 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 this is diagram, schematic diagram of the differential gearbox. So we will discuss how the differential gearbox will works before that. 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 automobile the differential gearbox is used because if you are taking the turns, 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 straight path at that time this whole unit acts as only the single because if the motion is transmitted from this to this there is no resistance occurs to the any of the wheel that is why the complete this spindle will not get rotated spites on because of that we are getting the straight line motion and the unit is considered as a whole single unit that is why the speed of the wheel these two wheels we can call this wheel P and this wheel Q. These two wheel P and Q rotates with the same speed on the straight 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 turn 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 this shaft P and the shaft P is get 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 gear 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 already we have seen in case of the epicyclic gear plane what is in by this stable formulation. So same thing is there. So in this case the gear B is fixed gear B is fixed and gear C rotates through the plus one revolution this gear C rotates with the plus one revolution that is why gear B is fixed gear C makes the plus one 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 beaver gear. So we are knowing and E by N C is inversely proportional to the number of T C by T therefore as the N C is equal to 1 we are getting and E is equal to T C by T therefore we are getting T C by T as both are rotated in the same plane that is why we have to take the same time convention. Now the C the gear E is mesh with the gear D now we have to calculate the relation between T C and T D therefore we are knowing that T C otherwise N D by N C is equal to N D is meshing with the D meshing with the E and N E meshing with the N C so E will get cancelled so we are getting N D by N C. So same things we are getting so it is equal to T C by T D so T C by T D we are getting as it is the equal number of the teeth so T C by T D is equal to 1 we are getting the ratio is equal to 1 that is 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 and anticlockwise are getting changed as the C rotates in the anticlockwise direction D will rotate in the clockwise direction that is why we have to put the minus sign. So the next one gear B is fixed gear C rotates through the plus X revolution 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 Y Y plus X into T C 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 as a twelfth so the pinion A having the propeller shaft as a twelfth teeth so T A is equal to twelfth and gears with the crown gear B having the sixtieth teeth T B is equal to sixtieth the shaft P and Q of the rear axle which the road will see that if the propeller shaft rotates at 1000 rpm this propeller shaft is keyed to the gear A that is 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 to find out the N P is equal to N C is equal to how much they are asked this so for that purpose we have to formulate one table in that table gear B is fixed so the speed of the B is fixed C makes the one rotation that is why one so E will makes T C Y 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 is why zero X minus sorry here it is rotated in the same plane that is why plus X into T C by T and here the minus X add Y rotation to each element and here the total rotation so this is the 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 TA as N A we are knowing 1000 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 1000 by 5 200 therefore N B we are getting 200 rpm so this is the speed of the wheel B as this condition the speed of the wheel B is why therefore we are getting Y is equal to 200 rpm and the next thing they are given us N 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 1 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 answer this rotates so by checking this equation we are getting speed of the P means wheel is attached to the speed V is 190 rpm and the Q speed as they are given us that is a 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 wheel Q rotates 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