 students. I am Bhagyaj Jaishmukh from Mechanical Engineering Department, Valchin Institute of Technology, Sulakpur. This session is on design of bevel gear. At the end of this session, the learner will be able to derive the expressions for beam strength of a bevel gear. Now, let us see a bevel gear. This is the part of a bevel gear we have considered. Here, this is the pitch angle gamma. This R represents d by 2 or it is a radius at the larger end of the tooth, pitch radius at larger end of the tooth. A naught is cone distance, B is the face width. We have considered a formative spur gear at the larger end. Here, the force is acting S B. You can think upon why we have considered formative spur gear over here as we are deriving the equation for the bevel gear. What could be the relation between spur gear and or the formative spur gear and the bevel gear? We are concentrating on this zone. We need to consider an elemental distance dx at a distance x from the cone apex and the radius of which is rx. Applying the Levy's equation to a formative spur gear at the distance x from the apex, this elemental zone which is at distance x from the center and the radius of which is rx and the width is dx. Delta S B equals mx Bx sigma B into y. These parameters are, delta S B is the elemental beam strength or the beam strength of the elemental section which is always expressed in Newton. mx is the module of the section it is in millimeter. Bx is the face width of the elemental section, this one this will be Bx, Bx equals dx. y is the Levy's form factor based on virtual number of teeth. You can think why this is based on virtual number of teeth which is very important. Let us consider this triangle which is having this height as rx and this is the other triangle where we can think that this radius if you check this radius is equal to r, capital R. From this figure we need to establish a relation between rx, r and corresponding distances for r the cone distance is a naught for rx cone distance is x. While we have done this similarity purpose we need to establish rx which is equal to x r upon a naught. Our interest is to obtain the radius rx in terms of the cone distance which is radius and the elemental cone distance. At the elemental section, this small section I need to check what is the value of mx. mx equals 2 rx upon z or we can rewrite it as 2 x r upon z a naught. Why we did this instead of rx? We put the value of rx x r upon a naught at the large end of the tooth this large end mx equals mx upon a naught since Bx equals dx. We can find out the relation delta SB equals m sigma by x dx upon a naught considering this zone only delta SB equals mx Bx sigma b into y. I can check this relation delta SB equals m sigma b y x dx upon a naught. I need to do some adjustment process it mathematically. If I substitute rx equals x r upon a naught what I can get rx if I multiply on the left hand end it is rx into delta SB which is equal to m b x is multiplied to this x it becomes x square r upon a naught into a naught a naught square. If I integrate on both the ends I can get this relation this left hand end it becomes empty it is a total torque on the right hand end these terms are unaffected and for this integral I need to put the limits a naught minus b to a naught and solving I can get m b y r upon a naught square x cube by 3 the limits are same a naught minus b to a naught the torque empty is given as m b sigma b y r into bracket 1 minus b by a naught plus b square upon 3 a naught square. Now for the same zone we are solving for delta SB mx Bx sigma b y with usual notations mt equals m b sigma b y r 1 minus b by a naught plus b square upon 3 a naught square parallel to this equation if we assume that this is the SB acting at the larger end of the tooth tangential force I can get another equation for torque mt mt equals SB into r as it is acting at the larger end of the tooth the radius is equal to capital R. I can write the equation for beam strength using these two equations if I put this mt equals SB into r r and r gets cancelled and we get the equation for SB equals m b sigma b y into bracket 1 minus b by a naught plus b square upon 3 a naught square as we know that b equals a naught by 3 this b is a naught by 3 and for this particular bracket b square upon 3 a naught square it happens that the last term in the bracket will never be more than 1 by 27 hence we neglect it. Therefore, this particular equation of SB or the beam strength for bevel gear simplifies and we get SB equals m b sigma b y into bracket 1 minus b by a naught this is the equation for beam strength of a bevel gear constrain y as it is based on virtual number of teeth m is the module b is the face width and sigma b is the bending stress which is obtained as SUT by 3 now you can see that delta SB equals mx bx sigma b y it was for the elemental strip which was considered and what we did to this elemental strip we have integrated put the limits and the equation of torque mt was established the equation for mt was m b sigma b y r into bracket 1 minus b by a naught plus b square upon 3 a naught square this was the first equation. Second equation we obtained as the beam strength is acting at the larger end of the tooth and hence we have obtained the relation as mt equals SB into r now comparing these two equation what we did is these mt are equal and hence this SB into r it was equated to m b sigma b y r into bracket 1 minus b by a naught plus b square upon 3 a naught square but we have seen that the effect of this last bracket is very negligible as b equals a naught by 3 and the last term is neglected as its effect is very small and finally we got the equation of beam strength as SB equals m b sigma b y which is similar to the spur gear however this y is based on virtual or formative number of teeth and some term we have got it as 1 minus b by a naught what is the term the term is called as bevel factor very important to remember it is the bevel factor 1 minus b by a naught which is called as a bevel factor thank you