Gear Design

20
Spur Gears Introduction ..... Standards ..... Terminology ..... Spur Gear Design ..... Materials ..... Basic Equations ..... Module ..... Pressure Angle ..... Contact Ratio ..... Forces- Torques etc ..... Strength Durability calcs ..... Design Process ..... Internal Gears ..... Table of Lewis Form Factors ..... Introduction Introduction Introduction Introduction Gears are machine elements used to transmit rotary motion between two shafts, normally with a constant ratio. The pinion is the smallest gear and the larger gear is called the gear wheel.. A rack is a rectangular prism with gear teeth machined along one side- it is in effect a gear wheel with an infinite pitch circle diameter. In practice the action of gears in transmitting motion is a cam action each pair of mating teeth acting as cams. Gear design has evolved to such a level that throughout the motion of each contacting pair of teeth the velocity ratio of the gears is maintained fixed and the velocity ratio is still fixed as each subsequent pair of teeth come into contact. When the teeth action is such that the driving tooth moving at constant angular velocity produces a proportional constant velocity of the driven tooth the action is termed a conjugate action. The teeth shape universally selected for the gear teeth is the involute profile. Consider one end of a piece of string is fastened to the OD of one cylinder and the other end of the string is fastened to the OD of another cylinder parallel to the first and both cylinders are rotated in the opposite directions to tension the string(see figure below). The point on the string midway between the cylinder P is marked. As the left hand cylinder rotates CCW the point moves towards this cylinder as it wraps on . The point moves away from the right hand cylinder as the string unwraps. The point traces the involute form of the gear teeth. The lines normal to the point of contact of the gears always intersects the centre line joining the gear centres at one point called the pitch point. For each gear the circle passing through the pitch point is called the pitch circle. The gear ratio is proportional to the diameters of the two pitch circles. For metric gears (as adopted by most of the worlds nations) the gear proportions are based on the module. m = (Pitch Circle Diameter(mm)) / (Number of teeth on gear). In the USA the module is not used and instead the Diametric Pitch d pis used d p = (Number of Teeth) / Diametrical Pitch (inches)

Transcript of Gear Design

Page 1: Gear Design

Spur Gears

Introduction..... Standards..... Terminology..... Spur Gear Design..... Materials..... Basic Equations..... Module..... Pressure Angle..... Contact Ratio..... Forces- Torques etc..... Strength Durability calcs..... Design Process..... Internal Gears..... Table of Lewis Form Factors.....

IntroductionIntroductionIntroductionIntroduction

Gears are machine elements used to transmit rotary motion between two shafts, normally with a constant ratio. The pinion is the smallest gear and the larger gear is called the gear wheel.. A rack is a rectangular prism with gear teeth machined along one side- it is in effect a gear wheel with an infinite pitch circle diameter. In practice the action of gears in transmitting motion is a cam action each pair of mating teeth acting as cams. Gear design has evolved to such a level that throughout the motion of each contacting pair of teeth the velocity ratio of the gears is maintained fixed and the velocity ratio is still fixed as each subsequent pair of teeth come into contact. When the teeth action is such that the driving tooth moving at constant angular velocity produces a proportional constant velocity of the driven tooth the action is termed a conjugate action. The teeth shape universally selected for the gear teeth is the involute profile. Consider one end of a piece of string is fastened to the OD of one cylinder and the other end of the string is fastened to the OD of another cylinder parallel to the first and both cylinders are rotated in the opposite directions to tension the string(see figure below). The point on the string midway between the cylinder P is marked. As the left hand cylinder rotates CCW the point moves towards this cylinder as it wraps on . The point moves away from the right hand cylinder as the string unwraps. The point traces the involute form of the gear teeth.

The lines normal to the point of contact of the gears always intersects the centre line joining the gear centres at one point called the pitch point. For each gear the circle passing through the pitch point is called the pitch circle. The gear ratio is proportional to the diameters of the two pitch circles. For metric gears (as adopted by most of the worlds nations) the gear proportions are based on the module.

m = (Pitch Circle Diameter(mm)) / (Number of teeth on gear).

In the USA the module is not used and instead the Diametric Pitch d pis used

d p = (Number of Teeth) / Diametrical Pitch (inches)

Page 2: Gear Design

Profile of a standard 1mm module gear teeth for a gear with Infinite radius (Rack ).

Other module teeth profiles are directly proportion . e.g. 2mm module teeth are 2 x this profile

Many gears trains are very low power applications with an object of transmitting motion with minium torque e.g. watch and clock mechanisms, instruments, toys, music boxes etc. These applications do not require detailed strength calculations.

Standards Standards Standards Standards

• AGMA 2001-C95 or AGMA-2101-C95 Fundamental Rating factors and Calculation Methods for involute Spur Gear and Helical Gear Teeth

• BS 436-4:1996, ISO 1328-1:1995..Spur and helical gears. Definitions and allowable values of deviations relevant to corresponding flanks of gear teeth

• BS 436-5:1997, ISO 1328-2:1997..Spur and helical gears. Definitions and allowable values of deviations relevant to radial composite deviations and runout information

• BS ISO 6336-1:1996 ..Calculation of load capacity of spur and helical gears. Basic principles, introduction and general influence factors

• BS ISO 6336-2:1996..Calculation of load capacity of spur and helical gears. Calculation of surface durability (pitting)

• BS ISO 6336-3:1996..Calculation of load capacity of spur and helical gears. Calculation of tooth bending strength

• BS ISO 6336-5:2003..Calculation of load capacity of spur and helical gears. Strength and quality of materials

If it is necessary to design a gearbox from scratch the design process in selecting the gear size is not complicated - the various design formulea have all been developed over time and are available in the relevant standards. However significant effort, judgement and expertise is required in designing the whole system including the gears, shafts , bearings, gearbox, lubrication. For the same duty many different gear options are available for the type of gear , the materials and the quality. It is always preferable to procure gearboxes from specialised gearbox manufacturers

Terminology Terminology Terminology Terminology ---- spur gears spur gears spur gears spur gears

• Diametral pitch (d p )...... The number of teeth per one inch of pitch circle diameter.

• Module. (m) ...... The length, in mm, of the pitch circle diameter per tooth.

• Circular pitch (p)...... The distance between adjacent teeth measured along the are at the pitch circle diameter

Page 3: Gear Design

• Addendum ( h a )...... The height of the tooth above the pitch circle diameter.

• Centre distance (a)...... The distance between the axes of two gears in mesh.

• Circular tooth thickness (ctt)...... The width of a tooth measured along the are at the pitch circle diameter.

• Dedendum ( h f )...... The depth of the tooth below the pitch circle diameter.

• Outside diameter ( D o )...... The outside diameter of the gear.

• Base Circle diameter ( D b ) ...... The diameter on which the involute teeth profile is based.

• Pitch circle dia ( p ) ...... The diameter of the pitch circle.

• Pitch point...... The point at which the pitch circle diameters of two gears in mesh coincide.

• Pitch to back...... The distance on a rack between the pitch circle diameter line and the rear face of the rack.

• Pressure angle ...... The angle between the tooth profile at the pitch circle diameter and a radial line passing through the same point.

• Whole depth...... The total depth of the space between adjacent teeth.

Page 4: Gear Design

Spur Gear DesignSpur Gear DesignSpur Gear DesignSpur Gear Design

The spur gear is is simplest type of gear manufactured and is generally used for transmission of rotary motion between parallel shafts. The spur gear is the first choice option for gears except when high speeds, loads, and ratios direct towards other options. Other gear types may also be preferred to provide more silent low-vibration operation. A single spur gear is generally selected to have a ratio range of between 1:1 and 1:6 with a pitch line velocity up to 25 m/s. The spur gear has an operating efficiency of 98-99%. The pinion is made from a harder material than the wheel. A gear pair should be selected to have the highest number of teeth consistent with a suitable safety margin in strength and wear. The minimum number of teeth on a gear with a normal pressure angle of 20 desgrees is 18. The preferred number of teeth are as follows

12 13 14 15 16 18 20 22 24 25 28 30 32 34 38 40 45 50 54 60 64 70 72 75 80 84 90 96 100 120 140 150 180 200 220 250

Materials used for gearsMaterials used for gearsMaterials used for gearsMaterials used for gears

Mild steel is a poor material for gears as as it has poor resistance 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 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.

Material Notes applications

Ferrous metals

Cast Iron Low Cost easy to machine with high damping

Large moderate power, commercial gears

Cast Steels Low cost, reasonable strength Power gears with medium rating to commercial quality

Plain-Carbon Steels Good machining, can be heat treated

Power gears with medium rating to commercial/medium quality

Alloy Steels Heat Treatable to provide highest strength and durability

Highest power requirement. For precision and high precisiont

Stainless Steels (Aust) Good corrosion resistance. Non-magnetic

Corrosion resistance with low power ratings. Up to precision quality

Stainless Steels (Mart) Hardenable, Reasonable corrosion resistance, magnetic

Low to medium power ratings Up to high precision levels of quality

Non-Ferrous metals

Aluminium alloys Light weight, non-corrosive and good machinability

Light duty instrument gears up to high precision quality

Page 5: Gear Design

Brass alloys Low cost, non-corrosive, excellent machinability

low cost commercial quality gears. Quality up to medium precision

Bronze alloys Excellent machinability, low friction and good compatability with steel

For use with steel power gears. Quality up to high precision

Magnesium alloys Light weight with poor corrosion resistance

Ligh weight low load gears. Quality up to medium precision

Nickel alloys Low coefficient of thermal expansion. Poor machinability

Special gears for thermal applications to commercial quality

Titanium alloys High strength, for low weight, good corrosion resistance

Special light weight high strength gears to medium precision

Di-cast alloys Low cost with low precision and strength

High production, low quality gears to commercial quality

Sintered powder alloys Low cost, low quality, moderate strength

High production, low quality to moderate commercial quality

Non metals

Acetal (Delrin Wear resistant, low water absorbtion

Long life , low load bearings to commercial quality

Phenolic laminates Low cost, low quality, moderate strength

High production, low quality to moderate commercial quality

Nylons No lubrication, no lubricant, absorbs water

Long life at low loads to commercial quality

PTFE Low friction and no lubrication Special low friction gears to commercial quality

Equations for basic gear relationshipsEquations for basic gear relationshipsEquations for basic gear relationshipsEquations for basic gear relationships

It is acceptable to marginally modify these relationships e.g to modify the addendum /dedendum to allow Centre Distance adjustments. Any changes modifications will affect the gear performance in good and bad ways...

Addendum h a = m = 0.3183 p

Base Circle diameter Db = d.cos α

Centre distance a = ( d g + d p) / 2

Circular pitch p = m.π

Circular tooth thickness ctt = p/2

Dedendum h f = h - a = 1,25m = 0,3979 p

Module m = d /n

Number of teeth z = d / m

Outside diameter D o = (z + 2) x m

Pitch circle diameter d = n . m ... (d g = gear & d p = pinion )

Page 6: Gear Design

Whole depth(min) h = 2.25 . m

Top land width(min) t o = 0,25 . m

Module (m) Module (m) Module (m) Module (m)

The module is the ratio of the pitch diameter to the number of teeth. The unit of the module is milli-metres.Below is a diagram showing the relative size of teeth machined in a rack with module ranging from module values of 0,5 mm to 6 mm

The preferred module values are

0,5 0,8 1 1,25 1,5 2,5 3 4 5 6 8 10 12 16 20 25 32 40 50

Normal Pressure angle Normal Pressure angle Normal Pressure angle Normal Pressure angle α

An important variable affecting the geometry of the gear teeth is the normal pressure angle. This is

generally standardised at 20o. Other pressure angles should be used only for special reasons and

using considered judgment. The following changes result from increasing the pressure angle

• Reduction in the danger of undercutting and interference

• Reduction of slipping speeds

• Increased loading capacity in contact, seizure and wear

• Increased rigidity of the toothing

• Increased noise and radial forces

Gears required to have low noise levels have pressure angles 15o to17.5

o

Contact RatioContact RatioContact RatioContact Ratio

The gear design is such that when in mesh the rotating gears have more than one gear in contact and transferring the torque for some of the time. This property is called the contact ratio. This is a ratio of the length of the line-of-action to the base pitch. The higher the contact ratio the more the load is shared between teeth. It is good practice to maintain a contact ratio of 1.2 or greater. Under no circumstances should the ratio drop below 1.1. A contact ratio between 1 and 2 means that part of the time two pairs of teeth are in contact and during the remaining time one pair is in contact. A ratio between 2 and 3 means 2 or 3 pairs of teeth are always in contact. Such as high contact ratio generally is not obtained with external spur gears, but can be developed in the meshing of an internal and external spur gear pair or specially designed non-standard external spur gears.

Page 7: Gear Design

contact ratio m =

[Rgo2 - Rgb

2 )1/2 + (Rpo2 - Rpb

2 )1/2 - a sin α] / p cos α

R go = D go / 2..Radius of Outside Dia of Gear R gb = D gb / 2..Radius of Base Dia of Gear R po = D po / 2..Radius of Outside Dia of Pinion R pb = D pb / 2..Radius of Base Dia of Pinion p = circular pitch. a = ( d g+ d p )/2 = center distance.

Spur gear Forces, torques, velocities & Powers Spur gear Forces, torques, velocities & Powers Spur gear Forces, torques, velocities & Powers Spur gear Forces, torques, velocities & Powers

• F = tooth force between contacting teeth (at angle pressure angle α to pitch line tangent. (N)

• F t = tangential component of tooth force (N)

• F s = Separating component of tooth force

• α= Pressure angle

• d 1 = Pitch Circle Dia -driving gear (m)

• d 2 = Pitch Circle Dia -driven gear (m)

• ω 1 = Angular velocity of driver gear (Rads/s)

• ω 2 = Angular velocity of driven gear (Rads/s)

• z 1 = Number of teeth on driver gear

• z 2 = Number of teeth on driven gear

• P = power transmitted (Watts)

• M = torque (Nm)

• η = efficiency

Tangential force on gears F t = F cos α

Separating force on gears F s = F t tan α

Torque on driver gear T 1 = F t d 1 / 2

Torque on driver gear T 2 = F t d 2 / 2

Speed Ratio =ω 1 / ω 2 = d 2 / d 1 = z 2 /z 1

Input Power P 1 = T1 .ω 1

Output Power P 2 =η.T 1 .ω 2

Spur gear Strength and durability calculations Spur gear Strength and durability calculations Spur gear Strength and durability calculations Spur gear Strength and durability calculations

Designing spur gears is normally done in accordance with standards the two most popular series are listed under standards above: The notes below relate to approximate methods for estimating gear strengths. The methods are really only useful for first approximations and/or selection of stock gears (ref links below). — Detailed design

Page 8: Gear Design

necessary guidance. Software is also available making the process very easy. A very reasonably priced and easy to use package is included in the links below (Mitcalc.com) The determination of the capacity of gears to transfer the required torque for the desired operating life is completed by determining the strength of the gear teeth in bending and also the durability i.e of the teeth ( resistance to wearing/bearing/scuffing loads ) .. The equations below are based on methods used by Buckingham..

Bending

The basic bending stress for gear teeth is obtained by using the Lewis formula

σ = Ft / ( ba. m. Y )

• F t = Tangential force on tooth

• σ = Tooth Bending stress (MPa)

• b a = Face width (mm)

• Y = Lewis Form Factor

• m = Module (mm)

Note: The Lewis formula is often expressed as

σ = Ft / ( ba. p. y )

Where y = Y/π and p = circular pitch

When a gear wheel is rotating the gear teeth come into contact with some degree of impact. To allow for this a velocity factor is introduced into the equation. This is given by the Barth equation for milled profile gears.

K v = 6,1 / (6,1 +V )

V = the pitch line velocity = d.ω/2 Note: This factor is different for different gear conditions i.e K v = ( 3.05 + V )/3.05 for cast iron, cast profile gears.

The Lewis formula is thus modified as follows

σ = K v.Ft / ( ba. m. Y )

Surface Durability

This calculation involves determining the contact stress between the gear teeth and uses the Herz Formula

σ w = 2.F / ( π .b .l )

σ w = largest surface pressure

Page 9: Gear Design

F = force pressing the two cylinders (gears) together l = length of the cylinders (gear) b = halfwidth =

d 1 ,d 2 Are the diameters for the two contacting cylinders.

ν 1, ν 2 Poisson ratio for the two gear materials

E 1 ,E 2 Are the Young's Modulus Values for the two gears To arrive at the formula used for gear calculations the following changes are made

F is replaced by F t/ cos α

d is replaced by 2.r l is replaced by W The velocity factor K v as described above is introduced. Also an elastic constant Z E is created

When the value of E used is in MPa then the units of Cp are √ MPa = KPa The resulting formula for the compressive stress developed is as shown below

The dynamic contact stress χc developed by the transmitted torque must be less than the allowable contact stress Se... Note: Values for Allowable stress values Se and ZE for some materials are provided at Gear Table

r1 = d1 sin α /2

r2 = d2 sin α /2

Important Note: The above equations do not take into account the various factors which are integral to calculations completed using the relevant standards. These equations therefore yield results suitable for first estimate design purposes only...

Design ProcessDesign ProcessDesign ProcessDesign Process To select gears from a stock gear catalogue or do a first approximation for a gear design select the gear material and obtain a safe working stress e.g Yield stress / Factor of Safety. /Safe fatigue stress

• Determine the input speed, output speed, ratio, torque to be transmitted

• Select materials for the gears (pinion is more highly loaded than gear)

• Determine safe working stresses (uts /factor of safety or yield stress/factor of safety or Fatigue strength / Factor of safety )

• Determine Allowable endurance Stress Se

• Select a module value and determine the resulting geometry of the gear

• Use the lewis formula and the endurance formula to establish the resulting face width

• If the gear proportions are reasonable then - proceed to more detailed evaluations

Page 10: Gear Design

• If the resulting face width is excessive - change the module or material or both and start again

The gear face width should be selected in the range 9-15 x module or for straight spur gears-up to 60% of the pinion diameter.

Internal Gears Internal Gears Internal Gears Internal Gears Advantages:

1. Geometry ideal for epicyclic gear design 2. Allows compact design since the center distance is less than for external gears. 3. A high contact ratio is possible. 4. Good surface endurance due to a convex profile surface working against a concave surface.

Disadvantages:

1. Housing and bearing supports are more complicated, because the external gear nests within the internal gear.

2. Low ratios are unsuitable and in many cases impossible because of interferences. 3. Fabrication is limited to the shaper generating process, and usually special tooling is required.

Lewis form factorLewis form factorLewis form factorLewis form factor.

Table of lewis form factors for different tooth forms and pressure anglesTable of lewis form factors for different tooth forms and pressure anglesTable of lewis form factors for different tooth forms and pressure anglesTable of lewis form factors for different tooth forms and pressure angles

No Load Near Tip of Teeth Load at Near Middle of Teeth

Page 11: Gear Design

Teeth 14 1/2 deg 20 deg FD 20 deg Stub 25 deg 14 1/2 deg 20 deg FD

Y y Y y Y y Y y Y y Y y

10 0,176 0,056 0,201 0,064 0,261 0,083 0,238 0,076

11 0,192 0,061 0,226 0,072 0,289 0,092 0,259 0,082

12 0,21 0,067 0,245 0,078 0,311 0,099 0,277 0,088 0,355 0,113 0,415 0,132

13 0,223 0,071 0,264 0,084 0,324 0,103 0,293 0,093 0,377 0,12 0,443 0,141

14 0,236 0,075 0,276 0,088 0,339 0,108 0,307 0,098 0,399 0,127 0,468 0,149

15 0,245 0,078 0,289 0,092 0,349 0,111 0,32 0,102 0,415 0,132 0,49 0,156

16 0,255 0,081 0,295 0,094 0,36 0,115 0,332 0,106 0,43 0,137 0,503 0,16

17 0,264 0,084 0,302 0,096 0,368 0,117 0,342 0,109 0,446 0,142 0,512 0,163

18 0,27 0,086 0,308 0,098 0,377 0,12 0,352 0,112 0,459 0,146 0,522 0,166

19 0,277 0,088 0,314 0,1 0,386 0,123 0,361 0,115 0,471 0,15 0,534 0,17

20 0,283 0,09 0,32 0,102 0,393 0,125 0,369 0,117 0,481 0,153 0,544 0,173

21 0,289 0,092 0,326 0,104 0,399 0,127 0,377 0,12 0,49 0,156 0,553 0,176

22 0,292 0,093 0,33 0,105 0,404 0,129 0,384 0,122 0,496 0,158 0,559 0,178

23 0,296 0,094 0,333 0,106 0,408 0,13 0,390 0,124 0,502 0,16 0,565 0,18

24 0,302 0,096 0,337 0,107 0,411 0,131 0,396 0,126 0,509 0,162 0,572 0,182

25 0,305 0,097 0,34 0,108 0,416 0,132 0,402 0,128 0,515 0,164 0,58 0,185

26 0,308 0,098 0,344 0,109 0,421 0,134 0,407 0,13 0,522 0,166 0,584 0,186

27 0,311 0,099 0,348 0,111 0,426 0,136 0,412 0,131 0,528 0,168 0,588 0,187

28 0,314 0,1 0,352 0,112 0,43 0,137 0,417 0,133 0,534 0,17 0,592 0,188

29 0,316 0,101 0,355 0,113 0,434 0,138 0,421 0,134 0,537 0,171 0,599 0,191

30 0,318 0,101 0,358 0,114 0,437 0,139 0,425 0,135 0,54 0,172 0,606 0,193

31 0,32 0,101 0,361 0,115 0,44 0,14 0,429 0,137 0,554 0,176 0,611 0,194

32 0,322 0,101 0,364 0,116 0,443 0,141 0,433 0,138 0,547 0,174 0,617 0,196

33 0,324 0,103 0,367 0,117 0,445 0,142 0,436 0,139 0,55 0,175 0,623 0,198

34 0,326 0,104 0,371 0,118 0,447 0,142 0,44 0,14 0,553 0,176 0,628 0,2

35 0,327 0,104 0,373 0,119 0,449 0,143 0,443 0,141 0,556 0,177 0,633 0,201

36 0,329 0,105 0,377 0,12 0,451 0,144 0,446 0,142 0,559 0,178 0,639 0,203

37 0,33 0,105 0,38 0,121 0,454 0,145 0,449 0,143 0,563 0,179 0,645 0,205

38 0,333 0,106 0,384 0,122 0,455 0,145 0,452 0,144 0,565 0,18 0,65 0,207

39 0,335 0,107 0,386 0,123 0,457 0,145 0,454 0,145 0,568 0,181 0,655 0,208

40 0,336 0,107 0,389 0,124 0,459 0,146 0,457 0,145 0,57 0,181 0,659 0,21

43 0,339 0,108 0,397 0,126 0,467 0,149 0,464 0,148 0,574 0,183 0,668 0,213

45 0,34 0,108 0,399 0,127 0,468 0,149 0,468 0,149 0,579 0,184 0,678 0,216

50 0,346 0,11 0,408 0,13 0,474 0,151 0,477 0,152 0,588 0,187 0,694 0,221

55 0,352 0,112 0,415 0,132 0,48 0,153 0,484 0,154 0,596 0,19 0,704 0,224

60 0,355 0,113 0,421 0,134 0,484 0,154 0,491 0,156 0,603 0,192 0,713 0,227

65 0,358 0,114 0,425 0,135 0,488 0,155 0,496 0,158 0,607 0,193 0,721 0,23

70 0,36 0,115 0,429 0,137 0,493 0,157 0,501 0,159 0,61 0,194 0,728 0,232

75 0,361 0,115 0,433 0,138 0,496 0,158 0,506 0,161 0,613 0,195 0,735 0,234

80 0,363 0,116 0,436 0,139 0,499 0,159 0,509 0,162 0,615 0,196 0,739 0,235

Page 12: Gear Design

90 0,366 0,117 0,442 0,141 0,503 0,16 0,516 0,164 0,619 0,197 0,747 0,238

100 0,368 0,117 0,446 0,142 0,506 0,161 0,521 0,166 0,622 0,198 0,755 0,24

150 0,375 0,119 0,458 0,146 0,518 0,165 0,537 0,171 0,635 0,202 0,778 0,248

200 0,378 0,12 0,463 0,147 0,524 0,167 0,545 0,173 0,64 0,204 0,787 0,251

300 0,38 0,122 0,471 0,15 0,534 0,17 0,554 0,176 0,65 0,207 0,801 0,255

Rack 0,39 0,124 0,484 0,154 0,55 0,175 0,566 0,18 0,66 0,21 0,823 0,262

http://www.ecs.umass.edu/mie/labs/mda/dlib/machine/gear/gear2.html

Page 13: Gear Design

Helical Gears

IntroductionIntroductionIntroductionIntroduction

Helical gears are similar to spur gears except that the gears teeth are at an angle with the axis of the gears. A helical gear is termed right handed or left handed as determined by the direction the teeth slope away from the viewer looking at the top gear surface along the axis of the gear. ( Alternatively if a gear rests on its face the hand is in the direction of the slope of the teeth) . Meshing helical gears

must be of opposite hand. Meshed helical gears can be at an angle to each other (up to 90o ). The

helical gear provides a smoother mesh and can be operated at greater speeds than a straight spur gear. In operatation helical gears generate axial shaft forces in addition to the radial shaft force generated by normal spur gears. In operation the initial tooth contact of a helical gear is a point which develops into a full line contact as the gear rotates. This is a smoother cycle than a spur which has an initial line contact. Spur gears are generally not run at peripheral speed of more than 10m/s. Helical gears can be run at speed exceeding 50m/s when accurately machined and balanced.

Standards Standards Standards Standards ... The same standards apply to helical gears as for spur gears

• AGMA 2001-C95 or AGMA-2101-C95 Fundamental Rating factors and Calculation Methods for involute Spur Gear and Helical Gear Teeth

• BS 436-4:1996, ISO 1328-1:1995..Spur and helical gears. Definitions and allowable values of deviations relevant to corresponding flanks of gear teeth

• BS 436-5:1997, ISO 1328-2:1997..Spur and helical gears. Definitions and allowable values of deviations relevant to radial composite deviations and runout information

• BS ISO 6336-1:1996 ..Calculation of load capacity of spur and helical gears. Basic principles, introduction and general influence factors

• BS ISO 6336-2:1996..Calculation of load capacity of spur and helical gears. Calculation of surface durability (pitting)

• BS ISO 6336-3:1996..Calculation of load capacity of spur and helical gears. Calculation of tooth bending strength

• BS ISO 6336-5:2003..Calculation of load capacity of spur and helical gears. Strength and quality of materials

Helical gear parametersHelical gear parametersHelical gear parametersHelical gear parameters

A helical gear train with parallel axes is very similar to a spur gear with the same tooth profile and proportions. The primary difference is that the teeth are machined at an angle to the gear axis.

Helix Angle ..

The helix angle of helical gears β is generally selected from the range 6,8,10,12,15,20 degrees. The larger the angle the smoother the motion and the higher speed possible however the thrust loadings on the supporting bearings also increases. In case of a double or herringbone gear β values

Page 14: Gear Design

the two sets of teeth cancel each other allowing larger angles with no penalty

Pitch /module ..

For helical gears the circular pitch is measured in two ways The traverse circular pitch (p) is the same as for spur gears and is measured along the pitch circle The normal circular pitch p n is measured normal to the helix of the gear. The diametric pitch is the same as for spur gears ... P = z g /dg = z p /d p ....d= pitch circle dia (inches). The module is the same as for spur gears ... m = dg/z g = d p/z p.... d = pitch circle dia (mm).

Helical Gear geometrical proportions Helical Gear geometrical proportions Helical Gear geometrical proportions Helical Gear geometrical proportions

• p = Circular pitch = d g. π / z g = d p. π / z p • p n = Normal circular pitch = p .cosβ

• P n =Normal diametrical pitch = P /cosβ

• p x = Axial pitch = p c /tanβ

• m n =Normal module = m / cosβ

• α n = Normal pressure angle = tan -1 ( tanα.cos β )

• β =Helix angle

• d g = Pitch diameter gear = z g. m

• d p = Pitch diameter pinion = z p. m

• a =Center distance = ( z p + z g )* m n /2 cos β

• a a = Addendum = m

• a f =Dedendum = 1.25*m

• b = Face width of narrowest gear

Herringbone / double crossed helical gearsHerringbone / double crossed helical gearsHerringbone / double crossed helical gearsHerringbone / double crossed helical gears

Crossed Helical GearsCrossed Helical GearsCrossed Helical GearsCrossed Helical Gears

When two helical gears are used to transmit power between non parallel, non-intersecting shafts, they are generally called crossed helical gears. These are simply normal helical gears with non-parallel shafts. For crossed helical gears to operate successfully they must have the same pressure angle and the same normal pitch. They need not have the same helix angle and they do not need to be opposite hand. The contact is not a good line contact as for parallel helical gears and is often little more than a point contact. Running in crossed helical gears tend to marginally improve the area of contact.

The relationship between the shaft angles Ε and the helix angles β 1 & β2 is as follows

Page 15: Gear Design

Ε = (Same Helix Angle) β 1 + β 2 ......(Opposite Helix Angle) β 1 - β 2

For gears with a 90o crossed axis it is obvious that the gears must be the same hand.

The centres distance (a) between crossed helical gears is calculated as follows

a = m * [(z 1 / cos β 1) + ( z 1 / cos β 1 )] / 2

The sliding velocity Vsof crossed helical gears is given by

Vs = (V1 / cos β 1 ) = (V 2 / cos β 2 )

Strength and Durability calculations for Helical Gear TeethStrength and Durability calculations for Helical Gear TeethStrength and Durability calculations for Helical Gear TeethStrength and Durability calculations for Helical Gear Teeth

Designing helical gears is normally done in accordance with standards the two most popular series are listed under standards above: The notes below relate to approximate methods for estimating gear strengths. The methods are really only useful for first approximations and/or selection of stock gears (ref links below). — Detailed design of spur and helical gears should best be completed using :

a) Standards. b) Books are available providing the necessary guidance. c) Software is also available making the process very easy. A very reasonably priced and easy to use package is included in the links below (Mitcalc.com)

The determination of the capacity of gears to transfer the required torque for the desired operating life is completed by determining the strength of the gear teeth in bending and also the durability i.e of the teeth ( resistance to wearing/bearing/scuffing loads ) .. The equations below are based on methods used by Buckingham..

Bending

The Lewis formula for spur gears can be applied to helical gears with minor adjustments to provide an initial conservative estimate of gear strength in bending. This equation should only be used for first estimates.

σ = Fσ = Fσ = Fσ = Fbbbb / ( b / ( b / ( b / ( baaaa. m. Y ). m. Y ). m. Y ). m. Y )

• Fb = Normal force on tooth = Tangential Force Ft / cos β

• σ = Tooth Bending stress (MPa)

Page 16: Gear Design

• ba = Face width (mm)

• Y = Lewis Form Factor

• m = Module (mm)

When a gear wheel is rotating the gear teeth come into contact with some degree of impact. To allow for this a velocity factor is introduced into the equation. This is given by the Barth equation for milled profile gears.

K v = 6,1 / (6,1 + V )

V = the pitch line velocity = PCD.ω/2 The Lewis formula is thus modified as follows

σ = Fσ = Fσ = Fσ = Fbbbb / (K / (K / (K / (K vvvv. b. b. b. baaaa. m. Y ). m. Y ). m. Y ). m. Y )

The Lewis form factor Y must be determined for the virtual number of teeth z' = z /cos3β The bending stress

resulting should be less than the allowable bending stress Sb for the gear material under consideration. Some sample values are provide on this page ef Gear Strength Values

Page 17: Gear Design

Surface Strength

The allowable gear force from surface durability considerations is determined approximately using the simple equation as follows

FFFFwwww = K = K = K = K vvvv d d d d pppp b b b b aaaa Q K / cos Q K / cos Q K / cos Q K / cos2222ββββ

Q = 2. dQ = 2. dQ = 2. dQ = 2. dgggg /( d /( d /( d /( dpppp + d + d + d + dpppp ) = 2.z ) = 2.z ) = 2.z ) = 2.zgggg /( z /( z /( z /( zpppp +z +z +z +zpppp ) ) ) )

Fw = The allowable gear load. (MPa) K = Gear Wear Load Factor (MPa) obtained by look up ref Gear Strength Values

Lewis Form factor for Teeth profile α = 20o , addendum = m, dedendum = 1.25m

Number of teeth

Y Number of teeth

Y Number of teeth

Y Number of teeth

Y Number of teeth

Y

12 0.245 17 0.303 22 0.331 34 0.371 75 0.435

13 0.261 18 0.309 24 0.337 38 0.384 100 0.447

14 0.277 19 0.314 26 0.346 45 0.401 150 0.460

15 0.290 20 0.322 28 0.353 50 0.409 300 0.472

16 0.296 21 0.328 30 0.359 60 0.422 Rack 0.485

Page 18: Gear Design

Material Properties Tables for Spur, Helical and Bevel Gears

Detailed gear designs should be based on more accurate information available using the relevant standards..

Suffix 1 relate to the driving gear (generally the pinion) Suffix 2 relate to the driven gear (generally the gear) Cp = Imperial elastic coefficient ZE = ISO elastic coefficient

Design factors for gear pairs

Material Poissons

Ratio (ν) Young's Modulus ( E )

K

for α = 20 deg

Cp Ze

Pinion Gear Pinion Pinion Pinion Gear Pinion Gear

Allowable Surface

Endurance Stress ( S e)

psi psi MPa MPa psi MPa psi MPa √psi √MPa

Steel.BHN Av=150 0.3 0.3 3.00E+07 3.00E+07 2.07E+05 2.07E+05 50000 345 41 0.281 2291 190

Steel.BHN Av=175 0.3 0.3 3.00E+07 3.00E+07 2.07E+05 2.07E+05 60000 414 59 0.404 2291 190

Steel.BHN Av=200 0.3 0.3 3.00E+07 3.00E+07 2.07E+05 2.07E+05 70000 483 80 0.550 2291 190

Steel.BHN Av=225 0.3 0.3 3.00E+07 3.00E+07 2.07E+05 2.07E+05 80000 552 104 0.719 2291 190

Steel.BHN Av=250 0.3 0.3 3.00E+07 3.00E+07 2.07E+05 2.07E+05 90000 621 132 0.910 2291 190

Steel.BHN Av=275 0.3 0.3 3.00E+07 3.00E+07 2.07E+05 2.07E+05 100000 689 163 1.123 2291 190

Steel.BHN Av=300 0.3 0.3 3.00E+07 3.00E+07 2.07E+05 2.07E+05 110000 758 197 1.359 2291 190

Steel.BHN Av=325 0.3 0.3 3.00E+07 3.00E+07 2.07E+05 2.07E+05 120000 827 235 1.617 2291 190

Steel.BHN Av=350 0.3 0.3 3.00E+07 3.00E+07 2.07E+05 2.07E+05 130000 896 275 1.898 2291 190

Steel.BHN Av=375 0.3 0.3 3.00E+07 3.00E+07 2.07E+05 2.07E+05 140000 965 319 2.201 2291 190

Steel.BHN Av=400 0.3 0.3 3.00E+07 3.00E+07 2.07E+05 2.07E+05 150000 1034 366 2.527 2291 190

Stl BHN=150 Cast Iron 0.3 0.211 3.00E+07 2.20E+07 2.07E+05 1.52E+05 50000 345 48 0.332 2077 172

Stl BHN=250 Cast Iron 0.3 0.211 3.00E+07 2.20E+07 2.07E+05 1.52E+05 70000 483 94 0.650 2077 172

Stl BHN=350 Cast Iron 0.3 0.211 3.00E+07 2.20E+07 2.07E+05 1.52E+05 90000 621 156 1.075 2077 172

Stl BHN=150 Phos Bros 0.3 0.38 3.00E+07 1.45E+07 2.07E+05 1.00E+05 59000 407 62 0.600 1888 157

Stl BHN=250 Phos Bros 0.3 0.38 3.00E+07 1.45E+07 2.07E+05 1.00E+05 65000 448 100 0.728 1888 157

Stl BHN=350 Phos Bros 0.3 0.38 3.00E+07 1.45E+07 2.07E+05 1.00E+05 85000 586 184 1.245 1888 157

Cast Iron Cast Iron 0.211 0.211 2.20E+07 2.20E+07 1.52E+05 1.52E+05 90000 621 264 1.240 1914 159

Cast Iron Phos Bros 0.211 0.38 2.20E+07 1.45E+07 1.52E+05 1.00E+05 83000 572 234 1.328 1763 146

Gear Materials PropertiesGear Materials PropertiesGear Materials PropertiesGear Materials Properties

Page 19: Gear Design

Material................ Specification.............. Ultimate Tensile Strength

Yield Tensile Strength

Tooth Hardness - Core

Tooth Hardness - Side

Allowable Endurance Stress

Allowable Bending Stress

Young's Modulus

Poison's Ratio

Rm Rp(0.2) VPN VPN Se Sb E

MPa MPa HV HV MPa MPa GPa

1 Grey Cast Iron BS EN 1561:1997 EN-GJL-200

200 100 200 200 340 95 91 0.25

2 Grey Cast Iron BS EN 1561:1997 EN-GJL-250

250 125 220 220 350 105 105 0.25

3 Grey Cast Iron BS EN 1561:1997 EN-GJL-300

300 150 240 240 360 120 113 0.25

4 Ductile Cast Iron BS EN 1563:1997 EN-GJS 600-2

600 370 190 190 430 315 169 0.2

5 Ductile Cast Iron BS EN 1563:1997 EN-GJS 700-2

700 420 230 230 510 325 169 0.2

6 Ductile Cast Iron BS EN 1563:1997 ENGJS 800-2

800 480 250 250 550 345 169 0.2

7 Carbon Cast Steel Normalised

BS 3100:1991 A3, A5 **

500 260 150 150 420 300 206 0.3

8 Carbon Cast Steel Normalised

BS 3100:1991 A3, A5 **

590 300 180 180 480 336 206 0.3

9 Alloy Cast Steel Normalised

36Mn5 (1,1167) 700 340 210 210 540 372 206 0.3

10 Alloy Cast Steel Heat Treated

36Mn5 (1,1167) 750 400 220 220 560 384 206 0.3

11 Alloy Cast Steel Normalised

BS EN 10213-2:1996 G17CrMo511

650 380 200 200 520 360 206 0.3

12 Alloy Cast Steel Heat Treated

BS EN 10213-2:1996 G17CrMo511

800 550 245 245 610 414 206 0.3

13 Structural Steel Untreated

BS EN 10025-1:2004 E295

490 295 150 150 370 330 206 0.3

14 Structural Steel Untreated

BS EN 10025-2:2004 S335J2G3

510 335 155 155 380 336 206 0.3

15 Structural Steel Untreated

BS EN 10025-2:2004 E335

588 335 175 175 420 360 206 0.3

16 Structural Steel Untreated

BS EN 10025-2:2004 E360

686 360 205 205 480 396 206 0.3

17 Carbon Structural Steel normalised

BS EN 10083-2 C45

540 325 155 155 430 356 206 0.3

18 Carbon Structural Steel heat treated

BS EN 10083-2:1991 C45

640 390 200 200 520 410 206 0.3

19 Carbon Structural Steel normalised

BS EN 10083-1:1991 C60/ER

660 380 200 200 520 410 206 0.3

20 Carbon Structural Steel heat treated

BS EN 10083-1:1991 C60E/R

740 440 235 235 590 452 206 0.3

21 Alloy Structural Steel Heat Treated

BS EN 10083-1:1991 37Cr4

883 637 285 285 690 512 206 0.3

22 Alloy Structural Steel Heat Treated

42CrV6 (1,7561)

980 850 300 300 720 530 206 0.3

23 Alloy Structural Steel Heat Treated

31NiCr14 (1,5755)

932 785 290 290 700 518 206 0.3

24 Carbon Cast Steel tooth face hardened

BS 3100:1991 A3,A5,AW2

590 300 180 600 1140 316 206 0.3

25 Carbon Cast Steel tooth face hardened

36Mn5 (1,1167) 700 340 210 600 1140 352 206 0.3

26 Carbon Structural Steel tooth face hardened

BS EN 10083-2:1991 C50

640 390 200 600 1140 390 206 0.3

27 Alloy Structural Steel tooth face hardened

BS EN 10083-1:1991 37Cr4

785 539 250 600 1140 450 206 0.3

28 Alloy Structural Steel 42CrV6 (1,7561) 980 850 315 600 1160 528 206 0.3

Page 20: Gear Design

tooth face hardened

29 Alloy Structural Steel tooth face hardened

BS EN 10083-1:1991 34CrNiMo6

965 750 300 600 1160 705 206 0.3

30 Alloy Structural Steel Nitrided

42MnV7 (1,5223) 800 620 250 550 930 580 206 0.3

31 Alloy Structural Steel Nitrided

30CrV9 800 600 250 800 1180 705 206 0.3

32 Alloy Structural Steel Nitrided

30CrMoV9 (1,7707) 800 600 250 800 1180 705 206 0.3

33 Alloy Structural Steel Nitrided

BS EN 10083-1:1991 34CrNiMo6

965 750 300 750 1180 730 206 0.3

34 Alloy Structural Steel Nitro Case hardened

BS EN 10083-1:1991 37Cr4

1570 1350 485 615 1288 740 206 0.3

35 Carbon Structural Steel case hardened

BS EN 10277-2:1999 C10

440 275 135 650 1210 500 206 0.3

36 Carbon Structural Steel case hardened

>C15E (1,1149 ) 495 295 150 650 1210 500 206 0.3

37 Alloy Structural Steel case hardened

16MnCr5 (1,7131) 785 588 250 650 1270 700 206 0.3

38 Alloy Structural Steel case hardened

35CrMo4 880 685 285 650 1270 700 206 0.3

39 Alloy Structural Steel case hardened

15NiCr6 880 635 285 650 1270 700 206 0.3

40 Alloy Structural Steel case hardened

14NiCr14(1.5732) 932 735 300 650 1270 700 206 0.3

41 Carbon Steel Nitro caburised

BS EN 10083-1:1991 C60E/R

740 440 235 235 800 650 206 0.3

42 Carbon Steel tooth face hardened

BS EN 10083-2 C50

640 390 200 600 1140 605 206 0.3

43 Alloy Steel tooth face hardened

BS EN 10083-1 37Cr4

900 700 250 600 1140 605 206 0.3

44 Bronze Sand Cast 276 207 39 100 0.38

45 Bronze Heat Treated 621 448 163 100 0.38