Elastohydrodynamic Lubrication Characteristics of Spiral ...downloads.hindawi.com/journals/mpe/2018/3043712.pdf · Model of Spiral Bevel Gear Infact,manyfactorswillin uencetheEHLoftoothsurface

Research ArticleElastohydrodynamic Lubrication Characteristics ofSpiral Bevel Gear Subjected to Shot Peening Treatment

Shuai Mo 1234 Ting Zhang12 Guoguang Jin1 Shengping Zhu12

Jiabei Gong12 and Jingyang Bian1

1School of Mechanical Engineering Tianjin Polytechnic University Tianjin 300387 China2The State Key Laboratory of Materials Processing and Die amp Mould TechnologyHuazhong University of Science and Technology Wuhan 430074 China3The State Key Laboratory of Mechanical Transmissions Chongqing University Chongqing 400044 China4School of Mechanical Engineering amp Automation Beihang University Beijing 100191 China

Correspondence should be addressed to Shuai Mo moshuai2010163com

Received 23 January 2018 Revised 25 March 2018 Accepted 24 April 2018 Published 10 June 2018

Academic Editor Fabrizio Greco

Copyright copy 2018 ShuaiMo et alThis is an open access article distributed under the Creative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

High speed and heavy load put forward demanding requirements for the bending strength of spiral bevel gear Shot peening as atechnological method can greatly improve the bending strength of gear During shot peening a large number of subtle projectilesbombard the tooth surface of spiral bevel gear at an extremely high speed which makes the latter produce plastic deformation andstrengthened layer Serving as the bombarded target spiral bevel gear inevitably leads to the change of tooth surfacesmicrostructureThis paper aims at revealing the coupling law between microstructure and lubrication characteristics and establishing a reasonablemodel that can describe the tooth surface microstructure of shot peened spiral bevel gear Besides this paper brings insight intothe lubrication characteristics of shot peened spiral bevel gears Based on the elastohydrodynamic lubrication (EHL) theory acomparative research on tooth surface lubrication characteristics has been conducted by various microscopic morphologies in thisstudy

1 Introduction

Shot peening is widely used for precision gear transmissionbut lubrication and fatigue characteristics become an increas-ingly significant issue for shot peened gear transmissionMany scholars obtained some academic achievements byresearching the lubrication and fatigue of gear transmissionDimitrov et al investigated the contact fatigue resistanceof gear teeth by experiment and theory and subjecting itto shot peening treatment [1] To present a new exper-iment for studying contact fatigue damage of gear teethsubjected to different load patterns Guo et al carried out aserial sectioning procedure on shot peened 12Cr martensiticstainless steel to determine the location of fatigue crackinitiation sites from pits of varying depths [2] These resultsindicate that common assumptions about crack initiationfrom pits in shot peened steel can be misleading Li and

Anisetti proposed a contact fatigue model for spur gearsoperating under a high speed condition where dynamicbehavior is evident [3] A comparison between tribodynamicand quasi-static life predictions is performed to demonstratethe important role of gear tribodynamics in the fatiguedamage The impacts of the input torque surface roughnessand lubricant temperature on the gear contact fatigue undertribodynamic condition were also investigated Zhou et alestablished a new normal stiffness model of oil film byviscous-elastic fluid between a spur gear drive equivalentand a massless spring element and proposed a tangentialstiffness model according to the hypothesis of equal shearstress on laminar element surfaces [4] It indicated that therational parameter match is valid in mesh impact reductionand stationarity enhancement Gallitelli et al presented amethodology capable of producing the relation between pro-cess parameters and the state of treated part taking a complex

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 3043712 12 pageshttpsdoiorg10115520183043712

2 Mathematical Problems in Engineering

geometry into consideration which include an initial statefrom previous manufacturing processes [5] The proposedapproach is consistent with industrial constraints in terms ofcomputation time The perspective is to complete the chain-ing process with fatigue life computations Lv et al studiedW6Mo5Cr4V2 steel gear tooth flank by laser irradiationAfterwards the gear tooth flank was treated by shot peeningwith shot particles of differentmaterials and peening time [6]The experimental results showed that the hardness of gearsincreased and the surface roughness of gears decreased aftershot peening treatment In addition the residual stress statein the near-surface layer of shot peened gears was changedfrom tensile stress to compressive stress Terrin et al analyzedsix case hardened sun gears damaged by pitting duringendurance tests [7] All analyzed gears belonged to planetaryfinal drives placed on a wheel hub of axles for off-highwayvehicles The aim of the analysis was to highlight the keyaspects of the morphology and the evolution of pitting dam-age on case hardened sun gears Pu et al conducted a com-prehensive analysis for gearing geometry kinematics mixedlubrication performance and friction and interfacial flashtemperature in spiral bevel and hypoid gears [8] This studywas developed based on a recently established mixed elasto-hydrodynamic lubrication model that was capable for han-dling practical cases with 3D machined roughness undersevere operating conditions considering the effect of arbi-trary entrainment angle Obtained results from sample casesshowed that the developed simulationmodel could be used asan engineering tool for spiral bevel and hypoid gears designoptimization and strength prediction Tang et al explored theeffect of plasma molybdenizing and shot peening on frettingwear and fretting fatigue behaviors of Ti6Al4V alloy [9] Theresults displayed that a beneficial residual compressive stressdistribution high surface hardness with suitable hardnessgradient distribution good apparent toughness relativelylow surface roughness and excellent surface integrity wasachieved AlMangour and Yang made an attempt at inducinggrain refinements through a shot peening process creatingsevere plastic deformation at the outer surface layers toimprove the physical andmechanical properties of 17-4 stain-less steel components produced by DMLS [10] The researchresult highlighted the efficiency and applicability of shotpeening treatment to practical cases This contribution ana-lytically derives a solution to the Reynolds equation to de-scribe the longitudinal fluid pressure distribution load capac-ity and coefficient of a MHPed surface structure friction[11] It was shown that the lateral effects in machine hammerpeened surface structures significantly increased the fluidpressure Therefore the analytical approach presented couldbe used to estimate the lower bound Simon raised an opti-mization methodology systematically to define the optimaltooth modifications introduced by head-cutter geometry andmachine-tool settings to minimize the influence of mis-alignments on the elastohydrodynamic (EHD) lubricationcharacteristics in face-hobbed spiral bevel gears [12] Its goalis to maximize the EHD load-carrying capacity of the oil filmand to minimize power losses in the oil film when differentmisalignments were inherent in the gear pair Shuai et aldeveloped the spiral bevel gear true tooth surface precise

modeling method based on machining adjustment parame-ters and conducted an experiment to prove that themodelingmethod is right and precise At the same time a research instar gear transmission system was also conducted [13 14]

Many of these researches have important effects only onelastohydrodynamic lubrication or shot peening until nowbut few literatures about elastohydrodynamic lubricationcharacteristics focus on shot peening In addition some evenassume that shot peening exerts few effects on elastohydrody-namic lubrication characteristics for gear pair Heretofore tothe best knowledge of the authors few published studies areavailable on the elastohydrodynamic lubrication characteris-tics of shot peened spiral bevel gear Based on the elastohydro-dynamic lubrication theory the paper making a comparativeresearch of the tooth surface lubrication characteristics ofvarious microscopic morphologies reveals the elastohydro-dynamic lubrication characteristics of spiral bevel gears aftershot peening process

2 Microtopography Description ofTooth Surface by Shot Peening

Different from the roughness of tooth surface after machin-ing the shot peened tooth surface roughness has certainregularity When spiral bevel gear gets into the stage ofelastohydrodynamic lubrication the oil film thickness oflubricating between the two contact tooth surfaces is in thesame order of magnitude with roughness at the morphologi-cal peak of shot peened tooth surface The materials used forshot peening mainly include ceramics cast steel glass castiron and so on To avoid concentrated stress points the shapeof projectile is spherical The projectile comes in differentsizes among which the smallest diameter is 01mmwhile themaximum reaches 2mm The size and material of projectileshould be selected according to strengthening requirementsof the sprayedworkpiece the surface roughness of workpieceand the mechanical properties of workpiece and so on Thestrengthening effect of shot peening is related tomany factorsfrom thematerial properties ofworkpiece and the contact dis-tance between nozzle andworkpiece to the angle of incidenceAdditionally there are shot peening speed coverage materialtype hardness shot peening time projectile size and othershot peening factors Numerous process parameters influ-ence the shot peening effect and the coupling mechanismsbetween them are very complex

21 Construction of Multiple Projectilesrsquo Offset Shot PeeningModel At present the finite element model in ABAQUS ofshot peening strengthening is a multiple projectilesrsquo offsetmodel As shown in Figure 1 this paper establishes the 49-pellet offset shot peening strengthening simulation model infour layers with high precision The model assumes that theprojectile is arranged in accordance with a certain law andhas a bias distribution and the target surface is vertically hitwhich reduces the blind area and increases the coverage rate

On the basis of the bias modeling method radius of thefirst layer of projectiles is 119877 and the center distance is 2119877 Theoffset distance of the second layer relative to the first layer is119877 in the119883 direction and the offset distance of the third layer

Mathematical Problems in Engineering 3

Figure 1 The finite element model of shot peening strengthening

Figure 2 The surface microtopography of target tooth surface afer shot peening in ABAQUS

relative to the first layer is 119877 in the 119884 direction Moreover theoffset distance of the forth layer relative to the first layer is 119877in the 119883 and 119884 direction respectively Finally the target sizeis 6119877 times 6119877 times 12119877 There are 9 pills in the first layer 12 pills inthe second layer 12 pills in the third layer and 16 pills in thefourth layer

The target material is made of high performance gearsteel 16Cr3NiWMoVNbE and the mechanical parametersare as follows elastic modulus 210GPa density 7800Kgm3Poissonrsquos ratio 03 initial yield stress 800MPa hardeningmodulus 1000MPa and coulomb friction coefficient 02Mechanical parameters of projectile are listed as followselastic modulus 210GPa density 7800Kgm3 Poissonrsquosratio 03 and initial velocity of shot peening 50ms Someconstitutive model can be used for shock problem such asCowper-Symonds model Johnson-Cook model and Zerilli-Armstrong model According to the speed of shot peeningthe Cowper-Symonds constitutive model was adopted Thismodel is suitable for the collision of high strain ratematerialsThe Cowper-Symonds constitutive model has considered thestrain rate in the FE model According to the practice ofengineering and references the friction coefficient is set as02

22 Construction of Surface Microtopography after Shot Peen-ing Because the roughness of high precisionmachined toothsurface has been completely destroyed after shot peening

the roughness peak of tooth surface caused by shot peeningis larger than that of grinding tooth surface The shotpeened morphology of tooth surface is the final shape Thesimulation results of shot peened spiral bevel gear and themicro topography of tooth surface are shown in Figure 2 Itcan be seen that projectiles collide with the target surfaceand bounce leaving many regular craters on tooth surfaceThese craters overlap each other and influence each otherwhich completely change the microscopic appearance ofspiral bevel gear tooth surface Therefore the orange peelpit morphology is formed on the shot peening strengthenedtooth surface In some industry situations the shot peeningstrengthening coverage is even bigger and the shot peeningintensity is stronger What is more the pits more irregularand smoother havemore influence on each otherThe surfacemicrostructure in this case can be used as the microstructureof shot peened gear tooth surface for studying the elastohy-drodynamic lubrication characteristics of shot peening gear

The numerical calculation of elastohydrodynamic lubri-cation needs to be analyzed in MATLAB but the FEA modelis built in ABAQUS so the FEA results are necessary to beused to reconstruct the surface topography after shot peeningin MATLAB The finite element model is converted into anumerical one extracting and exporting the deformationdata of target surface in FEA to form txt file According tothe number of surface nodes the micro topography data ofshot peened tooth surface are sorted to build a database


minus4S(Z

Disp

lace

men

t (m

m))

2

0

minus2

11 050 0minus1 minus05

4

minus2 minus1

times10minus4

Ｒ＜ＧＳ＜

Figure 3 Reconstruction surface topography of tooth surface aftershot peening in MATLAB

which includes all data of surface topography imported toMATLAB The three-dimensional matrix of surface topog-raphy for spiral bevel gear is established in MATLAB andreconstruction of the surface topography is carried outFinally a three-dimensional nephrogram is formed as shownin Figure 3

3 Equivalent Elastohydrodynamic LubricationModel of Spiral Bevel Gear

In fact many factors will influence the EHL of tooth surfaceIn this paper we just talk about EHL of tooth surfaceafter shot peening treatment Shot peening will change themicroscopic geometry and the microscopic geometry willhave a big influence on EHLThemicro geometry is the mostimportant factor for the research so the thermal factor ofEHL will be omitted During the meshing process of spiralbevel gears when lubricant oil film between the two contactsurfaces is in the same order ofmagnitudewith the roughnessof the machined surface the rough peak of local contactarea will be contacted At the same time the gears are inthe mixed lubrication state and the non-Newtonian fluidcharacteristics are produced by strong shear acting on thelubricating oil film Especially under the action of extremetorque the thickness of oil film decreases sharply and evenbreaks downTherefore the surface ofmetal comes in contactdirectly and the tooth surface enters the dry friction state Asone of the boundary conditions of the elastohydrodynamiclubrication model the load distribution determines the peakpressure in the contact area and the thickness of the oil film

31 Equivalent Cylindrical Gear Model of Spiral Bevel GearFrom the engineering practice the steady-state load distribu-tion model is applicable when the gear rotates at a nonhighspeed In order to establish the steady-state mixed elasto-hydrodynamic lubrication numerical model for long linecontact in the contact area the classical Hertz contact theoryand the elastohydrodynamic lubrication theory are needed

to establish the film thickness and Reynolds equation In theend the distribution of oil pressure and oil film thickness arefigured out and the elastohydrodynamic lubrication propertyof the gear pair is analyzed

Although the tooth surface shape of spiral bevel gearis complex it can be treated as equivalent cylindrical gearas long as operating conditions are the same which hasbeen proved in previous theoretical and experimental resultsFurthermore since this papermainly explores the lubricationcharacteristics of spiral bevel gear shot peened tooth surfaceclassical elastohydrodynamic lubrication theory can be usedA two-dimensional contact elastohydrodynamic lubricationequation for long line contact is established to solve theproblem of gear pair contact According to the requirementsthe entrainment velocity of contact point is regarded as119909-axisand the direction of the contact line is regarded as 119910-axis and119911-axis is the normal direction of the contact area In view ofthis themodel of this paper based on the equivalent cylindri-cal gear system is shown in Figure 4 Lubricating oil param-eters are viscosity coefficient 120572 = 2272 times 10minus2mmN anddynamicviscosity1205780 = 54times10minus8Nmmm2Gear drive param-eters are shown in Table 1

Take cylindrical gear pair as the equivalent gear pair of thespiral bevel gear the tooth number of equivalent cylindricalgear becomes the equivalent tooth number of the spiral bevelgear 119911V is the tooth number of equivalent cylindrical gear zis the tooth number of spiral bevel gear 120575 is the pitch angleof spiral bevel gear and 119889V is the pitch diameter of equivalentcylindrical gear for spiral bevel gear Based on the knowledgeof mechanical principle it can be easily obtained that 119911V =119911 cos 120575 so 119911V1 = 1199111 cos 1205751 119911V2 = 1199112 cos 1205752 119889V1 = 119898119911V1 and119889V2 = 119898119911V2 1198731119888 and 1198732119888 are the radius of curvature of twocontact cylinders in lubrication theory so 1198731119862 = 119889V1 sin120572and1198732119862 = 119889V2 sin12057232 Elastohydrodynamic Lubrication Control Equation

321 Reynolds Equation and Boundary Conditions In thispaper non-Newtonian fluid is chosen as lubricant and theReynolds equation suitable for unsteady state line contact ofelastohydrodynamic lubrication is presented as follows [15]

120597120597119909 ( 120588ℎ312120578lowast 120597119901120597119909) + 120597120597119910 ( 120588ℎ312120578lowast 120597119901120597119910)= 119906119909 120597 (120588ℎ)120597119909 + 120597120597119905 (120588ℎ)

(1)

In (1) 119901 and ℎ are respectively pressure and oil filmthickness 120578lowast is the equivalent viscosity of lubricating oil 120588is the density of lubricating oil 119905 is a time variable 120583x isthe tooth surface entrainment velocity the average value oftangential velocity of tooth surface and 120583x = (120583a + 120583b)2 120583119886and 120583119887 are the tangential velocity of tooth surface

Equation (1) is actually a flow balance equation fromthe physical point of view The first term on the left side ofthe equation is the pressure flow along the 119883 direction dueto the pressure gradient in the fluid lubrication oil film alsoknown as the Poiseuille flow while the first term on the


(driving)

(driven)

t tC

FＨ

FＨ

2

1

2

＞Ｐ1

2

＞Ｐ1

2４1

1

1

2

Figure 4 Spiral bevel gear meshing pair transferred to equivalent cylindrical gear meshing pair

Table 1 Parameters of equivalent cylindrical gear pair

Modulus Teeth Pitch angle Face wide Rotation rate Torque119898(m) z1z2 12057511205752 (∘) 119861 (m) 1198991 (119903min) 119879119901 (Nm)000483 933 15257475 00275 3000 200

right indicates the shear flow due to the surface velocity alsoknown as the Couette flow The second term on the right isthe flow change due to the squeezing motion which includesthe change in density

The boundary conditions of (1) are119901 (119909in 119905) = 119901 (119909out 119905) = 0119901 (119909 119905) ge 0 (2)

In (2) 119909in is the starting point coordinates of the calcu-lation area In this paper 119909in = minus4b b is the half-widthof Hertz contact 119909out is the end point coordinates of thecomputational domain and the value can only be determinediteratively in the numerical calculation process instead ofbeing set artificially

322 Oil Film Thickness Equation In the elastohydrody-namic lubrication problem the contact deformation in thenormal direction of the cylinder surface can affect the shapeof the gap as shown in Figure 5 Under the action of the con-tact zone load elastic deformation takes place in the normaldirection of the two cylinder surfaces which is manifestedas surface indentation Tooth surface 119886 is a rough surfacewhile tooth surface 119887 is a smooth one When two elasticcylinders contact the oil film thickness between the meshingteeth surface isℎ (119909 119910) = ℎ0 + Δℎ

+ 21205871198641015840 intint 119901 (1199091015840 1199101015840)radic(119909 minus 1199091015840)2 + (119910 minus 1199101015840)2 119889119909

10158401198891199101015840

minus 119878 (119909 119910)(3)

x

h(x)

R(x

) x22

R

ＢＩ

Figure 5 The gap shape for elastic deformation

In (3) ℎ0 is a constant related to the thickness of thecenter of the rigid body Δℎ = 11990922119877 is the variation of filmthickness with the change of tooth profile in the119883 direction119877 is the synthetic curvature radius of meshing point and 1198641015840represents the equivalent elastic modulus of a pair of gearteeth 119878(119909 119910) is the contact tooth surface roughness functionof tooth surface 119886 This function can be a random roughnessfunction simulated in this paper or a function of surfacetopography after shot peening

323 Load Balance Equation Under the condition of noexternal force the contact area of the two reverse cylinders isa line However when the external force is applied along thedirection of the generatrix the contact line will be extendedinto a contact area and the oil film will be pressed to produce


pressure distribution in the contact area Therefore the loadbalance equation can be written as follows

119862119908119908119899 = ∬119860

119901 (119909 119910) 119889119909 119889119910 (4)

In (4) 119860 is the total area of contact area 119862119908 is theload coefficient and 119908119899 is the load of each pair of meshinginstantaneous target teeth In steady-state 119862119908119908119899 = Fs

324 Rheological Model of Lubricating Oil Under the condi-tion of elastohydrodynamic lubrication the lubricant is usu-ally subjected to high pressure and high shear stress in smallcontact area where the lubricant filmwill withstandmomen-tary local high pressure Obviously under such a high pres-sure shear stress and temperature rise will lead to severenon-Newtonian characteristics of the lubricant As for the de-scription of rheological models Ree-Eyrin model J-Tmodeland B-Wmodel are generally accepted

At present Ree-Eyrinmodel is often used as the rheologi-cal model of lubricant in the elastohydrodynamic lubricationmodel which is applied to the unified Reynolds equation

120574 = 1205910120578 sinh( 1205911205910) (5)

In (5) 1205910 is reference shear stress 120574 is shear rate 120578is lubricating oil viscosity and 120591 is oil film shear stressLubricating oil viscosity 120578 uses Roelands relation

120578 = 1205780 exp (ln 1205780 + 967) [(1 + 51 times 10minus9119901)119911 minus 1] (6)

In (6) 119911 is the viscosity index and 1205780 is environmentalviscosity of lubricating oil The equivalent viscosity 120578lowast satis-fies the following equation

1120578lowast = 1120578 1205910120591 sinh( 1205911205910) (7)

The lubricating oil density 120588 uses the Dowson-Higginsonformula

120588 = 1205880 (1 + 06 times 10minus91199011 + 17 times 10minus9119901) (8)

In (8) 1205880 is initial density325 Nondimensionalize Equations In general the param-eters of each basic equation need to be nondimensionalizedbefore the operation Dimensionless parameter is to reducethe number of variables in the dimensionless equation sothat the calculation of the equation is more convenient and

easy The paper selects frequently used and dimensionlessparameter

119883 = 119909119887119898 119867 = ℎ1198771198981198872119898 119875 = 119901119901119898 120588 = 1205881205880 120578 = 1205781205780 119896119899 = 119871119887119887119898 119905 = 119906119898119905119887119898

(9)

In (9) 1205880 is environmental density of the lubricating oil1205780 is environmental viscosity of the lubricating oil 119877119898 isthe synthetic curvature radius of pitch point and 120583119898 is theentrainment velocity of pitch point The expressions of Hertzcontact area half-width and peak pressure of pitch pointunder load are as follows

119887119898 = radic41199081198981198771198981205871198641015840119871119887 119901119898 = 119908119898120587119887119898119871119887

(10)

Add formula (9) and formula (10) to (1) and to (4) andthe dimensionless elastohydrodynamic lubrication controlequation can be obtained119877 is the synthetic curvature radius of two contact surfacesand the curvature radius of surface 1 and surface 2 is 1198771 and1198772 The relation between them is

1119877 = 11198771 +11198772 (11)

1198641015840 is a synthetic elastic modulus of two surface solidmaterials and 1198641 and 1198642 are the elastic modulus of solid 1and solid 2 1205901 and 1205902 are Poissonrsquos ratioThe relation betweenthem is as follows

11198641015840 = 05 (1 minus 120590211198641 + 1 minus 120590221198642 ) (12)

For the steady-state the gear pair is equivalent to a pairof reverse disc rollers with different curvature radius at eachmeshing position and the load of each roller is differentWith steady load distribution as the boundary conditions ofelastohydrodynamic lubrication equation if the numericalcalculation of the lubrication state of the meshing gear atthe pitch circle is carried out the changing rule of pressure


H0 revision H0 = H0 + HΔH

Dimensionless parameters Input operating parameters

Oil pressure and oil film thicknessinitialization boundary conditions

Calculate viscosity densityelastic deformation

Solve Reynolds equation by iterative method and figure outoil pressure and oil film thickness

oil pressure and oil film thickness

Output the pitch point coordinates XY

Update initial pressure and film thickness

NO

YES

Distinguish convergence

Cw CuM L bm pH

Transfer surface topography S1 S2 or A

p w

Tp Ｈ1N

Before assignment of

m ２ 0 0 C1 and so onparameters

Figure 6 Flowchart of elastohydrodynamic lubrication analysis

distribution and oil film thickness at this instant will beobtained The lubrication state of the whole area is expressedby the state of the elastic fluid lubrication of the meshing gearat this position and then the elastohydrodynamic lubricationcharacteristics of the gear can be calculated

326 Calculation Flow Chart The Reynolds equation issolved by iteration method the Gauss-Seidel relaxationmethod is used Each cycle is determined by two convergencecriteria 120576119901 and 120576119908 to determine whether to continue the cycleor to jump out of the cycle to output pressure and filmthickness such as formula (13) Figure 6 provides a flowchartfor the calculation of two-dimensional finite line contactsteady-state hybrid elastohydrodynamic lubrication

120576119901sumsum 10038161003816100381610038161003816119901119896+1119894119895 minus 11990111989611989411989510038161003816100381610038161003816sumsum119901119896+1119894119895 le 1119890 minus 4

12057611990810038161003816100381610038161003816119862119908120587119896119899 minus ℎ119896119909ℎ119896119910sumsum11990111989611989411989510038161003816100381610038161003816sumsum119901119896119894119895 le 1119890 minus 4

(13)

4 Lubrication Characteristics Oil PressureDistribution and Oil Film Thickness

Elastohydrodynamic lubrication analysis of the multiple pro-jectilesrsquo offset shot peeningmodel canwell observe the impactof a single crater during lubrication and the effect studyof shot peening on elastohydrodynamic lubrication after thewhole micro surface is reshaped As the pits formed by shotpeening strengthening with 9-pellet model and 25-pelletmodel have no interaction it is more practical to study theelastohydrodynamic lubrication after shot peening strength-ening with the 49-pellet model With the increase of shotpeening coverage craters will affect each other and the bulgearound a single crater may be hit by a large number of pelletsThus the impact of bulge will be weakened

Figures 7 and 8 show the three-dimensional distributionprofile of elastohydrodynamic lubrication oil pressure and oilfilm thickness after shot peening in meshing tooth surfaceof spiral bevel gear Figure 7 is the oil pressure distributionin meshing tooth surface for spiral bevel gear during elasto-hydrodynamic lubrication 119883 direction is 119887119898 119887119898 stands for


Figure 7Oil pressure distribution inmeshing tooth surface of spiralbevel gear

Figure 8 Oil film thickness distribution in meshing tooth surfaceof spiral bevel gear

half-width of Hertz contact zone 119884 direction is 119871119887 119871119887 standsfor the length of contact line

According to the rule of Figures 7 and 8 in addition tothree basic rules of the elastic fluid lubrication oil film thick-ness and oil pressure distribution the orange peel pits areconnected with each other and the effect of pits on the elas-tohydrodynamic lubrication characteristics is more obviousOn the other hand along the direction of meshing line theinfluence of pits on the lubrication characteristics is smallerthan that in the 119909 direction which is the entrainment velocitydirection This is because the oil pressure and oil film thick-ness are greatly affected by the load and the entrainmentvelocity The transverse flow of the lubricant omitted inprinciple in the direction of meshing line is very small In the

direction of entrainment velocity the bulge formed aroundthe orange peel pit seriously hinders the flow of oil Pressureincreases here and local pressure value even exceeds thesecond peak value 121198750 and reaches the highest value 131198750while the oil film thickness decreases Furthermore it is veryeasy to produce oil film rupture resulting in rough peak con-tact which will increase tooth surface wear

Figure 9 shows the oil pressure and oil film thicknessdistribution in the119910 = 0119909 = 0 sectionThepurple dotted linestands for oil film thickness and the red solid line stands for oilpressure Meanwhile combined with Figures 9(a) and 9(b)these two sections are passed through the center of pit 119883direction is 119887119898 119887119898 stands for half-width ofHertz contact zone119884 direction is 119871119887 119871119887 stands for the length of contact line

As shown in Figure 9(a) along the 119909 direction thepressure is concentrated on the Hertz contact area Thereis a marked decrease of oil pressure in the contact areawhere peak pressure value falls from 081198750 on both sides to061198750 in the pit and the pressure drops about 25 The oilfilm thickness increases from 0551198670 to 061198670 and the filmthickness raises about 10

However as shown in Figure 9(b) along the 119910 directionpeak oil pressure value descends from 071198750 on both sides to0651198750 in the pit and the pressure falls about 7The oil filmthickness increased from 051198670 to 0581198670 and the oil filmthickness grows about 16

It is proved again that the pit has a greater influence on thepressure distribution in the direction of entrainment velocityand the influence on the oil film thickness is smaller than thatof the pressure But in general due to the local oil pressurearound the orange peel pit a large amount of lubricating oilflows to the orange peel pitTherefore with much lubricatingoil accumulated in the orange peel pit the oil film thicknessis increased which is favorable for lubrication

The roughness function is used to simulate tooth surfaceroughness of spiral bevel gear The distribution function oftooth surface roughness is 119878 = 119877119904 cos(2120587119909119908119909) cos(2120587119910119908119910)Figure 10 is the graph for effects on oil pressures distributionwith different roughness level surfaces S-P is short for shot-penning Figure 10(a) is 119910 = 0 section and Figure 10(b) is119909 = 0 section Figure 11 is the graph for effects on oil filmthickness distribution with different rough level surfaces S-Pis short for shot-penning Figure 11(a) is 119910 = 0 section andFigure 11(b) is 119909 = 0 section 119883 direction is 119887119898 119887119898 stands forhalf-width of Hertz contact zone 119884 direction is 119871119887 119871119887 standsfor the length of contact line

Intuitively asperity has great effects on both oil pressureand oil film thickness but the effect on oil pressure is rela-tively greater than that on oil film thickness The asperity ofshot peenedmicrosurface becomes less and those ridges thatprevent the lubrication also connect into one pieceThereforethe effect on the oil pressure and oil film thickness is relativelysmall

Due to pills at a certain speed shocking the tooth surfaceof spiral bevel gear continually shot peening will cause thedeformation of tooth surface and forming compressive stressin the subsurface of tooth A suitable shot peening treatmentwill increase the lubrication characteristics of gear trans-missionThemicrosurface lubrication characteristics formed


y=0

Film ThicknessPressure

050 15minus05minus15minus25 1minus2 minus1Ｒ＜Ｇ

0

05

1

15

2

25

Ｂ（

0

0

02

04

06

08

1Ｊ０

0

(a) 119910 = 0 section


x=0

0

02

04

06

08

1

Ｊ０

0

0

02

04

06

08

1

Ｂ（

0

0 05minus05 1minus1Ｓ＜

(b) 119909 = 0 section

Figure 9 Oil pressure and oil film thickness in meshing tooth surface of spiral bevel gear (119909 = 0 section)

y=0

S-P

050 15minus05minus15minus25 1minus2 minus1Ｓ＜

0

02

04

06

08

1

12

Ｊ０

0

２Ｍ=00２Ｍ=16

２Ｍ=32

(a) 119910 = 0 section

x=0


0

02

04

06

08

1

12

14

16

18

Ｊ０

0

S-P２Ｍ=00２Ｍ=16

２Ｍ=32

(b) 119909 = 0 section

Figure 10 Oil pressure distribution in meshing tooth surface of different surface roughnesses

by shot peening are much better than those of the machinedroughness model For instance for the cross section 119910 = 0compared with the 119877119878 = 32 120583m peak oil pressure value de-creases from 115 1198750 to 08 1198750 the minimum oil film thicknessincreases from 041198670 to 051198670 peak oil pressure drops about30 and the minimum oil film thickness grows about 25For the cross section 119909 = 0 oil pressure peak value dropsfrom 16 1198750 to 08 1198750 in the center contact area the minimumoil film thickness increases from 045 1198670 to 055 1198670 theoil pressure peak value falls about 100 and the minimumoil film thickness grows about 22 As pits formed by shotpeening this kind of effect becomes even more remarkable

From Figures 10 and 11 the effect of asperity on oil pres-sure and oil film thickness distribution reduces greatly asroughness gradually decreases When the machining accu-racy of the gear reaches the finishing machining the lubri-cation characteristics of the gear are closer to the curve of119877119878 = 00 120583m At this moment the effect of shot peeningon lubrication characteristics is either positive or negativemainly depending on the final effect of shot peening on thesurface roughness of the metal surface

There is no doubt that themicrostructure of tooth surfaceformed by shot peening decreases the effects of oil pressureand oil film thickness caused by asperity compared with the


y=0

0

05

1

15

2Ｂ（

0

05-15 0 15minus05minus25 1minus2 minus1Ｒ＜Ｇ

S-P２Ｍ=00２Ｍ=16

２Ｍ=32

(a) 119910 = 0 section

x=0

01

02

03

04

05

06

07

Ｂ（

0


S-P２Ｍ=00２Ｍ=16

２Ｍ=32

(b) 119909 = 0 section

Figure 11 Oil film thickness distribution in meshing tooth surface of different surface roughnesses

y=0

Row PressureOptimized PressureRow Film icknessOptimized Film ickness


0

01

02

03

04

05

06

07

08

09

1

Ｊ０

0

0

05

1

15

2

25

Ｂ（

0

(a) 119910 = 0 section

x=0

Optimized PressureRaw PressureRaw Film icknessOptimized Film ickness

0

01

02

03

04

05

06

07

08

09

1

Ｊ０

0

0 05minus05 1minus1Ｒ＜Ｇ

0

01

02

03

04

05

06

07

08

09

1

Ｂ（

0

(b) 119909 = 0 section

Figure 12 The comparison between pressure distribution and thickness distribution

machined tooth surface The less the flow resistance is theeasier the oil film forms and thus it obtained a nice lubrica-tion condition In the same situation the lubrication charac-teristics of shot peened spiral bevel gear are better than thoseof the machined gears

Figure 12 is the contrastive images about oil pressure andoil film thickness distribution of original shot peening modeland optimized shot-penning model on the cross section119910 = 0 119909 = 0 The 49-pellet model is the shot peening

process before optimization while the multiple-pellet modelis the shot peening process after optimization As shownin Figure 12 because of those pits and ridges the asperitybecomes finer and the fluctuation of oil pressure and oil filmthickness is more apparent With increasing intensity of shotpeening the overall roughness reduces on the gear surfaceand the asperity effect weakens

As shown in Figure 12(a) the first peak value on the crosssection 119910 = 0 falls from 08 1198750 to 063 1198750 and the pressure


drops about 21 At the pressure collapse on account ofmoreserried asperity the pressure of the collapse increases to someextent from 065 1198750 to 07 1198750 Overall oil pressure is moreevenly distributed Meanwhile as shown in Figure 12(b) theoil film thickness of cross section 119910 = 0 has a remarkable in-crease and the overall improvement is 051198670

In the central contact area the whole level of oil filmthickness improves gathering more lubrication oil thereFrom the cross section 119909 = 0 the increase of asperities makesthe fluctuation of pressure greater but it is beneficial forthe oil film thickness The oil film thickness becomes moresmooth and uniform and the thickness of oil film increases aswell which is good for the improvement of lubrication per-formances

5 Conclusion

Shot peening affects the roughness value and microstructureof tooth surface for spiral bevel gear This paper establisheda reasonable FEA model that can describe the shot peenedtreatment of spiral bevel gear and obtained the coordinateof microstructure for tooth surface after shot peening whichreveals the elastohydrodynamic lubrication characteristics ofspiral bevel gears after shot peening process by numericalmethod especially for oil pressure and oil film thicknessdistribution in micro tooth surface during lubrication fordifferent roughness conditions This method can be used topredict the lubrication characteristics of spiral bevel gear atdifferent surface roughness after shot peening treatmentThesuitable shot parameters are favorable for lubrication the oilfilm thickness can be increased and the oil film thicknessvariation can be decreased The oil pressure between theteeth can be increased and the oil pressure variation can bedecreased For the super smooth surface of spiral bevel gearshot peening can not increase the lubrication of gear trans-mission

After shot peening the tooth surface topography of spiralbevel gear is reshaped and the dent has a stronger effect on thelubrication characteristics for meshing tooth surface of gearpair based on 49-pellet model Pits are more likely to storelubricating oil which is conducive to lubrication At the sametime the bulge formed around the orange peel pits hindersthe flowing of lubrication and thus increases gear wearing

It can be predicted that if the shot peening coverage isfurther improved the bulge formed around the orange peelpits will be weakened and then pits will become more finelydividedWhat is more themain factors that affect lubricationwill become the pit instead of the bulge around the pit Solubrication effect will be improved with the increase of cover-age

For rough or semi-finish process the shot peened toothsurface is helpful When the intensity of shot peening in-creases gradually tool marks on the surface can be coveredHowever adopting shot peening process to treat gears is notappropriate for finishing machining tooth surface becauseshot peening will only make the gear surface rougher andincreasing roughness does no good for lubrication perfor-mance Consequently grinding process is necessary after shotpeening

Nomenclature

119860 Total area of contact area119861 Face wide119887 Half-width of Hertz contact119862119908 Coefficient of load1198641015840 Equivalent modulus of a pair of gear teethmaterial1198670 Dimensionless film thickness of the rigidbody centerℎ0 Constant related to the film thickness ofthe rigid body center119871119887 Half-length of the contact line119898 Modulus of gear119899 Rotation rate119875 Pressure119877 Comprehensive curvature radius119877119886 Surface roughness119877119904 Amplitude of the cosine function asperity119877119898 Comprehensive curvature119878 Surface topography119878 Dimensionless tooth surface morphologydata119879119901 Torque119905 Time variable119881 Elastic deformation119883 Node coordinates119909in Initial point coordinate of the calculationregion119909out End point coordinate of the calculationregion119884 Node coordinates

z Teeth number119911 Viscosity index120572 Viscosity coefficient120575 Pitch angle120578 Lubricating oil viscosity1205780 Environmental viscosity120578lowast Equivalent viscosity of lubricating oil120583119909 Entrainment velocity of the tooth face120583119898 Entrainment velocity120591 Oil film shear stress1205910 Reference shear stress120596119899 Target teeth carried load of each meshingmoment120574 Shear rate120588 Density of the lubricating oil1205880 Initial densityΔℎ Dimensionless thickness of the geometrygap120576119901 Convergence criterion120576119908 Convergence criterion

Data Availability

The data used to support the findings of this study are avail-able from the corresponding author upon request


Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research is supported by Natural Science Foundation ofTianjin (no 17JCQNJC04300) Open Funding of State KeyLaboratory of Materials Processing and Die amp Mould Tech-nology-Huazhong University of Science and Technology(P2019-022) Open Funding of The State Key Laboratoryof Mechanical Transmissions (no SKLMT-KFKT-201616)Fundamental Research Funds for the Tianjin Universities(nos 2017KJ083 TJPUZK20170118) National Natural Sci-ence Foundation of China (nos 51475330 U1733108) Nation-al Training Program of Innovation and Entrepreneurship forUndergraduates (no 201710058085) Applied Basic ResearchProject of China Textile Industry Association (no J201806)and the Program for Innovative Research Team in Universityof Tianjin (no TD13-5037) Thanks also are due to ProfessorHaruo Houjoh of Tokyo Institute of Technology ProfessorIzhak Bucher of Israel Institute of Technology Professor YiduZhang of Beihang University Mr Zhou Yao and Ms WeiHong for their constant assistances

References

[1] L Dimitrov D Michalopoulos C A Apostolopoulos and TD Neshkov ldquoInvestigation of contact fatigue of high strengthsteel gears subjected to surface treatmentrdquo Journal of MaterialsEngineering and Performance vol 18 no 7 pp 939ndash946 2009

[2] L Guo S Zhou L Crocker andA Turnbull ldquoInitiation sites forcracks developed from pits in a shot-peened 12Cr martensiticstainless steelrdquo International Journal of Fatigue vol 98 pp 195ndash202 2017

[3] S Li and A Anisetti ldquoA tribo-dynamic contact fatigue modelfor spur gear pairsrdquo International Journal of Fatigue vol 98 pp81ndash91 2017

[4] C Zhou Z Xiao S Chen and X Han ldquoNormal and tangentialoil film stiffness of modified spur gear with non-Newtonianelastohydrodynamic lubricationrdquo Tribology International vol109 pp 319ndash327 2017

[5] D Gallitelli V Boyer M Gelineau et al ldquoSimulation of shotpeening from process parameters to residual stress fields in astructurerdquoComptes Rendus (Doklady) de lrsquoAcademie des Sciencesde lrsquoURSS vol 344 no 4-5 pp 355ndash374 2016

[6] Y Lv L Lei and L Sun ldquoEffect of microshot peened treatmenton the fatigue behavior of laser-melted W6Mo5Cr4V2 steelgearrdquo International Journal of Fatigue vol 98 pp 121ndash130 2017

[7] A Terrin C Dengo andGMeneghetti ldquoExperimental analysisof contact fatigue damage in case hardened gears for off-highway axlesrdquo Engineering Failure Analysis vol 76 pp 10ndash262017

[8] W Pu J Wang R Yang and D Zhu ldquoMixed elastohydrody-namic lubrication with three-dimensional machined roughnessin spiral bevel and hypoid gearsrdquo Journal of Tribology vol 137no 4 Article ID 041503 pp 1ndash11 2015

[9] C-B Tang D-X Liu B Tang X-H Zhang L Qin and C-S Liu ldquoInfluence of plasma molybdenizing and shot-peeningon fretting damage behavior of titanium alloyrdquo Applied SurfaceScience vol 390 pp 946ndash958 2016

[10] B AlMangour and J-M Yang ldquoImproving the surface qualityand mechanical properties by shot-peening of 17-4 stainlesssteel fabricated by additive manufacturingrdquoMaterials and Cor-rosion vol 110 pp 914ndash924 2016

[11] D Trauth J Stanke A Shirobokov P Mattfeld and F KlockeldquoAnalysis of the fluid pressure load capacity and coefficient offriction of an ellipticmachine hammer peened surface structurein hydrodynamic lubricationrdquo Production Engineering Researchand Development vol 10 no 6 pp 539ndash550 2016

[12] VV Simon ldquoOptimal toothmodifications in face-hobbed spiralbevel gears to reduce the influence of misalignments on elasto-hydrodynamic lubricationrdquo Journal of Mechanical Design vol136 no 7 Article ID 071007 pp 1ndash9 2014

[13] M Shuai Z Yidu and W Qiong ldquoResearch on multiple-splitload sharing of two-stage star gearing system in consideration ofdisplacement compatibilityrdquo Mechanism and Machine Theoryvol 88 pp 1ndash15 2015

[14] M Shuai and Z Yidu ldquoSpiral bevel gear true tooth surface pre-cise modeling and experiments studies based on machiningadjustment parametersrdquo Proceedings of the Institution of Me-chanical Engineers Part C Journal of Mechanical EngineeringScience vol 229 no 14 pp 2524ndash2533 2015

[15] S H YuanH L Dong andX Y Li ldquoAnalysis of lubricating per-formance for involute gear based on dynamic loading theoryrdquoJournal of Mechanical Design vol 134 no 12 Article ID 121004pp 67ndash75 2012

Hindawiwwwhindawicom Volume 2018

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Submit your manuscripts atwwwhindawicom


geometry into consideration which include an initial statefrom previous manufacturing processes [5] The proposedapproach is consistent with industrial constraints in terms ofcomputation time The perspective is to complete the chain-ing process with fatigue life computations Lv et al studiedW6Mo5Cr4V2 steel gear tooth flank by laser irradiationAfterwards the gear tooth flank was treated by shot peeningwith shot particles of differentmaterials and peening time [6]The experimental results showed that the hardness of gearsincreased and the surface roughness of gears decreased aftershot peening treatment In addition the residual stress statein the near-surface layer of shot peened gears was changedfrom tensile stress to compressive stress Terrin et al analyzedsix case hardened sun gears damaged by pitting duringendurance tests [7] All analyzed gears belonged to planetaryfinal drives placed on a wheel hub of axles for off-highwayvehicles The aim of the analysis was to highlight the keyaspects of the morphology and the evolution of pitting dam-age on case hardened sun gears Pu et al conducted a com-prehensive analysis for gearing geometry kinematics mixedlubrication performance and friction and interfacial flashtemperature in spiral bevel and hypoid gears [8] This studywas developed based on a recently established mixed elasto-hydrodynamic lubrication model that was capable for han-dling practical cases with 3D machined roughness undersevere operating conditions considering the effect of arbi-trary entrainment angle Obtained results from sample casesshowed that the developed simulationmodel could be used asan engineering tool for spiral bevel and hypoid gears designoptimization and strength prediction Tang et al explored theeffect of plasma molybdenizing and shot peening on frettingwear and fretting fatigue behaviors of Ti6Al4V alloy [9] Theresults displayed that a beneficial residual compressive stressdistribution high surface hardness with suitable hardnessgradient distribution good apparent toughness relativelylow surface roughness and excellent surface integrity wasachieved AlMangour and Yang made an attempt at inducinggrain refinements through a shot peening process creatingsevere plastic deformation at the outer surface layers toimprove the physical andmechanical properties of 17-4 stain-less steel components produced by DMLS [10] The researchresult highlighted the efficiency and applicability of shotpeening treatment to practical cases This contribution ana-lytically derives a solution to the Reynolds equation to de-scribe the longitudinal fluid pressure distribution load capac-ity and coefficient of a MHPed surface structure friction[11] It was shown that the lateral effects in machine hammerpeened surface structures significantly increased the fluidpressure Therefore the analytical approach presented couldbe used to estimate the lower bound Simon raised an opti-mization methodology systematically to define the optimaltooth modifications introduced by head-cutter geometry andmachine-tool settings to minimize the influence of mis-alignments on the elastohydrodynamic (EHD) lubricationcharacteristics in face-hobbed spiral bevel gears [12] Its goalis to maximize the EHD load-carrying capacity of the oil filmand to minimize power losses in the oil film when differentmisalignments were inherent in the gear pair Shuai et aldeveloped the spiral bevel gear true tooth surface precise

modeling method based on machining adjustment parame-ters and conducted an experiment to prove that themodelingmethod is right and precise At the same time a research instar gear transmission system was also conducted [13 14]

Many of these researches have important effects only onelastohydrodynamic lubrication or shot peening until nowbut few literatures about elastohydrodynamic lubricationcharacteristics focus on shot peening In addition some evenassume that shot peening exerts few effects on elastohydrody-namic lubrication characteristics for gear pair Heretofore tothe best knowledge of the authors few published studies areavailable on the elastohydrodynamic lubrication characteris-tics of shot peened spiral bevel gear Based on the elastohydro-dynamic lubrication theory the paper making a comparativeresearch of the tooth surface lubrication characteristics ofvarious microscopic morphologies reveals the elastohydro-dynamic lubrication characteristics of spiral bevel gears aftershot peening process

2 Microtopography Description ofTooth Surface by Shot Peening

Different from the roughness of tooth surface after machin-ing the shot peened tooth surface roughness has certainregularity When spiral bevel gear gets into the stage ofelastohydrodynamic lubrication the oil film thickness oflubricating between the two contact tooth surfaces is in thesame order of magnitude with roughness at the morphologi-cal peak of shot peened tooth surface The materials used forshot peening mainly include ceramics cast steel glass castiron and so on To avoid concentrated stress points the shapeof projectile is spherical The projectile comes in differentsizes among which the smallest diameter is 01mmwhile themaximum reaches 2mm The size and material of projectileshould be selected according to strengthening requirementsof the sprayedworkpiece the surface roughness of workpieceand the mechanical properties of workpiece and so on Thestrengthening effect of shot peening is related tomany factorsfrom thematerial properties ofworkpiece and the contact dis-tance between nozzle andworkpiece to the angle of incidenceAdditionally there are shot peening speed coverage materialtype hardness shot peening time projectile size and othershot peening factors Numerous process parameters influ-ence the shot peening effect and the coupling mechanismsbetween them are very complex

21 Construction of Multiple Projectilesrsquo Offset Shot PeeningModel At present the finite element model in ABAQUS ofshot peening strengthening is a multiple projectilesrsquo offsetmodel As shown in Figure 1 this paper establishes the 49-pellet offset shot peening strengthening simulation model infour layers with high precision The model assumes that theprojectile is arranged in accordance with a certain law andhas a bias distribution and the target surface is vertically hitwhich reduces the blind area and increases the coverage rate

On the basis of the bias modeling method radius of thefirst layer of projectiles is 119877 and the center distance is 2119877 Theoffset distance of the second layer relative to the first layer is119877 in the119883 direction and the offset distance of the third layer










minus4S(Z

Disp

lace

men

t (m

m))

2

0

minus2


4

minus2 minus1

times10minus4

Ｒ＜ＧＳ＜











(1)




(driving)

(driven)

t tC

FＨ

FＨ

2

1

2

＞Ｐ1

2

＞Ｐ1

2４1

1

1

2









10158401198891199101015840

minus 119878 (119909 119910)(3)

x

h(x)

R(x

) x22

R

ＢＩ






119862119908119908119899 = ∬119860

119901 (119909 119910) 119889119909 119889119910 (4)




120574 = 1205910120578 sinh( 1205911205910) (5)




1120578lowast = 1120578 1205910120591 sinh( 1205911205910) (7)





119883 = 119909119887119898 119867 = ℎ1198771198981198872119898 119875 = 119901119901119898 120588 = 1205881205880 120578 = 1205781205780 119896119899 = 119871119887119887119898 119905 = 119906119898119905119887119898

(9)


119887119898 = radic41199081198981198771198981205871198641015840119871119887 119901119898 = 119908119898120587119887119898119871119887

(10)


1119877 = 11198771 +11198772 (11)


11198641015840 = 05 (1 minus 120590211198641 + 1 minus 120590221198642 ) (12)











NO

YES


Cw CuM L bm pH


p w

Tp Ｈ1N








(13)


















y=0



0

05

1

15

2

25

Ｂ（

0

0

02

04

06

08

1Ｊ０

0

(a) 119910 = 0 section


x=0

0

02

04

06

08

1

Ｊ０

0

0

02

04

06

08

1

Ｂ（

0


(b) 119909 = 0 section


y=0

S-P


0

02

04

06

08

1

12

Ｊ０

0

２Ｍ=00２Ｍ=16

２Ｍ=32

(a) 119910 = 0 section

x=0


0

02

04

06

08

1

12

14

16

18

Ｊ０

0

S-P２Ｍ=00２Ｍ=16

２Ｍ=32

(b) 119909 = 0 section






y=0

0

05

1

15

2Ｂ（

0


S-P２Ｍ=00２Ｍ=16

２Ｍ=32

(a) 119910 = 0 section

x=0

01

02

03

04

05

06

07

Ｂ（

0


S-P２Ｍ=00２Ｍ=16

２Ｍ=32

(b) 119909 = 0 section


y=0



0

01

02

03

04

05

06

07

08

09

1

Ｊ０

0

0

05

1

15

2

25

Ｂ（

0

(a) 119910 = 0 section

x=0


0

01

02

03

04

05

06

07

08

09

1

Ｊ０

0


0

01

02

03

04

05

06

07

08

09

1

Ｂ（

0

(b) 119909 = 0 section









5 Conclusion





Nomenclature



Data Availability





Acknowledgments


References























Journal of












Journal of







Volume 2018






Volume 2018

















minus4S(Z

Disp

lace

men

t (m

m))

2

0

minus2


4

minus2 minus1

times10minus4

Ｒ＜ＧＳ＜











(1)




(driving)

(driven)

t tC

FＨ

FＨ

2

1

2

＞Ｐ1

2

＞Ｐ1

2４1

1

1

2









10158401198891199101015840

minus 119878 (119909 119910)(3)

x

h(x)

R(x

) x22

R

ＢＩ






119862119908119908119899 = ∬119860

119901 (119909 119910) 119889119909 119889119910 (4)




120574 = 1205910120578 sinh( 1205911205910) (5)




1120578lowast = 1120578 1205910120591 sinh( 1205911205910) (7)





119883 = 119909119887119898 119867 = ℎ1198771198981198872119898 119875 = 119901119901119898 120588 = 1205881205880 120578 = 1205781205780 119896119899 = 119871119887119887119898 119905 = 119906119898119905119887119898

(9)


119887119898 = radic41199081198981198771198981205871198641015840119871119887 119901119898 = 119908119898120587119887119898119871119887

(10)


1119877 = 11198771 +11198772 (11)


11198641015840 = 05 (1 minus 120590211198641 + 1 minus 120590221198642 ) (12)











NO

YES


Cw CuM L bm pH


p w

Tp Ｈ1N








(13)


















y=0



0

05

1

15

2

25

Ｂ（

0

0

02

04

06

08

1Ｊ０

0

(a) 119910 = 0 section


x=0

0

02

04

06

08

1

Ｊ０

0

0

02

04

06

08

1

Ｂ（

0


(b) 119909 = 0 section


y=0

S-P


0

02

04

06

08

1

12

Ｊ０

0

２Ｍ=00２Ｍ=16

２Ｍ=32

(a) 119910 = 0 section

x=0


0

02

04

06

08

1

12

14

16

18

Ｊ０

0

S-P２Ｍ=00２Ｍ=16

２Ｍ=32

(b) 119909 = 0 section






y=0

0

05

1

15

2Ｂ（

0


S-P２Ｍ=00２Ｍ=16

２Ｍ=32

(a) 119910 = 0 section

x=0

01

02

03

04

05

06

07

Ｂ（

0


S-P２Ｍ=00２Ｍ=16

２Ｍ=32

(b) 119909 = 0 section


y=0



0

01

02

03

04

05

06

07

08

09

1

Ｊ０

0

0

05

1

15

2

25

Ｂ（

0

(a) 119910 = 0 section

x=0


0

01

02

03

04

05

06

07

08

09

1

Ｊ０

0


0

01

02

03

04

05

06

07

08

09

1

Ｂ（

0

(b) 119909 = 0 section









5 Conclusion





Nomenclature



Data Availability





Acknowledgments


References























Journal of












Journal of







Volume 2018






Volume 2018









minus4S(Z

Disp

lace

men

t (m

m))

2

0

minus2


4

minus2 minus1

times10minus4

Ｒ＜ＧＳ＜











(1)




(driving)

(driven)

t tC

FＨ

FＨ

2

1

2

＞Ｐ1

2

＞Ｐ1

2４1

1

1

2









10158401198891199101015840

minus 119878 (119909 119910)(3)

x

h(x)

R(x

) x22

R

ＢＩ






119862119908119908119899 = ∬119860

119901 (119909 119910) 119889119909 119889119910 (4)




120574 = 1205910120578 sinh( 1205911205910) (5)




1120578lowast = 1120578 1205910120591 sinh( 1205911205910) (7)





119883 = 119909119887119898 119867 = ℎ1198771198981198872119898 119875 = 119901119901119898 120588 = 1205881205880 120578 = 1205781205780 119896119899 = 119871119887119887119898 119905 = 119906119898119905119887119898

(9)


119887119898 = radic41199081198981198771198981205871198641015840119871119887 119901119898 = 119908119898120587119887119898119871119887

(10)


1119877 = 11198771 +11198772 (11)


11198641015840 = 05 (1 minus 120590211198641 + 1 minus 120590221198642 ) (12)











NO

YES


Cw CuM L bm pH


p w

Tp Ｈ1N








(13)


















y=0



0

05

1

15

2

25

Ｂ（

0

0

02

04

06

08

1Ｊ０

0

(a) 119910 = 0 section


x=0

0

02

04

06

08

1

Ｊ０

0

0

02

04

06

08

1

Ｂ（

0


(b) 119909 = 0 section


y=0

S-P


0

02

04

06

08

1

12

Ｊ０

0

２Ｍ=00２Ｍ=16

２Ｍ=32

(a) 119910 = 0 section

x=0


0

02

04

06

08

1

12

14

16

18

Ｊ０

0

S-P２Ｍ=00２Ｍ=16

２Ｍ=32

(b) 119909 = 0 section






y=0

0

05

1

15

2Ｂ（

0


S-P２Ｍ=00２Ｍ=16

２Ｍ=32

(a) 119910 = 0 section

x=0

01

02

03

04

05

06

07

Ｂ（

0


S-P２Ｍ=00２Ｍ=16

２Ｍ=32

(b) 119909 = 0 section


y=0



0

01

02

03

04

05

06

07

08

09

1

Ｊ０

0

0

05

1

15

2

25

Ｂ（

0

(a) 119910 = 0 section

x=0


0

01

02

03

04

05

06

07

08

09

1

Ｊ０

0


0

01

02

03

04

05

06

07

08

09

1

Ｂ（

0

(b) 119909 = 0 section









5 Conclusion





Nomenclature



Data Availability





Acknowledgments


References























Journal of












Journal of







Volume 2018






Volume 2018









(driving)

(driven)

t tC

FＨ

FＨ

2

1

2

＞Ｐ1

2

＞Ｐ1

2４1

1

1

2









10158401198891199101015840

minus 119878 (119909 119910)(3)

x

h(x)

R(x

) x22

R

ＢＩ






119862119908119908119899 = ∬119860

119901 (119909 119910) 119889119909 119889119910 (4)




120574 = 1205910120578 sinh( 1205911205910) (5)




1120578lowast = 1120578 1205910120591 sinh( 1205911205910) (7)





119883 = 119909119887119898 119867 = ℎ1198771198981198872119898 119875 = 119901119901119898 120588 = 1205881205880 120578 = 1205781205780 119896119899 = 119871119887119887119898 119905 = 119906119898119905119887119898

(9)


119887119898 = radic41199081198981198771198981205871198641015840119871119887 119901119898 = 119908119898120587119887119898119871119887

(10)


1119877 = 11198771 +11198772 (11)


11198641015840 = 05 (1 minus 120590211198641 + 1 minus 120590221198642 ) (12)











NO

YES


Cw CuM L bm pH


p w

Tp Ｈ1N








(13)


















y=0



0

05

1

15

2

25

Ｂ（

0

0

02

04

06

08

1Ｊ０

0

(a) 119910 = 0 section


x=0

0

02

04

06

08

1

Ｊ０

0

0

02

04

06

08

1

Ｂ（

0


(b) 119909 = 0 section


y=0

S-P


0

02

04

06

08

1

12

Ｊ０

0

２Ｍ=00２Ｍ=16

２Ｍ=32

(a) 119910 = 0 section

x=0


0

02

04

06

08

1

12

14

16

18

Ｊ０

0

S-P２Ｍ=00２Ｍ=16

２Ｍ=32

(b) 119909 = 0 section






y=0

0

05

1

15

2Ｂ（

0


S-P２Ｍ=00２Ｍ=16

２Ｍ=32

(a) 119910 = 0 section

x=0

01

02

03

04

05

06

07

Ｂ（

0


S-P２Ｍ=00２Ｍ=16

２Ｍ=32

(b) 119909 = 0 section


y=0



0

01

02

03

04

05

06

07

08

09

1

Ｊ０

0

0

05

1

15

2

25

Ｂ（

0

(a) 119910 = 0 section

x=0


0

01

02

03

04

05

06

07

08

09

1

Ｊ０

0


0

01

02

03

04

05

06

07

08

09

1

Ｂ（

0

(b) 119909 = 0 section









5 Conclusion





Nomenclature



Data Availability





Acknowledgments


References























Journal of












Journal of







Volume 2018






Volume 2018










119862119908119908119899 = ∬119860

119901 (119909 119910) 119889119909 119889119910 (4)




120574 = 1205910120578 sinh( 1205911205910) (5)




1120578lowast = 1120578 1205910120591 sinh( 1205911205910) (7)





119883 = 119909119887119898 119867 = ℎ1198771198981198872119898 119875 = 119901119901119898 120588 = 1205881205880 120578 = 1205781205780 119896119899 = 119871119887119887119898 119905 = 119906119898119905119887119898

(9)


119887119898 = radic41199081198981198771198981205871198641015840119871119887 119901119898 = 119908119898120587119887119898119871119887

(10)


1119877 = 11198771 +11198772 (11)


11198641015840 = 05 (1 minus 120590211198641 + 1 minus 120590221198642 ) (12)











NO

YES


Cw CuM L bm pH


p w

Tp Ｈ1N








(13)


















y=0



0

05

1

15

2

25

Ｂ（

0

0

02

04

06

08

1Ｊ０

0

(a) 119910 = 0 section


x=0

0

02

04

06

08

1

Ｊ０

0

0

02

04

06

08

1

Ｂ（

0


(b) 119909 = 0 section


y=0

S-P


0

02

04

06

08

1

12

Ｊ０

0

２Ｍ=00２Ｍ=16

２Ｍ=32

(a) 119910 = 0 section

x=0


0

02

04

06

08

1

12

14

16

18

Ｊ０

0

S-P２Ｍ=00２Ｍ=16

２Ｍ=32

(b) 119909 = 0 section






y=0

0

05

1

15

2Ｂ（

0


S-P２Ｍ=00２Ｍ=16

２Ｍ=32

(a) 119910 = 0 section

x=0

01

02

03

04

05

06

07

Ｂ（

0


S-P２Ｍ=00２Ｍ=16

２Ｍ=32

(b) 119909 = 0 section


y=0



0

01

02

03

04

05

06

07

08

09

1

Ｊ０

0

0

05

1

15

2

25

Ｂ（

0

(a) 119910 = 0 section

x=0


0

01

02

03

04

05

06

07

08

09

1

Ｊ０

0


0

01

02

03

04

05

06

07

08

09

1

Ｂ（

0

(b) 119909 = 0 section









5 Conclusion





Nomenclature



Data Availability





Acknowledgments


References























Journal of












Journal of







Volume 2018






Volume 2018

















NO

YES


Cw CuM L bm pH


p w

Tp Ｈ1N








(13)


















y=0



0

05

1

15

2

25

Ｂ（

0

0

02

04

06

08

1Ｊ０

0

(a) 119910 = 0 section


x=0

0

02

04

06

08

1

Ｊ０

0

0

02

04

06

08

1

Ｂ（

0


(b) 119909 = 0 section


y=0

S-P


0

02

04

06

08

1

12

Ｊ０

0

２Ｍ=00２Ｍ=16

２Ｍ=32

(a) 119910 = 0 section

x=0


0

02

04

06

08

1

12

14

16

18

Ｊ０

0

S-P２Ｍ=00２Ｍ=16

２Ｍ=32

(b) 119909 = 0 section






y=0

0

05

1

15

2Ｂ（

0


S-P２Ｍ=00２Ｍ=16

２Ｍ=32

(a) 119910 = 0 section

x=0

01

02

03

04

05

06

07

Ｂ（

0


S-P２Ｍ=00２Ｍ=16

２Ｍ=32

(b) 119909 = 0 section


y=0



0

01

02

03

04

05

06

07

08

09

1

Ｊ０

0

0

05

1

15

2

25

Ｂ（

0

(a) 119910 = 0 section

x=0


0

01

02

03

04

05

06

07

08

09

1

Ｊ０

0


0

01

02

03

04

05

06

07

08

09

1

Ｂ（

0

(b) 119909 = 0 section









5 Conclusion





Nomenclature



Data Availability





Acknowledgments


References























Journal of












Journal of







Volume 2018






Volume 2018






















y=0



0

05

1

15

2

25

Ｂ（

0

0

02

04

06

08

1Ｊ０

0

(a) 119910 = 0 section


x=0

0

02

04

06

08

1

Ｊ０

0

0

02

04

06

08

1

Ｂ（

0


(b) 119909 = 0 section


y=0

S-P


0

02

04

06

08

1

12

Ｊ０

0

２Ｍ=00２Ｍ=16

２Ｍ=32

(a) 119910 = 0 section

x=0


0

02

04

06

08

1

12

14

16

18

Ｊ０

0

S-P２Ｍ=00２Ｍ=16

２Ｍ=32

(b) 119909 = 0 section






y=0

0

05

1

15

2Ｂ（

0


S-P２Ｍ=00２Ｍ=16

２Ｍ=32

(a) 119910 = 0 section

x=0

01

02

03

04

05

06

07

Ｂ（

0


S-P２Ｍ=00２Ｍ=16

２Ｍ=32

(b) 119909 = 0 section


y=0



0

01

02

03

04

05

06

07

08

09

1

Ｊ０

0

0

05

1

15

2

25

Ｂ（

0

(a) 119910 = 0 section

x=0


0

01

02

03

04

05

06

07

08

09

1

Ｊ０

0


0

01

02

03

04

05

06

07

08

09

1

Ｂ（

0

(b) 119909 = 0 section









5 Conclusion





Nomenclature



Data Availability





Acknowledgments


References























Journal of












Journal of







Volume 2018






Volume 2018









y=0



0

05

1

15

2

25

Ｂ（

0

0

02

04

06

08

1Ｊ０

0

(a) 119910 = 0 section


x=0

0

02

04

06

08

1

Ｊ０

0

0

02

04

06

08

1

Ｂ（

0


(b) 119909 = 0 section


y=0

S-P


0

02

04

06

08

1

12

Ｊ０

0

２Ｍ=00２Ｍ=16

２Ｍ=32

(a) 119910 = 0 section

x=0


0

02

04

06

08

1

12

14

16

18

Ｊ０

0

S-P２Ｍ=00２Ｍ=16

２Ｍ=32

(b) 119909 = 0 section






y=0

0

05

1

15

2Ｂ（

0


S-P２Ｍ=00２Ｍ=16

２Ｍ=32

(a) 119910 = 0 section

x=0

01

02

03

04

05

06

07

Ｂ（

0


S-P２Ｍ=00２Ｍ=16

２Ｍ=32

(b) 119909 = 0 section


y=0



0

01

02

03

04

05

06

07

08

09

1

Ｊ０

0

0

05

1

15

2

25

Ｂ（

0

(a) 119910 = 0 section

x=0


0

01

02

03

04

05

06

07

08

09

1

Ｊ０

0


0

01

02

03

04

05

06

07

08

09

1

Ｂ（

0

(b) 119909 = 0 section









5 Conclusion





Nomenclature



Data Availability





Acknowledgments


References























Journal of












Journal of







Volume 2018






Volume 2018









y=0

0

05

1

15

2Ｂ（

0


S-P２Ｍ=00２Ｍ=16

２Ｍ=32

(a) 119910 = 0 section

x=0

01

02

03

04

05

06

07

Ｂ（

0


S-P２Ｍ=00２Ｍ=16

２Ｍ=32

(b) 119909 = 0 section


y=0



0

01

02

03

04

05

06

07

08

09

1

Ｊ０

0

0

05

1

15

2

25

Ｂ（

0

(a) 119910 = 0 section

x=0


0

01

02

03

04

05

06

07

08

09

1

Ｊ０

0


0

01

02

03

04

05

06

07

08

09

1

Ｂ（

0

(b) 119909 = 0 section









5 Conclusion





Nomenclature



Data Availability





Acknowledgments


References























Journal of












Journal of







Volume 2018






Volume 2018











5 Conclusion





Nomenclature



Data Availability





Acknowledgments


References























Journal of












Journal of







Volume 2018






Volume 2018











Acknowledgments


References























Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








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