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# Dynamics AE 2220

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This 0 page Class Notes was uploaded by Demond Hoppe on Monday November 2, 2015. The Class Notes belongs to AE 2220 at Georgia Institute of Technology - Main Campus taught by Joseph Saleh in Fall. Since its upload, it has received 11 views. For similar materials see /class/234300/ae-2220-georgia-institute-of-technology-main-campus in Aerospace Engineering at Georgia Institute of Technology - Main Campus.

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Date Created: 11/02/15

amel Guggenheim I u n D Georgia SchooloiAeros pace Engmeenng I Tech AE 2220 Dynamics The most important course in your engineering education there is no engineering Without dynamics NEWTON S LAWS THE FOUNDATIONS OF MECHANICS AND PARTICLE DYNAMICS Pan 3 Work and Energy Joseph H Saleh Georgia Institute of Technology jsalehgatechedu Fall 2013 PoIential energy Kmetwc energy Geor ia quot Teh OUTLINE Daniel Guggenheim School of Aerospace Engineering Integrating the equation of motion The Work Energy Theorem Potential energy Potential energy force and stability Nonconservative forces Power Particle collisions quot z Daniel Guggenheim 5 School of Aerospace Engineering MOTIVATION WHY INTEGRATE THE EQUATION OF MOTION In a previous lecture we integrated the equation Of motion with respect tO time and what did we get MOTIVATION WHY INTEGRATE THE EQUATION OF MOTION Daniel Guggenheim School of Aerospace Engineering In a previous lecture we integrated the equation of motion with respect to time and we obtained tfinal f ext APIOIal tinitial That is the change in momentum between two time instances is equal to the impulse or integral of external forces between these two instances Q is impulse a scalar or a vector Daniel Guggenheim School of Aerospace Engineering MOTIVATION WHY INTEGRATE THE EQUATION OF MOTION Re ection what did we gain from integration the equation of motion with respect to time 1 This time integration led to the de nition of a new concept impulse 2 It teased out a result that was embedded in Newton s laws but it made it more explicit namely the relationship between change in momentum and impulse 3 it identi ed the very important case of conservation of momentum for isolated systems which we saw can be very helpful in solving some types of problems where a direct application of Newton s laws can be quite involved and unnecessary for the results and parameters sought Georgia T s ail MOTIVATION WHY INTEGRATE THE EQUATION OF MOTION quota DanielGuggenheim t V SchoolofAerospace Engineering So to answer the question what did we get from integration the equation of motion with respect to time Lots of stuff It was pretty helpful Let s apply the same procedure an integration of the equation of motion except let s do it now with respect to the position vector not time and let s see what we can gain from that Preview of coming attractions we will nd similar or parallel bene ts to the previous three bullet points We will encounter two new concepts energy and work we will tease out a result that was embedded in Newton s laws and make it more explicit and we will have a new very helpful way to solve some types of problems GL3 MOTIVATION WHY INTEGRATE THE E EQUATION OF MOTION There s a special motivation for attempting an integration with respect to position Force is sometimes known as a function of position not time think spring or gravity But in Newton s 2nd law force is a function of time Finding the motion of a particle is theoretically easy we have to integrate with respect to time twice the acceleration vector to obtain the position vector The problem is that in some many situations force is not known as a function of time but as a function of space We ll work out examples shortly and you ll see in practice what this means So the integration of the equation of motion with respect to the position vector will help bridge this gap as we ll see shortly Side note integration can be thought Of as a lter it eliminates certain details and derives sort of cumulative or macro results There are advantages and disadvantages in the process which we ll discuss in our derivations J Geor ia l Teh Daniel Guggenheim V School of Aerospace Engineering INTEGRATING THE EQUATION OF MOTION IN 1D Let s start with the problem in one dimension 1 D that is the force and acceleration are in one dimension and let s assume m constant dV F x mdt We start by multiplying both sides by an in nitesimal displacement dx Fx39dxmd Vdx dt What is dxdt Georgia I Tech Ll Daniel Guggenheim School of Aerospace Engineering INTEGRATING THE EQUATION OF MOION IN 1D Let s start with the problem in one dimension 1D dx Fxdxmd VdxmdV Fx dxmVdV dt dt Let s now integrate this equation between two locations xl where velocity is V1 and x2 where V2 V2 xdxfmVdV V1 V2 x2 szV1 Fxdx ltgt lsz lszjchx39dx 2 2 2 1 X1 l Daniel Guggenheim l School of Aerospace Engineering Georgia Tech ill E INTEGRATING THE EQUATION OF MOTION IN 1D Let s start with the problem in one dimension 1D Embedded in Newton s second law is the fact that when integrated with respect to position 1D we have this result 1 1 x2 EmVZ2 mV12 Fx dx We will generalize it shortly to 3D and we will examine each term carefully Notice rst that in form this is a bit similar to a previous result change in momentum is equal to integral of force over time in this case we have change in some new quantity is equal to the integral of force over position What is this quantity Georgia I Tech Q Daniel Guggenheim School of Aeros ace Engineering INTEGRATING THE EQUATION OF MOON N 1D Let s start with the problem in one dimension 1 D 1 1 mV2 2 2 mV2 2F x dx 2 1 You probably know this quantity 1zmZ is the kinetic energy of the object and the left hand side of the above equation is the change in kinetic energy between two positions In the previous time integration we had the change in momentum on one side of the equation now we have the change in kinetic energy This has an interesting historical story behind it Initially proposed by Leibniz 1646 1716 and it was called Vis Viva Latin for the living force Initially it did not have the coef cient 12 Leibniz noticed that the quantity EmVIZ for a system of particles was sometimes conserved This was vehemently opposed by the Newtonian camp who advocated quotmomentumquot It wasn t until the early 1800 that vis viva started to be known as energy and later in the 18205 and 18305 that it was recalibrated to include the coef cient 12 by GaspardGustave Coriolis 1792 1843 whom we have seen in discussing acceleration and polar coordinates and we ll revisit again later in this course I amel Guggenheim Geo ia ngch D School of Aerospace Engineering INTEGRATING THE EQUATION OF MOTION IN 1D I think vis viva is cute and wish it were de obsoleted and brought back into usage in dynamics the 1690 s Daniel Guggenheim Geor ia I Teh I School of Aerospace Engineering INTEGRATING THE EQUATION OF MOTION IN 1D Examples An object is thrown straight up from a height h 0 with an initial velocity V0 How high does it go Assume gravitational force constant and neglect drag The maximum height is reached when V 0 ms gt 1 X2 O EmVOZ Fx ltgt 1mV2 mgh 2 0 ltgt h Voz max 2g hmx dx f mgdx mgh 0 So what s the difference between using this change in kinetic energy result and directly using Newton s 2nd law Let s see Daniel Guggenheim School of Aerospace Engineering Georgia I Tech irri INTEGRATING THE EQUATION OF MOION IN 1D So what s the difference between using this change in kinetic energy result and directly using Newton s 2nd law Let s see Working with the position integral of the equation of motion the quotEnergyquot approach Working directly With the equation of motion dV 1 2 hm F mgm O mV0 f mgdx mgh dt 2 0 gt 3 dV gdt 1 EmVO2 mgh 3 V V0 g I lt9 gt V02 1 max 2g hV0t gt2 We have height as a function of time To nd the maximum height we rst set V O and nd the time needed to reach this condition Then we plug this time in the height equation Both lead of course to the same result but working directly with Newton s 2nd law was a bit more involved and required additional steps and it provided us with additional information or result the time dependence of height But the problem statement did not ask for this information i Dame Guggenheim 39 quot Georg39a School of Aerospace Engineering 3 Techi 7 iii A THE WORK ENERGY THEOREM IN 1 We have already derived this theorem 1 1 x2 EmVZ2 mV12 3 Fxa x 1 EmV2 kinetic energy x2 fFX dx gt Work done by the force as the object is moved from a to b x1 i Danie Guggenheim 39 Georg39a School of Aerospace Engineering Tech iii i THE WORK ENERGY THEOREM IN 1 We have already derived this theorem 1 1 x2 EmVZ2 mV12 3 Fxdx The work energy theorem in 1D states that the change in kinetic energy between two positions is equal to the work of the force between these two positions 599g x THE WORK ENERGY THEOREM IN 1D The units of energy and work are of course the same and in the SI it is joule lsz L2Lkgm2 s2 aJoule J F0rceLNmkgms2 mkgm2s2 aJoule J unit

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