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A comparative linear and nonlinear dynamic analysis of compliant cylindrical journal bearings

Matthew Cha a,?,Evgeny Kuznetsov b ,Sergei Glavatskih a,c

a Machine Design,KTH Royal Institute of Technology,Stockholm,Sweden

b Comsol AB,Stockholm,Sweden

Mechanical Construction and Production,Ghent University,Ghent,Belgium

a r t i c l e i n f o a

b s t r a

c t

Article history:

Received 21June 2011

Received in revised form 11December 2012Accepted 17January 2013

Available online 27February 2013Dynamic behaviour of compliant cylindrical journal bearings is investigated using linear and nonlinear numerical approaches.Journal motion orbits based on linearized dynamic coefficients are compared to the journal trajectories obtained by the nonlinear transient analysis.Obtained results are presented in terms of orbit amplitude,shape and location.The influence of compliant liner thickness,viscoelastic properties and deformation model is also investigated.A linear model is found to deliver acceptable results at a relatively small shaft unbalance under low to average loads.However,with a journal amplitude motion greater than 37%of the bearing diametral clearance,the linear model should not be used to analyse journal transient motion.Plane strain hypothesis is found to be a proper substitute for a full deformation model when a compliant liner is thinner than 2mm (for the bearing geometry used in this study).It was also shown that the liner viscoelasticity should be taken into account whenever a compliant liner is relatively thick (in our case,2mm).Viscoelasticity of the liner decreases journal amplitude compared to a pure elastic liner.

Keywords:

Compliant bearing Viscoelasticity Nonlinear analysis

1.Introduction

Hydrodynamic cylindrical journal bearings are widely used in large rotating machinery due to their simple design,low cost and operating characteristics.Such bearings can be found in turbines,electric motors,generators,pumps,etc.They provide greater damping compared to rolling element bearings and are easier to install due to the split design.Journal bearings are typically coated with white metal to prevent shaft damage when the journal comes into contact with the bearing surface,such as during start-ups and shut-downs.A hydrostatic system can also be used to support the shaft at low angular speeds if the bearing is subjected to heavy loads.However,introduction of an additional high pressure system increases installation complexity and maintenance costs.

It has been shown that a polymer composite liner in tilting pad thrust bearings can operate in more severe conditions than white metal without a hydrostatic system [1].Such a liner is therefore a good candidate as a white metal replacement even in journal bearings as it will reduce break-away friction [2].A thermohydrodynamic (THD)analysis of compliant journal bearings shows [3]that a thin layer (1mm)of polytetrafluoroethylene (PTFE)like material can improve bearing characteristics such as oil film thickness and load carrying capacity and reduce maximum pressure by up to 40%in comparison to conventional white metal bearings.

While a compliant liner was shown to improve bearing steady state performance and reduce breakaway friction,its influence on bearing dynamic behaviour should also be studied.Fluid film bearings often provide enough damping to reduce synchronous

Mechanism and Machine Theory 64(2013)80–92

?Corresponding author.Tel.:+46(0)87906805.E-mail address:match@kth.se (M.

Cha).

https://www.sodocs.net/doc/7618454213.html,/10.1016/j.mechmachtheory.2013.01.008

Contents lists available at SciVerse ScienceDirect

Mechanism and Machine Theory

j ou r n a l h o m e p a g e :w w w.e l s e v i e r.c o m /l o c a t e /m e c h mt

vibrations in the system,but the cross-coupling stiffness terms make this type of bearing unstable at certain operating conditions. Newkirk and Taylor[4]were the first to demonstrate that at a certain speed and load,the journal centre did not remain fixed as predicted by the steady state Reynolds equation,but moved around the equilibrium position at a speed approximately equal to half the rotational speed.They called this phenomenon oil whip and also presented guidelines to minimise or even prevent shaft vibration.In1965,Lund[5]carried out an extensive analysis on the dynamic behaviour of fluid film bearings presenting linearized dynamic coefficients and stability parameters for different types of bearings.Akers et al.[6]investigated journal centre trajectory by solving the Reynolds and motion equations.It was shown that inclusion of the friction term representing fluid shearing improves stability of the bearing by increasing the amount of time required for the journal to reach the extremities of its clearance. Singh et al.[7]investigated journal motion trajectories in solid and porous bearings for small initial disturbances(0.1%of radial clearance).It was shown that a linear method is very close to the nonlinear method.In the latter,bearing reaction forces in the motion equation are derived from the Reynolds equation solved at each point of the journal trajectory.

In1978,Lund and Thomsen[8]published a numerical algorithm based on small shaft perturbations to calculate stiffness and damping coefficients.In1987,Lund[9]used linearized coefficients to obtain bearing reaction forces in the equation of motion to find the limit cycle of the journal orbit.At the same time,Zhang et al.modified Lund's approach for an isothermal compliant liner bearing[10].It was shown that the influence of the pressure perturbation on the compliant liner deformation(so-called dynamic deformation or perturbation of the deformation)plays a crucial role in the linear analysis.Journal critical mass and whirl ratio of the rotor-bearing system with two identical compliant bearings were found to be better than those with the white metal bearings. The same result was shown in[11].The difference between Lund's model and the modified approach that accounts for deformation perturbation of the liner was analysed in[12]using a2-axial groove journal bearing.The compliant liner was found to increase journal critical mass if deformation perturbation is considered and decrease it otherwise.The effects of shaft misalignment and coupled stress fluids on compliant bearing performance were investigated in[13].The compliant liner was shown to reduce the negative effects of shaft misalignment generated at a bearing edge.High maximum pressure peak found at a rigid bearing's edge was significantly reduced and smoothed out in the case of a compliant bearing.

A journal limit cycle obtained using linearized dynamic coefficients agrees well with the nonlinear analysis for small shaft perturbations.However,at large journal amplitudes,a linearized approach does not predict journal motion with acceptable accuracy. An isothermal nonlinear analysis was used by[14]to investigate journal motion trajectories.It was concluded that deformation of the compliant liner influences bearing dynamic performance to such an extent that the linear analysis could not be used even for small shaft displacements.Damping was improved when an elastic liner was considered.Van de Vrande[15]investigated short and long compliant journal bearings using a nonlinear isothermal analysis.Critical journal speed was shown to decrease for short bearings and increase for long bearings with more compliant liners.

Another aspect to consider among compliant liner properties is viscoelasticity.Linear dynamic analysis with viscoelastic effects included was carried out by Zhang and Jiang[16].They obtained coefficients for a cylindrical bearing with a compliant liner made of viscoelastic material.It was found that the stability threshold speed of the rotor-compliant bearing system was increased compared to that with conventional white metal bearings.Kulkarni et al.[17]investigated the nonlinear dynamic response of a rotor-bearing system with viscoelastic support observing a reduction of the unbalance response and increased damping.Sun et al.

[18]used Rayleigh damping to handle the damping matrix in the equation of elasticity.The model assumed plain stress and no effect of stiffness matrix on the damping https://www.sodocs.net/doc/7618454213.html,ing short bearing theory it was shown that decreasing damping factor reduced shaft eccentricity in the Y-direction(perpendicular to the load vector).

Differences in numerical models used(e.g.[7,10,12,14])lead to the differences in results obtained.A question about applicability and accuracy of the assumptions made therefore appears.This paper provides a comparative analysis of plain cylindrical journal bearings to investigate the applicability of the following aspects used in the journal bearing analysis:linear approach vs.nonlinear method;plane strain hypothesis vs.3D liner deformation model;and liner elastic strain vs.viscoelastic deformation.

2.Numerical model

In this study,a rotor-bearing system with a rigid shaft supported by two identical plain journal bearings is considered. Lubricant flow is assumed to be incompressible,isothermal,and laminar.Newtonian behaviour of the lubricant is assumed.The numerical model includes the equations of motion(shaft trajectory analysis),the Reynolds equation(journal response analysis) and deformation equation(compliant liner deformation analysis).

A nonlinear model is used to obtain bearing reaction forces and journal centre position at each time step by simultaneously solving the Reynolds and motion equations.

A linear model provides linearized stiffness and damping coefficients[12]to solve the equations of motion.Journal orbits calculated from linearized coefficients are compared to the journal trajectories obtained by the nonlinear model.

2.1.Reynolds equation

The Reynolds equation in polar coordinates(cylindrical domain)is given as

1 R ?ρh3

?ρh3

?6U

?ρh

t12

?ρh

e1T

81 M.Cha et al./Mechanism and Machine Theory64(2013)80–92

Oil film thickness can be represented as the sum of the journal displacement and the compliant liner deformation terms.

h ?C r 1te cos θeTtδ

e2T

Deformation,δ,which is a function of pressure and compliant liner properties,increases film thickness in the mid-plane of the bearing [3]and affects its dynamic response.The differences in the liner deformation models used are discussed in Sections 2.4and 2.5.

2.2.Boundary conditions

The boundary conditions for Eq.(1)are set to be periodic in the sliding direction.Pressure is set to be equal to atmospheric pressure at the bearing edges.The Reynolds cavitation model is used in the linear analysis while a density –pressure cavitation model [19]is used in the nonlinear analysis in order to satisfy the flow continuity condition.Lubricant density,ρ,is governed by the following expression:

ρ?ρ0;if p >p sat ρ03p p sat

2?2p p sat 3

;if

p ≤p sat

><>:e3T

Saturation pressure is set to 20kPa in both,linear and nonlinear,analyses.2.3.Dynamic analysis

Equations of motion are solved in the linear and nonlinear analyses to obtain journal orbits:

m €x

?F x tm ξΩ2

sin Ωt eTm €y

?F y tm ξΩ2cos Ωt eT?W e4T

Since the viscous friction forces are so small compared to other forces,they have very small effect on the final results and can

be ignored.If the rotor centre of gravity does not coincide with its geometric centre,an unbalance force,m ξΩ2,is produced.The rotor unbalance is almost an inevitable element in rotor-bearing systems due to manufacturing and assembly tolerances.In order to keep the same unbalance force for all the cases studied in this paper,ξhas been adjusted for different journal angular speeds.

In the linear approach,bearing reaction forces in the equations of motion are expressed using linearized stiffness and damping coefficients.Small perturbations are applied to the shaft resulting in an additional component in the initial film thickness h 0so that the complete film thickness can be represented as follows:

h ?h 0th P tδ

e5T

Oil film pressure can be written in the following way:

p ?p 0tp x Δx tp y Δy tp ′x

Δ_x tp ′y Δ_y e6T

By substituting Eqs.(5)and (6)in Eq.(1)and solving the obtained equation,the unknown pressure components can be found.

Stiffness and damping coefficients are obtained by integration of the pressure component.Dynamic deformation of the compliant layer is considered as shown in [12,20,21].In the nonlinear analysis dynamic deformation of the compliant layer is already taken into account as the bearing reaction forces are obtained by solving the Reynolds equation at each time step along the journal orbit.2.4.Elastic deformation

Two deformation models are used in the simulations to calculate changes in the oil film thickness caused by the compliant liner reaction to the applied pressure.In the linear analysis,bearing liner deformation is calculated using a plane strain hypothesis [12],also known as a Winkler (or column)model.According to this model the liner can be represented by a set of independent springs-columns (that can be pressed only in one direction).Therefore,deformation in each point is evaluated according to the pressure acting on this point,regardless of pressure acting on other points in the bearing.In the nonlinear analysis,deformations are evaluated by using a more realistic model based on a 3D elasticity matrix [22].A zero displacement condition is used at the interface between the steel sleeve and the compliant liner.2.5.Viscoelastic deformation

The deformation of some polymer materials can be represented as a combination of Hooke's law (elastic component)and Newton's law (viscous component).The generalized Maxwell model is used to combine these laws,which is a common approach

82M.Cha et al./Mechanism and Machine Theory 64(2013)80–92

to describe solid body viscoelastic properties [23].The model is represented by an elastic spring and a series of elastic springs and viscous dashpots as shown in Fig.1[22,23].

3.Numerical simulations

We consider a cylindrical plain journal bearing of finite length.Fig.2illustrates the cross-section of the journal bearing.Table 1contains lubricant properties and geometry of the bearing while compliant liner properties are given in Table 2.

The linearized model is solved on a 256×33mesh grid (8448degrees of freedom)using finite differences and successive over-relaxation (SOR)techniques with a convergence criterion set to 10?6.A finite element method (FEM)is used to solve Eqs.(1)and (6)with a convergence criterion set to 10?6for the Reynolds equation and 10?3for the equations of motion in the nonlinear analysis.A second-order triangular mesh with 17,136degrees of freedom is used.A further increase in degrees of freedom resulted in a negligible difference compared to the selected mesh size.Convergence criteria are chosen in a way that a smaller value results in no visible discrepancies in the results,while a larger value produces considerably higher deviation in the results.A comparison of different convergence criteria for the viscoelastic simulations has been https://www.sodocs.net/doc/7618454213.html,rge differences were found between 10?3,10?5and 10?7.At the same time,10?8shows no visible discrepancy if compared to 10?7.Therefore a convergence criterion of 10?7is used for the viscoelastic simulations.

4.Results

4.1.Model validation

Results obtained by the commercial software package were compared to a steady state pressure profile [24]measured in a plain cylindrical journal bearing without oil supply grooves,L/D ratio of 4/3,at a 48.1rad/s shaft speed.Experimental points are located at a 10%distance from the bearing centre plain.Fig.3a shows a very good agreement between the numerical and experimental data.Small discrepancies are due to the cavitation algorithm used and a small number of measured points.The difference in the calculated load carrying capacity is less than 1%.The isoviscous linear model was verified in terms of the stiffness and damping coefficients for cylindrical and lemon bore bearings using Lund's results [12,21].No differences were observed between the models.Verification of the motion equation model is accomplished by comparison with Lund's dynamic simulation [9].A plain cylindrical journal bearing with two oil supply grooves,L/D ratio of 1,and a rotational speed of 157.1rad/s was considered in [9].Fig.3b shows a good agreement.

https://www.sodocs.net/doc/7618454213.html,parison of the deformation models

We investigate applicability of the plane strain hypothesis to predict deformation of the compliant bearing liner.Fig.4shows comparison of pressure profiles obtained with plane strain hypothesis (in-house code)and 3D deformation (software package)models.The white metal liner is unaffected by the pressure considered and therefore shows almost perfect match in pressure profiles (Fig.4a).The only visible difference is due to different cavitation algorithms used in the linear and nonlinear models.As the liner compliance increases,the difference between deformation models becomes more visible.Thus,the 3D deformation model produces a lower maximum pressure than the plane strain model if a 2mm compliant lined bearing is analysed for a fixed relative eccentricity of 0.6(Fig.4b).A further increase in compliant liner thickness increases deviation between results produced by the plane strain and 3D deformation models (Fig.4c).Therefore,the plane strain hypothesis can only be used for relatively small deformations,thin liners or under low loads.

The plane strain model has also been accomplished in the software package and compared to the in-house code.A very good agreement has been observed for the compliant liner thickness in the range from 1to 4mm.Slight discrepancies have been observed in the cavitation

zone.

Fig.1.Generalized Maxwell model.

M.Cha et al./Mechanism and Machine Theory 64(2013)80–92

4.3.Linear vs.nonlinear

We begin dynamic analysis for a small unbalance eccentricity.White metal and compliant liner bearings are compared at a rotational speed of 2000rpm in a static load range from 500N to 2500N.Linearized stiffness and damping coefficients are obtained by the in-house code and then used to plot journal orbits.Journal transient motion trajectories are obtained by using the software package.Fig.5a shows a very good agreement between the limit cycles obtained by the linear (solid line)and nonlinear (asterisk)models.Orbits tend to be more circular at low loads becoming more elliptical at higher loads.Some discrepancies between linear and nonlinear models can be seen at eccentricities below 0.5.Similar to the white metal case,transition of the orbit shape from circular to pure elliptical is observed for the compliant bearing,Fig.5b.Amplitudes produced by the linear model are slightly higher than those produced by the nonlinear approach.The difference is within few percents.A comparison between compliant and white metal bearings reveals a decrease in the attitude angle due to deformation of the compliant liner,which was also shown in [3].

Journal trajectories at large unbalance forces are shown in Fig.6.Orbits become larger when unbalance eccentricity increases from 0.5to 1.8mm.Orbit shape transformation from elliptical to oval can also be observed.Table 3presents eccentricity ratios and attitude angles at static equilibrium,which is represented by “cross ”symbol in Fig.7.To account for different speeds,unbalance eccentricity is adjusted as shown in Table 4.Journal eccentricity obtained by the software package is used in our numerical code to calculate linearized stiffness and damping https://www.sodocs.net/doc/7618454213.html,ing these coefficients,limit cycles are evaluated [25].For journal motion amplitudes greater than 40%of bearing diametral clearance,the linear and nonlinear analyses show quite different results.Fig.7depicts the limit cycles for different rotational speeds.The amplitude of orbits produced by linear and nonlinear models are comparable for 2000rpm (green colour),while their size and position are different,Fig.7a.This result is similar to [9].As the rotational speed decreases,the differences become more pronounced for 500rpm (red)and 1000rpm (blue).Amplitudes are still comparable but differences in the orbit locations become essential.

Fig.7b shows results for a 1mm compliant liner bearing in the speed range from 500rpm to 3000rpm.At the highest speed,the “linear ”and “nonlinear ”orbits have comparable shapes and amplitudes,while at 2000rpm the linear model produces

Fig.2.Journal bearing geometry.

Table 1

Journal bearing and lubricant properties.Radial clearance,C r 50.8[μm]Rotor mass,m

100[kg]Journal diameter,D 50[mm]Bearing length,L 50[mm]

Lubricant viscosity,μ

6.895×10?3[Pas ]Initial lubricant density,ρ0900[kg /m 3]Journal angular speed,Ω

52–314[rad /s ]

84M.Cha et al./Mechanism and Machine Theory 64(2013)80–92

orbit with significantly higher amplitude.The difference in orbit amplitudes decreases with the speed while the difference in location increases.Fig.7c shows journal orbits for the 2mm compliant liner bearing.Linear analysis for 2000rpm and 3000rpm produces at least 20%larger orbits,if the orbit size is measured in terms of the maximum amplitude.At the same time,orbits produced using linearized coefficients for 500rpm and 1000rpm show a physical contact between shaft and the bearing.By increasing compliant liner thickness to 4mm we observe higher discrepancies between linear and nonlinear models as shown in Fig.7d.

4.4.Viscoelasticity

In the generalized Maxwell model,relaxation time constants and relative stiffness determine how a compliant liner would behave.The compliant liner used in this study has the viscoelastic properties of PTFE.A tensile strength machine was used to measure PTFE viscoelastic properties.4MPa load was applied to the 40mm×40mm×4mm PTFE sample block.The rate of the displacement of the PTFE sample was recorded and the tests were repeated three times to assure measurement consistency.Table 5shows viscoelastic properties obtained in the tests.Figs.8and 9present the limit cycles for different elastic and

Table 2

Polymer liner properties.Young's modulus,E 0.11×109[Pa ]Poisson's ratio,ν0.46[??]Thickness

1,2,3,4

[mm]

Fig.3.Validation of nonlinear model,steady state case (a)and dynamic case (b).

M.Cha et al./Mechanism and Machine Theory 64(2013)80–92

18161412

1086420-21410128

6420-2

1210

86240-2

x 105x 105x 105

P r e s s u r e [P a ]

100

150

200

250

300350

Circumferential angle [deg]

050100150200250300350

Circumferential angle [deg]

050100150200250300350

Circumferential angle [deg]

White metal bearing, eccentricity ratio 0.6

Compliant liner 2mm, eccentricity ratio 0.6

Compliant liner 4mm, eccentricity ratio 0.6

3000rpm 3D 2000rpm 3D 1000rpm 3D 3000rpm PS 2000rpm PS 1000rpm PS

Fig.4.Pressure distribution at 0.6eccentricity (solid line –plane strain and dot –3D)white metal bearing (a)complaint liner thickness of 2mm (b)complaint liner thickness of 4mm (c).

86M.Cha et al./Mechanism and Machine Theory 64(2013)80–92

viscoelastic liners at 500rpm (Fig.8)and 2000rpm (Fig.9).It can be seen that viscoelasticity reduces the journal motion amplitude compared to pure elastic deformation.The influence of the damping component increases as the liner thickness increases.

If a stiffer viscoelastic material is used,the limit cycle is expected to approach the limit cycle for the white metal bearing.Fig.10shows a compliant bearing with viscoelastic properties.Visco1corresponds to the PTFE liner and Visco2is obtained by multiplying relative stiffness coefficients in Table 5by 50,which has material properties similar to polyetheretherketone (PEEK).We can clearly see that the Visco2limit cycle approaches the white metal case.Thus,it is shown that viscoelasticity increases the orbit amplitude in the circumferential direction and decreases it in the radial direction compared to the purely elastic

case.

Fig.5.Small amplitude journal motions for white metal (a)and compliant bearings (b)2mm thickness.(solid line –linear analysis and asterisks –nonlinear analysis).

270

Attitude angle

Fig.6.Journal transient motion at 500rpm with different unbalance eccentricities.

M.Cha et al./Mechanism and Machine Theory 64(2013)80–92

5.Discussion

As shown by the numerical analysis,plane strain hypothesis can be used as an accurate-enough simplification if the elastic liner deformation is relatively small,which is the case for the thin liner (up to 2mm in our case).Differences between plane strain and 3D deformation models are within a few percent.In general,a combined effect of bearing geometry,maximum oil film pressure (bearing load)and material properties (Young's modulus)should be considered.

Table 3

Eccentricity ratio and attitude angle at different rotational speeds.White metal

Speed,rpm

500100020003000Eccentricity ratio,ε

0.8340.7110.538–Attitude angle,?41.95262.8–Compliant 1mm

Speed,rpm

500100020003000Eccentricity ratio,ε

0.8390.7150.5450.428Attitude angle,?38.549.360.366.4Compliant 2mm

Speed,rpm

500100020003000Eccentricity ratio,ε

0.8510.7220.5470.432Attitude angle,?36.146.95863.9Compliant 3mm

Speed,rpm

500100020003000Eccentricity ratio,ε

0.8650.7320.5540.439Attitude angle,?34.244.955.961.6Compliant 4mm

Speed,rpm

500100020003000Eccentricity ratio,ε

0.8810.7430.5620.562Attitude angle,?

32.7

43.2

53.9

59.4

Fig.7.Limit cycles of white metal (a),1mm (b),2mm (c),and 4mm (d)compliant bearings (red is 500rpm,blue is 1000rpm,green is 2000rpm,cyan is 3000rpm,solid line –linear analysis and asterix –nonlinear analysis).

88M.Cha et al./Mechanism and Machine Theory 64(2013)80–92

Table 4

Unbalance eccentricities at different rotational speeds.Speed,rpm Unbalance eccentricity 5001800μm 1000450μm 2000110μm 3000

50μm

Table 5

Viscoelastic properties.Branch Relaxation time constant (s)Relative stiffness 110.1272260.11093400.003342510.0408515850.02336

10,000

0.0270

Viscoelasticity @ 500rpm with compliant liner 1mm

Attitude angle

E c c e n t r i c i t y r a t i o

Viscoelasticity @ 500rpm with compliant liner 4mm

Attitude angle

E c c e n t r i c i t y r a t i o

Fig.8.Limit cycles for white metal and compliant https://www.sodocs.net/doc/7618454213.html,pliant liner thickness 1mm (a)and 4mm (b),500rpm.

M.Cha et al./Mechanism and Machine Theory 64(2013)80–92

Viscoelasticity @ 2000rpm with compliant liner 1mm

Attitude angle

E c c e n t r i c i t y r a t i o

Viscoelasticity @ 2000rpm with compliant liner 4mm

Attitude angle

E c c e n t r i c i t y r a t i o

Fig.9.Limit cycles for white metal and compliant https://www.sodocs.net/doc/7618454213.html,pliant liner thickness 1mm (a)and 4mm (b),2000rpm.

Viscoelasticity @ 500rpm with compliant liner 4mm

0.2

0.4

0.6

0.8

Attitude angle

E c c e n t r i c i t y r a t i o

Fig.10.Viscoelasticity effect of the compliant liner on the limit cycles.

90M.Cha et al./Mechanism and Machine Theory 64(2013)80–92

A comparison of journal orbits obtained by linear and nonlinear models shows their agreement for small journal amplitudes (less than 40%of bearing diametral clearance)in the white metal and thin compliant liner bearings.The only difference between the linear and nonlinear models,observed for small journal amplitudes,is the location of the static equilibrium position caused by different cavitation algorithms.However,Jain et al.[14]showed inapplicability of the linear model to compliant bearings even for small journal amplitudes (0.5%of bearing radial clearance).Such a conclusion can be due to a pressure perturbation component (see [10,12])missing in the Jain et al.'s model.

Journal amplitude motion over 40%of the bearing diametral clearance (80%of the radial clearance)shows large differences between results obtained by the linear and nonlinear models.Differences between amplitudes increase with higher compliance (Fig.7d against Fig.7c).Differences in orbit location depend more on the relative eccentricity.The linear model produces orbits of elliptical shape while orbits produced by the nonlinear model become more oval,gradually transforming to a banana-like shape.Such a shape transformation can be explained as a result of the combined influence of high loading (shaft weight)from the top and oil pressure build-up from the bottom of the bearing.From the theoretical analysis of the steady state problem,it is well known that load carrying capacity grows exponentially as a function of relative eccentricity,especially in the vicinity of 1.Therefore,the oil pressure component becomes the main (and only)factor determining the shape of the orbit whenever the shaft approaches the bearing surface.At the same time,as the shaft moves away from the bearing surface,the pressure influence gets weaker and the shaft weight (or any loading)prevails.

The linear model on the other hand is based on the stiffness and damping coefficients,and therefore produces an orbit that is always symmetric with respect to the shaft equilibrium position.That,in particular,leads to the visible contact between shaft and the bearing for low journal rotational speeds in Fig.7c and d.Nonlinear orbits show no contact for the same load and unbalance eccentricity.Oil film pressure forms a pocket shape indentation in the compliant liner where oil is trapped,[3].This means that if the shaft approaches the bearing surface,contact will take place only at the bearing edges.A small amount of liner material would most likely be worn out forming an alternative bearing geometry and eliminating the contact.This self-regulation property of the compliant liner can be considered as an advantage over the conventional white metal.6.Conclusions

Linear and nonlinear dynamic models have been used for white metal and compliant liner bearings.The influence of the unbalance force,load,eccentricity,viscoelastic property and compliant liner thickness on the journal orbit has been studied.The shape of the limit cycles obtained by the nonlinear analysis become more oval with an increase in eccentricity and/or unbalance forces.The comparison of linear and nonlinear models has also shown that:

?Results obtained by the models are in a good agreement for small journal amplitudes (motion less than 37%of the bearing diametral clearance,that is 74%of the radial clearance)for both white metal and compliant liner bearings.Orbits obtained using linearized dynamic coefficients are not accurate enough at large journal amplitudes (motion greater than 37%of the bearing diametral clearance).

?A plane strain hypothesis can be used by both linear and nonlinear models when deformations are small.For example,in our case a 3D deformation model has to be used if the compliant liner is thicker than 2mm.This thickness value will be different if geometry,load or material properties are changed.

?Viscoelasticity should be taken into account whenever a compliant liner is relatively thick (more than 2mm in the present paper).Viscoelasticity was also found to decrease journal orbit amplitude by reducing the amount of deformation in comparison to the pure elastic case.

Nomenclature

C r

radial clearance [m]F X ,F Y

oil film force [N]G i

relative stiffness coefficients [?]R radius of journal [m ]

T x ,T y

viscous friction force [N]U 0

surface velocity in tangential direction of the shaft [m/s]W static load [N]h total oil film thickness [m]

h 0

initial oil film thickness [m]h p

oil film perturbation [?]m mass of rotor [kg]p pressure [Pa]

p 0

steady-state pressure [Pa]p sat

saturation pressure [Pa]p x ,p y

components of the pressure connected to the displacement of the shaft in x and y directions [Pa]p ′x ;p ′

components of the pressure connected to the velocity of the displacement in x and y directions [Pa]t time [s]X ,Y Nondimensional coordinates [?]

M.Cha et al./Mechanism and Machine Theory 64(2013)80–92

92M.Cha et al./Mechanism and Machine Theory64(2013)80–92

x,y,z coordinates of the journal centre[m]

Δx,Δy displacement of the shaft in x and y directions[m]

Δ_x;Δ_y velocity of the displacement in x and y directions[m/s]

€x;€y acceleration of the journal centre[m/s2]

Ωrotational speed of journal[rad/s]

δdeformation of the liner[m]

εeccentricity ratio[?]

ηi relative damping coefficients[?]

μdynamic viscosity of lubricant[Pas]

θcircumferential coordinate[deg]

ρlubricant density[kg/m3]

τi relaxation time constants[s]

ξunbalance eccentricity[m]

Acknowledgements

The authors would like to gratefully acknowledge the financial support provided by the Swedish Hydropower Centre(SVC). SVC has been established by the Swedish Energy Agency,Elforsk and Svenska Kraftn?t in partnership with academic institutions. References

[1]S.B.Glavatskih,G.A.Paramonov,PTFE-faced bearing technology:advantages and practical examples,Waterpower XVI Conference Proceedings,Pennwell

Corporation,Tulsa,Okla,2009.

[2] A.Golchin,G.F.Simmons,S.B.Glavatskih,Break-away friction of PTFE materials in lubricated conditions,Tribology International48(2012)54–62.

[3] E.Kuznetsov,S.B.Glavatskih,M.Fillon,THD analysis of compliant journal bearing considering liner deformation,Tribology International44(2011)

1629–1641.

[4] B.I.Newkirk,H.D.Taylor,Shaft whipping due to oil action in journal bearing,General Electric Review(1925)559–568.

[5]J.W.Lund,Rotor Bearing Dynamics Design Technology—Part III:Design Hand Book for Fluid Film Bearings,Tech.Report AFAPL TR-65-45,Dayton,Ohio,USA,

1965.

[6] A.Akkers,S.Michaelson,A.Cameron,Stability contours for a whirling finite journal bearing,Journal of Lubrication Technology93(1971)77–190.

[7] D.V.Singh,R.Sinhasan,S.P.Tayal,Theoretical prediction of journal center motion trajectory,Journal of Lubrication Technology98(1976)619–629.

[8]J.W.Lund,K.K.Thomsen,A calculation method and data for the dynamic coefficients of oil-lubricated journal bearings,Topics in Fluid Bearing and Rotor

Bearing System,New York,1978,pp.1–28.

[9]J.W.Lund,Review of the concept of dynamic coefficient for fluid film journal bearings,Journal of Tribology109(1987)37–41.

[10]Z.Zhang,Q.Mao,H.Xu,The effect of dynamic deformation on dynamic properties and stability of cylindrical journal bearings,Tribology Series11(1987)

262–266.

[11]https://www.sodocs.net/doc/7618454213.html,hmar,A.Haddad,D.Nicolas,Elastohydrodynamic analysis of one-layered journal bearings,Proceedings of the Institution of Mechanical Engineers;Part

J:Journal of Engineering Tribology,212,1998,pp.193–205.

[12] E.Kuznetsov,S.B.Glavatskih,Dynamic characteristics of a2axial groove journal bearing with a PTFE lining,Proceedings of the16th International

Colloquium Tribology,Germany,January2008.

[13]H.Boucherit,https://www.sodocs.net/doc/7618454213.html,hmar,S.Boud-Said,Misalignment effects on steady-state and dynamic behaviour of compliant journal bearings lubricated with couple

stress fluids,Lubrication Science20(2008)241–268.

[14]S.C.Jain,R.Sinhasan,S.C.Pilli,A study on the dynamic response of compliant shell journal bearings,Tribology Transactions32(1989)297–304.

[15] B.Van de Vrande,Nonlinear dynamics of elementary rotor systems with compliant plain journal bearings.PhD Thesis,Eindhoven University of Technology,

2001.

[16]Z.Zhang,X.Jiang,Analysis of cylindrical journal bearing with viscoelastic bush,Journal of Tribology112(1990)442–446.

[17]P.Kulkarni,S.Pannu,B.C.Nakra,Unbalance response and stability of a rotating system with viscoelastically supported bearings,Mechanism and Machine

Theory28(1993)427–436.

[18]Y.T.Sun,B.Bou-Said,B.Fantino,Influence of viscoelastic parameters on coated bearing behaviour,Tribology Series17(1990)389–396.

[19]S.Cupillard,Thermohydrodyanmics of sliding contacts with textured surfaces.PhD Thesis,Lulea University of Technology,2009.

[20]https://www.sodocs.net/doc/7618454213.html,hmar,S.Ellagoune,B.Bou-Said,Elastohydrodynamic lubrication analysis of a compliant journal bearing considering static and dynamic deformations

of the bearing liner,Tribology Transaction53(2010)349–368.

[21] E.Kuznetsov,S.B.Glavatskih,Stability analysis of a compliant lemon bore journal bearing,ESDA2008-59589Proceedings of the9th Biennial ASME

Conference on Engineering Systems Design and Analysis,Haifa,Israel,2008.

[22]COMSOL Multiphysics User's Guide,2008.

[23]V.Popov,Contact Mechanics and Friction:Physical Principles and Applications,Springer,2010.

[24] B.Jakobsson,L.Floberg,The finite journal bearing considering vaporization,Chalmers Tekniska Hogskolas Handlingar190(1957)1–116.

[25] D.Childs,Turbomachinery Rotordynamics:Phenomena,Modeling,and Analysis,Wiley-Interscience,1993.

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