Now, let’s discuss about torque calculations in epicyclic gear trains for planetary gear trains This is such a topic which is very easy, very direct and likely to be asked in GATE or any other examination But, the thing is that many students, I know many students who have learnt the process of solving the questions using torque calculation but conceptually they don’t understand what is to be done and why it is to be done So, in this discussion I am sure you will learn a couple of such crucial concepts who will help you in solving any and all types of questions regarding power calculation not necessarily in epicyclic gear train but also in any type of simple or compound gear train But, firstly let’s focus on planetary or epicyclic gear train Look, how many different components a planetary or epicyclic gear train has? That, we have already covered thoroughly This part here, the central part is called as Sun gear This gear which rotates about the sun gear is called as planet gear and this internal gear, the largest one out of all is called as ring gear or annular gear This is what we are covered Now, we know that the axis of Sun gear is passing centrally like this It is passing centrally through the system And it is a fixed Axis Its Axis does not change with time Its location does not change, it stays here The sun gear keeps on rotating about its own center and at the axis of this gear does not change, that is something that we already know So, if it’s a fixed Axis here If I were to show its connection at if it is connected to some shaft, this is how I can roughly show a shaft which is connected to the sun gear So, if any torque is to be transmitted to the sun or from the sun gear, it can be done directly from that shaft You can apply any torque on this shaft that will be received by the sun because they are directly connected and the axis of the shaft does not have to change because suns location is not changing Similarly, if you look at the ring gear or the annular gear, this gear also its axis coincides with the axis of the sun The axis of ring gear also does not change So, this ring here as I have already shown you and I have already told you earlier, that it can be connected to any other external shaft via some connection which can transmit the torque to this ring or to this annular gear But, what about the planet gear? Planet gear access is currently, (at this instant) is passing from the center of the planet gear But, planet gear keeps on rotating Planet gear does not have a constant access The planet gear will keeps on rotating, revolving around the sun gear So, its central axis also will keep on rotating around the sun gear It does not have a fixed axis That is why whenever torque is to be transmitted to or from the planet gear, it is done with the help of an Arm You can see this region, this shaded portion that I have shown here which is bluish in color This region is called as Arm or it is also called as carrier This is also something that we already know But, the thing to note here is that whenever we are going to do the torque calculation, in that case instead of planet, we are going to consider the arm for calculating the top which is very much obvious because planets keep on rotating not only about their own axis but also about the sun and they do not have a fixed Axis about which their torque can be calculated Hence, the torque transmission for planets is facilitated, is done with the help of arm, as I have already told you. and about which Axis does arm rotate? You can clearly see that arm rotates about the same central axis This whole arm is rotated round and round about the same central axis about which the sun was rotating, about which the ring gear was rotating

Let us visualize this even better with the help of one more diagram Now, let’s have a look at this diagram on the left and compare it with the diagram on the right side Conceptually, both of them are similar but there is just one difference between them at on this diagram on the right side we were having a single planet but in the diagram on the left side, we have two planets You can see that, one this yellow colored gear, this is the planet gear and this is also a planet gear And again both of these planet gears are connected together via an arm This shading that I am doing right now, this link is the arm or body carrier On the right side, there was just a single planet been held by the arm On the left side there are 2 planets held by the arm Similarly, you can have 3 planets also, 4 planets also As we know, the shape of the arm can change, the shape of the carrier can change in case of three planets, it may look something like this One will be held here, other will be held here, 3rd will be held there or it can have a totally different shape all together The shape of arm, listen it carefully, the shape of a arm does not matter What matters is that arm is going to connect or carry all the planets together and this arm will rotate about which Axis, about the central axis In this diagram also you can clearly see that arm rotates about the central axis Look, practically you understand this that planets do not have a fixed axis and there can be more, one or more planets This planet on the top has the Axis passing from here This planet has the axis passing from here In such case, if you want to extract, if you want to transmit the torque to this planet How will you do it? Their access is not fixed and they have multiple access So, it cannot be done practically, unless you connect them using a setup like an arm and to the single arm definitely you can transmit the torque You can see that there is a shaft, the shaft connected to the arm to which we can transfer the torque And even if the number of planets increase, it will not be affected since a single arm will be carrying all of them The reason why I am stressing on this point is that when you do the torque calculation, you do not consider the torque of planets Because planets as such do not have a fixed Axis about which you can apply the torque from which you can receive the torque The planets are joined together, are connected together with the help of arm and that arm has a fixed central axis and on that axis, about that axis you can transfer the torque That is why in torque calculation, we do not consider the torque of planets rather we consider torque of arm So, what are the three components of which we are going to consider the torque? First is sun, other is ring gear and third is the arm These are the three components which will be involved in torque calculation and not the planet And in case you had any confusion regarding torque transmission through Sun or through ring gear, you can see this diagram You can see that this shaft is connected to the sun, to the central sun gear To which you can easily apply the torque You can see that they have shown a handle to which the torque can be applied And the shaft will rotate and the sun gear will rotate Similarly, on the ring gear also you can see that there is a handle connected and torque can be transmitted via this handle Now, the setup can be different depending upon the different gear trains There are more than one ways to do it But, I hope you got the point that the sun gear is rotating about the central axis This ring gear also is rotating about the central axis and this arm also which is connected to a central shaft is rotating about the central axis All these three this rotation, this rotation, and this rotation are happening about the same Central axis And this point is quite unique about the planetary or epicyclic gear train And now since we have done this whole discussion and we have understood that what is going on I want you to write two points in your notes: – the first point is that I’ve already explained you that in the torque calculation, Sun gear is considered, ring air is considered and arm is considered but planets are not considered I’ve already explain this point to you This is the first point that you need to write and the second point is that all the three components the sun gear, the ring gear and the arm, they all rotate about the same Central axis These two are very important points Theoretically also they are very important and you will see that numerically also in

torque calculation, these two concepts are extremely useful Now, in the study of gear trains, we know that there are different rotating components involved, different gears involved And whatever be the type of gear train, we consider that all such members are rotating with a constant value of angular velocity or constant Omega That is why angular the acceleration for all these members is equal to zero Now, moving ahead with the same discussion, we know that the summation of force is equal to mass multiplied by acceleration and for rotational motion we can write it like submission of torque is equal to mass moment of inertia multiplied by angular acceleration This is something that we have discussed already in mechanics So, this is the expression that we are going to apply about which axis? About this Central axis This is the Funda that you need to understand And we have taken this central axis, this axis about which all the rotations are happening Some of the rotations may be happening clockwise Some of them maybe happening counterclockwise They may have different magnitudes but all the respective rotations of sun, of ring and of arm are happening about the same axis, about the same central axis So, about this central axis, what we are doing that we have applied this expression that net torque about the central axis is equal to I multiplied by angular acceleration about this axis Now, if you look individually at all the components which are rotating, as I told you that they have the same constant value of angular velocity So, the value of angular acceleration for them is equal to zero Hence, if the value of Alpha is equal to zero then the value of net torque submission of all torques will also be zero That is what we have written here Now, let’s see what are the different types of torques inward Although we have already seen that sun is going to have a torque, the arm is going to have torque and the ring gear is going to have a torque But, let’s see that how they are practical used Look, in the analysis, in the speed analysis of epicyclic gear train, we have already discussed about the different types of function that these gears, that these different gears perform Any one of them can be a input gear, input can be provided to that Any other can be a gear from which we are taking output and any one of them can be a holding or breaking gear which is fixed Basically, it can be a fixed gear This discussion I am not doing here right now again as we have already done that But, I am just briefing you about it So, any of these three components of sun, of arm and of ring can act as input or output or fixed gear So, suppose that Sun gear is the input gear The output is taken from the arm and the ring gear or the annular gear is the fixed gear If you look at it from the torque perspective then input torque needs to be provided to the sun Some torque will be provided to the sun, input torque about the central axis, whatever be the direction of that And the output from the arm is also going to be received about the same Central axis because arm is rotating about the same central axis And, similarly 1 gear out of 3 gears is going to be fixed which is not going to rotate And, let’s say if that gear is ring gear then the net torque that you need to apply to the ring gear in order to hold it, in order to break it is called as holding or braking torque So, in that case also a torque needs to be provided Don’t think that if ring gear is to be set fixed then you don’t need to provide any torque to it Because, when all of these gears are rotating, so they will try to rotate the ring gear as well So, you need to hold this ring or annular gear I have shown a hand here ok This represents as if we are trying to hold the ring, so that it does not move because we need to fix this gear So, in such a case, input will be provided to the Sun, arm will be rotating in a particular direction with a particular speed and that arm will be connected to some output as I showed you in one of the diagrams And this ring or annual gear will be held in its place It will be prevented from rotation And all these torques which we are applying either as input or output or to hold it are going to get applied about the same Central axis So, summation of torque, when we say that summation of torque is equal to zero, so what all torques do we need to consider- the torque of Sun, the torque of arm, the torque of ring which are nothing but input, output and holding torques And, this is just one of the example configuration that I told you

It may be possible that input torque is to be given to the arm and output is taken from the ring and maybe sun here is fixed So, different configurations are possible but ultimately what we know is that when we are talking about summation of torque then the torque of son, of arm and of ring are to be considered and that summation is to be equated with zero The torque of planet is not going to come So, now we have written all the individual torques and added them up – the torque on sun, the torque on arm and the torque on ring And, don’t confuse this A with the annular a ok Because generally speaking arm is represented by ‘a’ and capital A represents the annular gear What we have done is that the ring gear we are using as a notation so R is representing this ring and A is representing this arm But, generally arm is represented by a But, since you can decide notations accordingly, so this is the notation that we have followed So, 3 individual members who are transmitting the torque, their net torque should be zero about this central axis Now, notice something interesting here In case of speed analysis of gear trains, let me give you the example of tabular method We used to make the table and out of the table that we make, there were three rows of speed that we used to consider And out of those 3, 2 were known to us In the end we used to get some expressions in the terms of x and y And, 2 of them were known to us using and solving those 2 equations We used to find out the value of x and y and we used to calculate the third speed using the x and y This was the funda for Speed analysis of gear trains 2 speeds should be known to you in order to find out the desired speed But, coming back to the discussion of torque, if two torques are known to you already, you will be able to calculate the third one directly using this equation And, they don’t want things to be easy for you That is why they will just give you one value of torque It’s a standard way of defining it that for Speed analysis, two speeds are known and for torque analysis only one torque is known and rest 2 are not known to you You need to calculate them Now, obviously you cannot calculate these two values or any two values of torque from the single equation because you have two unknowns So, you need to have one more equation of torque in order to solve it So, merely this first equation is not going to get you the desired answer You need to have one more equation including the different torques which you can use to solve these two equations and get the answer And, to get the second equation, our favorite law is going to come to our help- law of conservation of energy I have already told this number of times, be it in thermodynamics, be it in mechanics, be it in heat transfer, be it in strength of materials, be it in theory of machines, everywhere this law finds its use in one form or the other Now, what exactly these gear trains are doing? They are receiving some energy at the input, rotational energy at the input which is having some torque and some angular velocity and it is giving again some rotational energy as output with some other torque, with some other angular velocity Basically, the energy is also getting transmitted from input to the output and for ideal cases, the cases that we are studying, we will not consider any of the loss in energy Any mechanical laws or any other type of laws, we are not going to consider we will assume that all the energy provided remains conserved throughout this process So, let’s apply that So, considering this whole gear train as a single system to which we are providing some energy as input and getting some energies as output The net summation of rotational energy should be equal to zero Whatever is the energy given, should be received at the output Obviously, its speed will be different Its torque will be different and that is what the gear train is doing It is modifying the motion It’s modifying the angular velocity, the torque of rotation But, net energy, it is not going to change if we neglect all the frictional losses in the bearings or between the teeth If we ignore all those losses then whatever energy we supply as input, as a particular value of torque and angular velocity then the same energy will be received at the output with some other particular value of torque or some other particular value of angular velocity But, net transaction of rotational energy, the submission of net energy will be equal

to zero Because whatever energy receives as input, same amount of energy it gives out in the form of output So, net transaction of energy input and output will be equal to zero So, let’s apply the conservation of energy principle Since, it is a rotational motion so the energy is defined per unit time As it was in the case of linear motion, we can directly define the energy as the work done, as this much force was applied and this much distance it moved So, the multiplication of them is going to give you work But, in the case of rotational motion, it rotates continuously about the same point So, the angular displacement is not the best way to define the energy That is something that we have already discussed in mechanics So, we observe the motion per unit time That is why we basically apply the conservation of energy per unit time that in unit time how much energy went in is equal to how much energy went out in the same unit time Basically, we are applying the conservation of power, energy per unit time is power And power for rotational motion we know is given by torque multiplied by angular velocity, just like for linear motion- it is force multiplied by linear velocity So, for rotational motion it is torque multiplied by angular velocity This net transaction has to be equal to zero which means that net rotational energy of Sun plus net rotational energy of arm plus net rotational energy of the ring gear should be equal to zero And don’t confuse that how their sum is going to be zero, because some of the torque will be positive, some of the torque will be negative In a similar way, some of the Omega can be positive or some of the Omega can be negative depending upon the direction, depending upon whether that is rotating clockwise or counterclockwise about this axis So, the signs will take care of it and they will get cancelled such that the net summation will come out to be zero Now, as we know that in this setup, one member is input member, one is output member and third is fixed member So, obviously for fixed member, the angular velocity is going to be zero Suppose in this case, suppose that the ring gear is the fixed gear So, if the ring gear is fixed gear so the torque applied on the ring gear will be called as breaking torque or holding torque But, the value of angular velocity of ring gear will be zero, since it will be addressed So, automatically this whole term will become zero and if this term becomes equal to zero, what will you be having? This submission is equal to zero That’s it You will be having just these two values equated to 0 Now, in our previous videos, we have already discussed the tabular method to determine the different velocities, angular velocities and I told you that two of them are given and third one of any of these three is to be calculated So, angular velocity Omega S and Omega A is not the issue These values either will be known to us or we will be able to calculate them So, angular velocity will be known us One value of torque will also be known to us and it will be known to you and it will be given to you in the question in such a manner that ultimately you will be having two equations with two unknowns You don’t have to bother about it that what values will be given? How to calculate these values? the given values of torque and angular velocity is in the question will be such that using these two equations is going to get you the required answer, whatever question is going to ask you So, you don’t have to bother about it but you should know to apply these two equations And, I know many aspirants, good aspirants as well that they just learn these two expressions and specially this one, without understanding that why the summation of torque is equal to zero, without understanding the concept of this axis, without understanding the concept that planets do not contribute directly in rotation about this axis but it is done with the help of arm And, even if the number of planets increase, the number of planets are 2 or 3 it will not matter, the analysis will not change for us because there is no expression for the torque of the planet or the angular speed of the planet because that is completely facilitated with the help of this arm And, don’t try to apply this Funda of summation T=0 in any other gear train randomly It is happening for epicyclic gear train because it is such a configuration that is allowing the torque about the same axis But, in the case of let’s say a simple gear train, this gear is transporting torque about this axis The second gear is transporting torque about this Axis So, both these axis are different In this case you don’t try to apply summation T=0 for torque calculation

In this case, you simply apply the second conservation of energy principle that T1 Omega 1 will be equal to T2 Omega 2 and using that analysis only you calculate the value of torque So, it is a very simple funda that you need to keep in mind in torque calculation and you can solve any type of gear train The funda is that observe that about which Axis the gears are rotating If that axis is different and it is in mesh just like this case, just like this of simple gear train and apply just the conservation of rotational energy funda But, if the two gears are rotating about the same axis for example in compound gear train also there will be cases when two gears are mounted on the same shaft So, in that case if you want to apply the torque calculation or in this case you want to apply the torque calculation then definitely apply the concept that summation of torque about the central axis is going to be zero I am sure this confusion is cleared to all the aspirants that why submission of torque is coming out to be zero in the first case and how to solve the questions of torque analysis or calculation for any type of gear train Now, let’s solve questions on this concept