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ADDENDUM (a) is the height by which a tooth projects beyond the pitch circle or pitch line.
BASE DIAMETER (D/b )is the diameter of the base cylinder from which the involute portion of a tooth profile is generated.
BACKLASH (B) is the amount by which the width of a tooth space exceeds the thickness of the engaging tooth on the pitch circles. As actually indicated by measuring devices, backlash may be determined variously in the transverse, nor- mal, or axial-planes, and either in the direction of the pitch cir- cles or on the line of action. Such measurements should be corrected to corresponding values on transverse pitch circles for general comparisons.
BORE LENGTH is the total length through a gear, sprocket, or coupling bore.
CIRCULAR PITCH (p) is the distance along the pitch circle or pitch line between corresponding profiles of adjacent teeth.
CIRCULAR THICKNESS (t) is the length of arc between the two sides of a gear tooth on the pitch circle, unless otherwise specified.
CLEARANCE-OPERATING (c) is the amount by which the dedendum in a given gear exceeds the addendum of its mat- ing gear.
CONTACT RATIO (m/c) in general, the number of angular pitches through which a tooth surface rotates from the begin- ning to the end of contact.
DEDENDUM (b) is the depth of a tooth space below the pitch line. It is normally greater than the addendum of the mating gear to provide clearance.
DIAMETRAL PITCH (P) is the ratio of the number of teeth to the pitch diameter.
FACE WIDTH (F) is the length of the teeth in an axial plane. FILLET RADIUS (r/f) is the radius of the fillet curve at the base of the gear tooth.
FULL DEPTH TEETH are those in which the working depth equals 2.000 divided by the normal diametral pitch.
GEAR is a machine part with gear teeth. When two gears run together, the one with the larger number of teeth is called the gear.
HUB DIAMETER is outside diameter of a gear, sprocket or coupling hub.
HUB PROJECTION is the distance the hub extends beyond the gear face.
INVOLUTETEETH of spur gears, helical gears and worms are those in which the active portion of the profile in the trans- verse plane is the involute of a circle.
LONG- AND SHORT-ADDENDUM TEETH are those of engaging gears (on a standard designed center distance) one of which has a long addendum and the other has a short addendum.
KEYWAY is the machined groove running the length of the NORMAL DIAMETRAL PITCH (P/n) is the value of the bore. A similar groove is machined in the shaft and a key fits into this opening. diametral pitch as calculated in the normal plane of a helical gear or worm.
NORMAL PLANE is the plane normal to the tooth surface at a pitch point and perpendicular to the pitch plane. For a helical gear this plane can be normal to one tooth at a point laying in the plane surface. At such point, the normal plane contains the line normal to the tooth surface and this is normal to the pitch circle.
NORMAL PRESSURE ANGLE in a normal plane of helical tooth. OUTSIDE DIAMETER (D/0) is the diameter of the addendum (outside) circle.
PITCH CIRCLE is the circle derived from a number of teeth and a specified diametral or circular pitch. Circle on which spacing or tooth profiles is established and from which the tooth proportions are constructed.
PITCH CYLINDER is the cylinder of diameter equal to the pitch circle.
PINION is a machine part with gear teeth. When two gears run together, the one with the smaller number of teeth is called the pinion.
PITCH DIAMETER (D) is the diameter of the pitch circle. In parallel shaft gears, the pitch diameters can be determined directly from the center distance and the number of teeth.
PRESSURE ANGLE (?) is the angle at a pitch point between the line of pressure which is normal to the tooth surface, and the plane tangent to the pitch surface. In involute teeth, pressure angle is often described also as the angle between the line of action and the line tangent to the pitch circle. Standard pressure angles are established in connection with standard gear-tooth proportions.
ROOT DIAMETER (D) is the diameter at the base of the tooth Space. PRESSURE ANGLE?TM)OPERATING () is determined by the center distance at which the gears operate. It is the pressure angle at the operating pitch diameter. TIP RELIEF is an arbitrary modification of a tooth profile whereby a small amount of material is removed near the tip of the gear tooth.(D)
UNDERCUT is a condition in generated gear teeth when any part of the fillet curve lies inside a line drawn tangent to the working profile at its point of juncture with the fillet.(D) WHOLEDEPTH (h/t) is the total depth of a tooth space, equal to addendum plus dedendum, equal to the working depth plus variance.
WORKING DEPTH (h/k ) is the depth of engagement of two gears; that is, the sum of their addendums.
Gear teeth could be manufactured with a wide variety of shapes and profiles. The involute profile is the most commonly used system for gearing today, and all Boston spur and helical gears are of involute form.
An involute is a curve that is traced by a point on a taut cord unwinding from a circle, which is called a BASECIRCLE. The involute is a form of spiral, the curvature of which becomes straighter as it is drawn from a base circle and eventually would become a straight line if drawn far enough.
An involute drawn from a larger base circle will be less curved (straighter) than one drawn from a smaller base circle. Similarly, the involute tooth profile of smaller gears is considerably curved, on larger gears is less curved (straighter), and is straight on a rack, which is essentially an infinitely large gear.
Involute gear tooth forms and standard tooth proportions are specified in terms of a basic rack which has straight-sided teeth, for involute systems.
All stock gears are made in accordance with the diametral pitch system. The diametral pitch of a gear is the number of teeth in the gear for each inch of pitch diameter. Therefore, the diametral pitch determines the size of the gear tooth.
Pressure angle is the angle at a pitch point between the line of pressure which is normal to the tooth surface, and the plane tan- gent to the pitch surface.The pressure angle, as defined in this catalog, refers to the angle when the gears are mounted on their standard center distances. Boston Gear manufactures both 14-1/2?and 20?PA, involut full depth system gears. While 20?PA is generally recognize as having higher load carrying capacity, 14-1/2?PA gears hav extensive use. The lower pressure angle results in less change in backlash due to center distance variation and con- centricity errors. It also provides a higher contact ratio and consequent smoother, quieter operation provided that under- cut of teeth is not present.
For convenience, Tooth Proportions of various standard diametral pitches of Spur Gears are given below.
Stock spur gears are cut to operate at standard center dis- tances. The standard center distance being defined by: Standard Center Distance = (Pinion PD + Gear PD)/2
When mounted at this center distance, stock spur gears will have the following average backlash:
An increase or decrease in center distance will cause an increase or decrease in backlash.
Since, in practice, some deviation from the theoretical stan- dard center distance is inevitable and will alter the backlash, such deviation should be as small as possible. For most appli- cations, it would be acceptable to limit the deviation to an increase over the nominal center distance of one half the aver- age backlash. Varying the center distance may afford a practi- cal means of varying the backlash to a limited extent. The approximate relationship between center distance and backlash change of 14-1/2?and 20?pressure angle gears shown below:
For 14-1/2?(R)CChange in Center Distance = 1.933 x Change in Backla For 20?(R)CChange in Center Distance = 1.374 x Change in Backla From this, it is apparent that a given change in center dis- tance, 14-1/2?gears will have a smaller change in backlas than 20?gears. This fact should be considered in cases wher backlash is critical.
When the number of teeth in a gear is small, the tip of the mating gear tooth may interfere with the lower portion of the tooth pro- file. To prevent this, the generating process removes material at this point. This results in loss of a portion of the involute adjacent to the tooth base, reducing tooth contact and tooth strength.
On 14-1/2?PA gears undercutting occurs where a number of teeth is less than 32 and for 20?PA less than 18. Since thia condition becomes more severe as tooth numbers decrease, it is recommended that the minimum number of teeth be 16 for 14-1/2?PA and 13 for 20?PA
In a similar manner INTERNAL Spur Gear teeth may interfere when the pinion gear is too near the size of its mating internal gear. The following may be used as a guide to assure proper operation of the gear set. For 14-1/2?PA, the difference in tooth numbers between the gear and pinion should not be less than 15. For 20?PA the difference in tooth numbers should no be less than 12.
Gear failure can occur due to tooth breakage (tooth stress) or surface failure (surface durability) as a result of fatigue and wear. Strength is determined in terms of tooth-beam stresses for static and dynamic conditions, following well established for- mula and procedures. Satisfactory results may be obtained by the use of Barth°Os Revision to the Lewis Formula, which consid ers beam strength but not wear. The formula is satisfactory for commercial gears at Pitch Circle velocities of up to 1500 FPM. It is this formula that is the basis for all Boston Spur Gear ratings. METALLIC SPUR GEARS
W=SFY/ P(600/(600+V))
W=Tooth Load, Lbs. (along the Pitch Line)
S =Safe Material Stress (static) Lbs. per Sq. In. (Table II)
F=Face Width, In.
Y =Tooth Form Factor (Table I)
P=Diametral Pitch
D=Pitch Diameter
V=Pitch Line Velocity, Ft. per Min. = .262 x D x RPM For NON-METALLIC GEARS, the modified Lewis Formula shown below may be used with (S) values of 6000 PSI for Phenolic Laminated material.
W=SFY/p(150/(200 + V) + .25)
The information contained in the Spur Gear section is also pertinent to Helical Gears with the addition of the following:
HELIX ANGLE () is the angle between any helix and an element of its cylinder. In helical gears, it is at the pitch diameter unless otherwise specified.
LEAD (L) is the axial advance of a helix for one complete turn, as in the threads of cylindrical worms and teeth of helical gears.
NORMAL DIAMETRAL PITCH (P/n ) is the Diametral Pitch as calculated in the normal plane.
HAND (R)C Helical Gears of the same hand operate at righ angles, see Fig. 1
Helical Gears of opposite hands run on parallel shafts. Fig. 2
All Boston Helicals are cut to the Diametral Pitch system, resulting in a Normal Pitch which is lower in number than the Diametral Pitch.
INVOLUTE?TM)The Helical tooth form is involute in the plane o rotation and can be developed in a manner similar to that of the Spur Gear. However, unlike the Spur Gear, which may be viewed as two-dimensional, the Helical Gear must be viewed as three-dimensional to show change in axial features.
Helical gears offer additional benefits relative to Spur Gears, those being:
Improved tooth strength due to the elongated helical wrap- around.
Increased contact ratio due to the axial tooth overlap.
Helical Gears thus tend to have greater load carrying capac- ity than Spur Gears of similar size.
Due to the above, smoother operating characteristics are apparent.
When Helical gears are operated on other than Parallel shafts, the tooth load is concentrated at a point, with the result that very small loads produce very high pressures. The sliding velocity is usually quite high and, combined with the concentrated pressure, may cause galling or excessive wear, especially if the teeth are not well lubricated. For these reasons, the tooth load which may be applied to such drives is very lim- ited and of uncertain value, and is perhaps best determined by trial under actual operating conditions. If one of the gears is made of bronze, the contact area and thereby the load carrying capacity, may be increased, by allowing the gears to ?run in°± in their operating position, under loads which graduall increase to the maximum expected.
As a result of the design of the Helical Gear tooth, an axial or thrust load is developed. Bearings must be adequate to absorb this load. The thrust load direction is indicated below. The magnitude of the thrust load is based on calculated Horsepower.
Axial Thrust Load =126,050 x HP/ (RPM x Pitch Diameter)
Boston Helicals are all 45?Helix Angle, producing a tangentia force equal in magnitude to the axial thrust load. A separating force is also imposed on the gear set based on calculated Horsepower.
Separating Load = Axial Thrust Load x .386 Above formulae based on Boston 45?Helix Angle and 14-1/2 Normal Pressure Angle.
Gear geometry for both straight and spiral tooth Miter and Bevel gears is of a complex nature and this text will not attempt to cover the topic in depth.
The basic tooth form is a modification to the involute form and is the common form used in production today. All Boston stan- dard stock Miter and Bevel gears are manufactured with a 20? Pressure Angle. Bevel gears are made in accordance with A.G.M.A. specifications for long and short Addendum system for gears and pinions (pinion is cut long Addendum) which serves to reduce the amount of pinion tooth undercut and to nearly equalize the strength and durability of the gear set.
NOMENCLATURE Nomenclature may best be understood by means of graphic representation depicted below:
Similar in nature to Helical gearing, Spiral Miters and Bevels must be run with a mating pinion or gear of opposite hand.
The teeth of a Left Hand gear lean to the left when the gear is placed on a hori- zontal surface.
The teeth of a Right Hand gear lean to the right when the gear is placed flat on a horizontal surface.
All Boston Spiral Miter and Bevel gears are made with 35?spi ral angles with all pinions cut left hand.
Straight tooth bevel (and miter) gears are cut with generated tooth form having a localized lengthwise tooth bearing known as the ?Coniflex tooth form. The superiority of these gears over straight bevels with full length tooth bearing, lies in the control of tooth contact. The localization of contact permits minor adjustment of the gears in assembly and allows for some displacement due to deflection under operating loads, without concentration of the load on the end of the tooth. This results in increased life and quieter operation.
Boston Gear Bevel and Miter Gears will provide smooth, quiet operation and long life when properly mounted and lubricated. There are several important considerations in mounting these gears.
1.All standard stock bevel and miter gears must be mounted at right angles (90?) for proper tooth bearing
2.Mounting Distance (MD) is the distance from the end of the hub of one gear to the center line of its mating gear. When mounted at the MD specified, the gears will have a proper backlash and the ends of the gear teeth will be flush with each other (see drawings).
3.All bevel and miter gears develop radial and axial thrust loads when transmitting power. See page 145. These loads must be accommodated by the use of bearings.
If Mounting Distance of one or both gears is made less than dimension specified, the teeth may bind. Excessive wear or breakage can result. Drawing below shows gears mounted incorrectly with the Mounting Distance too short for one gear.
If Mounting Distance of either gear is made longer than dimension specified, as shown in drawing below, the gears will not be in full mesh on a common pitch line and may have excessive backlash. Excessive backlash or play, if great enough, can cause a sudden impulse or shock load in starting or reversing which might cause serious tooth damage. WORMS ANDWORM GEARS
Boston standard stock Worms and Worm Gears are used for the transmission of motion and/or power between non-inter- secting shafts at right angles (90?). Worm Gear drives ar considered the smoothest and quietest form of gearing when properly applied and maintained. They should be considered for the following requirements:
General nomenclature having been applied to Spur and Helical gear types, may also be applied to Worm Gearing with the addition of Worm Lead and Lead Angle, Number of Threads (starts) and Worm Gear Throat diameter.
As is true with Helical and Bevel gearing, Worm gearing, when operating, produces Thrust loading. The Chart below indicates the direction of thrust of Worms and Worm Gears when they are rotated as shown. To absorb this thrust loading, bearings should be located as indicated.
The efficiency of a worm gear drive depends on the lead angle of the worm. The angle decreases with increasing ratio and worm pitch diameter. For maximum efficiency the ratio should be kept low.
Due to the sliding action which occurs at the mesh of the Worm and Gear, the efficiency is dependent on the Lead Angle and the Coefficient of the contacting surface. A com- mon formula for estimating efficiency of a given Worm Gear reduction is:
For a Bronze Worm Gear and hardened Steel Worm, a Coefficient of Friction in the range of .03/.05 may be assumed for estimated value only.
There is often some confusion as to the self-locking ability of a worm and gear set. Boston worm gear sets, under no condition should be considered to hold a load when at rest. The statement is made to cover the broad spectrum of variables effecting self-locking characteristics of a particular gear set in a specific application. Theoretically, a worm gear will not back drive if the friction angle is greater than the worm lead angle.
However, the actual surface finish and lubrication may reduce this significantly. More important, vibration may cause motion at the point of mesh with further reduction in the friction angle. Generally speaking, if the worm lead angle is less than 5? there is reasonable expectation of self-locking. Again, no guarantee should be made and customer should be advised. If safety is involved, a positive brake should be used.
This is the converse of self-locking and refers to the ability of the worm gear to drive the worm. The same variables exist, making it difficult to predict. However, our experience indicates that for a hardened worm and bronze gear properly manufac- tured, mounted and lubricated, back-driving capability may be expected, if the lead angle is greater than 11?. Again, no guar antee is made and the customer should be so advised.
The high rate of sliding friction that takes place at the mesh of the Worm and Gear results in a more complex method of rating these Gears as opposed to the other Gear types. Material factors, friction factors and velocity factors must all be consid- ered and applied to reflect a realistic durability rating.

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