[10]studied the lubrication problem in HD bearing with alternate rotational motion. In that work, Reynolds’ lubrication theory is presented taking into account the local acceleration of the fluid film due to the motion of the slider in the mathematical model. Moreover, in the field of AE, Aggelis et al.[11]presented some preliminary results on the AE monitoring during fatigue of aluminium coupons. It is concluded that AE parameters are sensitive to the damage process and should be further studied.

Fig.1 Coordinate system and journal bearing geometry

The originality of this work is the application of Reynolds’ lubrication theory taking into account the flexibility of the bearing liner and the compressibility of the fluid film. A comparative study between the hydrodynamic and EHD theories is also a scientific contribution. Influence of some design parameters on the noise of oil lubricated journal bearings is presented.

1. Governing equations

The coordinate system and journal bearing geometry of oil lubricated journal bearings are shown in Fig.1. It is assumed that the journal and the housing are circular and rigid. The bearing load is applied in the x direction. For the EHD lubrication, the bearing liner is a finite length cylinder subjected to hydrodynamic loading due to the fluid film pressure on its internal surfaces. The distribution of fluid film pressure is such that it deforms the bearing in all directions.

Using the classical lubrication hypothesis, it is assumed that the flow is laminar and that inertia is neglected. The fluid is Newtonian, the density, specific heat, thermal conductivity and heat transfer coefficients are assumed to be constant.

The Reynolds equation, with fluid compressibility effects governing the pressure distribution of the oil film in a finite width bearing under unsteady conditions can be written as follows[3]

where g is a switch function becoming unity in the full film zone and zero in the cavitated zone x and z are the coordinates of the lubricating plane, h is the oil film thickness, U is the surface velocity in the x direction, t is the time, θ is the angular coordinate and β is the fluid bulk modulus.

The fluid fractional film content θcis defined as[3]

where ρ is the oil density andcρ is the cavitations density.

In the angular coordinate system as shown in Fig.1, the oil film thickness, in hydrodynamic lubrication, can be expressed as

where C is the radial clearance of the bearing.

When the elastic deformation of the bearing liner is taken into account, Eq.(3) becomes

where hais the additional thickness caused by the displacement of bearing liner in the radial direction.

In the case of EHD lubrication, the Reynolds equation (Eq.(1)) is resolved with the fluid film thickness computed in Eq.(4).

The pressure in the full film region can be expre-

where pis the oil film pressure.

The following boundary conditions for the oil pressure in the fluid film are adopted according to the geometric configuration and the periodic condition[12-14]

The bearing liner is discretized into 3 168 elements (72 element in the circumferential direction, 22 in the axial direction, and 2 in the radial direction). Eightnoded hexahedral isoparametric elements are used, in which displacements are assumed to vary linearly[15,16].

Fig.2 Coordinate system and journal bearing geometry

The discretization of the bearing liner is shown in Fig.2.

The displacement vector｛d｝, which is to be determined for the bearing liner is[15]

where uθ, urand uzare displacements of bearing liner respectively in the circumferential, radial and axial directions.

The following boundary condition is used for bearing liner analysis. The bearing shell is assumed to be contained in a rigid housing as shown in Fig.1.

where n is the global number of nodes on the bearing and rigid housing interface.

Using the potential energy theorem, an algebraic equation is obtained in terms of nodal displacement vector ｛｝δ for the displacement field of the bearing liner[15]

where ｛F｝, the force vector, is the result of surface traction force caused by hydrodynamic pressure acting on the fluid and bearing liner interface, and [k]is the stiffness matrix.

Solution of Eq.(9) satisfying the boundary conditions (Eq.(8)) gives the nodal deformations which define the bearing surface deformation and the resulting additional fluid film thickness ha.

For the steady state response computed by applying an effective numerical methodology, the amplitude of the pressure fluctuation in the root mean squared value can be written as[3]

where T is the period of the steady state response and the mean pressure of the oil, pm, is defined as[3]

The sound pressure level of the fluid film radiated at a certain location of the bearing can be written as

where N the sound pressure level of the fluid film, prmsis the pressure fluctuation in the root mean squared, prefis the reference sound pressure, standardized at 10-6N/m2[3].

The different parameters of the bearing are presented in Table 1.

Table 1 Specification and parameter values

2. Results and discussions

The finite element method is used to solve the constitutive equations. The Gauss-Seidel iterative scheme with over-relaxation is employed for the resolution of Reynolds equation. Boundary conditions are used to compute the pressure field for a rigid bearing. For the EHD analysis, an iterative process is repeated until the required convergence is achieved. The converged nodal pressures are then used to calculate the nodal displacements. The film thickness is modified by considering the radial component of the nodal displacements to get the solution of the nodal pressures. Iterations are also required to obtain performance characteristics for a wide range of values of the deformation coefficient which take into account the flexibility of the bearing liner. For a given set of operating conditions, direct numerical integration of the global equation (Eq.(1)) was carried out using a software package. The finite element method with an iteration scheme was employed to solve both the Reynolds equation and the three-dimensional elasticity equation representing the displacement field in the bearing shell. The software package utilizes also a variable-step continuous solver based on a two-dimensional Newton-Raphson search technique to compute the static equilibrium position of the rotor. Starting from this position, the transient response of the journal centre was obtained by numerical integration of its acceleration.

Fig.3 Time response of journal centre

Fig.4 Orbit of the journal centre

Figure 3 shows the instantaneous state of the journal centre in the oil film space in the x and y directions for HD and EHD lubrications. The journal rotates at 5 000 rpm and its rotational frequency is 83.33 Hz. Figure 4 shows the journal’s orbits from a disturbed position. In both case of lubrication, the journal’s centre converges and is reduced gradually to a point that corresponds to the static equilibrium position. But we can notice that the time required for the journal center converges is less important in EHD lubrication. It reveals that bearings with some given disturbance can still converge to its static equilibrium position if the journal rotational speed is under the threshold speed of the system.

In the case of the steady state response, freque-ncy spectra of the whirl amplitude of the journal centre in the x directions are shown in Fig.5. The means of spectral analysis show that all the theoretically predicted vibration frequencies actually appear with a dominant 1×running speed component.

Fig.5 Frequency spectra of the journal centre in x direction

Fig.6 Pressure fluctuation and sound pressure level distribution for HD lubrication

The pressure fluctuation and the sound pressure level distributions for HD and EHD analysis are shown in Figs.6 and 7. We note that in the EHD lubrication, the pressure ranges from 45oto 180owhile the HD lubrication, it covers only about 50o. The pressure distribution indicates that peak pressure decreases in EHD lubrication. This is due to the deformation of the bearing liner. A similar trend was observed by Sukumaran and Prabhakaran[17]in their investigation for three-lobe journal bearing. These results also show that the elastic deformation of the bearing liner decreases the sound pressure level of the bearing. This is readily explained by the increase of the film fluid in the clearance space of the bearing caused by a decrease in the pressure fluctuation.

Fig.7 Pressure fluctuation and sound pressure level distribution for EHD lubrication

Fig.8 Sound pressure level with respect to viscosity of the lubricant

Figure 8 shows the effects of the viscosity of lubricants on the sound pressure level of the bearing for various operational speeds for HD and EHD lubrications. The results show that high viscosity of the lubricant decreases the sound pressure level of the bearing, but its effects are relatively low at speeds above 6 000 rpm.

Sound pressure level is plotted against various typical bearing liner materials for HD and EHD lubri-cations in Fig.9. The results show that for the same material of the bearing liner, the pressure level is higher in the HD theory, this is clearer in the case of Brass. Moreover, when Young’s modulus of the bearing liner decreases, the pressure level decreases.

Fig.9 Sound pressure level for typical bearing liner materials

Fig.10 Sound pressure level with respect to the applied load

The sound pressure level changes with respect to the applied load W for various rotational speed of the journal are shown in Fig.10. The sound pressure level of HD and EHD lubrications increases with the rotational speed of the journal and the applied load. It means that the oil film pressure occurred between the journal and the rigid housing or the bearing liner, increases as the rotational speed and the load applied to journal increases.

Figure 10 also shows the increase of the sound pressure level with respect to the applied load becomes higher at low speed than at higher speed.

3. Conclusion

Acoustic properties and vibration analysis of HD and EHD journal bearings have been numerically investigated. The following results are summarized from the analysis. HD and EHD analysis show that the journal’s centre converges and is reduced gradually to a point that corresponds to the static equilibrium position. However, the elastic deformation of the bearing liner gives some disturbances which increase the vibratory level of the journal’s centre. The frequency spectra of the journal centre show a dominant peak corresponds to 1 × running speed of the journal. The results also show that the elastic deformation of the bearing liner decreases the sound pressure level of the bearing. The bearing noise may decrease due to the EHD lubrication because the film thickness will be greater than that of HD lubrication. However, in both HD and EHD lubrications, for a low viscosity of lubricants and a higher Young’s modulus of the bearing liner, the sound pressure level increases. The bearing noise increases with the rotational speed of the journal and the applied load. This increase becomes higher at low speed than at high speed. It is expected that such work could aid in the evaluation and understanding of the acoustical properties of oil lubricated journal bearings. Furthermore, bearing acoustic properties could provide diagnostic information on abnormal phenomena of rotor-bearing system.

[1]DIANGUI H. Experiment on the characteristics of torsional vibration of rotor to stator rub in turbo machinery[J]. Tribology International, 2000, 32(2): 75-79.

[2]RHO B. H., KIM K.W. A study on nonlinear frequency response analysis of hydrodynamic journal bearings with external disturbances[J]. Tribology Transactions, 2002, 45(1): 117-121.

[3]RH Byoung-Hoo, KIM Kyong-Woong. Acoustical properties of hydrodynamic journal bearings[J]. Tribology International, 2003, 36(1): 61-66.

[4]MIETTINEN J., ANDERSSON P. Acoustic emission of rolling bearings lubricated with contaminated grease[J]. Tribology International, 2000, 33(11): 777-787.

[5]MIRHADIZADEH S. A., MBA D. Observations of acoustic emission in a hydrodynamic bearing[J]. Journal of Quality in Maintenance Engineering, 2009, 15(2): 193-201.

[6]RHO Byoung-Hoo, KIM Dae-Gon and KIM Kyung-Woong. Effects of design parameters on the noise of rotor-bearing systems[J]. Tribology International, 2004, 37(8): 599-605.

[7]BAN Jong-Eok, RHO Byoung-Hoo and KIM Kyung-Woong. A study on the sound of roller bearings operating under radial load[J]. Tribology International, 2007, 40(1): 21-28.

[8]RHO Byoung-Hoo,KIM Dae-Gon and KIM Kyung-Woong. Noise analysis of oil lubricated journal bearings[J]. Journal of Mechanical Engineering Science, 2003, 217(3): 365-371.

[9]GOBERT M.-L., EHRENSTEIN U. and ASTOLFI J. A. et al. Nonlinear disturbance evolution in a two-dimensional boundary-layer along an elastic plate and induced radiated sound[J]. European Journal of Mechanics B/ Fluids, 2010, 29(2): 105-118.

[10]BANNWAR A. C., CAVALCA K. L. and DANIEL G. B. Hydrodynamic bearings modelling with alternate motion[J]. Mechanics Research Communucations, 2010, 37(6): 590-597.

[11]AGGELIS D. G., KORDATOS E. Z. and MATIKAS T. E. Acoustic emission for fatigue damage characterization in metal plates[J]. Mechanics Research Communucations, 2011, 38(2): 106-110.

[12]BOUAZIZ S., ATTIA HILI M. and MAATAR M. et al. Unbalance and gear mesh effects on the dynamic behaviour of a hydrodynamic journal bearings[J]. Advance in Computer Science and Engineering, 2007, 1(2): 169-187.

[13]BOUAZIZ S., MAATAR M. and FAKHFAKH T. et al. Angular misalignment effect on hydrodynamic journal bearings dynamical behaviour[J]. International Journal of Engineering Simulation, 2007, 8(1): 3-10.

[14]BOUAZIZ S., ATTIA HILI M. and MAATAR M. et al. Dynamic behaviour of hydrodynamic journal bearings in presence of rotor spatial angular misalignment[J]. Mechanism and Machine Theory, 2009, 44(8): 1548-1559.

[15]ATTIA HILI Molka, BOUAZIZ Slim and MAATAR Mohamed et al. Hydrodynamic and elastohydrodynamic studies of a cylindrical journal bearing[J]. Journal of Hydrodynamics, 2010, 22(2): 155-163.

[16]RAO S. S. The finite element method in enginee- ring[M]. Fourth Edition, Elsevier Inc., 2005.

[17]SUKUMARAN NAIR V. P., PRABHAKARAN NAIR K. Finite element analysis of elastohydrodynamic circular journal bearing with micropolar lubricants[J]. Finite Elements in Analysis and Design, 2004, 41(1): 75-89.

September 7, 2011, Revised Devember 22, 2011)

* Biography: BOUAZIZ Slim (1979-), Male, Ph. D., Assistant Professor