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A Numerical Model to Calculate Elastohydrodynamic (EHL) Properties in Involute Spur Gears


20IO International Conference on Mechanical and Electrical Technology (ICMET 2010)

A

Numerical Model to Calculate Elastohydrodynamic (EHL) Properties in Involute Spur Ge

ars
dept. mech. eng Urmia University Urmia, Iran Sh.khalilarya@urmia.ac.ir dept. mech. eng Urmia University Urmia, Iran enpim@yahoo.com

Shahram Khalilarya

dept. mech. eng Urmia Univ. of Technology Urmia, Iran measmm@uut.ac.ir

Mahdi Mohammadpour

Iraj Mirzaee

Davood Jalali Vahid Sahand Univ. of Tech. Tabriz, Iran davoodjalali@yahoo.co.uk
lubrication developed newly in compare with other lubrication methods. Recently, many studies have been done on EHD theory and its applications. After development of EHD theory, presentation of numerical solution for line and point contacts followed by Dowson and Whitaker [1] in 1966, ranger [2] in 1974, Tapley [3] in 1976, evans and sindle [4] in 1982, Carlson [5] in 1984 and Mostofi and Gohar [6] in 1984. Generally two types of EHD contacts may be composed in applied cases, line and point contacts. When the intersection of contacting bodies is a thin rectangle it assumed as a line and named line contact and if the intersect be a circular area it can be considered as a point and named point contact. In gear applications, a line contact can be considered and investigating the composed film properties along the path of contact (POC) is very important. To get such properties, the inputs of EHD models are required for any position along POCo The main input parameters of EHD models is load, relative velocities of teeth, sliding velocity, radius of curvature of contacting surfaces and lubricant properties. All of them can be calculated using geometrical relationships of gear profiles because of their standard geometries. Only load distribution requires running numerical models to calculation and application of it. In gear design, the calculation of load distribution along the path of contact is very important. This distribution obtained simultaneously with transmission error and contact stiffuess. A finite element analysis can be alternative method to get this error in any point of contact through the line of contact. Litvin et al [7] in 1996 and Lundvall and Klarbring [8] in 200 1 have presented a full fmite element based method to calculate the load share, tooth deformation and real contact ratio. He et al [9] in 2008 used a single? degree-of-freedom, linear time-varying system model and then developed it to a six-degree-of-freedom model to get the load distribution and tooth deflection and dynamic TE. He and Singh [ 10] in 2008 presented a method to get the Dynamic TE using one-degree-of-freedom stiffuess function. Tamminana et al [ 1 1] in 2007 have used a finite element method to get the relationship between Dynamic factor and transmission error. But, the time consuming and difficulty to give the correct contact condition, beside the dept. mech. eng

Abstract- This paper presents a modified numerical model for

Elastohydrodynamic lubrication (EHL) calculations of involute spur gears. This model considers the fluid as a Newtonian fluid. The main study performs on gear EHL properties along path of contact thickness and maximum pressure value as well complete film geometry and pressure distribution could be obtained for various applicable purposes. instead of This study also uses a full approximation or numerical model for both gear load calculations and EHL calculations experimental, analytical-experimental equations. Numerical method lets the model to calculate all required data without any limitations about the validity rang of In experimental order and analytical? the process experimental equations. to explain

(POC).

Using this study, the minimum film

completely, paper firstly reviews a mathematical model to calculate the load distribution, single contact stiffness and meshing stiffness, time. the Then a transmission error and path of contact method is presented to to solve for the any under load without using finite element method and in shortest numerical This model mathematical model using a double iteration flowchart to close problem. flexible adapt modification in spur gear profile geometry. In addition to the gear load model, a modified and accelerated EHL numerical method reviewed to be utilized in this paper. The model for load distribution presents a realistic calculation for current study and modified EHL model helps to do calculations more exactly in shortest time. Finally the results of full model (that utilizes the two old models) will discuss and compared with previous methods.
Keywords-Design Theory and Methodology, Gear Design, Lubrication, Elasto-Hydro-Dynamics

INTRODUCTION Lubrication of gears is the most important field of gears design that affects the design quality significantly. Lubrication importance is because of the great effect of composed oil film between teeth on gear wear, transferring of generated heat between teeth and damping of the impact effects. Lubrication method of gears is Elasto-Hydro? Dynamics (EHD). Such lubrication can be considered anywhere that a hydrodynamic pressure distribution composed because of geometrical properties and nature of lubricated surfaces and this pressure cause to elastic deformation of surfaces. Investigation of this branch of
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20iO international Conference on Mechanical and Electrical Technology (ICMET 20iO)

difficulty of solve a finite element method (FEM) in iteration model, give sufficient reasons to develop a Non? FEM to calculate the TE in simpler and fast way. These methods usually require a method to calculate the contact stiffuess and load distribution. Many works have been done about calculations of load distribution and contact stiffuess (are in relation with each other). Church [ 12] in 1957 has presented a torsion model to calculate the meshing stiffuess. This model has been known as Holzer model and is very simplified and has considered only the stiffuess of tooth. Nestorides [ 13] in 1958 has presented a model that describes the tooth as a beam and is very simplified and rough to get the exact results. Tobe and Takatsu in 1973 [14] have presented another model that is simpler, but is very similar to Nestorides method. Spotts [ 15] presented a similar method that is simpler than previous models. Hoechst AG [ 16] presented a mathematical model that was different from previous models and was specialist for polymer gears. Some experimental parameters have used in this model. A parallel experimental model presented with Marchek [ 17] for plastic gears. All of these models considered the gear body rigid and many effects neglected in these models. A more perfect model is Peterson [ 18] model to get the contact stiffuess as the start point of calculations. In this model, the teeth normal compression has been neglected and the surface compression follows the hertz method. The first section of this paper, firstly reviews a mathematical model to get the load distribution and meshing stiffuess along the path of contact. Such model presented previously by Mohammadpour et al. [19]. This model is based on a statically un-determined problem between two gears and requires some assumptions. Teeth normal compression considered in this paper and teeth surface compression solved using a numerical method that similar models are usual in Elasto-Hydro-Dynamic Numerical models. This case gives the opportunity to couple this model to an EHD model or used the Hydrodynamic pressure distribution to solve the surface normal compression. Then, a numerical method will be reviewed [ 19] because of iteration requirements to solve the mathematical model completely. The model requires solving by double iteration because of double relation between the problem variations (load? stiffuess and load-position). After calculation of input data of EHD model, the EHD numerical model can be used for any position along POC to calculate required information of oil film geometry as minimum film thickness, maximum value of pressure distribution, complete film geometry and pressure distribution and temperature of any point of contact. Because of great time that the numerical EHD model requires to get the EHD properties, a modified numerical model must be used to get the demanded properties in shortest time and high accuracy for any point of POCo Such modifications have been presented by Mohammadpour et al [20]. These modifications can be used for any protocol of EHD software. For example, Newtonian and Non? Newtonian model may be used with these modifications or such modifications may be used for rough or smooth

surfaces. In the second section of this paper, an adapted model of this work for gear application reviewed. Finally a full numerical model that used previously discussed gear load model and modified and accelerated EHD model presented in this paper to calculate all parameters of oil film in Spur gears. The resultant values for all point along the POC compose the film properties charts. A comparison between old methods and presented model in results and between modified and Non-modified numerical model will be illustrated. II GEAR LOAD MODEL

Transmission error is the deflection of a pair of gear and gear body along path of contact or equivalent rotational angle of it. This causes to a difference between unloaded (un-deformed) position of contact and loaded position. In fact, TE is the reason of differences between path of contact and path of contact under load. TE is an important factor in gear design because of its great effect on gear noise. TE can be calculated using finite element method (FEM), but this method is time consumer and difficult to consider correct boundary condition. In gear design, a simplified and accurate method can be developed for calculation of load sharing, contact stiffuess and TE because the gear geometry is known and standard. The method that has been used in this paper is based on works of Mohammadpour et al [ 19]. It is reviewed here to explain the process obviously.

To develop a mathematical model, we must consider a statically un-determined problem in any position of contact. According to some assumptions, this problem will change to a fully defined problem. The method that used in this paper is energy conservation and CASTIGLIANO method [ 19]. According to any position, a single or double contact must be considered and then energy equations must be written. To derivation of the energy equations, we must consider all matters that cause to deformation in a pair of gear. These matters are illustrated in Fig.I. Four of these effects can be written as equations in Table I. For gear tooth normal compression, when a pair of teeth is in contact, the contacting interface is a long and thin rectangle that can be assumed a line. It named line contact. To get the deformation or stiffuess of the gear contact surfaces, the contacting area (the rectangle area) dimensions must be known in this method. To calculate the dimensions,
Tooth normal compression

A. Mathematical model

Tooth Bending and shearing

\
Gear body tiHing

Surface normal compression

I

\

Figure 1. Schematic view of gear deformation causes

232

2010 International Conference on Mechanical and Electrical Technology (ICMET 2010)

TABLE 1: EQUATIONS FOR CALCULATION OF GEAR DEFORMATION.

Deflection type Gear tooth bending

Mathematical equation

Gear tooth shearing

In application, a discretised form of this equation will be used. To get the result, the contacting area must be divided in to several meshes. To get the deflection of any point, effect of all of the pressure elements must be considered. This method to get the deflection is usual in Elastohydrodynamic calculations. In complete mathematical model of this paper, the maximum value of these calculations is considered as the contacting compression deflection.
In

this paper, it is namedDj.

To obtain the complete energy of any tooth, the energy equations must be calculated and summed for the entire domain. The deflection of each tooth can be calculated using Castigliano method as the following equation shows:

Gear tooth normal compression

BEL BE2 IJ BE3 DJ + _+ + BPtij BPtij BPnij 2 (5) Considering the fact that Pt. and Pn. are functions of ?. , J de/;
lj
= __

1J

__

__

1J

_

therefore one can write:

J

de/ij = de/ij (?)

(6)

Gear body tilting

This method requires some assumptions to convert the problem to a fully defined statistical problem, as the loads are unknown. The assumptions are stated as follows:

1.
the applied load is required. Therefore, load and stiffuess are function of each other and must be calculated simultaneously. The contacting area dimensions can be calculated according to following equations [21].

The total deflection of root of any pair of tooth in contact is the sum of deflection of both of teeth. Based on this assumption, following relations can be derived.

length = Min(bi) width
=

de/rotalj (?) = I defu (?)
2.

(1 ) (2 )

(7)

length x Er

After calculation of the contacting area dimensions, an elliptic pressure distribution must be considered to get the deflection and related stitfuess. The pressure boundary is the contact boundaries and the ellipse maximum value can be calculated using following equation [21]:

The deflections of root of two pair of teeth are equal. This assumption means that the deflection gradient of body along the angular direction of gear body is zero. Because of this assumption, following equation can be obtained.

3.

The normal load to the profile is the sum of the loads exerted on each individual tooth in contact. The resultant equation is:
j

wJ =!

7r R}ength

(3)

(9)

This elliptic pressure must be used to get the deformation of each point in coordinates x, y based on following equation [22].

Using (8) and (9), the load in any pair can be obtained. Using a reverse method, the deflection and stiffuess can be obtained using following equations.

1 PY1)dxldYl D{x, Y)=_Er H14? P{X _ {X XJ2+(Y _ yJ2

(4)

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2010 International Conference on Mechanical and Electrical Technology (ICMET 2010)

?=

de/latal2 p dej/olal1 + de/latal2
? = p - ? (11)

(10) U(X)
=

(12)
V Xc osrtJ

{

(_1 __1 )
1'1 " 1 R"2 Rr.1

2SinrtJ

Consider that when the contact point is between LPSTC and HPSTC points, the load is carried by means of a single pair otherwise; the load must be calculated using previous equations. To obtain the meshing stiffuess, the model illustrated in Fig. 2-a. must be used. In this model, the stiffuess between LPSTC and HPSTC points is considered as single contact and in out of this section, Fig 2-b must be used. Note that in single contact stiffuess and meshing stiffuess calculations, the gear body stiffuess must be considered as well.
A.

hU(X)
IV

=

VXc o s rtJ

( ? __1 )
l'l1>2

}
(14)

(13)

MODIFIED EHD MODEL TO CALCULATE OIL FILM PARAMETERS ALONG THE PATH OF CONTACT

To calculate the outputs of this model, a numerical computational method must be used. III OTHER PARAMETERS THAT ARE REQUIRED TO RUN THE EHD MODEL

Numerical model

run

In addition to the load, some parameters are required to the EHD model. These parameters are local tooth surface radii of curvature, relative velocities of surfaces and sliding velocities (velocities differences) in contacting point. These parameters can be obtained using gears geometries and can be calculated using following equations [2 1].
\ Bas

An EHD model generally must solve two main domains simultaneously to get the required data. These domains are fluid domain to get the pressure distribution and elastic solid domains to get the body's deformation under calculated Hydrodynamic pressure. Using a perfect numerical model, the film thickness can be calculated as the final result. In gear applications, Newtonian model can be considered. In this paper, a modified and accelerated model utilized to get required results in acceptable time. Such modification presented previously [20]. To explain this model more clearly, the general properties of this model and its modifications presented below. A.

Stiffness OfTooth Pair

Stiffness OfQ?il.!:'?Rll.l; ,&,.,v .

? ? ?
I

This model uses a finite difference Newtonian model to obtain pressure distribution. It is the Reynolds equation that after some manipulations has been utilized in this model. In computational process, a normalized model has been used to get required data. Half-Somerfield boundary condition has been assumed in this model. This boundary considered expressed as ( 15) [20].
x

Fluidflow Model

tiffness of Gear Body

I

B. Boundary Condition

Stiffness ofTooth Pair 2

y ?CI),p ? 0
Figure 2-a. stiffness model for double pair

?CI),p ? 0

( 15)

Base Circle of Gear

Stiffness of Tooth Pair Stiffness of Gear Body 2

Stiffness of Gear Body

I

This boundary is usual in EHD (Elasto-Hydro-Dynamic) line contacts. Physical infmity (boundary domain) has been considered 4 times greater than contact dimension along rolling direction and parallel with contact actual boundary. In direction along the assumed line of contact, the boundary assumed equal to the actual line dimensions. Contact dimensions firstly calculate using elastostatic (Hertz base) equations and then modified using fmalized and converged EHD results.
A.

Base Circle of Gear 2

Figure 2-b. stiffness model for single pair

( 16) [2 1] has been used to achieve value of film thickness in per node. The parameters of this equation

Fluid Film Model

234

2010 International Conference on Mechanical and Electrical Technology (ICMET 2010)

illustrated in Fig. 3. This thickness and pressure distribution must iterate in an iteration loop to converge.

h (x,y)= h cD +h u(x,y)+w(x,y)

( 16)

The first term is minimum film thickness. The second term in right hand side is the non-deformed shape of bodies. In this model, it is assumed that this shape is quadratic. The third term of ( 16) is elastic deformation under calculated pressure distribution. The deflection can be obtained using (4) [22]: To start the computational process, initial estimations of film properties are required. Current model uses Elasto? Static method to have realistic initial guesses. Minimum Film thickness is required to start calculations too. This value is calculated using Grubin equation [23].

B. Elastic Deformation Model

C. initial Estimation

A complete model for calculations has been presented in [22].
A.

D. Complete Numerical Model Model modifications

c) We can cut the second loop before third loop and the mInImum film thickness and pressure distribution converged simultaneously. In fact, pre converging of minimum film thickness doesn't carry out and it converged using any new pressure distribution. Generally, when the load loop is before than pressure loop, calculations consume less time. In addition to, pre? convergence of load help the calculation to complete faster. Because usually it is independent of pressure distribution and will not repeated inside pressure loop. d) As another modification, a new iteration method presented to get the converged film thickness in the shortest time. Changes of load error against iteration are mainly linear when we adjust the minimum film thickness by uniform steps (equal amount of minimum film thickness). Therefore, we can estimate the required amount of modification for minimum film thickness. In fact, this error is in deal with our unrealistic estimation of minimum film thickness in any problem condition. For this purpose, firstly a little modification must be made in minimum film thickness. Then, the slope of changes in load error against iteration (or minimum film uniform modifications) could be calculated. Considering the whole load error and obtained slope, the modification amount can be obtained using following equation.
ho new - ho

a) As it described [20], firstly the minimum film thickness must be converged and then the pressure distribution as an initial estimation will converged. In [ 19], firstly the effect of changes in this arrangement investigated. In fact, change of load and pressure loop position will be discussed. b) As second discussion, as it illustrated in [20], firstly the film thickness converged and the fmalized minimum film thickness used for converging the pressure distribution. Inside pressure loop, the minimum thickness will checked in any iteration (integrated load loop). When the minimum film thickness converged, usually it isn't required to be iterated inside pressure loop iterations. In fact, the minimum film thickness is independent of pressure distribution and only is dependent on pressure integration (applied load). This result is not valid when pressure converged firstly. In fact, pressure distribution is not independent of minimum film thickness. Therefore, the integrated pressure loop will carry out in per load iteration.
z .I
I

_

omJ) old [(Errorc Slope
x

I - -'----'-'-

Usually, our demanded modification can't be done completely. This is because of non- complete linearity between load error and iteration. It is better that we modify about 98% of whole load error and then iterate remain amount using little and uniform steps. In this case, it is important to select steps amount to avoid of exceed of load error tolerance. e) Considering the grid independency in minimum film thickness, as the final modification, we can run this model firstly using a coarse grid. Then, the model can be started using fine grid and results of coarse grid as realistic initial guess. V RESULTS OF COMPLETE NUMERICAL MODEL TO CALCULATE EHD PROPERTIES AND LOAD SIMULTANEOUSLY As it said, to calculate film properties along POC a full numerical model is required to get demanded parameters. For this purpose, it is required to consider some points along POC to do the numerical process on them. These points are rolling steps of meshed gears on POCo After considering such positions, using ( 12), ( 13) and ( 14) required geometrical and kinematical parameters could be calculated for any position. To investigate the outputs of model, a sample problem considered to use in this model. The problem conditions are illustrated in table 2.

--+-?"""" "-...1 " 7
,.,. o ..c

01 ?

: ./,1

.-{

.,.'

...... . Undistorted .. paranoia

\

,I

.I

I

x
Figure 3. Film thickness parameters

235

2010 International Conference on Mechanical and Electrical Technology (ICMET 2010)

Number of teeth in gear 1 Number of teeth in gear 2 Center distance load Rotational velosity Pressure angle

TABLE2· SAMPLE PROBLEM CONDITION

20 50 100 mm 3000 N 300 rad/s 20 0

The load value that a pair must carry on any point can be obtained using presented numerical model. Complete outputs of EHD model can be used for various purposes as dynamic calculations of gears contact as a damping effect or determine required gear surface quality. Presented EHD model gives required outputs in shortest time using explained modifications. Without such modifications, the model consumes a large time and cost because of points number along POc. To investigate differences between outputs of Grubin Equation and presented numerical model, a comparison illustrated in Fig.4. In these two models, the load distribution that calculated using presented model in [19] has been used. Beside EHD properties, the surface stresses can be calculated more accurately using this model. These stresses are related with local contact pressure that usually considered as Elasto? Static (ES) maximum pressure of contact [21] and then maybe modified to set for oil film conditions. Such contacts assumed without oil film between contacting bodies. Using EHL pressure distribution on any contact point a more realistic contact pressure can be obtained. Fig.5 gives the differences between calculated values of contact pressure using ES method and EHL method that presented here.
VI

calculation consumes more time in compare with little differences (because the initial estimation is the old method and requires more iterations). But using presented method, all conditions have been done in approximately same time because of optimized methods. POC is divided in to two regimes. In the first regime, the numerical model gives thinner film thickness and in second regime, the old method results are littler. d) Using the results of numerical EHL model, the realistic contact pressure can be obtained too. This result illustrated in Fig.5. This figure shows that in EHL contacts the contact pressure is low in compare with ES method. NOMENCLATURE
Number of gear (1, 2)

2)

j

Number of tooth in contact (forwarded tooth is 1 and the other one is

Re

Pt j bi Thi

i}

Contact radius of gear i and tooth j Tangential force of toothj (load shared between tooth 1 and 2) Tooth wide of gear i

(r) 7rMn 2
=

X

?
R bi

-

2r(Inv(a))

Tooth thickness

12110
"

I.

SUMMARY AND DISCUSSION

...... Presented numerical method ....... Grubin qualk>n

a)

b)

c)

To get accurate and realistic results of EHL properties for gear contacts along POC, a good calculation method for load distribution is required. This load distribution must consider all effective causes of gear deformation to get the exact and accurate results. Such model presented in [19] and reviewed here. To do the EHL calculations, the time saving is important because such calculations are time consumer. Such method can be obtained using modified and optimized numerical model that presented in [20]. Using the presented EHL numerical model instead of experimental or analytical model gives more realistic results. This calculation can be performed in an acceptable time using modified model. Related results illustrated in Fig.4 in compare with old method. The differences are because of limitations about experimental or analytical validity range. In high loads and velocities, the difference between numerical and old method is considerable. In a non-modified EHL numerical model, when the difference is very large, the

..

- .?

. ? , -, . ,---!" -!----!" Rei dlstMte 1m)
numerical model

.1.

Figure4: Comparison between Grubin method and presented
3110'

FigureS: Comparison between ES and EHL (using presented model) contact pressure (related with contact stress)

236

2010 International Conference on Mechanical and Electrical Technology (ICMET 2010)

Gi

Shearing module of gear i Normal force of toothj (load shared between tooth I and 2)

[2] [3] [4] [5] [6] [7] [8] [9]
pac pac

Ranger. A, P. , Phd thesis. Imperial College, University of London, 1974. Tapley. F., "Pressure distribution on the contact face of a rigid indenter," MSc thesis. Imperial College,University of London, 1976. Evans. H. P., Sindle. R. W.," The ehl of point contacts at heavy loads," Proc. R. Soc. Lond. A382, pp. 183-199,1982. Carlson. S.," A mathematical theory and solution for the non? Newtonian ahl of cylinders. private communication,",1984. Mostofi. A, Gohar. R.," The use of various types of pressure elements in some contact problems," Proc. 196,1984. Litvin, F.

I s the reduced young module

Vi

The poison coefficient of gear i The contacting force of tooth j The curvature radius of toothj of gear 1,2 Is the applied torque on any gear

P b .'
,

Pj R1j & R2j 1; = P Rbi

L Mech. E. 198, pp. 189-

Analysis for Determination of Load Share, Real Contact Ratio, Precision of Motion, and Stress Analysis," ASME, Journal of Mechanical Design,118 (4),1996. gears as a consequence of wear," Mechanics of structures and machines, 29,pp. 431-449,2001. He, S., Rook, T., Singh,R.,"Construction of Semianalytical Solutions to Spur Gear Dynamics Given Periodic Mesh Stiffness and Sliding Friction Functions," ASME, Journal of Mechanical Design, \30 (12), 2008. Lundvall, 0., Klarbring, A,"Prediction of transmission error in spur

1.., Chen, J. S., Lu, J.,"Application of Finite Element

The total applied load The gear body thickness The Normal load to the profile
X

P R U v

de!/oIGlj = ? de!rolGlj
Local radii of curvature for any position on

Local relative velocity of teeth for any position on

X 1

!.!. U
?

Local tooth surface velocity of any gear for any position on Local velocity difference between teeth for any position on Roll distance Number of gear (1,2) Pressure angle Base circle radius of gear Local velocity along x axis of local coordinate Normal direction to surface of local coordinate Shear stress of fluid Local viscosity Local density Local film thickness Local pressure Film thickness in center of contact area Un-deformed film thickness Deformation of body due to pressure

pac

[10] [11] [12] [13] [14] [15] [16] [17] [18]
Y

He, S., Singh, R. , 2008, "Dynamic Transmission Error Prediction of Helical Gear Pair Under Sliding Friction Using Floquet Theory," ASME,Journal of Mechanical Design, 130 (5). Tamminana, V. K., Kahraman, A, Vi jayakar, S.,"A Study of the Relationship Transmission Between Error of the Dynamic Gear Factors Pairs," and the Dynamic of Spur ASME, Journal

pac

Mechanical Design,129

(I), 2007.

U
Z

R i b

Church, A H.,"Elementary mechanical vibration," Pitman Publishing Corporation,New York,1957. Nestorides, E. Cambridge University Press,1958.

l, "A Handbook on Torsional Vibration," Ist edition,

Tobe, T., Takatsu, N.," Dynamic Load on Spur Gear Teeth Caused by Teeth Impact," JSME,16,(96),pp. 1031-1037,1973. Spotts, M. F., Shoup, T. E, Hornberqer, L. E. , "Design of Machine Elements," 7th edition,Prentice-Hall,New Jersey,1997. Hoechst AG, I st edition, 1987. Hoechst AG, "Technical plastics calculations design applications," Marchek, C. P., " Determination of the Dynamic Gear Meshing Stiffness of an Acetal Copolymer," AGMA Technical Papers,1995. Peterson, D. , "Auswirkung der Lastverteilung auf die Zahnfusstragfahigkeit von hochiiberdeckenden Stirnradpaarungen," Dissertation, TU Braunschweig, 1989. Mohammadpour M, Mirzaee I, Khalilarya Sh "A Mathematical? ASME,

Body curvature on plane that is normal to contact and through
X

[19] [20] [21]

deformation

1.,)'1.

Numerical Model to Calculate Load Distribution, Contact Stiffness and Transmission Error in Involute Spur Gears," 2009, IDETC/PTG2009 Mohammadpour, M. , Mirzaee, Numerical Method for Calculations of Film Geometry of Elasto? Hydro-Dynamic line Contacts," ICMET,2010,un published. Gohar, R., "Elastohydrodynamics," 2nd edition, Imperial College Publication, London, 2001. Hill,New York,1970. Grubin. 1949. Timoshinko, S. P. , Goodier, l N., "Theory of Elastisity," McGraw?

Position of point that its pressure will be used to calculate

Errorcomp

Total error that must be removed from calculation

i., Khalilarya, Sh., "A Modified

Sfope

=

Mrror
---

!1hco

!!.Error

Difference of value of

h

0 in primary modification

[22] [23]

Difference of calculated error due to primary modification of

lubrication of heavy loaded cylindrical surfaces," Book 30, Moscow,

A N., "Fundamentals of the hydrodynamic theory of

REFERENCES

[1]

Dowson. D., Whitaker. A V., "A numerical procedure for the solution of the ehd problem of rolling and sliding contacts lubrication by a Newtonian fluid," Proc.

L Mech. E., 1966,180 (3b), pp. 57.

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