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Bevel GearsBevel gears are cut on conical blanks to be used to transmit motionbetween intersecting shafts. The simplest bevel gear type is the straighttooth bevel gear or straight bevel gear as can be seen from Figure 1. Asthe name implies, the teeth are cut straight, parallel to the cone axis, likespur gears.Fig.(1) Bevel gearsStraight Bevel GearsStraight bevel gears are the most economical of the various bevel geartypes. These gears are used primarily for relatively low-speedapplications with pitch-line velocities up to 1000 fpm, where smoothnessand quietness are not significant considerations. However, with the use ofa finishing operation (e.g., grinding), higher speeds have beensuccessfully handled by straight bevel gears.451

GeometryThe geometry of bevel gears is shown in Figure (2).The size and shape ofthe teeth is defined at the large end on the back cones. They are similar tothose of spur gear teeth. Standard straight bevel gears are cut by using a20 pressure angle and full-depth teeth, which increase the contact ratioand the strength of the pinion.The diametral pitch refers to the back- cone of the gear. Therefore, therelationships between the geometric quantities and the speed for bevelgears are given as follows:Fig.(2) Bevel gear .455

whered the pitch diameterP the diametral pitchN the number of toothα the pitch angleω the angular speedrs the speed ratio.It is to be noted that, for 20o pressure angle straight bevel gear teeth, theface width (b) should be made equal to:whichever is smaller, The uniform clearance is given by the followingformula:The quantities L and c represent the pitch cone length and clearance,respectively (Figure 2).Virtual Number of Teeth:The virtual number of hypothetical spur gear can be calculated from thefollowing equation:This may be written in the following convenient form:451

in which (rb ) is the back cone radius and (N) represents the actualnumber of teeth of bevel gear.Tooth Loads of Straight Bevel GearsThe forces at the midpoint of the bevel gear tooth can be shown infigure(3).Fig.(3) forcer at the midpoint of bevel gearThe transmitted tangential load or tangential component of the appliedforce, acting at the pitch point P, is thenHereT represents the torque appliedravg is the average pitch radius of the gear under consideration.451

The resultant force normal to the tooth surface at point P of the gear hasvalueFn Ftsecϕ(Figure 3.b). The projection of this force in the axial plane, Ft tan ϕ, isdivided into the axial and radial componentswhereFt the tangential forceFa the axial forceFr the radial forceϕ the pressure angleα the pitch angleEx:A set of 20 pressure angle straight bevel gears is to be used to transmit20 hp from a pinion operating at 500 rpm to a gear mounted on a shaftthat intersects the shaft at an angle of 90 (Figure 4). Calculatea. The pitch angles and average radii for the gearsb. The forces on the gearsc. The torque produced about the gear shaft axisFig.(4)451

SolutionOr()b. Through the use of the power equation()(451)()()()()

Fig.(5) Forces on bevel gear and pinionc.()()Bevel Gear Tooth Bending and Wear StrengthsThe allowable bending load is given by:The factor Y for a gear of N′ virtual number of teeth.Buckingham Equation:Due to the difficulty in achieving a bearing along the entire face width b,about three quarters of b alone is considered as effective. So theallowable wear load can be expressed as:Wheredp the diameter measured at the back of the toothN' the virtual tooth number the pitch angleK the wear load factor.For the satisfactory operation of the bevel gear sets, the usual requirementis that411

Fb FdandFw Fd(Q)A pair of bevel gears is to transmit 15 hp at 500 rpm with a speed ratioof (0.5). The 20 pressure angle pinion has an 8 in. back cone pitchdiameter, 2.5 in. face width, and a diametral pitch of 7 teeth/in. Calculatethe axial and radial forces acting on each gear. If the gears are made ofSAE 1020 steel (WQ&T), will they be satisfactory from a bending pointof view? Employ the Lewis equation and Kf 1.4.Solution:OrHence414

()(())()To discuss the suitability of the chosen material from the bendingstrength point of view:The virtual number of teeth must be calculated as follows:Take Lewis form factor for a gear with 50tooth()SinceOrSinceThe chosen material is suitable from the bending strength point of view.411

(Q) A pair of 20 pressure angle bevel gears of N1 30 and N2 60 hasa module m of 8.5 mm at the outside diameter. Determine the powercapacity of the pair, using the Lewis and Buckingham equations.The gears have face width of 70 mm, Kf 1.5, and the pinion rotates at720 rpm. The gears are made of steel SAE1040 and hardened to about200 Bhn.Solution()To select the Lewis form factor the virtual number of teeth must becalculated:Chose Y 0.358411

To calculate the pitch circle diameter of the pinion can be evaluated asK 0.545MPaTo calculate the dynamic load()The value of the tangential force transmitted must be calculated accordingto the lowest value of Fb and Fw.So thatOr411

between intersecting shafts. The simplest bevel gear type is the straight-tooth bevel gear or straight bevel gear as can be seen from Figure 1. As the name implies, the teeth are cut straight, parallel to the cone axis, like spur gears. Fig.(1) Bevel gears Straight Bevel Gears Straight bevel gears are the most economical of the various bevel gear