 Hello students, I am Dr. Bhargesh Deshmukh, Professor in Mechanical Engineering Department, Valchin Institute of Technology, Sulapu. This session is on design of spur gear. At the end of this session, we will be able to derive the Levis beam strength equation for a spur gear design. In order to derive the equation, first we need to know what is the nomenclature of a spur gear. In the spur gear, the important term is a circular pitch, then the addendum circle, the outermost pitch circle, the imaginary circle where the contact happens, the base circle, and the deadendome circle. Another important terms are the whole depth, then the working depth, remaining part is the clearance, the pitch circle, the tooth thickness, the tooth space and the fillet radius. Next part is the module of a gear. For a gear, the module design, we need to evaluate the beam strength. The modules are given as m equals to 3, 4, 5. All these values are in millimeter. If you look towards the sizes of module, choice 1, 1, 1.25, 1.5, 2, these are preferred choices. The choice 2 is intermediate value in between 2, 1.25, 1.5, the intermediate value is available in choice 2. Now the forces on gear tooth. A typical gear system involves the pressure line along which the normal force PN acts at an angle alpha. If I consider the driven gear, this is the direction of the normal force acting at pitch point. For the driven gear at the pitch point, tangential force Pt and radial force Pr are acting. Our interest is the tangential force, the Pt equals 2 times the torque divided by d dash is the Pcd. The radial force is given as Pr equals Pt tan alpha. Let us think of a cantilever beam, a cantilever beam with a point load Pt, span of the beam is L. The cross section of this cantilever beam is taken as Bt. The maximum bending stress at the support, this end can be written as sigma b equals mb y by i. In this case, we can recall the bending of a cantilever beam and write first is the bending moment for this beam and the section modulus of the beam. If we know the permissible bending stress sigma b, if we know bending moment y, can we calculate the area of cross section. Now let us consider the gear tooth as a cantilever beam. This is the gear tooth, it is as a cantilever beam. Tangential force Pt is distributed along the length of the phase width B. Height of the tooth is H and thickness is T. The same system is represented in rotated form and a 2D plane. The tangential force Pt, this tangential force is given as 2mt upon d dash. Tangential force PR is given as PR, this vertical force which is directed towards the center of the gear is given as PR equals Pt tan alpha. The assumptions in the theory, the effect of radial force PR is neglected. This PR, the effect is neglected. Tangential component Pt is uniformly distributed over the phase width, this is the phase width over PR is acting which is neglected but Pt is assumed to be uniformly distributed. The effect of stress concentration is neglected and it is assumed that at a time only one pair is in contact and takes the entire load. At the weakest section in this gear tooth, section xx, we can write the bending moment equals Pt into H, this force multiplied by the height. Section modulus for the section is 1 by 12 Bt cube and we can write y equals t by 2. With these values, we can state that bending stress sigma B equals mB y by i, substitute these values which are calculated right now and simplifying we can get Pt equals B sigma B t square upon 6H. This is the equation for tangential force Pt. If we multiply and divide this equation by m which is the module, we can get Pt equals mB sigma B t square upon 6Hm. Now in this equation, t square upon 6Hm is called as Levy's form factor, Levy's form factor y equals t square upon 6Hm. What is the contribution of Levy's? Levy's has calculated the value of y for a typical gear system which involves t, H, m and he has given the values for typical number of teeth. For say z number of teeth, what is the value of y, Levy's form factor is the contribution of Levy's. Now the tangential force the equation is changed Pt equals mB sigma B and last term is y which is nothing but the Levy's form factor. Now the beam strength, we have right now calculated the beam strength in the form of tangential force. Why I call beam strength as the tangential force? The reason is the maximum value of tangential force Pt is called as the beam strength Sb which is equal to mB sigma B y. This is also called as Levy's beam strength equation. See what are the terms involved in this equation? Sb is the beam strength which is called as the maximum value of tangential force Pt, m is the module which is expressed in millimeter, B is the face width, sigma B is the permissible bending stress which can be obtained by SUT by 3 and y is the Levy's form factor, Sb is the beam strength, sigma B is the permissible bending stress. The reference for this book, Design of Machine Element, thank you.