ELEMENTS OF METRIC GEAR TECHNOLOGY

Transcription

1 ELEMENS OF MEC GE ECHNOLOGY SECON SPU GE CLCULONS PHONE:..00 FX:.. Stanar Spur Gear 0 0 Figure - shows the meshing of stanar spur gears. he meshing of stanar spur gears means pitch circles of two gears contact an roll with each other. he calculation formulas are in able -. a b b f a O O 0 Moule Pressure ngle Number of eeth Center Distance Pitch Diameter Base Diameter enum Deenum Outsie Diameter oot Diameter Fig. - he Meshing of Stanar Spur Gears ( = 0, z =, z =, x = x = 0) No. tem Symbol Formula 0 able - he Calculation of Stanar Spur Gears m z, z * a zm cos.00m.m + m.m *he subscripts an of z an z enote pinion an gear. b h a h f a f (z + z )m * Pinion Example Gear ll calculate values in able - are base upon given moule (m) an number of teeth (z an z ). f instea moule (m), center istance (a) an spee ratio (i ) are given, then the number of teeth, z an z, woul be calculate with the formulas as shown in able -. -

2 ELEMENS OF MEC GE ECHNOLOGY PHONE:..00 FX:.. able - he Calculation of eeth Number No. tem Symbol Formula Example Moule Center Distance Spee atio Sum of No. of eeth Number of eeth m a i z + z z, z a m i (z + z ) (z + z ) i + i Note that the numbers of teeth probably will not be integer values by calculation with the formulas in able -. hen it is incumbent upon the esigner to choose a set of integer numbers of teeth that are as close as possible to the theoretical values. his will likely result in both slightly change gear ratio an center istance. Shoul the center istance be inviolable, it will then be necessary to resort to profile shifting. his will be iscusse later in this section.. he Generating Of Spur Gear nvolute gears can be reaily generate by rack type cutters. he hob is in effect a rack cutter. Gear generation is also accomplishe with gear type cutters using a shaper or planer machine. Figure - illustrates how an involute gear tooth profile is generate. t shows how the pitch line of a rack cutter rolling on a pitch circle generates a spur gear. ack Form ool 0 0 sin b 0 O Fig. - he Generating of a Stanar Spur Gear ( = 0, z = 0, x = 0) -

3 ELEMENS OF MEC GE ECHNOLOGY. Unercutting PHONE:..00 FX:.. From Figure -, it can be seen that the maximum length of the line-of-contact is limite to the length of the common tangent. ny tooth aenum that extens beyon the tangent points ( an ') is not only useless, but interferes with the root fillet area of the mating tooth. his results in the typical unercut tooth, shown in Figure -. he unercut not only weakens the tooth with a wasp-like waist, but also removes some of the useful involute ajacent to the base circle. 0 0 a a r b r a W B' B Z WZ = Length-of-ction B'Z = B = Base Pitch ' ( b Fig. - Geometry of Contact atio 0 - Fig. - Example of Unercut Stanar Design Gear ( eeth, 0 Pressure ngle) From the geometry of the limiting length-of-contact (-', Figure -), it is evient that interference is first encountere by the aena of the gear teeth igging into the matingpinion tooth flanks. Since aena are stanarize by a fixe value (h a = m), the interference conition becomes more severe as the number of teeth on the mating gear increases. he limit is reache when the gear becomes a rack. his is a realistic case since the hob is a rack-type cutter. he result is that stanar gears with teeth numbers below a critical value are automatically unercut in the generating process. he conition for no unercutting in a stanar spur gear is given by the expression: mz Max aenum = h a sin an the minimum number of teeth is: (-) z c sin a his inicates that the minimum number of teeth free of unercutting ecreases with increasing pressure angle. For. the value of z c is, an for 0 it is. hus, 0 pressure angle gears with low numbers of teeth have the avantage of much less unercutting an, therefore, are both stronger an smoother acting.

4 ELEMENS OF MEC GE ECHNOLOGY. Enlarge Pinions PHONE:..00 FX:.. Unercutting of pinion teeth is unesirable because of losses of strength, contact ratio an smoothness of action. he severity of these faults epens upon how far below z c the teeth number is. Unercutting for the first few numbers is small an in many applications its averse effects can be neglecte. For very small numbers of teeth, such as ten an smaller, an for high-precision applications, unercutting shoul be avoie. his is achieve by pinion enlargement (or correction as often terme), where in the pinion teeth, still generate with a stanar cutter, are shifte raially outwar to form a full involute tooth free of unercut. he tooth is enlarge both raially an circumfe ren tially. Comparison of a tooth form before an after enlargement is shown in Figure Fig. - Comparison of Enlarge an Unercut Stanar Pinion ( eeth, 0 Pressure ngle, Fine Pitch Stanar) Pitch Circle Base Circle. Profile Shifting s Figure - shows, a gear with 0 egrees of pressure angle an 0 teeth will have a huge unercut volume. o prevent unercut, a positive correction must be introuce. positive correction, as in Figure -, can prevent unercut. x m ack Form ool 0 sin b O Fig. - Generating of Positive Shifte Spur Gear ( = 0, z = 0, x = +0.) -

5 ELEMENS OF MEC GE ECHNOLOGY PHONE:..00 FX:.. Unercutting will get worse if a negative correction is applie. See Figure -. he extra fee of gear cutter (xm) in Figures - an - is the amount of shift or correction. n x is the shift coefficient. x m ack Form ool 0 0 b O 0 Fig. - he Generating of Negative Shifte Spur Gear ( = 0, z = 0, x = -0.) he conition to prevent unercut in a spur gear is: zm m xm sin (-) he number of teeth without unercut will be: ( x) z c = (-) sin he coefficient without unercut is: z c x = sin (-) Profile shift is not merely use to prevent unercut. t can be use to ajust center istance between two gears. f a positive correction is applie, such as to prevent unercut in a pinion, the tooth thickness at top is thinner. able - presents the calculation of top lan thickness. -

6 ELEMENS OF MEC GE ECHNOLOGY Pressure angle at outsie circle of gear Half of top lan angle of outsie circle op lan thickness able - a θ s a PHONE:..00 FX:.. he Calculations of op Lan hickness No. tem Symbol Formula Example. Profile Shifte Spur Gear b cos ( ) a π x tan + + (inv inv a) z z (raian) θ a m =, = 0, z =, x = +0., =, b = 0.00 a =. a =.0 inv a = 0.0 inv = 0.00 θ =. (0.0 raian) s a =.0 Figure - shows the meshing of a pair of profile shifte gears. he key items in profile shifte gears are the operating (working) pitch iameters ( w )an the working (operating) pressure angle ( w ). hese values are obtainable from the operating (or i.e., actual) center istance an the following formulas: z w = a x z + z z w = a x (-) z + z b + b w = cos ( ) a x a x 0 0 w b w f b w a 0 O w O Fig. - he Meshing of Profile Shifte Gears ( = 0, z =, z =, x = +0., x = +0.) -

7 0 ELEMENS OF MEC GE ECHNOLOGY PHONE:..00 FX:.. n the meshing of profile shifte gears, it is the operating pitch circles that are in contact an roll on each other that portrays gear action. he stanar pitch circles no longer are of significance; an the operating pressure angle is what matters. stanar spur gear is, accoring to able -, a profile shifte gear with 0 coefficient of shift; that is, x = x = 0. No. 0 Moule Pressure ngle Number of eeth Coefficient of Profile Shift nvolute Function Working Pressure ngle w Center Distance ncrement Factor Center Distance Pitch Diameter Base Diameter Working Pitch Diameter enum Whole Depth Outsie Diameter oot Diameter able - he Calculation of Positive Shifte Gear () tem Symbol Formula m z, z x, x inv w w y a x b w h a h a h a f x + x tan ( )+ inv z + z Fin from nvolute Function able z + z cos ( cos w ) z + z ( + y)m zm cos b cos w ( + y x )m ( + y x )m [. + y (x + x )]m + h a a h able - is the inverse formula of items from to of able -. Pinion 0 0 Example 0 Gear No. able - he Calculation of Positive Shifte Gear () tem Symbol Formula Example Center Distance Center Distance ncrement Factor Working Pressure ngle Sum of Coefficient of Profile Shift Coefficient of Profile Shift a x y w x + x x, x a x z + z m (z + z )cos cos [ ] y + z + z (z + z ) (inv w inv ) tan

8 No. ELEMENS OF MEC GE ECHNOLOGY PHONE:..00 FX:.. here are several theories concerning how to istribute the sum of coefficient of profile shift, (x + x ) into pinion, (x ) an gear, (x ) separately. BSS (British) an DN (German) stanars are the most often use. n the example above, the tooth pinion was given sufficient correction to prevent unercut, an the resiual profile shift was given to the mating gear.. ack n Spur Gear able - presents the metho for calculating the mesh of a rack an spur gear. Figure -a shows the pitch circle of a stanar gear an the pitch line of the rack. One rotation of the spur gear will isplace the rack (l) one circumferential length of the gear's pitch circle, per the formula: l = πmz Figure -b shows a profile shifte spur gear, with positive correction xm, meshe with a rack. he spur gear has a larger pitch raius than stanar, by the amount xm. lso, the pitch line of the rack has shifte outwar by the amount xm. able - presents the calculation of a meshe profile shifte spur gear an rack. f the correction factor x is 0, then it is the case of a stanar gear meshe with the rack. he rack isplacement, l, is not change in any way by the profile shifting. Equation (-) remains applicable for any amount of profile shift. able - he Calculation of Dimensions of a Profile Shifte Spur Gear an a ack tem Symbol Formula Moule m Pressure ngle Number of eeth z Coefficient of Profile Shift x Height of Pitch Line H Working Pressure ngle w zm Center Distance a x + H + xm Pitch Diameter zm Base Diameter b 0 Working Pitch Diameter w enum h a Whole Depth h Outsie Diameter a oot Diameter f cos b cos w m ( + x ).m + h a a h (-) Spur Gear 0. Example 0 ack