 Hello students, I am Bhargesh Dishmukh from Valchana Institute of Technology Mechanical Engineering Department. This session is on design of bevel gear. We have derived the beam strength of a bevel gear. At the end of this session, we will be able to derive the wear strength of a bevel gear. Let us recall the terminology of bevel gears. In bevel gear, we have seen a pitch cone. Pitch cone is an imaginary cone, the surface of which contains all the pitch lines of the teeth in the bevel gear. It is represented by this cone, then the back cone. The back cone is an imaginary cone and its all elements are perpendicular to the elements of pitch cone. We can see that this triangle, red color triangle represents a back cone. Pitch cone is represented by this triangle and back cone is represented by this particular triangle. Now, let us see a formative spur gear in the bevel gear. There is a back cone radius Rb. When we project this larger end of the tool, we rotate it and then project, we get this particular spur gear. It is called as a formative spur gear in a bevel gear. The back cone distance Rb, it is also called as back cone radius. It is the length of the back cone element. Back cone is from point C to this one, this pitch point A. A is representing pitch point at this particular tooth. If I take this radius and draw a pitch circle, it represents the radius Rb representing a formative spur gear. This will be the pitch circle of the formative spur gear. It is also called as equivalent spur gear. This is the real dimension available, capital D, which is of the bevel gear. This is called as the pitch circle diameter D, but this pitch circle diameter D is of bevel gear. Now, cone distance, the pinion of bevel is shown by this zone. The diameter of the pinion is given by Dp. The gear is shown with the diameter Dg. It is the common point apex of both the bevel gears. A0 is the cone distance. Yama represents the respective pitch angle. The formative number of teeth for a bevel gear are given by Z dash equal to Rb upon m. The actual number of teeth on the bevel gear, I need to take the actual dimensions. Z is the actual number of teeth on bevel gear, D is the actual diameter of bevel gear and m is its module. I need to divide these two equations in order to establish the relation between Z dash and Z. Z dash upon Z is hence equal to 2Rb upon D. What we did over here is, we have established a relation of formative number of teeth, actual number of teeth, radius of formative spur gear, multiplied by 2 represents the diameter of the spur gear or the formative spur gear and capital D represents the diameter of actual bevel gear. Now, let us do this construction. OC is the line, then AB join and point A join to C. Now, in this triangle BCA, sign of that angle is AB upon AC or we can write that sign of 90 minus gamma equals D by 2 upon Rb or we can say that backbone radius equals D upon 2 cos gamma. Let us put this value of Rb in the equation of Z dash upon Z. The equation then simplifies and we can get Z dash equals Z upon cos gamma. It is the relation between virtual number of teeth or formative number of teeth and actual number of teeth which is related by 1 upon cos gamma. When we derive a beam strength of a bevel gear, it is derived at the larger end of the tooth. We need to consider delta SB equals mx Bx sigma B into y, where delta SB represents beam strength of the elemental section in Newton, mx is the module of the section in millimeter, Bx is the phase width of the elemental section and y is the Levy's form factor based on virtual number of teeth or formative number of teeth. In the torque, it is obtained as mB sigma B y into R into bracket 1 minus B by a naught plus B square upon 3 a naught square. What we will assume that beam strength SB is the tangential force at the large end of the tooth and hence mT equals SB into R. From these two equations what we can obtain is SB equals mB sigma B y 1 minus B by a naught plus B square upon 3 a naught square. However, we know that B equals a naught by 3. Therefore, this last term in the equation of SB, it will be never having the value more than 1 by 27. Hence, we neglected it. Therefore, SB equals mB sigma B y into bracket 1 minus B by a naught. Here, we can define that the factor 1 minus B by a naught is a Bevel factor. Taking all this methodology further, let us derive the Weir strength equation of a Bevel gear. The Bevel gear, we need to consider it as equivalent formative spur gear in the plane which is perpendicular to the tooth at the large end. This is the assumption that we did. For a formative spur gear SW equals BQ dP dash k. All the terms have usual meaning. B is the face width of the gear in millimeter, Q is the ratio factor, dP dash is the pitch circle diameter of the formative pinion in millimeter and k is the material constant in Newton per mm square. Pitch circle diameter of the formative pinion is given as dP dash equals 2RB or we know that RB equals D upon 2 cos gamma. Using these equations dP dash equals dP upon cos gamma, where dP is the pitch circle diameter of the pinion at the large end of the tooth. The Weir strength hence is given by SW equals BQ dP k upon cos gamma. Either the pinion or the gear in Bevel is generally overhanging. Only three quarters of the face width is hence effective. Therefore, we need to introduce 0.75 in the equation and hence equation SW changes to 0.75 BQ dP k upon cos gamma. The ratio factor Q is given by 2 ZG dash upon ZG dash plus ZP dash. K is given by 0.16 BH and upon 100 bracket square. Q can be also obtained by 2 ZG upon ZG plus ZP tan gamma. This represents actual number of teeth on Bevel. As we have seen that there is lesser contact available in the face width. Therefore, mounting methods of Bevel gears are established. First is overhang mounting of both shaft of pinion, shaft of gear. Support is given at only one end of the shaft. There is no support beyond the gear zone here. There is a saddle mounting of either pinion or gear. One can be having two bearings as shown. One bearing on one end, another bearing as like overhang. But additional bearing is used which support this shaft and deflection can be avoided for that shaft. Definitely it is going to contribute to the cost. Therefore, designer has to design in such a way that overhang mounting method shall satisfy the design requirements. But sometimes we need to restrict the bending of the shaft. Third method is both pinion and saddle for gear also, shaft of pinion, shaft of gear. First two bearing represents overhang mounting of both. Now, disappearance represents saddle mounting of pinion. Another bearing is used on the shaft of the gear so that gear shaft is also mounted as straddle. There is very least chance or no chance to bend the shaft. Thank you.