 Hello students, I am Bhagyash Deshmukh, Mechanical Engineering Department, Valchin Institute of Technology, Solapur. This session is on design of bevel gears. At the end of this session, you will be able to derive the cone distance using the terminology of bevel gear. Let us first see what are the mechanical drives. Mechanical drives includes belt, chain and gear drives. A mechanical drive is a mechanism which is intended to transmit mechanical power over a certain distance. Usually involves a change in speed and torque. To attend this, we may need to use belt, chain or gear drives. The drive is required between the prime mover and the part of the operating machine. We can consider the example of motor and a floor mill. We need to connect a belt drive in between. Likewise, for different applications, we need a different drive. Let us see the gear drives. The gears are defined as the tooth wheels or multi-lobe cams which transmit power and motion from one shaft to another by means of successive engagement of the teeth. Advantages of gear trains or gear drives are positive drives, compact in construction, transmit very large power, transmit motion at very low speed and possesses high efficiency. For spur gear, you can say that it is having efficiency above 99%. Different types of gears are used in power transmission. First is spur and helical. These are used when the shafts are parallel. Worm gears are used for the perpendicular and non-intersecting shafts. This provides very high speed reduction. Now the bevel gears. Bevel gears, pinion and the gear, pinion and the gear. These are used for perpendicular intersecting shafts. In bevel gears, there are different types. State bevel gear, spiral bevel gear, miter gears where transmission ratio, speed is 1 is to 1, number of teeth on pinion equals number of teeth on gear. Then crown and pinion generally used in differential. SQ bevel gears. Now the terminology of the bevel gear. In bevel gear, we can see that this line represents the boundary of the bevel gear. We need to understand the terminology. There is the pitch angle from the center line to the pitch line. This is the pitch line. We can say that it is pitch angle gamma. Then there is the phase width. It is back cone. We will see what is back cone. It is the pitch cone at the front end. These two lines represent the pitch lines. Bevel gears, pinion and gear are shown over here. This bevel gear, if I say that there is RB, back cone radius, how it is drawn. If I extend these lines back and we get a point C. With C as center and C as radius, if I draw a circle, it is having radius RB. Let us see the terminology in detail. First is the pitch cone. This cone which is representing these all lines is called as the pitch cone. It is a surface which contains all the pitch lines of all teeth in the bevel gear. Next is the cone center. Cone center is indicated over here. It is the apex of the cone, imaginary cone. Next is the cone distance which is important. A naught. This is the cone distance from the apex to the larger end of the tooth. That distance is called as A naught. Back cone. It is another imaginary cone. If I extend this line up to the back end, up to the center line, we get the back cone apex C and this cone is called as back cone. Similar form is shown over here. Pitch cone and the back cone. Pitch cone, imaginary cone, then a back cone. Back cone represents this back zone of the bevel gear. Now, if I say that this is RB, what is RB? It is back cone distance RB or back cone radius. It is the length of the back cone element which is also called as back cone radius. This represents a virtual or a formative spur gear. You may think upon why we have considered a formative spur gear in the bevel gear. This equivalent spur gear called as formative spur gear. Now, this is the true diameter of the gear which is indicated by capital D. It is the pitch circle diameter capital D. Now, let us try finding out the cone distance of a bevel gear. This is DG which is the actual diameter of the gear bevel gear and DP is the actual pitch diameter of the bevel pinion. Small gamma and capital gamma are the respective pitch angles. A naught is the cone distance. O represents the apex. The formative spur gear number of teeth are given by Z dash which is equal to 2 RB upon M. It is similar to diameter of the gear upon module. Actual number of teeth of the bevel gear are given by Z equals D by M. If I take the ratio of both, I will get the relation between virtual number of teeth and actual number of teeth. Let us draw this line. Let us find out point C by doing this construction. This point represents the apex of the back cone. In this triangle B, A and C, this is B. This point is A. This point is C. I can write down the equation for AB and AC. It is angle sine of BCA equals AB by AC. I can put this as capital D by 2 and denominator is RB. It is back cone radius. I can get the relation as RB equals D upon 2 cos gamma. If I put this equation in the relation of virtual number and actual number of teeth, I can get the relation simplified as Z dash equals Z upon cos gamma. This is the relation between virtual number of teeth and actual number of teeth. Then about the cone distance, I need to get the cone distance. B, P and DG are the pitch diameter, pitch circle diameter of the pinion and gear respectively. Now gamma, you can see that it is over here. Pitch angle for the pinion, pitch angle for the gear. Now in the triangle OAB, triangle OAB, OAB. This is the triangle under consideration. You can see that if I take this tan gamma which is AB, AB and OB, OB. It is equal to DP by 2 since AB is the radius of the pinion. It is equal to DP by 2. Next is DG by 2 OB. What is that? OB, OB. This represents the radius of the gear. Therefore, I can say that it is DG by 2. Canceling out the common factor DP by DG. I can get the ratio and as this is the actual diameter of the pinion and DG is the actual diameter of the gear, I can substitute for DP as MZP and for DG I can put it as MZG. I can get tan gamma equals ZP by ZG. This gamma is the pitch angle for the pinion. I can rewrite the equation in the form of pitch angle of gear ZG by ZP. You can think upon why this gamma plus gamma equals pi by 2. The cone distance A0 is hence given by A0 equals OAOAA which is the square root of AB square plus OB square or A0 equals square root of DP by 2 square plus DG by 2 square.