外文文獻(xiàn)翻譯--使用靜壓軸承減輕齒輪嚙合頻率噪聲【中文4186字】 【PDF+中文WORD】,中文4186字,PDF+中文WORD,外文文獻(xiàn)翻譯,使用靜壓軸承減輕齒輪嚙合頻率噪聲【中文4186字】,【PDF+中文WORD】,外文,文獻(xiàn),翻譯,使用,靜壓,軸承,減輕,齒輪,嚙合,頻率,噪聲,中文 Mitigation of Gear Mesh-FrequencyNoise Using a Hydrostatic BearingZamir A.Zulkefli1Faculty of Engineering,Mechanical and Manufacturing Engineering,Universiti Putra Malaysia,43400 UPM,Serdang,Selangor,Malaysiae-mail:zamirdinupm.edu.myMaurice L.Adams,Jr.Case School of Engineering,Mechanical and Aerospace Engineering,Case Western Reserve University,Cleveland,OH 44106-71222e-mail:maurice.adamscase.eduA proposed solution to reducing gear mesh-frequency vibrationsin a gear-set involves the utilization of hydrostatic bearingsplaced in series,load wise,with the main support bearing.Thehydrostatic bearings are expected to utilize its low pass filteringeffect of the vibrational energies to prevent its transmission fromthe shaft to the gear housing where it would be emitted as noise.The present investigation examines the frequency response of asingle-recess circular hydrostatic bearing under applied sinusoi-dal loads.The results show that as the driving frequencyincreases,the filtering effect of the hydrostatic bearing increases.The exhibited behavior is similar to the behavior of a low passfilter:negligible filtering effect at low frequencies,the filteringeffect increasing from 0%to 90%over the midfrequencies rangeand the filtering effect remaining at the maximum value as thefrequencies of the applied signals continue to increase.Thisobserved behavior is expected to play a central role in the pro-posed gear mesh-frequency vibration mitigation system.DOI:10.1115/1.4029613IntroductionThe origin of gear mesh-frequency acoustic noise is recognizedas the ever present residual manufacturing imperfections,toothelasticity,and sliding friction that preclude perfect conjugateactions between the gears 1,2.It is also known that most gear-set generated acoustic noise first passes as mesh-frequency vibra-tions primarily through the shaft support bearings to the housing,which then emits the vibrational energy as acoustic noise 3.Gear-set generated noise is concentrated at the mesh-frequency,which is typically quite acoustically objectionable and producesmore intense gear mesh-generated vibrations as the pitch linevelocity increases.Whether the gear shaft support bearings are ofthe rolling-element or hydrodynamic fluid-film type,high bearingstiffness is an obvious requirement to maintain the needed gearcenterline positioning accuracy so that the level of precision man-ufactured into the gear-set is realized during its performance.However,high bearing stiffness facilitates the transmission of thegear mesh-frequency vibrations.So therein lies the dilemma ofattempts at gear noise attenuation measures,reduction in the bear-ing stiffness jeopardizes the required gear centerline positioningaccuracy while maintaining high bearing stiffness facilitates thetransmission of the gear mesh-frequency vibrations.A proposed design to address gear mesh-frequency vibrationsconsists of hydrostatic bearings placed in series,load wise,withthe primary shaft support bearing,configured to ensure that thehigh overall static stiffness normally required from a gear-set ismaintained as much as possible.The hydrostatic bearings mean-while act as a low pass filter for the vibratory energy.Employinghydrostatic bearings alone as the main bearings could accomplishthe same noise attenuation objective as the proposed attenuationsystem,but with a potential for lower operational reliability dueto the lower reliability of the hydrostatic bearing.Additionally,Zaretsky 4 relates how rolling bearing life predictions thataccount only for static load can be considered optimistic by notincludingtheadditionalcontributionofvibrationinduceddynamic loads.Thus,the potential for rolling-element bearingfatigue life extension from the hydrostatic backup is alsosuggested.Moreover,significant attenuation of mesh-frequency noise is adesirable design objective for gear-sets.However,the trade-offsin actual gear-set designs will generally discourage making theelimination of mesh-frequency noise the only design objective.Inthat context,the proposed vibration mitigation system,even withthe addition of supporting systems,facilitates its incorporationinto current gear-set designs with minimal effect on the perform-ance of the gear-sets.Thus,the advantages of the proposedsystem:vibration mitigation and the preservation of the gearperformance,are expected to outweigh the disadvantages of incor-porating the proposed system in the gear-set design.Previous work on hydrostatic bearings has focused on itsdynamic behavior,specifically on the ability of the bearing to ful-fill high stiffness and high damping requirements 57.Rohdeand Ezzat meanwhile investigated the effect of lubricant compres-sibility on the dynamic behavior of hydrostatic bearings 8.Thefindings based on full numerical solutions of the Reynolds lubrica-tion equations determined that under the influence of the fluidscompressibility,the dynamic behavior of the bearing is character-ized by a“break frequency”above which the bearing stiffnessincreases sharply in conjunction with a sharp decrease in the bear-ing damping.Similar results were reported by other researchers912.These works,however,have typically assumed highlycompressible fluids as the working fluid.The present investiga-tion,instead examines the frequency response of the hydrostaticbearing when the working fluid,typically assumed to be incom-pressible,is now instead assumed to be weakly compressible.Thecompressibility of this working fluid is quantified via its finitebulk modulus,instead of infinite for a truly incompressible fluid.Control Volume Approach for Modeling the LowPass Filtering EffectThe development of the model in the present investigation fol-lows the work presented in detail by Zulkefli for a simple single-recess hydrostatic bearing 13.The hydrostatic bearing consistsof two main parts:the bearing pad and bearing runner where theformer is made up of a bearing recess and relatively thin bearingsills while the latter is made up of a flat surface that totally enclo-ses the bearing recess.During operation of the bearing,externallypressurized fluid is pumped into the recess,filling the availablespace.As the fluid continues to be pumped into the recess,thefluid pressure increases until the pressure is high enough to sepa-rate the bearing pad from the bearing runner,allowing the fluid toflow out over the bearing sills.In formulating the low pass behavior,it is assumed that the fluidpressure in the trapped fluid volume within the recess,V,thoughnot constant in time,is uniform throughout V at all times.Addi-tionally,V is selected to be a constant-volume control volumedefined by the enclosed bearing recess.At static load condition,the volumetric inflow and outflow into the control volume areequal.Under dynamic conditions,the inflow and outflow are not1Corresponding author.Contributed by the Technical Committee on Vibration and Sound of ASME forpublication in the JOURNALOFVIBRATIONANDACOUSTICS.Manuscript receivedSeptember 17,2014;final manuscript received December 2,2014;published onlineFebruary 20,2015.Assoc.Editor:Philippe Velex.Journal of Vibration and AcousticsJUNE 2015,Vol.137/034502-1CopyrightVC2015 by ASMErequired to be instantaneously equal.Fluid inertia is neglectedassuming that only viscous effects dominate.The three recognized methods for flow compensation in hydro-static bearings are orifice,capillary,and constant flow 14.Forthe analysis,constant flow compensation is assumed.The bulkmodulus,b,of the fluid can be defined in terms of V,the incre-mental change in the fluid volume,DV,and the incrementalchange in the fluid pressure Dp asb?DpVDV(1)Since the control volume is defined to be constant,the incrementalchange in the control volume,DV is zero.Instead,the incrementalchange in the trapped fluid mass,Dm,is used,and the relationshipbetween the two determined from the expression for the fluid den-sity is:DV?Dm=q0.Here,q0is the nominal fluid density andis assumed to be very much bigger than the incremental change inthe fluid density.The bulk modulus is thenb DpVq0Dm(2)The instantaneous mass inflow rate,mass outflow rates,and thetime rate of change of the mass can be written in terms of the vol-umetric inflow,Qin,the volumetric outflow coefficient,Qout,theoutflow coefficient,C,and the nominal fluid pressure,p0as_ min?_ moutdmdt q0Qin?Qout q0Qin?C p0 Dp?(3)The value of the outflow coefficient at static load condition isC Qin=p0.Integrating Eq.(3)and employing the expression forthe bulk modulus in Eq.(2)yields the following equation:DpVb Qint?Cp0t?Ct0Dpds(4)The instantaneous load transmitted through the hydrostatic bear-ing film is instantaneously proportional to the hydrostatic bearingpressure,hence the transmitted dynamic load is proportional toDp.Integrating Eq.(4)and rewriting it as a first-order linearordinary differential equation givesDp0bCVDp?bVQin?Cp0 0(5)The nondimensional pressure and time are chosen asDP?DpDp1;T?QinV?t(6)The nondimensional form of Eq.(5)is then rewritten asddTDP BDP?B 0(7)where B is the design factor and is defined as B?bC=Qin b=p0.An exact solution of Eq.(7)was found to beDP 1?e?BT(8)Equation(8)is observed to be the theoretical pressure response toa step change in the outflow.To determine the time response to aspecified shaft vibration signal,a convolution integral was utilizedto determine the response of the dynamic pressure,and thereforedynamic force,transmitted through the bearing under the actionof a harmonic input.The dimensionless harmonic shaft vibrationand dimensionless frequency are chosen as follows:X sinXT and X xVQin(9)The convolution integral for the transmitted dynamic pressure,DPXis thenDPXT0dXdsDP T?sds XT01?e?B T?s?cosXsds(10)From Ref.15,Eq.(10)was integrated into the following:DPX sinXT?X2sinXT XBcosXTB2 X2?Be?BTX(11)The steady state portion of which isDPX sinXT?X2sinXT XBcosXTB2 X2(12)The single peak amplitude of the harmonically varying dynamicpressure at a frequency X is thus expressible asDPXjj 1?X2B2 X2?2XBB2 X2?2()1=2(13)Examination of Eq.(13)shows that as the frequency,X is madesmaller,the amplitude DPXjj will approach that of the unity,indi-cating that all the applied dynamic force is transmitted across thehydrostatic bearing.Instead,when X is increased,DPXjj willinstead approach zero,indicating that all the applied dynamicforces are not transmitted across the hydrostatic bearing.Thisbehavior,no filtering at low frequencies and total filtering at highfrequencies,is similar to the behavior expected from a low passfilter.The present investigation shows that a similar behavior isobserved in the data.Experimental ResultsThe frequency response of the hydrostatic bearing was deter-mined using the setup described in detail in Refs.13,16.Thesetup consists of a single-recess circular hydrostatic bearing underthe action of applied dynamic loads,placed in a materials testingmachine.The loads are applied on the bearing pad,and the trans-mitted loads were measured at the bearing runner.The hydrostaticbearing used in the experiment was sized using the methodsoutlined in Ref.14 for constant flow compensation scheme.Theimportant system parameters used to size the bearing were:driving frequency range of 1100 Hz,lubricant fluid flow of0:95?10?560:16?10?5m3s?1(0:15060:025 gpm),supplypressure of 2:76?10560:03?105Pa(4060:5 psi),and anominal applied load range of 300?50062 N.The appliedloads consist of a sinusoidal signal at a constant frequency with anamplitude that is ten percent(10%)of the applied nominal loads.The dimensions of the hydrostatic bearing used for the setup are:recess diameter of 0:076260:0025 m(3:0060:01 in.),recessdepth of 0:007660:0025 m(0:3060:01 in.),and sill thicknessof 0:003260:0025 m(0:1360:01 in.).The working fluid usedwas shell spindle oil ten.The frequency response of the transmitted loads for differentvalues of the applied load is shown in Fig.1.It is observed fromthe figure that for a driving frequency between 1 Hz and 40 Hz,the transmitted loads show no significant change in value from theapplied input loads.For a driving frequency between 40 Hz and70 Hz,the transmitted loads show a steady decrease to around50 N regardless of the value of the applied loads.For a driving034502-2/Vol.137,JUNE 2015Transactions of the ASMEfrequency between 70 Hz and 100 Hz,the transmitted loadsremain fairly stable at around 50 N.A similar filtering behavior isobserved from the normalized frequency response of the transmit-ted loads as shown in Fig.2.It is clearly observable from thefigure that for all values of the nominal applied loads,the hydro-static bearing is able to filter out almost 90%of the applied loads.Furthermore,the figure shows that the filtering occurs over thesame frequency range for all three load conditions.Therefore,thelow pass filtering behavior observed from the frequency responseof the hydrostatic bearing is similar to the expected behaviorgiven by Eq.(13).Furthermore,examination of Eq.(13)shows that the equationprovides a relationship between V and the frequency response,ofthe hydrostatic bearing.Thus,by choosing an appropriate value ofV,the frequency range over which the filtering occurs can bedetermined.Conversely,by choosing the frequency range overwhich the filtering is expected to occur,V can be determinedallowing the hydrostatic bearing to be sized to the frequency rangeof interest.For the present investigation,the maximum drivingfrequency for the experimental setup was limited to 100 Hzprimarily due to safety concerns and machinery limitations.How-ever,it is expected that the same general low pass filteringFig.1Frequency response of the transmitted loadsFig.2Normalized frequency response of the transmitted loadsJournal of Vibration and AcousticsJUNE 2015,Vol.137/034502-3behavior predicted by Eq.(13)and observed in Figs.1 and 2 willbe observed for higher driving frequency values.Similarly,it isexpected that the same behavior predicted and observed in thepresent investigation will be observed when the value of theapplied loads is higher.Ultimately,it is expected that the low passfiltering behavior observed from the present investigation will besuccessfully incorporated in the proposed gear mesh-frequencyvibration mitigation system to disrupt the transmission ofvibratory energy from gear-sets.ConclusionThe present investigation shows that a hydrostatic bearing isable to prevent the transmission of applied vibratory loads at cer-tain frequency ranges.This observed behavior indicates the abilityon the part of the hydrostatic bearing to act as a low pass filter ofvibratory loads.It is expected that the same general low pass fil-tering behavior will be observed when the driving frequencies andsystem parameters are changed to values that are typicallyencountered in an actual gear-set.The proposed vibration mitiga-tion system is expected to utilize this observed low pass filteringbehavior of the hydrostatic bearing to prevent the transmission ofgear mesh-frequency vibrations through the gear-set to the hous-ing where it would be emitted as noise.References1 Smith,J.D.,1983,Gears and Their Vibration:A Basic Approach to Under-standing Gear Noise,Macmillan Press,New York,pp.170.2 Litvin,F.L.,and Fuentes,A.,2004,Gear Geometry and Applied Theory,2nded.,Cambridge University Press,New York,pp.800.3 Houser,D.R.,2007,“Gear Noise and Vibration Prediction and ControlMethods,”Handbook of Noise and Vibration Control,M.J.Crocker,ed.,Wiley,Hoboken,NJ,pp.847856.4 Zaretsky,E.V.,1999,STLE Life Factors for Rolling Bearings,2nd ed.,Societyof Tribologists and Lubrication Engineers,Park Ridge,IL.5 Brown,G.M.,1961,“The Dynamic Characteristics of a Hydrostatic ThrustBearing,”Int.J.Mach.Tool Des.Res.,1(1),pp.157171.6 Adams,M.L.,and Shapiro,W.,1969,“Squeeze Film Characteristics in FlatHydrostaticBearingsWithIncompressibleFlow,”Tribology,12(3),pp.183189.7 Bouzidane,A.,and Thomas,M.,2007,“Equivalent Stiffness and DampingInvestigation of a Hydrostatic Journal Bearing,”Tribol.Trans.,50(2),pp.257267.8 Rohde,S.M.,and Ezzat,H.A.,1976,“Dynamic Behavior of Hybrid Journal-Bearings,”ASME J.Tribol.,98(1),pp.9094.9 Ghosh,M.K.,andViswanath,N.S.,1987,“RecessVolumeFluidCompressibilityEffectontheDynamicCharacteristicsofMultirecessHydrostatic Journal Bearings With Journal Rotation,”ASME J.Tribol.,109(3),pp.417426.10 Ghosh,M.K.,Guha,S.K.,and Majumdar,B.C.,1989,“RotordynamicCoefficientsofMultirecessHybridJournalBearings,”Wear,129(2),pp.245259.11 Andres,L.A.S.,1991,“EffectsofFluidCompressibilityontheDynamicResponseofHydrostaticJournalBearings,”Wear,146(2),pp.269283.12 Pollmann,E.,and Vermeulen,M.,1989,“Compressibility and Inertia Effectson the Dynamic Behavior of Recessed Hydrostatic Bearings,”Tribol.Int.,22(3),pp.166176.13 Zulkefli,Z.A.,2013,“Mitigation of Gear Mesh-Frequency Vibrations Using aHydrostatic Bearing,”Ph.D.thesis,Case Western Reserve University,Cleve-land,OH.14 Rippel,H.C.,1963,“Cast Bronze Hydrostatic Bearing Design Manual,”CastBronze Bearing Institute,Cleveland,OH,pp.75.15 Thomas,G.B.,1957,Calculus,Addison-Wesley,Reading,MA,pp.692.16 Zulkefli,Z.,and Adams,M.L.,2014,“Experimental Investigation of the LowPass Filtering Effect of a Hydrostatic Bearing,”SAE Technical Paper No.2014-01-1758.034502-4/Vol.137,JUNE 2015Transactions of the ASME