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CR MATH 45 - The Spring Pendulum

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IntroductionDerivation of the Mathematical Model of the SystemUsing Ode45 to Model the Spring PendulumExamplesHome PageTitle PageContentsJJ IIJ IPage 1 of 15Go BackFull ScreenCloseQuitThe Spring PendulumNick WhitmanMay 16, 2000AbstractThe spring pendulum is analyzed in three dimensions using differentialequations and the Ode45 solver in Matlab. Newton’s s ec ond law is usedto write second order differential equations that describe the path of thespring pendulum. The spring pendulum has aspects of both the spring-mass oscillator and the simple pendulum.Home PageTitle PageContentsJJ IIJ IPage 2 of 15Go BackFull ScreenCloseQuitContents1 Intro duction 32 Derivation of the Mathematical Model of the System 43 Using Ode45 to Model the Spring Pendulum 94 Examples 12Home PageTitle PageContentsJJ IIJ IPage 3 of 15Go BackFull ScreenCloseQuit1. IntroductionA spring p endulum is simply a spring with a mass on the end. If it has no initialhorizontal displacement it acts like a simple spring mass system, oscillating upand down. If it has an initial displacement to the side it will oscillate fromside to side as well as up and down. In this case its motion will have elementsof both a simple pendulum and a simple harmonic oscillator. If the horizontalcomponent of its initial displacement is parallel to the horizontal component ofits initial velocity it will move only in two dimensions. If these components arenot parallel the mass will move in all three dimensions.Home PageTitle PageContentsJJ IIJ IPage 4 of 15Go BackFull ScreenCloseQuit2. Derivation of the Mathematical Model of the SystemThe mathematical model of the system can be determined by analyzing theforces acting on the mass at the end of the spring. Several assumptions will bemade to simplify the model of the system. They are: the spring is ideal, thereis no air resistance acting on the spring, the magnitude of the force due to airresistance acting on the mass at the end of the spring is directly proportional toits speed, and the mass’s only loss of energy is due to this air resistance.There are three forces acting on the mass: the force due to gravity, the springforce, and the force due to air resistance. The z-axis will be the vertical axiswith pos itive z being upwards. The spring will be hung from the origin.Fgforce due to gravityFsspring forceFrforce due to air resistanceL relaxed spring lengthK spring constantM mass of object at end of springG acceleration due to gravityR air resistance constantFnet= Fg+ Fs+ FrThe force due to gravity is equal to mass times the acceleration of gravity andit is in the downward direction:Fg= −M GkHome PageTitle PageContentsJJ IIJ IPage 5 of 15Go BackFull ScreenCloseQuit−2002040−100102030−40−30−20−10010xyzFigure 1: A model of the spring pendulum.Home PageTitle PageContentsJJ IIJ IPage 6 of 15Go BackFull ScreenCloseQuitThe spring force is in the direction of the displacement of the mass and itsmagnitude is the negative of the spring constant times the magnitude of itsdisplacement from relaxed spring length. The displacement vector is:xi + yj + zkThe magnitude is of the displacement is:px2+ y2+ z2Dividing the displacement vector by its magnitude you get a unit vector in thedirection of the spring force:us=1px2+ y2+ z2(xi + yj + zk)Now we need to find the magnitude of the spring force. This is the negative ofthe spring constant times the displaceme nt from the natural length of the spring.If the spring is stretched past its natural length the spring force will be towardthe origin, so to get the displacement form the natural length of the spring wetake the distance from the origin of the mass:px2+ y2+ z2And subtract from this the natural length of the spring to get the displacementfrom the natural length of the spring:px2+ y2+ z2− LHome PageTitle PageContentsJJ IIJ IPage 7 of 15Go BackFull ScreenCloseQuitMultiplying this by the negative of the spring constant we get the magnitude ofthe spring force:Fs= −Kpx2+ y2+ z2− LBy multiplying the magnitude of the spring force by the unit vector in its direc-tion we get a vector that is the spring force:Fs= −Kpx2+ y2+ z2− L1px2+ y2+ z2(xi + yj + zk)This can then be simplified a little:Fs= K Lpx2+ y2+ z2− 1!(xi + yj + zk)We now have one more force to find, the force due to air resistance. Thisforce is in the opposite direction of the velocity and its magnitude is directlyproportional to the velocity. The velo city is:x0i + y0j + z0kMultiplying this by the negative of the air resistance constant we have the airresistance in vector form:Fr= −Rx0i + y0j + z0kWe now have an expression for all three forces and can add the to get the netforce:Fnet= Fg+ Fs+ Fr= −MGk+K Lpx2+ y2+ z2− 1!(xi + yj + zk) − Rx0i + y0j + z0kHome PageTitle PageContentsJJ IIJ IPage 8 of 15Go BackFull ScreenCloseQuitAdding the components along the different axis together and separating theminto a force along each axis we have:Fx="Kx Lpx2+ y2+ z2− 1!− Rx0#iFy="Ky Lpx2+ y2+ z2− 1!− Ry0#jFz="Kz Lpx2+ y2+ z2− 1!− Rz0− MG#kIf we divide both sides of each of these equations by the mass we get expressionsfor the acceleration along each axis due to Newton’s second law:Fnet= Maa =FnetMAt the same time we can drop the vector notation and express each accelerationas a scalar:x00=KLxMpx2+ y2+ z2−KxM−Rx0My00=KLyMpx2+ y2+ z2−KyM−Ry0Mz00=KLzMpx2+ y2+ z2−KzM−Rz0M− GThese e quations completely describe the motion of the spring pendulum.Home PageTitle PageContentsJJ IIJ IPage 9 of 15Go BackFull ScreenCloseQuit3. Using Ode45 to Model the Spring PendulumThese equations have no solutions, so x, y, and z can not be written as func-tions of t. Another method is needed to analyze the system. Matlab’s ode45routine can be used find solutions to the system that can then be graphed inthree dimensions to show the path of the spring pendulum. To do this I willfirst change the above system of three second order differential equations into asystem of six first order differential equations. Let:x0= vxy0= vyz0= vzWe can then write:x0= vxv0x=KLxMpx2+ y2+ z2−KxM−RvxMy0= vyv0y=KLyMpx2+ y2+ z2−KyM−RvyMz0= vzv0z=KLzMpx2+ y2+ z2−KzM−RvzM− GThe following Matlab commands when put into an m-file will make “proj” afunction that describes the spring pendulum.Home PageTitle PageContentsJJ IIJ IPage 10 of 15Go BackFull ScreenCloseQuitfunction


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