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The Nonlinear Pendulum - pgccphy.net

The Nonlinear Simpson, of Physical Sciences and EngineeringPrince George s Community CollegeDecember 31, 20101 The Simple Plane PendulumAsimple plane pendulumconsists, ideally, of a point mass connected by a light rod of lengthLto a frictionlesspivot. The mass is displaced from its natural vertical position and released, after which it swings back and are two major questions we would like to answer:1. What is the angle of the Pendulum from the vertical at any timet?2. What is the period of the motion?For such a simple system, the simple plane Pendulum has a surprisingly complicated solution. We llfirstderive the differential equation of motion to be solved, thenfind both the approximate and exact Differential Equation of MotionTo derive the differential equation of motion for the Pendulum , we begin with Newton s second law in rotationalform: DI DId2 dt2;(1)where is the torque,Iis the moment of inertia, is the angularacceleration, and is the angle from the the case of the Pendulum , the torque is given by D mgLsin ;(2)and the moment of inertia isIDmL2:(3)Substituting

The Nonlinear Pendulum D.G. Simpson, Ph.D. Departmentof PhysicalSciencesandEngineering Prince George’s Community College December31,2010 1 The Simple Plane Pendulum

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Transcription of The Nonlinear Pendulum - pgccphy.net

1 The Nonlinear Simpson, of Physical Sciences and EngineeringPrince George s Community CollegeDecember 31, 20101 The Simple Plane PendulumAsimple plane pendulumconsists, ideally, of a point mass connected by a light rod of lengthLto a frictionlesspivot. The mass is displaced from its natural vertical position and released, after which it swings back and are two major questions we would like to answer:1. What is the angle of the Pendulum from the vertical at any timet?2. What is the period of the motion?For such a simple system, the simple plane Pendulum has a surprisingly complicated solution. We llfirstderive the differential equation of motion to be solved, thenfind both the approximate and exact Differential Equation of MotionTo derive the differential equation of motion for the Pendulum , we begin with Newton s second law in rotationalform: DI DId2 dt2;(1)where is the torque,Iis the moment of inertia, is the angularacceleration, and is the angle from the the case of the Pendulum , the torque is given by D mgLsin ;(2)and the moment of inertia isIDmL2:(3)Substituting these expressions for andIinto Eq.

2 (1), we get the second-order differential equation mgLsin DmL2d2 dt2;(4)which simplifies to give the differential equation of motion,d2 dt2D gLsin (5)13 Approximate Equation of MotionThe easy way to solve Eq. (5) is to restrict the solution to cases where the angle is small. In that case, we canmake thelinearapproximationsin ;(6)where is measured inradians. In this case, Eq. (5) becomes the differential equation for a simple harmonicoscillator,d2 dt2D gL :(7)The solution to this differential equation is .t/D 0cos rgLtC ;(8)as may be verified by direct substitution. Here 0and are arbitrary constants that depend on the initial angle 0is called theamplitudeof the motion, and is the maximum displacement of the Pendulum from thevertical.

3 The constant is called thephase constant, and represents where in its motion the Pendulum is at PeriodEq. (8) implies that the angular frequency of the motion is!Dpg=L; since the periodTD2 =!,wefind theperiod for small amplitudes to beT0D2 sLg:(9)4 Exact SolutionWhile the small-angle approximate solutionto Eq. (5) is fairly straightforward,finding an exact solutionfor anglesthat arenotnecessarily small is considerably more difficult. We won t go through the derivations here we ll justlook at the results. Here we ll assume the amplitude of the motion 0< , so that the Pendulum doesnotspin incomplete circles around the pivot, but simply oscillates back and Equation of MotionWhen the amplitude 0is not necessarily small, the angle from the vertical at any timetis found to be.

4 T/D2sin 1 ksn t0/Ik :(10)where aJacobian elliptic functionwith moduluskDsin. 0=2/. The timet0is a time at which thependulum is vertical ( D0).2 The Jacobian elliptic function is one of a number of so-called special functions that often appear in math-ematical physics. In this case, the function defined as a kind of inverse of an integral. Given t2/.1 k2t2/;(11)the Jacobian elliptic function is defined as the inverse :(12)Values of be found in tables of functions or computed by specialized mathematical software PeriodEq. (9) is really only an approximate expression for the period of a simple plane Pendulum ; the smaller theamplitude of the motion, the better the approximation. Anexactexpression for the period is given t2/.

5 1 k2t2/;(13)which is a type of integral known as acomplete elliptic integral of thefirst integral in Eq. (13) cannot be evaluated in closed form, but itcanbe expanded into an infinite series. Theresult isTD2 sLg(1C1 XnD1 .2n 1/ .2n/ 2sin2n 02 )(14)D2 sLg(1C1 XnD1 .2n/ /2 2sin2n 02 )(15)We can explicitly write out thefirst few terms of this series; the result isTD2 sLg 1C14sin2 02 C964sin4 02 C25256sin6 02 C122516384sin8 02 C396965536sin10 02 C533611048576sin12 02 C1840414194304sin14 02 C414092251073741824sin16 02 C1477440254294967296sin18 02 C213342372168719476736sin20 02 C :(16)If we wish, we can write out a series expansion for the period in another form one which does not involvethe sine function, but only involves powers of the amplitude 0.

6 To do this, we expand sin. 0=2/into a Taylorseries:sin 02D1 XnD1. 1/nC1 2n 1022n 1/ (17)D 02 3048C 503840 70645120C 90185794560 11081749606400C (18)3 Figure 1: Ratio of a Pendulum s true periodTto its small-angle periodT0 DpL=g, as a function of amplitude 0. For small amplitudes, this ratio is near 1; for larger amplitudes, the true period is longer than predicted by thesmall-angle substitute this series into the series of Eq. (14) and collect terms. The result isTD2 sLg 1C116 20C113072 40C173737280 60C229311321205760 80C1319183951268147200 100C2335264632009078326886400 120C2673857519265928913086054400 140C399595918503714493134915501975142400 0 160C879711629097500310999194273148835148 5952000 180C487253231701972813366875101180744917 7034588160000 200C :(19)5 Plot of Period vs.

7 AmplitudeShown in Fig. 1 is a plot of the ratio of the Pendulum s true periodTto its small-angle periodT0(T=.2 pL=g/)vs. amplitude 0for values of the amplitude between 0 and180 , using Eq. (15). As you can see, the ratio is1for small amplitudes (as expected), and increasingly deviates from1for large amplitudes. The true period willalways be longer than the small-angle References1. Fulcher and Davis, Theoretical and experimental study of the motion of the simple Pendulum , ,44, 51 (1976).2. Nelson and Olsson, The Pendulum Rich physics from a simple system ,Am. J. Phys.,54,112 (1986).3. Whittaker,A Treatise on the Analytical Dynamics of Particles and Rigid Bodies(Cambridge, NewYork, 1937), 4th ed., p. Baker and Blackburn,The Pendulum : A Case Study in Physics(Oxford, New York, 2005).

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