Gears Theory and Design Notes

Evolution of Dynamic Loads in Steel Spur GearsS. Amaldhasan1* and S. PonPaul2

1,2 Department of Mechanical Engineering, Bharath Institute of Science and Technology, Bharath University, Chennai-600073, Tamilnadu; [email protected], [email protected]

*Corresponding author:S. Amaldhasan ([email protected])

Indian Journal of Science and Technology

AbstractThe Complete study of Literature and reporting on how Spur gears are designed using steel-steel alloys and dynamic load is calculated for the spur gears. The analytical model is developed to simulate the load sharing characteristics through a mesh cycle. The model takes into account the main internal factors of dynamic load as time-varying mesh stiffness and composite tooth profile errors. Comparative different types of dynamic load processors study are included, which shows the different processors of the dynamic load. Pro E is an interactive CAD/CAM system, a fully 3-D double precession system that allows accurate description of almost any geometric shape. Pro E modeling provides capabilities to help the design engineer to perform conceptual and detailed deigns. It is a feature and constrained based solid modeler that allows users to create and edit complex solid models interactively. MSC Nastran is a powerful general purpose, Finite element analysis solution for small to complex assemblies. A standard tool, in the field of structural analysis for over four decades, Nastran provides a wide range of analysis capabilities including linear, statics, dynamic, displacement, strain, stress, vibration, heat transfer and more. Nastran can handle any material type from plastic and metal to composites and hyper elastic materials.Keywords: Pro E , MSC Nastran, Spur Gears.

1. IntroductionAny toothed member designed to transmit motion to another one, or received motion from it, by means of suc-cessively engaging tooth is called a (toothed) gear. Gears are a means of changing the rate of rotation of a machinery shaft. They can also change the direction of the axis of rota-tion and can change rotary motion to linear motion. Gears are of several categories and can be combined in a mul-titude of ways, some of which are meshing circular spur gears, rack and pinion spur gears, and worm gears. Helical and herringbone gears utilize curved teeth for efficient, high-capacity power transmission. Worm gears, driven by worms transmit motion between non-intersecting right-angle axes. Gears mate or mesh via teeth with a very specific

geometry. Gears are used for two basic purposes; increase or decrease of rotation speed and increase or decrease of power or torque. Torque is a measure of a force to produce torsion and rotation about an axis. To increase speed and reduce torque a large drive gear is coupled to a smaller driven gear.

1.1 Gear Manufacturing MaterialsMild steel is a poor material for gears as it has poor resis-tance to surface loading. The carbon content for unhardened gears is generally 0.4 %(min) with 0.55 %(min) carbon for the pinions. Dissimilar materials should be used for the meshing gears - this particularly applies to alloy steels. Alloy steels have superior fatigue properties compared to carbon

Supplementary Article

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Evolution of Dynamic Loads in Steel Spur Gears4590

Indian Journal of Science and Technology | Print ISSN: 0974-6846 | Online ISSN: 0974-5645www.indjst.org | Vol 6 (5S) | May 2013

steels for comparable strengths. For extremely high gear loading case hardened steels are used the surface hardening method employed should be such to provide sufficient case depth for the final grinding process used. While selecting a gear material depends on its cost, strength, wear resistance, noise level etc. has to be compromised. Materials such as ferrous metals Non-Ferrous metals, Non metals.

1.2 Law of GearingIn order for two meshing gearing to maintain a constant angular velocity ratio, they must satisfy the fundamen-tal law of gearing. The shape of the teeth of a gear must be such that the common normal at the point of contact between two teeth must always pass through a fixed point on the line of centers. Conjugate action on spur gears: - The gear-tooth profile that will transmit through each other uniform rotary motion. The action between such teeth is called conjugate gear-tooth action

2. Literature Review The principal advances in the engineering of gears have been in measurement, inspection and manufacturing techniques. These, and a new lubrication theory, have contributed to the improved performance required from modern high-speed transmissions. Unfortunately, gear case design has not kept up with the design of gears themselves [1]. In a theoretical analysis of a single pair of high-class gears three internal sources of vibration are considered. Dynamic tooth forces and amplitudes of vibration were found from photo-elastic stress patterns of model gears with both small and large errors [2]. Spur gear tooth force analysis, Spur gear - tooth stresses, Tooth bending stress Lewis equation, Tooth bending stress AGMA procedure, Bending fatigue strength AGMA procedure, Permissible bending stress, Buckingham equation for dynamic load on gears [3]. The model takes into account the main internal factors of dynamic load as time- varying mesh stiffness and composite tooth profile errors. The complicated phenom-enon of contact tooth pairs alternation during mesh cycle is integrated in this dynamic load modeling [4].

3. Spur Gear DesignThe formulae and parameters are used from a Hand Book of Gear Design by G.M.Maitra and Machine Design Data Book P.S.G. College of Technology.

3.1 Dimensions for Standard Spur GearPitch circle diameter for pinion, d1 = m*z1 mmPitch circle diameter for Gear, d2 = m*z2 mmBase circle diameter for Pinion, db1 = d1*cos mmBase circle diameter for Gear, db2 = d2*cos mmTip circle diameter for pinion, da1 = d1 + 2m mmTip circle diameter for gear, da2 = d2 + 2m mmRoot circle diameter for pinion, df1 = d1 2*1.25 mmRoot circle diameter for gear, df2 = d2 2*1.25 mmTooth thickness on pitch circle, S = m/2 mmClearance c = 0.25mmCentre distance, a = (d1 + d2)/2 or m (z1+z2)/2Total Depth, h = 2.25*m mmCircular pitch, p = m mmFace width, b = 3**m mmRatio of gear width to center distance y = b/aGear ratio i = z1/z2 = n1/n2 = 1/2 Module m = d/z mmDiametral pitch pd = z/d or 1/mPower P = 2 n1 Mt/60 KWGear life L = 60n1LH revolutionsPinion Torque Mt = P*60/(2 n1) N mm

3.2 Design Bending Stress b = (1.4 kbl -1) /(n k).for rotation in one direction b = (kbl -1) /(n k).for rotation in both directionskbl = life factork = fillet stress concentration factor-1 = endurance limit in reversed bendingn = factor of safety,

3.3 Induced Bending Stress b = [Mt]Y is form factor taken from Ref

3.4 Design Torque [Mt] = k0* k * kd * Mtk0 = shock loading conditionk = load concentration factor kd = dynamic load factor Mt = pinion Torque

3.5 Design Contact Stress[c]=CB*HB*Kcl N/mm2[c]=CR*HRC*Kcl N/mm2

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S. Amaldhasan and S. PonPaul 4591


Table 3.1. Proportions of Standard Gear Teeth

200 Full Depth Involute 200 Stub Involute

Addendum M 0.8mMinimum dedendum

1.25m M

Whole depth 2.25m 1.8mClearance 0.25m 0.2m

Table 3.2. Stress Concentration Factor for Fillet, K

Material and Heat Treatment Ksteel, normalised, surface hardened 1.5steel, case hardened (low carbon steels) 1.2

cast iron 1.2

Table 3.3. Factor of Safety, N

Material Mode of Manufacture

Heat Treatment Factor of Safety, n

Steel Cast Iron

Cast No heat treatment tempered or normalised

2.52.0

Steel Cast or forged Forged

Case hardened Surface hardened Normalised

2.02.52.0

Table 3.4. Life Factor For Surface (Contact Compressive) Strength, Kcl

Material Surface Hardness HB

Life In Number Of Cycles, N

Life Factor Kcl

Steel

Carbon

350

>350

10 7



Table 3.9. Allowable Peripheral Speed of Gears

IS quality Preferred quality

Speed of gears in m/s

Cylindrical gears

Straight bevel gears

High precision

3 & 4 4 Above 15 Upto 9

Precision 5 & 6 6 8-15 Upto 6Medium 7,8,& 9 8 1-8 Upto 3Coarse 10 & 12 10,12 Upto 1 Upto 2

4. Design Calculation Standard gear teeth-200 stub involute Material = Steel Poissons Ratio = 0.27 to 0.30Youngs Modulus E = 2.1*105 N/mm2Module m = 10 Speed of pinion n1 = 1800 rpm Gear ratio i = 1:3Pinion no. of teeth Z1 = 16Life of teeth LH= 20000 hrPressure angle () = 200Power P = 18 KW

Calculation :-Speed of gear n2 = 1800/3= 600 rpmGear no. of teeth Z2 = 12*3 = 36Pitch circle diameter for pinion,d1 = m*z1 = 10*12 = 120 mmPitch circle diameter for Gear,d2 = m*z2 =10*36 = 360 mmCenter distance of gear & pinion a = (z2+z1)m/2 = (36+12)10/2 = 240 mmBase circle diameter for Pinion,db1 = d1*cos =120*cos200 =112.76 mmBase circle diameter for Gear,db2 = d2*cos = 360*cos200 = 338.3 mmTip circle diameter for pinion,da1 = d1 + 2m =120+2*10 = 140 mmTip circle diameter for gear,da2 = d2 + 2m=360+2*10 = 380 mmRoot circle diameter for pinion,df1 = d1 2*1.25m = 120-2*1.25*10 = 95 mmRoot circle diameter for gear,df2 = d2 2*1.25m = 360-2*1.25*10 = 335 mm

Tooth thickness on pitch circle,S = m/2=*10/2=15.71 mmClearance c = 0.25m =0.25*10=2.5 mmTotal Depth, h = 2.25*m=2.25*10=22.5 mmCircular pitch, p = m= *10=3.14*10=31.4 mmFace width, b = 3**m =3**10=94.25 mmDiametrical pitch pd = z/d or 1/m =1/10=0.1Pitch line velocityV = *d1*n1/(60*1000)=11.31 m/secRatio of gear width to center distance y = b/a =0.3Face width b = y*a=0.3*240=72 mmPinion life L = 60 n1LH = 60*1800*20000 = 2160000000 revolutions = 216*107 rev.Pinion Torque Mt = P*60 / (2 n1) = 10000*60/(2*3.14*1800) = 95.5 N m

a) Pinion 1. Material : Case hardening steel is used. Assume surface hardness 55 RC and core hardness greater than 350 BHN. (Table 3.6, Table 3.9)Design stressesDesign bending stress b = -1-1 = 0.25( u+y)+50 (for cast steel)y=879 N/mm2 -1=0.25 (1050+879)+50=532.25 N/mm2n = 2.5 (for surface hardened steel)K=1.5 (for surface hardened steel)For steel, surface hardness >350 BHN, core hardness > 350 BHN and life N is 25*10 7, Kbl=0.7 b = -1 = (1.4*0.7*532.25) / (2.5*1.5) = 139.1N/mm2Design Contact Stress [c]

[c] =CR*HRC*Kcl N/mm2For surface hardened alloy steel CR=23For steel with surface hardness > 350 BHN and N 25 * 107, Kcl = 0.585Therefore, [c] = 23*55*0.585 = 740.025 N/mm2Design Torque [Mt]

[Mt] = k0* k * kd * Mtwhere,K0= 1.5 for medium shock yP= b/d1 = 72/120 = 0.6

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Assume IS quality 6 gears (Table 3.2), for pitch line veloc-ity 11.31 m/s.

k = 1.03 kd = 1.3

[Mt]=1.5*1.03*1.3*53.08=191.79NmTo Calculate The Induced Stressesb = [Mt]Form factor for z1 = 16, y=0.355 (Table). For stud tooth,

Y = 0.355/0.6 =0.592)b = [(3+1)(191.79*1000)]/(240*10*72*0.592)=7.5 N/mm2b = 7.5 N/mm2 < [b] =139.1 N/mm2

c = 0.74 c = 0.74 ((3+1)/240)[((3+1)/(3*72))(2.1*105 *191.79*103)]c = 336.83N/mm2c = 336.83 N/mm2 < [c] = 740.025 N/mm2The design is satisfactory.

b) GearGear life L = 60*600*20000 = 720000000 rev. = 72*107 rev.Gear Torque Mt = P*60/(2 n2) = 286.48 Nm Material : CS 85, case hardness 55 RC and core hardness greater than 350 BHN. Design stressesDesign bending stress b = -1-1= 0.22( u+ y)+50 (for cast steel)u = 850 N/mm2 y = 710 N/mm2 -1 = 0.22 (850+710)+50=393.2 N/mm2n = 2.0 (for case hardened steel) K = 1.2 (for case hardened steel) For steel, surface hardness >350 BHN, core hardness > 350 BHN and lifeN is 25*10 7, Kbl=0.7 b = -1 = (1.4*0.7*393.2) / (2.0*1.2) = 160.556 N/mm2Design Contact Stress [c][c] = CR*HRC*Kcl N/mm2For case hardened alloy steel CR=28 For steel with surface hardness > 350 BHN and N 25*107, Kcl = 0.585Therefore, [c] = 28*55*0.585 = 900.9 N/mm2Design Torque [Mt]

[Mt] = k0* k * kd * MtWhere,K0= 1.5 for medium shock

yP = b/d1 = 72/360 = 0.2assume IS quality 6 gears (Table 1.21), for pitch line veloc-ity 11.31 m/s.

k = 1.0 kd = 1.3

[Mt]=1.5*1.0*1.3*286.48=558.636 NmTo Calculate The Induced Stressesb = [Mt](Form factor for z1 = 48, y=0.475 (Table For stud tooth, Y= 0.475/0.2 =2.375)b = [(3+1)(558.636*1000)]/(240*10*72*2.375) = 5.445 N/mm2b = 5.445 N/mm2 < [b] =160.556 N/mm2

c = 0.74 c = 0.74 ((3+1)/240)[((3+1)/(3*72)) (2.1*105 *558.636*103)]c = 574.854 N/mm2c = 574.854 N/mm2 < [c] = 901 N/mm2The design is satisfactory.

5. Dynamic LoadThe level of vibration and noise of the gear pairs is in corre-lation with the characteristics of dynamic load. Calculation of dynamic loads and determination of their variation dur-ing a mesh cycle for spur gears pairs has been considered a major aspect of gear design. An analytical model that cov-ers the main influence factors with sufficient accuracy is not currently available.

This paper presents an analytical approach used in order to predict the characteristics of dynamic loads by considering the time-varying mesh stiffness and variable tooth profile errors. The calculus procedure of the mesh stiffness is found by using an exact analytical model.

5.1 Dynamic Load Procedure Dynamic load Fd = e)... eq.1Here e is the sum of the errors, inches, of the two mating teeth, t is the time, seconds, during which the error is act-ing, k is the spring constant, pounds per inch, of the two mating teeth, and me is the effective mass, lb sec2/in., of the two gears.Time t for the passage of a single pitch, then ist = 60/(n1N1) sec eq.2Spring Constant and Effective Mass

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K = b/9 [(E1E2)/(E1+E2)]. eq.3Where E1 and E2 are the module of elasticity for the materi-als of the two gears.When both gear & pinion are steel, E1 = E2 =30,000,000 psi, = 2.1*105 N/mm2 the value of k for a meshing pair becomes K=1,670,000b lb/in. = 11500b N/mmWhen one gear is steel, E1=30,000,000 psi and the other gear is cast iron or bronze, E2 = 16,000,000 psi, = 1.1*105 N/mm2K =1,160,000b lb/in. = 8000b N/mm b = width of face of the gears.The moment of inertia of a gear can be found by approxi-mating it as a solid circular cylinder of diameter equal to the pitch diameter of the gear, and axial length equal to the face width of the teeth. The mass moment of inertia I, fig.3-1(a), then is

I = m (r2/2)Where m is the mass, with dimensions of lb sec2 /in. as found from Newtons second law of motion, mass = force/acceleration. Mass is thus equal to the weight divided by the gravitational constant g .A concentrated mass m, located at the pitch circle, has a moment of inertia I about the axis of

I = m r2 Suppose m were of such size as to make I and I equal. Then

m = m*1/2The dynamical system of the two gears is considered as the masses m1 and m2 Concentrated at the pitch circles connected by a spring comprising the two teeth. For Such a system, the effective mass me is given by the following equation.1/me = 1/m1 + 1/m2. eq. 4 Mass m1 of the pinion is equal to m1 = m1(1/2) = ( r12 b 1)/ (2g) .. eq.5where 1 is the weight per cubic inch of the material.Mass m2 of the gear is given by a similar equation.m2 = m2(1/2) = ( r22 b 2)/ (2g) eq.6where 2 is the weight per cubic inch of the material com-posing the gear.Substitution into Eq.1 gives 1/me = 2g/( b) [(1/ (r12 1))+(1/( r22 b 2))] eq.7

5.2 Design Equations for Dynamic Load steel pinion steel gear .here 1 = 2 =0.283 lb/in.3 = 0.0000768 N/mm3 and acceleration g due to gravity is taken as 386 in./sec2. = 9806.6 mm/sec2 Equation (7) becomes

1/me = ((1/r12) + (1/r22) )

= ((r12 + r22)/(r12 r22)) mm/N sec2At all values substitute in eq.1Fd = e)= (e*n1*N1*b*r1*r2)/(2520(r12 + r22)) Newtons Steel pinion cast iron gear . hear 1 = 0.283 lb/in3, and 2 = 0.256 lb/in.3 or 0.0000695 N/mm3 1/me = (2g / b2) ((r12 + (2 / 1) r22)/(r12 r22)) = ((r12 + 0.9r22)/(r12 r22) )At all values substitute in eq.1Fd = e)= (e*n1*N1*b*r1*r2)/( 3180(r12 + 0.9r22)) N

5.3 Dynamic Loada) Process 1Dynamic LoadFd = e)= (e*n1*N1*b*r1*r2 )/(2520(r12 + r22)) (N=z no. of teeth)e = 0.0048+0.0050=0.0098 in. = 0.2489mm (1 in = 25.4 mm)Fd = 0.2489*1800*16*72*60*180/ (2520 (602 *+1802)) = 11657.9 NFw = d1bQKWhere, d1 = 120mm, b = 72mmQ = 2z2 / (z1 +z2)= 2*48 / (16+48) = 1.5K = 651psi = 4.48842N/mm2 Fw = 120*72*1.5*4.48842 = 58.1699 KNFt = Fw- Fd = 58.1699-11.679 = 46.4909 KN

b) Process 2Buckinghams Dynamic LoadFd = Ft +[0.164 Vm (c*b + Ft)/[0.164 Vm + 1.486(c*b+Ft)]]Where, vm = pitch line velocity, =11.31m/s or 678.6 m/minVm = 678.6 m/minb = face width = 72mm = 0.072 mc = factor depending on machining error = 12300e = 12300*0.03 = 369Ft = transmitted load, kgfFt = HP*75/vm = 24.14*75/11.31

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P = 24.14 HP = 160.08 kgf or 1570.4 NFd = 160.08 + [0.164*678.6 [369*0.072+160.08]/[0.164* 678.6 + 1.486[369*0.072 + 160.08]]] = 160.08+[20988.5/131.6] = 319 kgf or 3134.95 N = 3.135 KN

c) Process 3Lewis Equation of dynamic loadFd = Ft* Cv Ft = HP*75/ vmvm = 11.31 m/sP = 24.14 hpCv =(6+vm)/6 = (6+11.31)/6 = 2.885Ft = 24.14*75/11.31 = 160.08 kgf Fd = 160.08*2.885 = 461.831 kgf = 4530.6 N or 4.531 KN

6. ConclusionsDesign of spur gears using steel-steel alloys is done and the dynamic loads on spur gears are calculated by using com-plete study of literature and reporting.

In phase -2 the design of spur gears using Sintered materials is yet to be done. The dynamic load characteristic is to be calculated manually then the model is created in CAD softwares.

An analytical investigation of the dynamic loads fluc-tuation in spur gears is presented. The dynamic model permits to predict the individual tooth load characteristics during the meshing cycle.

7. References 1. Winter H. (1961). Gear Tooth Strength of Spur Gears, British

Gear Manufacturers Association. 2. A.S.M.E (1931). Dynamic Loads on Gear Teeth. 3. http://nptel.iitm.ac.in/courses/IIT-MADRAS/Machine_

Design_II/pdf/2_7.pdf 4. Vedmar L, Henriksson B (1998). A general approach for

determining dynamic forces in Spur gears, Journal of Mechanical Design, Transaction of the ASME, 593598.

5. Prabhu T J (2005). Design of Transmission Elements, 6. Maitra G M (1989). Handbook of Gear Design,140 7. Lingaiah K (2007). Machine Design Data Handbook, vol 1. 8. P.S.G. College of Technology (1988). Design Data Book. 9. Merrit H E (1992). Gear Engineering, 15-53,90 Dr. Ulrich

Kissling, Effects of Profile corrections on peak to peak transmission error, Gear Technology.

10. Kiss Soft (2008). Gear Optimizing with advanced calcula-tion method. www.kisssoft.ch/english/downloads/pdf/Gear_ Optimizing

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