Actual Output Speed (Gear)

Spur gears

Spur gears have teeth that are straight and arranged parallel to the axis of the shaft that carries the gear. The curved shape of the faces of the spur gear teeth has a special geometry called an involute curve. This shape makes it possible for two gears to operate together with smooth, positive transmission of power. The shafts carrying gears are parallel.

Spur gear design

 Actual output speed (gear) n p nG  VR nP = rotational speed of the pinion VR = gear ratio N VR  G N P NG ,N P = number of gear, pinion teeth. The spreadsheet computes the approximate number of gear teeth to produce nGd the desired speed from NG  N P ( nGd = desired output speed). But, of nP course, the number of teeth on any gear must be an integer, and the actual value of NG is selected by the designer.

Spur gear geometry For full depth involute teeth in the diametral pitch system

 Pitch diameter N D  Pd  Diametral Pitch N Pd  D  Outside diameter N  2 Do  Pd  Addendum 1 a  Pd  Dedendum if Pd  20 1.25 b  Pd if Pd  20 1.2 b   0.002 Pd  Clearance

1 if Pd  20 0.25 c  Pd if Pd  20 0.2 c   0.002 Pd  Root diameter DR  D  2b  Base circle diameter Db  D cos   Circular pitch  D p  N  Whole depth ht  a  b  Working depth hk  2a  Tooth thickness  t  2Pd  Center distance D  D C  G P 2

Bending geometry factor, J, is dependent on the number of teeth of gear for which geometry factor is desired and on the number of teeth in mating gear. Values can be found from AGMA 908-B89(R1995).

Pitting geometry factor, I, is dependent on the tooth geometry and on gear ratio. Values can be found from AGMA Standard 218.01.

Force and speed factors

 Pitch line speed DP nP Vt  12  Tangential force 33000  (P ) Wt  Vt or 126000  (P ) Wt  nD where: P = transmitted power

 Radial force

2 Wr  Wt tan   Normal force Wt Wn  cos   Expected bending stress Wt Pd St  KoK s K mK B Kv F  J where: J = bending geometry factor K o = overload factor K s = size factor K m = load-distribution factor K B = rim thickness factor Kv = dynamic factor.

Expected bending stress formula for SI units is given by

σb = (Wt / F.m.J) (KO.KS.Km.KB.KV) Where;

σb = bending stress [MPa], m = module [mm], F = face width [mm],

Wt = transmitted load [N]

The AGMA indicates that the size factor KS can be taken to be 1.00 for most gears. But for gears with large-size teeth or large face widths, a value greater than 1.00 recommended. Gear design programs compute the size factor automatically.

The determination of load-distribution factor Km is based on many variables in the design of the gears themselves as well as in the shafts, bearings, housings, and the structure in which the gear drive is installed. Therefore, it is one of the most difficult factors to specify. Much analytical and experimental work is continuing on the determination of values for K m . We will use the following equation for computing the value of the load- distribution factor: K m  1.0  C pf  C ma where: C pf = pinion proportion factor is dependent on face width and pitch diameter C ma = mesh alignment factor.

The dynamic factor, Kv , accounts for the fact that the load is assumed by a tooth with some degree of impact and that the actual load subjected to the tooth is higher than the transmitted load alone. The value of Kv

3 depends on the accuracy of tooth profile, the elastic properties of tooth, and the speed with which the teeth come into contact.

AGMA Standard 2001-C95 gives recommended values for Kv based on the AGMA quality number, Qv , and the pitch line velocity.

. Gears in typical machine design would have AGMA quality ratings of 5 through 7, which are for gears made by hobbing or shaping with average to good tooling. . If the teeth are finish-ground or shaved to improve the accuracy of the tooth profile and spacing, quality numbers in the 8 - 11 range should be used. Under very special conditions where teeth of high precision are used in applications where there is little chance of developing external dynamic loads, higher quality numbers can be used. . If the teeth are cut by form milling, factors lower than those found from QV = 5 should be used. . Note that the quality 5 gears should not be used at pitch line speed above 12.7 m/s (2500 ft/min). Note that the dynamic factors are approximate.

Expected contact stress

Wt KoK s K mKv Sc  C p FDp I where:

C P = elastic coefficient that depends on the material of both the pinion and the gear.

C P = 2300 for two steel gears. The design program automatically selects the appropriate value after the user specifies the materials.

Procedure for selecting materials for bending stress

K R SF  St  Sat YN where: K R = reliability factor SF = factor of safety YN = stress cycle factor for bending.

AGMA Standard 2001-C95 allows the determination of the life adjustment factor, YN , if the teeth of the gear being analyzed are expected to experience a

4 number of cycles of loading much different from 107 . Note that the general type of material is a factor for the lower number of cycles. For the higher number of cycles, a range is indicated by a shaded area.

Expected number of cycles of loading Nc  (60)(L)(n)(q ) where: L = design life in hours n = rotational speed in rpm q = number of load applications per revolution.

Procedure for selecting materials for contact stress K R SF  Sc  S ac Z N where: Z N = pitting resistance stress cycle factor.

AGMA Standard 2001-C95 specifies the determination of the stress cycle factor, ZN . If the teeth of the gear being analyzed are expected to experience a number of cycles of loading much different from 107 , a factor should be used. The user specifies the desired life for the system in hours and the program computes the values for YN and Z N .

After computing the values for allowable bending stress number, Sat , and for allowable contact stress number, S ac , you should go to the data in AGMA Standard 2001-C95, to select a suitable material. Consider first whether the material should be steel, cast iron, bronze, or plastic. Then consult the related tables of data.

5 Pressure angle

The pressure angle is the angle between the tangent to the pitch circles and the line drawn normal (perpendicular) to the surface of the gear tooth

The line normal to the gear teeth is sometimes referred to as the line of action. When two gear teeth are in mesh and are transmitting power, the force transferred from the driver to the driven gear tooth acts in a direction along the line of action. Also, the actual shape of the gear tooth depends on the pressure angle.

Standard values of the pressure angle are established by gear manufacturers, and the pressure angle of two gears in mesh must be the same. Current 1 ° standard pressure angles are, although the 14 tooth form is considered to be 2 obsolete. Although it is still available, it should be avoided for new designs. The 20 degrees tooth form is the most readily available at this time. The advantages and disadvantages of the different values of pressure angle relate to the strength of the teeth, the occurrence of interference, and the magnitude of forces exerted on the shaft.

6 7 Face width

The face width can be specified once the diametral pitch is chosen. Although a wide range of face widths is possible, the following limits are used for general machine drive gears: 8 16  F  Pd Pd 12 Nominal value of F  Pd For SI System of Units,

3 π m< F < 5 π m

Nominal value of F = 4 π m

Also, the face width normally is not greater than pitch diameter of the pinion.

An upper limit is placed on the face width to minimize problems with alignment. A very wide face width increases the chance for less than full face loading of the teeth. When the face width is less than the lower limit, it is probable that a more compact design can be achieved with different pitch.

F Notice that  2.00 is recommended. D p

Number of pinion teeth

In the selection of the number of teeth on the pinion, the designer must be aware of potential interference. For certain combinations of numbers of teeth in a gear pair, there is interference between the tip of the teeth on the pinion and the fillet or root of the teeth on the gear. Obviously this cannot be tolerated because the gears simply will not mesh. The probability that interference will occur is greatest when a small pinion drives a large gear, with the worst case being a small pinion driving a rack. A rack is a gear with a straight pitch line; it can be thought of as a gear with an infinite pitch diameter.

It is the designer’s responsibility to ensure that interference does not occur in a given application. The surest way to do this is to control the minimum number of teeth in the pinion.

8 - If a designer wants to be sure that there will not be interference between 1 ° any two gears when using the 14 , full-depth, involute system, the pinion 2 of the gear pair must have no fewer than 32 teeth. - For the 20° , full-depth, involute system, using no fewer than 18 teeth will ensure that no interference occurs. - For the 25° , full-depth, involute system, using no fewer than 12 teeth will ensure that no interference occurs.

Design life

Design life is, indeed, a design decision based on the application. As a guideline, this set of data was created for use in bearing design and is used here for gearing:

Application Design life L (h) Domestic appliances from 1000 to 2000 Aircraft engines from 1000 to 4000 Automotive from 1500 to 5000 Agricultural equipment from 3000 to 6000 Elevators, industrial fans, multipurpose gearing from 8000 to 15,000 Electric motors, industrial blowers, general industrial from 20,000 to 30,000 machines Pumps and compressors from 40,000 to 60,000 Critical equipment in continuous 24-h operation from 100,000 to 200,000

Number of load applications per revolution, q

The normal number of load applications per revolution for any given tooth is typically, of course, one. But consider the case of an idler gear that serves as both a driven and driving gear in a gear train. It receives two cycles of load per revolution: one as it receives power from and one as it delivers power to its mating gears. Also, in certain types of gear trains, one gear may deliver power to two or more gears mating with it. Gears in a planetary gear train often have this characteristic.

Rim thickness

The rim thickness factor, K B , accounts for a rim that may be too thin. The basic analysis used to develop the Lewis equation assumes that the gear tooth behaves as a cantilever attached to a perfectly rigid support structure at its base. If the rim of the gear is too thin, it can deform and

9 cause the point of maximum stress to shift from the area of the gear-tooth fillet to a point within the rim.

The key geometry parameter is called the backup ratio, mB , where t R mB  ht t R = rim thickness; ht = whole depth of the gear tooth.

For mB >1.2, the rim is sufficiently strong and stiff to support the tooth, and K B =1.0.

For mB <1.2, rim thickness factor determined:  2.242    K B  1.6 ln   mB 

When a solid gear blank is used, input a large value (say tR > 1.0 inch) for rim thickness. The resulting value is K B  1 .

Gear application

The mesh alignment factor, Cma, is dependent on the application of the gear pair.

Open gearing refers to drive systems in which the shafts are supported in bearings that are mounted on structural elements of

10 machines with the expectation that relatively large misalignments will result.

In commercial-quality enclosed gear units, the bearings are mounted in a specially designed housing that provides more rigidity than for open gearing, but for which the tolerances on individual dimensions are fairly loose.

The precision enclosed gear units are made to tighter tolerances.

Extra-precision enclosed gear units are made to exacting precision and are often adjusted at assembly to achieve excellent alignment of the gear teeth.

Elastic coefficient

The value of this coefficient depends on the material of the pinion and the gear. For instance if the pinion and the gear are both steel the elastic coefficient Cp =2300. Simply click on the database icon and you will be prompted to input the material type for the pinion and the gear, and the appropriate value will be inserted for calculations. For a complete listing of values, AGMA standard 2001-C95 can be referenced.

Overload factor, Ko

Overload factors consider the probability that load variations, vibrations, shock, speed changes, and other application-specific conditions may result in peak loads greater than Wt being applied to the teeth during operation. A careful analysis of actual conditions should be made, and the AGMA Standard 2001-C95 gives no specific values for overload factor.

The primary considerations are the nature of both the driving power source and the driven machine. An overload factor of 1.00 would be applied for a perfectly smooth electric motor driving a perfectly smooth generator through a gear type speed reducer. Any rougher conditions call for a value of overload factor Ko greater than 1.00. For power sources we will use the following:

- Uniform: Electric motor or constant-speed gas turbine - Light shock: Water turbine, variable speed drive - Moderate shock: Multi-cylinder engine

Examples of the roughness of driven machines include the following: - Uniform: Continuous-duty generator - Light shock: Fans and low-speed centrifugal pumps, liquid agitators, variable-duty generators, uniformly loaded conveyors, rotary positive displacement pumps

11 - Moderate shock: High-speed centrifugal pumps, reciprocating pumps and compressors, heavy-duty conveyors, machine tool drives, concrete mixers, textile machinery, meat grinders, saws - Heavy shock: Rock crushers, punch press drives, pulverizers, processing mills, tumbling barrels, wood chippers, vibrating screens, railroad car dumpers

Factor of safety

The factor of safety may be used to account for the following: - Uncertainties in the design analysis - Uncertainties in material characteristics - Uncertainties in manufacturing tolerances

It may also be used to provide an extra measure of safety in critical applications.

No general guidelines are published, and designers must evaluate the conditions of each application. Note, however, that many of the factors often considered to be a part of a factor of safety in general design practice have already been included in the calculations for the allowable stresses. Therefore, a modest value for factor of safety should suffice, between 1.00 and 1.50.

Hardness ratio factor, CH

Good gear design practice calls for making the pinion teeth harder than the gear teeth so that the gear teeth are smoothed and work-hardened during operation. This increases the gear capacity with regard to pitting resistance and is accounted for by the factor C H .

Data for C H for through-hardened gears (AGMA Standard 2001-C95) depend on the ratio of the hardness of pinion and the hardness of gear, expressed as NG the Brinell hardness number, and on the gear ratio where mG  . NP Use the given curves in the graphic help for hardness ratios between 1.2 and 1.7. For hardness ratios under 1.2, use C H =1.00. For hardness ratios over 1.7, use the value of C H for 1.7, as no substantial additional improvement is gained.