The Nonlinear Pendulum

Transcription

1 The Nonlinear Pendulum D.G. Simpon, Ph.D. Department of Phyical Science and Enineerin Prince Geore ommunity ollee December 31, 1 1 The Simple Plane Pendulum A imple plane pendulum conit, ideally, of a point ma connected by a liht rod of lenth to a frictionle pivot. The ma i diplaced from it natural vertical poitionand releaed, after which it win back and forth. There are two major quetion we would like to anwer: 1. What i the anle of the pendulum from the vertical at any time t?. What i the period of the motion? For uch a imple ytem, the imple plane pendulum ha a urpriinly complicated olution. We ll firt derive the differential equation of motion to be olved, then find both the approximate and exact olution. Differential Equation of Motion To derive the differential equation of motion for the pendulum, we bein with Newton econd law in rotational form: D I D I d dt ; (1) where i the torque, I i the moment of inertia, i the anular acceleration, and i the anle from the vertical. In the cae of the pendulum, the torque i iven by D m in ; () and the moment of inertia i I D m : (3) Subtitutin thee expreion for and I into Eq. (1), we et the econd-order differential equation m in D m d dt ; which implifie to ive the differential equation of motion, d dt D in (4) (5) 1

2 3 Approximate Solution 3.1 Equation of Motion The eay way to olve Eq. (5) i to retrict the olution to cae where the anle i mall. In that cae, we can make the linear approximation in ; (6) where i meaured in radian. In thi cae, Eq. (5) become the differential equation for a imple harmonic ocillator, d dt D : (7) The olution to thi differential equation i r.t/ D co t ı ; (8) a may be verified by direct ubtitution. Here and ı are arbitrary contant that depend on the initial condition. The anle i called the amplitude of the motion, and i the maximum diplacement of the pendulum from the vertical. The contant ı i called the phae contant, and repreent where in it motion the pendulum i at time t D. 3. Period Eq. (8) implie that the anular frequency of the motion i! D p =; ince the period T D =!, wefind the period for mall amplitude to be T D : (9) 4 Exact Solution While the mall-anle approximate olution to Eq. (5) i fairly traihtforward, findin an exact olution for anle that are not necearily mall i coniderably more difficult. We won to throuhthe derivationhere we lljut look at the reult. Here we ll aume the amplitude of the motion <, o that the pendulum doe not pin in complete circle around the pivot, but imply ocillate back and forth. 4.1 Equation of Motion When the amplitude i not necearily mall, the anle from the vertical at any time t i found to be r.t/ D in 1 k n.t t /I k : (1) where n.xi k/ i a Jacobian elliptic function with modulu k D in. =/. The time t i a time at which the pendulum i vertical ( D ).

3 The Jacobian elliptic function i one of a number of o-called pecial function that often appear in mathematical phyic. In thi cae, the function n.xi k/ i defined a a kind of invere of an interal. Given the function Z y dt u.yi k/ D p.1 t /.1 k t / ; (11) the Jacobian elliptic function i defined a the invere of u: y D n.ui k/: (1) Value of n.xi k/ may be found in table of function or computed by pecialized mathematical oftware librarie. 4. Period Eq. (9) i really only an approximate expreion for the period of a imple plane pendulum; the maller the amplitude of the motion, the better the approximation. An exact expreion for the period i iven by Z 1 dt T D 4 p.1 t /.1 k t / ; (13) which i a type of interal known a a complete elliptic interal of the firt kind. The interal in Eq. (13) cannot be evaluated in cloed form, but it can be expanded into an infinite erie. The reult i (.n 1/ŠŠ ) T D 1 in n (14) D ( 1.n/ŠŠ.n/Š n.nš/ ) in n (15) We can explicitly write out the firt few term of thi erie; the reult i T D in 9 64 in in in in in in in in in14 : (16) If we wih, we can write out a erie expanion for the period in another form one which doe not involve the ine function, but only involve power of the amplitude. To do thi, we expand in. =/ into a Taylor erie: in D. 1/ n1 n 1 n 1.n 1/Š D (17) (18) 3

4 Fiure 1: Ratio of a pendulum true period T to it mall-anle period T D p =, a a function of amplitude. For mall amplitude, thi ratio i near 1; for larer amplitude, the true period i loner than predicted by the mall-anle approximation. Now ubtitute thi erie into the erie of Eq. (14) and collect term. The reult i T D : (19) 5 Plot of Period v. Amplitude Shown in Fi. 1 i a plot of the ratio of the pendulum true period T to it mall-anle period T (T=. p =/) v. amplitude for value of the amplitude between and 18 ı, uin Eq. (15). A you can ee, the ratio i 1 for mall amplitude (a expected), and increainly deviate from 1 for lare amplitude. The true period will alway be loner than the mall-anle period T. 4

5 6 Reference 1..P. Fulcher and B.F. Davi, Theoretical and experimental tudy of the motion of the imple pendulum, Am.J.Phy., 44, 51 (1976).. R.A. Nelon and M.G. Olon, The pendulum Rich phyic from a imple ytem, Am. J. Phy., 54, 11 (1986). 3. E.T. Whittaker, A Treatie on the Analytical Dynamic of Particle and Riid Bodie (ambride, New York, 1937), 4th ed., p G.. Baker and J.A. Blackburn, The Pendulum: A ae Study in Phyic (Oxford, New York, 5). 5