 In this video I want to talk about planetary gear systems. Now I'm not going to go into a lot of terrible depth about planetary gear systems, but I do want to talk about some key features and especially one key analysis method that I think is really useful for planetary gear sets. So the basic principle behind a planetary gear system is that it has two degrees of freedom rather than a single degree of freedom. So what I mean by that is I have if I have a basic gear train so say just two gears right if I specify one input on one gear that fully specifies what's happening on the second gear that's mating with it right. With a planetary gear system which in this arrangement I have an output gear over here for example an input gear over here might be the sun and the and the planet gear you know given given the naming convention. I could input the rotation of the sun gear as one input but then I also have rotation of the arm so this gear being attached to the same axis of the sun gear by this arm means that the planetary gear can move around its outer orbit I guess sticking with the planet metaphor. So that means there's two inputs right I can specify two different things in order to get different different output configurations. So that's kind of a unique principle of the planetary gear set. So again if I have an omega in I can have an omega of the arm and I might get a rotational velocity of the output which is something different. So the sun gear in this case treating that as my input and planet gear as my output. Now oftentimes I'm going to take that configuration now and mount it inside of a ring gear and this really allows me then to extract the input or excuse me the resulting output and make use of it right. I can attach this ring gear to something else it's geared on the inside with teeth the gear teeth are on the inside of the ring and I can make use of that output which again is taking into account my two potential inputs. Now in order to be able to analyze this we need to use something called the relative velocity equation now this may be familiar to you from any content or course in dynamics and the way this is written is that the velocity of a gear is equal to the velocity of the arm plus the velocity of a gear relative to the arm. So I'm factoring in this relative velocity in there you know most of the time when we talk about velocities we're defining it as relative to earth right which is like our fixed coordinate system. But in this case we're talking about something relative to something else which is itself also moving and it's in this case the arm. So this equation becomes useful if I want to you know go ahead and analyze the system so I'm going to go ahead and write just a quick example and I'm going to say suppose we have a sun gear which has 40 teeth we have a planet with 20 teeth and a ring with 80 teeth. Now I'm going to specify my two inputs and if I didn't specify two inputs I wouldn't really know what's going on because I'd have too many degrees of freedom that aren't defined and I wouldn't be able to actually you know solve for anything. So I've specified that my arm is going to rotate clockwise be driven clockwise at 200 rpm my sun is going to be driven clockwise at 100 rpm and then I want to figure out what's what's going on here. So the way I'm going to do this is I'm going to set up a table and I'm going to set it up with basically this equation that I've written above which is W gear equals excuse me not W omega omega gear equals omega arm plus omega gear relative to arm where gear is defined by whatever shows up over here in the left hand column which is the sun the planet and the ring. So I can start filling in a couple things one we're going to define I'll just write it up here we're going to define counterclockwise positive because that's kind of a standard convention and we're going to fill in what we know first we know that omega arm is 200 rpm counterclockwise so that's negative 200 and I have an arm column here which means I can fill in negative 200 in that entire column. I've also defined what my sun gear rotation is and that is minus 100 and it goes there because I've already defined that. So now what I can see is that I can go ahead and define or solve this equation because I only have one unknown so far and I can see that this has to be a positive 100 rpm in order to satisfy this relative velocity equation. Now it gets a little confusing because our remaining two equations both have two unknowns so we have some trouble solving those but what we can do is we can go ahead and specify our gear ratio since we know something about that and I'm going to write it off to the side here and the gear ratio remember is omega p over omega g equals ng over np and I know the number of teeth in a couple of things so between my sun and my planet I know that I have a gear ratio of 40 over 20. However one thing I want to be careful of is I need to add a negative sign in here because those two rotate opposite directions of each other. On the relationship between my planet and my ring I have 20 over 80 and this isn't going to be negative because if you look at my diagram up here I can see that if my planet gear rotates clockwise by itself it's going to push the the gear in the same direction clockwise so it's the relative ratio is not negative it's actually positive and what this does for me is I can multiply this ratio across these rows in order to calculate what the other relative gearing velocity should be so if I multiply my positive 100 by negative 40 over 20 I get negative 200 and if I multiply my negative 200 by 20 over 80 I get negative 50 great so that gives me the ability to solve the remaining pieces of my equation so I have negative 200 plus negative 200 negative 400 and I have negative 200 plus negative 50 is negative 250 so if I take this and kind of look at what's going on up here what I'm determining or figuring out is that this gear rotates this direction at 100 rpm and my arm rotates that direction same direction at 200 rpm and as a result it causes my planet to rotate in the same clockwise direction at 400 rpm and my ring to rotate in the same direction at 250 rpm now sometimes this can be a little bit confusing because the ratios of the way these things move can be really challenging to to work out in your head so in that case I have sometimes some trouble figuring that out but this table to me has always been a really useful way to kind of walk through the planetary gear system and figure out what I'm expecting to happen so we can get a sense but between taking the numbers that we pull from here and applying them to our you know poorly drawn planetary system up here and see what we expect to happen for how fast these things are going to be rotating and planetary gear systems are really useful because they give us the ability to to achieve relatively high gear ratios in a in a fairly small package or in a at least a comparably small package so really helpful from that standpoint all right thank you