Spur GearsIntroduction..... 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 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)
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.
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 - 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
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.
Spur 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 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.
Notes Ferrous metals
applications Large moderate power, commercial gears Power gears with medium rating to commercial quality Power gears with medium rating to commercial/medium quality Highest power requirement. For precision and high precisiont Corrosion resistance with low power ratings. Up to precision quality
Low Cost easy to machine with high damping Low cost, reasonable strength
Good machining, can be heat treated
Heat Treatable to provide highest strength and durability
Stainless Steels (Aust)
Good corrosion resistance. Nonmagnetic
Stainless Steels (Mart)
Low to medium Hardenable, Reasonable corrosion power ratings Up to resistance, magnetic high precision levels of quality Non-Ferrous metals Light duty instrument gears up to high precision quality
Light weight, non-corrosive and good machinability
Low cost, non-corrosive, excellent machinability
low cost commercial quality gears. Quality up to medium precision
For use with steel Excellent machinability, low friction power gears. and good compatability with steel Quality up to high precision Light weight with poor corrosion resistance Ligh weight low load gears. Quality up to medium precision Special gears for thermal applications to commercial quality
Low coefficient of thermal expansion. Poor machinability
Special light weight High strength, for low weight, good high strength gears corrosion resistance to medium precision Low cost with low precision and strength Low cost, low quality, moderate strength Non metals High production, low quality gears to commercial quality High production, low quality to moderate commercial quality Long life , low load bearings to commercial quality High production, low quality to moderate commercial quality Long life at low loads to commercial quality Special low friction gears to commercial quality
Sintered powder alloys
Wear resistant, low water absorbtion Low cost, low quality, moderate strength No lubrication, no lubricant, absorbs water Low friction and no lubrication
Equations 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