 Hello everyone, Myself Sachin Rathod working as assistant professor in Mechanical Engineering department from VW Technology Solapur. Today we are dealing with the chapter or the bit differential gearbox. So under this bit the learning outcome is at the end of this session student will able to calculate the speed ratio of the differential gearbox. So the differential gearbox is used for giving or transmitting the motion from engine to the rear axle. This is the first application and the second and very important application of this differential gearbox is to vary the speed from inner wheel and the outer wheel. Means suppose the speed of the inner wheel if you are taking a turn the speed of the inner wheel should be less than that of the outer wheel because the inner wheel will get lesser space or lesser distance than that of the outer wheel. The outer wheel covers the maximum distance if it is taking in this direction. So there is a speed variation in this two wheel that can be achieved by using the differential gearbox. So the construction of this differential gearbox is before that what is the application of the differential gearbox. So you can think about this, this is very simple in atomobiles the differential gearbox is used because if you are taking a turn the outer wheel takes more radius or the distance than that of the inner wheel at that time we required the speed variation between the two wheel that is the inner wheel and the outer wheel that can be achieved by using the differential gearbox. So we will see the construction of this differential gearbox this is a profiler shaft A which is connected to the engine from the engine the profiler shaft is going to rotate. So from the profiler shaft the bevel gear A is mounted on the profiler shaft which is keyed then A from the bevel gear A it is transmitted to the bevel gear B then the B is connected by the arms as the B will get rotated the arm will get rotated. So if suppose we are moving on a state path at that time this whole unit acts as only the single because if the motion is transmitted from this to this there is a no resistance occurs to the any of the wheel that's why the complete this spindle will not get rotated by its own because of that we are getting the state line motion and the unit is considered as a whole single unit that's why the speed of the wheel these two wheels we can call this wheel P and this wheel Q this two wheel P and Q rotates with the same speed on the state path. So next this is a arms the arms and the gear E as connected by this spindle gear C is mesh with the gear D gear D mesh with the sorry gear C mesh with the gear E gear E mesh with the gear D the number of teeth on the gear C and D are the same as well as the number of teeth on the E and F are the same. So if you are taking a turn left or right term at that time if you are taking the turns in this direction at that time the resistance is occurs to this wheel P so as the resistance is occurs to this wheel P this is connected to the shaft P and the shaft P is keyed to this gear C so at that time the spindle will revolve about its own axis so that we are getting the different speed for the C and this gear D. So the gears the speed of the gear C is different and the gear speed of the gear D is different as the rotation this rotates if we are taking in this turn the speed of this is increases and the speed of this wheel P decreases increases and decreases so this will take the more time like this construction is there so we should know how to calculate the speed or the speed ratio of this differential gearbox or the speed of every element used in the differential gearbox so in this case the gear B is fixed gear fixed gear B is this gear B is fixed and gear C rotates through the plus 1 revolution this gear C rotates with the plus 1 revolution that's why gear B is fixed gear C makes the plus 1 revolution so we have to find out the speed of the E now we are knowing the C and gear C and E are external image by the B wear gear so we are knowing and E pi and C is inversely proportional to the number of TC by T therefore as the NC is equal to 1 we are getting and E is equal to TC by T therefore we are getting TC by T as both are rotated in the same plane that's why we have to take the same same convention now the C the gear E is mesh with the gear D now we have to calculate the relation between TC and TD therefore we are knowing that TC otherwise and D by NC is equal to and D is meshing with the D meshing with the E and and E meshing with the NC so E will get cancelled so we are getting ND by NC so same things we are getting so it is equal to TC by TD so TC by TD we are getting as it is the equal number of the teeth so TC by TD is equal to 1 we are getting the ratio is equal to 1 that's why we are getting the 1 but this acts in the same plane as the C rotates in the clockwise direction D will rotate in the anticlockwise means the motion of the rotation of clockwise anticlockwise are getting changed as the C rotates in the anticlockwise direction D will rotate in the clockwise direction that's why we have to put the minus sign so the next one gear B is fixed gear C rotates through the plus X revolution so as the gear C makes the X revolution just multiply with the X C by T and it is a minus X add Y revolution to each element then total motion Y X plus 5 Y plus X into TC by T Y minus X this is the table of the motion now we will see one example so in this example I will write down the given data the pinion A on the propeller shaft has a 12 teeth so the pinion A having the propeller shaft has a 12 teeth so TA is equal to 12 and gears with the crown gear B having the 60 teeth TB is equal to 60 the shaft P and Q of the rare axles which which the road will see that if the propeller shaft rotates at 1000 rpm this propeller shaft propeller shaft is key to the gear A that's why they are given a speed of the A is equal to 1000 rpm and the road will attach the Q has speed of 210 rpm this Q is attached to gear D so they are given as N Q is equal to N D is equal to 210 rpm find the speed of the P so they are find out the speed of the P is nothing but the speed of the C so they are asked find out the NP is equal to NC is equal to how much they are asked so for that purpose you have to formulate one table so in that table gear B is fixed up so the speed of the B is fixed C makes the one rotation that's why one so E will makes TC by T as we have seen in the previous and D will makes this makes the plus one rotation this will make the minus one rotation so as the gear B is fixed gear C rotates through the plus one rotation that's why 0x minus it is rotated in the same plane that's why plus x into TC by T and here the minus x add y rotation to each element and here the total rotation so this is a table of the motion so just you have to check this condition under this given data we have to find out the speed of the wheel this wheel P so right now they are given as the speed of the A so A is meshing with the B so we are knowing that N A by N B it is inversely proportional to the number of TB by T as N A we are knowing thousand divided by N B is equal to number of the teeth on the B is 60 divided by TA is 12 so it is a 5 therefore N B is equal to thousand by 5 200 therefore N B we are getting 200 rpm so this is a speed of the wheel B as this condition the speed of the wheel B is y therefore we are getting y is equal to 200 rpm and the next thing they are given us and q is equal to N D is equal to 210 rpm so here the speed of the D y minus x y minus x is equal to 210 rpm as we are going the value of the y is 200 200 minus x is equal to 210 therefore 200 minus 210 is equal to x therefore x is equal to minus 10 rpm so this is the value of the x now they are asked find out the speed of the P or speed of the C so speed of the C is equal to x plus y so speed of the C is equal to x plus y the value of x is minus 10 and the y is 200 so we are getting 190 so N C is equal to N P so the speed of the P is equal to 190 rpm this is the answer this rotates so by checking this equation we are getting speed of the P means we will attach to the speed V is 190 rpm and the q speed as they are given us that is the q speed so here q speed is 210 rpm and N P we are getting the 190 means this rotates at 210 rpm simultaneously this will q rotate 190 rpm in the same way so we are getting the difference in the speed of these two wheel so I have taken these two references