the following content is provided under a Creative Commons license your support will help MIT open courseware continue to offer highquality educational resources for free to make a donation or to view additional materials from hundreds of MIT courses visit MIT open courseware at ocw.mit.edu I let's get on with some Dynamics so the place the place I'm going to begin is just a comment about mechanical engineering courses the first and you may have heard this already in classes you'll be taking subjects 21 if you're course two major through 29 and if you're 2A most of the odd ones but the subjects 21 through 25 are really basically engineering science subject are all foundational to mechanical engineering and they all have a common sort of property to them and that is that we make observations of the world and we we ask you know we try to understand them we Poe problems you know why you know 400 years ago is the sun the center of the solar system or not and we try to produce models that explain the problem so here's the problem the question of the day we've tried to produce models to describe it and we make observations measurements to see if our models are correct and we feed that information back into the models we try out the models we test it against more OB observations and you go round and round and this is kind of the fundamental this is the way the all of these basic first five subjects use basically this method of inquiry so in two3 the way this system works my kind of mental conception of this modeling process is three three things and this applies to you you have a homework problem how do you attack a homework problem you're going to need to describe the motion you're going to need to choose the physical laws pick I'll call it because it's short the physical law that you want to apply like f f equals ma conservation of energy conservation of momentum you got to know which physical laws to apply and then finally third you need to apply the correct math and that's really most most dynamic problems can be broken down this way this is the way I like to conceptually break them down you might have another model but this is the way I'm going to teach it so you need to describe the motion pick the correct physical laws to apply to the problem and apply the be able to do the correct math solving the equation of motion for example and this is all this is what fits in our models box and we test it against observations and measurements and improve those things over time so I'm going to give you how many of you like history I find Tech History and history of Technology kind of fun and interesting so I'm going to throw a little bit history into giving you a little quick course outline of how what we're going to do in this subject this term because the history of Dynamics and what we're going to do in the course actually F Track one another remarkably closely so if I gave you a bunch of names like Galileo Kepler decart Newton cernic Oiler lrange and Brae which one come comes first take a guess good cernus so cernus was polish and the story starts long before then but you know in about 1500 cernus said what the sun's a center what is or or the Earth is the center which did he say yeah so to me you know back around 130 ad said well the sun's the Earth the center of the of the solar system Cernic's came along and said nope I think that in fact the sun's the center of the solar system and for the next 100 years more than a hundred years a couple hundred years it was a really raging controversy about that so cernus Brae Kepler so I'm putting him in rough chronological order here now I'm going to run out of board oh well um Galileo decart I'm going to cheat okay decart Newton Oiler and lrange so we're going to talk and say a little bit about each of them and now that I'm like I told you I haven't used this classroom before so I got to learn how to play this game I need to be able to reach this for a minute so Brae he was along about 1600 Brae was the mathematician the RO the Imperial mathematician to the emperor in Prague and he did 20 years of observations and he was out to prove that the Earth was the center of the solar system and then Kepler actually worked with him as a mathematician and then took over as the Imperial mathematician and he took bra's data 20 years of astronomical data without the use of a telescope and used it to come up with the three laws of planetary motion and so his first and second laws were put out about 16009 and you know one of the laws is like equal area swept out in equal time time have you heard that one that's actually turns out to be a statement of conservation of angular momentum which we'll talk about quite a bit about in the course then came Galileo and I'm put not putting their birth and death dates here I'm kind of putting in Period of time in which kind of important things happened around him so 401 years ago a really important thing happened Galileo in 16009 turned a telescope on Jupiter and saw what four moons right and that then they really started having some data with which to really argue against the toan view of the world of the solar system the um deart is is an important figure to us and in the period of about 1630 to 1644 in that period deart uh began what is today known as analytic geometry he was a geometer he studied uclid a lot but then he came up with the cartisian coordinate system XYZ and the beginnings of analytic geometry which is essentially algebra coordinates and geometry all put together and that we are going to make great use of analytic geometry in this course then came Newton kind of in his uh uh his actual lifespan 16 19 43 is kind of interesting he spans these people and in about 1666 is when he first the first statement of the three laws of motion okay then Oiler and he's 1707 to 1783 that's his life span Oiler came up you know Newton never talked about angular momentum he mostly talked about particles Oiler put Newton's three laws into mathematics Oiler taught us about angular momentum and torque being dhdt in most cases and did he a most prolific mathematician of all time solve all sorts of important problems and then finally is lrange and lrange in about 1788 uses an energy method energy and the concept of work to give us equations of motion so the course 203 stands on the shoulders of all these people but with decart we start with kinematics really really this is analytic geometry and that's where we're going to start today is with kinematics and very soon therea we're going to review Newton the three laws and what we call the direct method for finding equations of motion conservation of momentum fact that Force sum of the forces on a object equals mass time acceleration or it's a Time derivative of its linear momentum so that's and we use that to derive equations of motion so we're going to go kinematics into doing the direct method to getting equations of motion and we go from there into angular momentum and what Oiler gave us the same thing torque we're going to do quite a lot with angular momentum because I know you know a lot about fals Ma and you've done lots of problems in 801 applying that done some problems use on Rigid body rotations but I think you there's a lot more you need to understand about this and we'll spend quite a bit of time on it and excuse me and then near the last third of the course we shift because lrange said that if you just write down expressions for energy kinetic and potential energy without any consideration of of Newton's laws and the direct method you can derive the equations of motion that's pretty remarkable so there are actually two independent roots to coming up with equations of motion and in this course so in about the last third of the course we're going to you teach you about lrange and all then all of these things are going to be one of the applications that are important to Engineers is a study of vibration so we'll be looking at vibration examples as we go through the course uh and applying these different methods to First modeling and then solving interesting vibration problems which brings ah I have a question for you so how many of you were in this classroom last May with Professor Haynes Miller and I showed up one day and we talked about vibration how many remember I told you I was going to ask this question right great okay it's good to see see you here again and we will talk about vibration in this course so there's kind of the subject outline built on the shoulders of these people in history that made important contributions to Dynamics any questions about the history if you if you want to know one of my Tas compiled a pretty neat little summary maybe I will see if I can go back and find this I just printed it out and send it how many of would you like to know a little bit more about the history these like like two liners on each person anybody want is it worth my time to send this out okay it's kind of fun so let's um do an example of this modeling describing the motion picking physical laws applying the math and that'll get us uh get us launched in the course and we'll do it using Newton and the direct method so last May Hayes Miller and I talked about vibration and so let's start I'm going to start with a vibration problem and I brought one so here's my couple of lead weights and a couple of Springs so really I just want to talk about this is the problem I want to talk about now you've done this problem before hannes Miller and I did it last last May and you've no doubt done it in other classes okay it's a system which has a spring a mass it exhibits something called a natural frequency but let's go let's see what it takes to just initially begin to follow follow this modeling method to arrive an equation of motion for this problem so what do I mean by when I say describe the motion really what that boils down to is you have to assign a coordinate system so that you can actually say where the object's moving and I'm going to pick one here so here's coordinate system can be really important in this course and I'll give us an XYZ cartisian coordinate system and I'm going to try to adopt the habit for the most part during the course that this o marks this origin but it also names the frame so we we have we're going to talk about things that are reference frames and and and most important one that we need to know about in the course is an inertial reference frame and when you can use it and when a system is inertial and is not so I'm going to say that this is inertial it's fixed to the Earth it's not moving and we're going to use this coordinate X to describe the motion of this mass and the motion is going to be this x is from the zero spring Force position it's actually quite important that you pick you have to say you know what's the condition in the spring of this system when you when X is zero so we're going to say it's when there's no force in the spring means it's not stretch that's where zero is so we've established a coordinate system second we need to apply physical laws now I'm going to do this problem by fals Ma Newton's Second Law sum the external forces is equal to math mass times the acceleration so that's the law I'm going to apply sum of the external forces it's a vector but we're just doing the X component only so we don't have to area along Vector notation is equal to this case mass time acceleration so that's the law we're going to apply and then finally the um the math that solve the equation of motion that we find that'll be the third piece but part of applying the physics in order to do this now we need what I call an fbd what do you suppose that is free body diagrams you've run you've used these many times before so we're going to do those and free body diagrams and I'm going to teach you uh at least the way I go about doing free body diagrams as things get more and more complicated you have to you're going to have to be more and more sophisticated in the way that you do these things so I just have some simple little rules to do free body diagrams that keep you from getting hung up up on sign conventions I think the thing people make most mistakes about is they get confused about signs so I'll try I'll show you how I do it so first you draw the you draw forces that you know basically in the direction in which they act seems obvious so when you know the direction so this is a really trivial problem but the method here is is very specific see in so what's an example well gravity so we'll start our free body diagram gravity acts at the center of mass hits downward this is what I mean by the direction in which it acts and it has magnitude mg okay now the other forces aren't so obvious the force that's put on by the stiffness and the damper in the spring which way do you draw them what's a sign what's the sign convention so the convention the way I go about doing these things is I assume positive positive values for the deflections and velocities so in this case x and x dot you just require that the the deflections that you're going to work with are positive and then from the positive deflection you say which way is the resulting Force so if the deflection in this is downwards which direction is the force that the spring applies to the mass up right what about if the velocity is downwards which which direction is the force that the damper puts on the mass also up right okay so this this allows this gives us so here's F spring and here's the F damper and are there any other forces on this Mass so spring Force damper force and the gravitational force whoops and so third you deduce the signs it's basically from the direction of the arrows first we need what's called your constitutive relationship so so the spring Force FS well you've made X positive so it keeps things nice the spring constants a positive number so FS is k x FD is BX Dot and now we write the statement that the sum of the forces in the X direction we look at up here we say well that's going to be FS Plus FD minus mg so that's whoops I moved the wrong way around minus minus plus because I'm plus downwards right well spring minus FS is minus KX minus BX dot plus mg equal MX Dot and I rearrange this to put all the motion variables on one side MX dot plus BX dot plus KX equals mg so there's my equation of motion but with a method for doing the free body diagrams which will work with multiple bodies so you have two bodies with springs in between them this is when the confusion really comes up two bodies of a spring trapped between them what's the sign convention you do the same thing both Bodies exhibit positive motions the the force that results is proportional to the difference and you work it out and you'll get the science right okay so here's our equation of motion arrived at by doing the direct method and just [Music] to we could you if we went on to the third step which we're not going to do today and that is apply the math it might be because I want you now to describe the motion for me get solve for the motion means solving the differential equation and that's what we did last May in Hanes Miller's class we'll come back to this later on but for today's purposes we don't need to we don't need to go there we got something else much more important to get to about kinematics but I want to show you one thing and that is just just a little tiny introductory taste to this point so I've derived the equation of motion of this by saying Say by Newton's laws but I'm going to ignore Newton now and say I'm going to drive the equation of motion by another way and it's an energy technique and that is well let's talk about the total energy of this system it's going to be the sum of a kinetic energy and a potential energy okay and un and we'll find that even with lrange there's a problem with forces on systems that are what we call non-conservative things that either take energy out of or put it energy into the system and the dash poot does that Dash poot generates heat and takes energy out of the system so I'm going to have to ignore it for the moment so the sum of the kinetic and the potential energies in this problem is a half K x^2 for the potential of the spring plus a half m x dot squar for the kinetic energy of the mass and minus m g x for the potential energy that is due to the object moving in the gravitational field right and that's the total energy of the system now and I have my problem I've allowed no forces there's no excitation on here this is just free vibration only it's all we're talking about maybe initial displacement and it vibrates if there's no damping what can you say about the total energy of the system say it again I heard it over it's got to be constant right all right well so it must this must be constant therefore the time derivative of U my system it better be zero the energy is constant take its time deriv it's got to be zero apply that to the right hand side of this I get k x x dot plus m x dot X double dot minus mgx dot = Z and I now cancel out the common x dot terms go away and I'm left with and I've essentially solved for the equation of motion of this system without ever looking at conservational momentum Newton's Laws only by energy considerations okay so that's a very simple example of that you can use energy to derive equations of motions but you then have you have to go back and fix it to account for the the loss term the damping term and that you still have to consider it as a force we'll find out even with L Grange you have to go back and consider the work done by external forces okay so this you've just kind of seen the whole course we've described the motion we've applied Newton's Laws the physics to the direct method to derive the equations of motion we have gone to an energy method and have derive the equations of motion that way and that's basically what you're going to do in the course but now we're now you're going to do it with much more sophisticated tools you'll have multiple degree of Freedom systems the description describing the motion is maybe going to be for some of you the most challenging part of the course and this is a topic we call kinematics and that's what we'll turn to next so reference frames and vectors that's the topic this is now that we're talking about kinematics and this is all about describing the motion so decart gave us the car Ian coordinate system and we'll start there so imagine this is a fixed frame we'll talk about what makes an inertial frame than the next lecture but here we have an inertial frame and it's the frame we'll call O XY o XYZ or o for short and in this frame maybe this is me and up here is a dog and I'm going to call this point a and this point B and I'm going to describe the positions of these two points by vectors this one will be R and the notation that I'm going to use is the point and its measure meas with respect to something well it's with respect to this point O in this inertial Frame so this is a with respect to O is the way to read this there's another Rector here this is r b with respect to a and finally R of B with respect to O they're all vectors on the board I'll try to remember to underline them in the textbooks and things they're usually vectors are no noted as with bold letters and Vector allows to say the following that R the position of the dog and the reference with respect to O is the sum of these other two vectors R of a with respect to o plus r of B with respect to a and mostly to do Dynamics were really interested in things like velocities and accelerations so to get to velocities and accelerations we have to take a Time derivative of rbo DT and that's going to give us what we'll call the velocity I'll usually write it as V and it' be the velocity of point B with respect to O and no surprise it'll be the velocity of point A Plus the velocity of B with respect to a and finally if we take two derivatives dt^ 2 we'll get the acceleration of B with respect to O and that'll be the sum of a the acceleration of a with respect to o plus the acceler eration of B with respect to a all again vectors now just to look ahead you know this seems all really triv trivial you guys are going to sleep on me right if these are rigid bodies this is a rigid body that is moving and maybe rotating and B is on it and a is on it and O isn't on it it starts getting a little tricky and this the derivative of a vector that's attached to the body somehow has to account for the the fact that if I'm the observers on the body this other Point's on the body say it's I'm on this asteroid and there's and I've got a dog out there and the dog's running away from me the speed of the dog with respect to me I can measure but if I'm down here looking at it it'll look different because it's rotating so how do you account for all that so taking these derivatives of vectors in moving frames is where the the devil's in the details and that's part of what I have to that's part of what I'm going to be teaching you okay I'm still learning how to how to maximize you know going to optimize my board use I haven't got it perfect yet because I'm having to move around a lot here and improvise but we'll persevere you need to know a remember a couple things about vectors how to add [Music] them uh dot products if you've had forgotten these things you need to go back and review them really quickly there's usually a little review section in the book you need to practice that sort of thing couple other little facts you need to remember so the derivative of the sum of two vectors is just the sum of the derivatives and quite importantly we're going to make use of this one a lot is the derivative of a product of two things one of them being a vector some function maybe of time and a here is derivative of f with respect to T * a plus the derivative of a with respect to T time F that will make a lot of use of so just your basic calculus so now I want to take up let's talk about the simplest form of being able to do these derivatives and calculate these velocities when everything's described cribed in terms of cartisian coordinates now I'm going to give you a little look ahead because I don't I don't want to uh I going to try to avoid confusion as much as much as possible here the hardest problem is when you're the you have a rigid body you got the dog on it you got the Observer on it it's rotating and translating uh and to take this derivative you end up with a number of terms the simplest problem is just something in a fixed cartisian coordinate system so we're going to start with the simple one and build our way up to the complicated one okay but let's now we're going to do the really the the simplest one we're going to do velocity and acceleration in cartisian coordinates and basically I should say fixed cartisian coordinates not moving all right so now let's consider my the dog out here and his position in the cartisian coordinate system and I could write that and I you'll without any loss of generality here you'll know what I mean if I say RB X component and I'm going to stop writing the SL O's because this is now all in this fixed reference frame and it's in the I hat Direction and I've got another component RB y in the J hat and an RBZ in the K hat and I want to take the time derivative I'm looking for the velocity I want to calculate the velocity so the velocity here of B in O is d by DT of r b and now this is now the product of two things so I got to use that formula over here product one term times the other so forth so I I go to these and I say okay so this is r dot BX time I plus r dot b y * J plus r do bz * k and then the other the flip side of that is I have to take the derivatives of the I time RBX the derivative J and so forth but what's the derivative of let's say I I is the capital I is my unit Vector in the fixed reference frame my o XYZ frame so it's a constant it is unit length and it points in a direction that is fixed so what's its derivative okay it's going to have a zero derivative so the second part of this second bits of that is zero and that's so that's the Velocity in cartisian coordinates of my dog out there on running around okay and the acceleration in a similar way you get now to get the acceleration you take another derivative of this and again you'll have to take derivatives of i j and k and again they're going to be zero so you will find that the acceleration then is just r do X term in the I plus r dot by y in the J plus r dot bz in the K it'll be our acceleration term and it's easy now imagine that we were doing this in polar coordinates unit vectors in polar coordinates you actually let me check last year the students told me that m in your physics courses you use unit vectors R hat Theta hat and K is that right so I'll use those unit vectors so they look familiar because in polar coordinates people use lots of different things but think about it in polar coordinates Theta you know it's a fixed maybe coordinate system but now you know Theta goes like this and R moves with Theta right so the unit Vector is pointing here but over time it might move down to here and unit Vector has changed Direction and its derivative in time is no longer zero so it starts getting messy as soon as the unit vectors change in time and so that's one our objectives here is to get to that point and describe how you handle those cases [Music] okay so quick point about velocity need to really understand what we mean by velocity so here's our car system here's this point out here B now this is the dog running around and the path of the dog you know might have been you know like this and right in here he's going this direction and in a little time in delta T he moves by an amount Delta r b with respect to O and that's what this is he's move this little bit in time delta T he happens to be going off in that direction so this then is our Prime I'll call it of B with respect to O and this is our original r b with respect to O So we can say that his new position you know r b with respect to O Prime is rbo plus delt R and these are you know all vectors and the velocity of B with respect to O is just equal to this limit of Delta rbo over delta T as T goes to zero so what direction is the velocity velocity is in the direction of the change not the original Vector but it's in the direction of the change and in fact if the path of the dog is like this at the point the instant you compute the velocity you're Computing the tangent to the path of the dog so that's what velocity is at any instant in time is a tangent to the path and that's a good concept to remember so we're still in this fix cartisian space and I have a couple of points I'll make it really trivial here here's B and here's a and the velocity of B where's my number we'll make this 10 feet per second and it's in the J at Direction and a the this is the velocity of B in O the velocity of a and o we'll say is uh 4T per second also in the J Direction and I want to know what's the velocity of B with respect to a so now I'm chasing the dog he's running at 10 I'm running it at four and I what's my how do I perceive the speed of the dog well to do this in vectors which is the point of the exercise here is we have the uh Expressions we start started with over there and we're going to use these a lot in the course so the velocity of B with respect to O is the velocity of a with respect to o plus the velocity of B with respect to a and you can just if I want to know velocity B with respect to a I just solve this so velocity B with respect to O minus velocity of a with respect to O and in this case that's 10 - 4 is 6 in the J point of the exercise is to manipulate the vector expressions like this so take whatever known quantities you have and solve for the unknown one in this case I want to know the relative velocity between the two and it's this now if I that's if I'm here and I'm watching the dog that's how I perceive the speed of the dog relative to me right 6 feet per second in the J Direction what's the speed of the dog from the point of view of over here the speed of the dog relative to me so it's again the velocity of B with respect to a but from a different position in this fixed reference frame really important Point actually this is a really important conceptual Point somebody be bold what's what's the speed with respect to O the velocity of a b with respect to a at seen from o it's computed from o measured from o got radar down there and you're tracking them in what direction yeah it's a same the point is it's the same if you're in a fixed reference frame velocity of some of a a vector of velocity is the same as seen from any diff any point in the frame any fix point the velocity is always the same and in fact in this case the velocity this is IM moving point and the velocity of him with respect to me is is the differ 6 feet per second and I from here say the velocity of that guy with respect to this guy is still 6 feet per second any place in the in that frame or even any any point moving at constant velocity you're going to see the same answer okay so it doesn't matter where you are to compute the velocity of B with respect to a that's the that's the important point okay okay we got to pick up with and I may not quite finish but I am going to introduce the next complexity okay so the what I we just what we just derived on a minute ago is that the velocity is seen from o is the same as the velocity is seen from a and a is me and I'm moving I'm chasing the dog so I'm a moving reference frame I'm what's called a translating reference frame so now we're going to step the take the next step we had a fix reference frame before purely and now I want to talk about having the idea concept of having a moving reference frame within a fixed one so this is a reference frame o capital x YZ and this little reference frame now is attached to me and it's a and I call it X Prime y Prime so just so you can it's going to be hard to tell my this X from this x if I don't do something like a prime so that this is the concept of a translating coordinate system attached to a body like a rigid body for example we can do of rigid body Dynamics here and within this coordinate system I can compute the velocity of B with respect to a and I'll get exactly the same answer I get that six feet per second in the J Direction so it's as if so this concept of being able to have a reference frame attached to a body and translating with it you can measure things within it get the answer and then convert that answer to here if you're using you know different coordinate you know you know one you could use polar coordinates here and rectangular here but they still can be related to one another we'll do problems like that okay so now I what I'm what I'm doing is I told you like in the readings the endgame is to be able to talk about translating rotating bodies and do Dynamics in three dimensions with translating and rotating objects and we're going to get there somewhat step by step but I want to show but I want you to understand the endgame so you know where we're going and you need to have a couple of Concepts in mind so so the first concept is that this is a rigid body now and you can describe the rigid the motion of rigid bodies by the summation the combination of a translation and a rotation and the rigid body if you can describe its translation and you can describe its rotation you have the complete motion so you got to understand what do we mean by what's it really the definition of translation so translation so I've got this call it Amer around we'll we'll use it we'll use using a Maro example in a minute and your observers in a fixed inertial Frame Up Above This Maro looking down on it okay but so you can see it I got to I got to turn it on its side so here's my marar on and if it's not rotating but let's say it's on a sitting on a train on a flat bed and moving along it's translating and when you say a body translates any two points on the body body move in parallel paths so two points my thumb and my finger if I'm just going along with this those two paths are traveling parallel to one another uh if the body does you know I got y pointing up if the body does this is it rotating or translating are any two points on it moving in parallel paths right okay so this called when you when it goes through curve things it's called curval linear translation but it's still just translation okay so I'll stop and Hold Steady the train stopped and the thing let it uh rotate so that's pure rotation and the thing to remember about pure rotation is that anywhere on the body rotates at the same rate this is going around once a second the rotation rate is one rotation per second 360 Degrees 2 pi radians per second is its rotation rate every point on the body experiences the same rotation rate that's a really important one to remember now this if I'm if I'm holding still marar round's going round and round it has a fixed axis of rotation right but do rotating bodies have to have fixed axes of rotation so if I throw that up in the air there's I'm not hanging on to it it's got gravity acting on it it's rotating what's it rotate about Center of mass okay is the center of mass moving ah so this is clearly this is an example of rotation plus translation it rotates about an axis but the axis can move that's another important concept that we we have to allow in order to be able to do these problems but this is now General motion It's a combination of translation and rotation and you we figure out each of those two pieces then we can describe the complete motion of the system okay all right the uh where we'll pick up next time is then doing that and and this is and it would help actually if you go read that reading especially up to chapter 16 we have to get into being taking derivatives of vectors which are rotating and come up with a general formula that allows us to do velocities and accelerations under those conditions see you on Tuesday next