 Hello students, I am Dr. Bhagyaj Deshmukh, Professor in Mechanical Engineering Department, Valchin Institute of Technology, Saulabu. This session is on design of spur gear. At the end of this session, you will be able to determine the condition for maximum power transmission of spur gear. You can recall the concept of beam strength. The beam strength is given by SB equals MB sigma BY, where SB is the beam strength of the gear tooth in Newton. Sigma B is the permissible bending stress in Newton per mm square. Weir strength SW equals BQ dP dash K, where SW is the weir strength of the gear tooth in Newton. Sigma C is the surface endurance strength of the material in Newton per mm square. Q equals 2 ZG upon ZG minus ZP, where ZG is the number of teeth on gear, ZP is the number of teeth on pinion and it gives you the ratio factor K. It is given as 0.16 BHN upon 100 the bracket square. The dynamic load is also given as PD equals 21V CEB plus PT upon 21V plus square root of CEB plus PT. In this case, C is the deformation factor, E is the sum of the error in pinion and gear respectively, B is the phase width and PT is the tangential force and V is the pitch line velocity. Dynamic load is also given by the spots equation PD equals E NP ZP BE R1 R2 divided by 2530 square root of R1 square plus R2 square, where PD is the dynamic load, E is again the sum of errors between two bashing teeth, NP is the speed of pinion in rpm, R1 and R2 are the pitch circle radii of the pinion and gear respectively. Recall maxima and minima, you have studied it earlier years. To obtain the maximum value of a given function, what you did? Derivative of the function, you have derived the equation, derivative it and equate it to 0 in order to get the maxima. Can we use it for the maximum power transmitted by gear design? Let us see how we can do it. The spur gear can be designed to transmit maximum power. There are typical assumptions for this derivation. The failure occurs due to pitting and wear strength is the criteria of design. It is the assumption, we do not consider that the failure happens due to bending, rather it is assumed that the failure occurs due to pitting and wear strength is the criteria of design. The service factor CS and the factor of safety are considered as 1. Then how to begin with? The effective load P effective is given as P effective equals CS PT plus PD. This is the exact estimation of the dynamic load and P effective. P effective is also given as P effective equals SW into FS. We have two equations of P effective. As per the assumptions, we know that CS equals 1 and factor of safety equals 1. This equation becomes as CS equals 1, P effective equals PT plus PD. As the factor of safety is 1, this equation is simplified and P effective equals SW. From these two equations, we can say that SW equals PT plus PD or in other words my interest should be PT that equals SW minus PD. Now the dynamic load, we know that by the spots equation PD equals E NP ZP BR1 R2 divided by 2 phi 3 0 square root of R1 square plus R2 square if the pair of gear is steel material. If you check this equation E ZP BR1 R2 2 phi 3 0 this square root term all contributes to a constant C1, variable is NP therefore PD equals C1 NP. Tangential force we have already calculated it as PT equals SW minus PD. If I put this value of PD in the equation of tangential force I can get PT equals SW minus C1 NP. Then if I want to get the power I need to get the torque but right now I have only the force tangential force torque MT is given as PT into dP dash by 2. I can use this equation MT equals put the value of PT this value SW minus C1 NP dP dash divided by 2 torque we have obtained then power equals kilowatt that equals 2 pi NP MT upon 60 into 10 to the power 6. Here I need to put the value of torque if I put this value of torque the equation of power is obtained like this I am going to simplify this equation I am taking out the terms which can be represented as constant outside rearranging the terms for NP the maximum power capacity I need to find out how the equation of power I need to differentiate this equation. If I differentiate and equate this equation to 0 I can get the condition for maximum power transmission. The equation is differentiated and equated to 0 SW minus C1 to NP equals 0 I can get from this differentiation if I rearrange the terms SW minus 2 PD equals 0 if I simplify and rearrange the terms SW equals 2 PD. In other words PT equals PD equals SW by 2 this is the condition for maximum power transmission by a spur gear pair thank you.