AC : USING ELLIPTIC INTEGRALS AND FUNCTIONS TO STUDY LARGE-AMPLITUDE OSCILLATIONS OF A PENDULUM

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1 AC 007-7: USING ELLIPTIC INTEGRALS AND FUNCTIONS TO STUDY LARGE-AMPLITUDE OSCILLATIONS OF A PENDULUM Josue Njock-Libii, Idiaa Uiversity-Purdue Uiversity-Fort Waye Josué Njock Libii is Associate Professor of Mechaical Egieerig at Idiaa Uiversity-Purdue Uiversity Fort Waye, Fort Waye, Idiaa, USA. He eared a B.S.E i Civil Egieerig, a M.S.E. i Applied Mechaics, ad a Ph.D. i Applied Mechaics (Fluid Mechaics) from the Uiversity of Michiga, A Arbor, Michiga. His areas of iterests are i mechaics, particularly fluid mechaics, applied mathematics, ad their applicatios i egieerig, sciece, ad educatio. America Society for Egieerig Educatio, 007

2 Usig elliptic itegrals ad fuctios to study large-amplitude oscillatios of a pedulum Abstract The solutio to the oscillatios of a pedulum that icludes large amplitudes is preseted for the purpose of comparig it to that for small amplitudes. Such a compariso allows for the determiatio of the limits of applicability of the liearized equatio. It is show that, i both cases, the agle of swig is a periodic fuctio of time but that the ature of the fuctios ivolved varies with the amplitude of motio. For small agular displacemets, the period of oscillatio is a costat ad the esuig agle of swig ca be represeted accurately by meas of circular fuctios. However, for large amplitudes, the period is represeted by Jacobi s complete elliptic itegral of the first kid ad varies with the iitial amplitude, while the correspodig agle of swig is represeted by elliptic fuctios of Jacobi. It is show that the period of the liearized motio is always smaller tha, or equal to, that from the oliear motio. The errors iduced by the liearizatio process are determied aalytically ad represeted graphically. It is demostrated that those i the magitude ad phase of swig vary with time ad the iitial amplitude of the pedulum. Cosequetly, as a geeral rule, it is iaccurate to use the error i the agle as a estimate of the accuracy of how well the liearized solutio approximates the actual motio.. Itroductio The motio of a pedulum is studied i the first college physics course; ad its goverig differetial equatio is amogst the first oes that are solved i a itroductory course o ordiary differetial equatios. This equatio is ecoutered agai ad agai i courses such as dyamics, cotrols, vibratios, ad acoustics. I all these cases, however, it is liearized by assumig that the amplitude of oscillatio is small. As a cosequece, studets do ot see what happes to the oscillatio of a pedulum whe the amplitudes are large ad the restorig force becomes oliear. More importatly, they do ot kow the limits of applicability of the liearized solutio they have studied. I this article, we preset the solutio to the oscillatios of a pedulum that icludes large amplitudes ad compare the geeral solutio to that which is valid oly for small

3 amplitudes. This allows oe to determie the errors iduced by ad the limits of applicability of the liearized equatio.. The Basic equatio Cosider a rigid body that is suspeded from a poit O about which it oscillates i the vertical plae. Let the agular displacemet about the vertical axis be deoted by, measured i radias. After applyig either Newto s secod law of motio, or the coservatio of mechaical eergy, it is foud that udamped oscillatios about poit O ca be obtaied by solvig the equatio si( ) 0, () I geeral, the coditios at the startig time, t = t s, are give by t t, ( t ), ( t ). (a) s s s s s I these equatios, the dots represet differetiatio with respect to time t ad the quatity, which has uits of rad/s, is related to the atural frequecy of the system. As a example, for a compoud pedulum swigig i the vertical plae about a horizotal axis that goes through poit O, mtotal gd J 0, (b) where, m total is the total mass of the pedulum; g is the acceleratio of gravity; d is the distace betwee poit O ad the ceter of mass of the pedulum; ad J 0 is the (polar) mass momet of iertia of the body about poit O. It ca be see that is a physical parameter that does ot deped o time. 3. The solutio for small agles: circular fuctios For small amplitudes, it is covetioal to liearize Eq.() by expadig the si ito a power series as show below ( ) si( ) () 3! 5! 7! ( )!

4 ad replacig the si with, the first term i that series. Doig so gives 0 (3) This is the equatio that is used i all the courses metioed above. Its solutio is () t Asi( t) Bcos( t) (4) I this case, is the circular frequecy of the motio expressed i radias per secod. After the iitial coditios give i Eq (a) are used i Eq (4), the costats A ad B are foud to be give, respectively, by s A ssi( ts) cos( t s) (5) s B cos( t ) si( t ) s s s I order to obtai a solutio with a simple mathematical form, it is covetioal to let be the maximum amplitude of oscillatio ad set t s 0, s 0, s. Icorporatig these assumptios ito Eq. (5) leads to A ad B 0 ; ad Eq (4) becomes () t si( t) Here the period of oscillatio,, is related to the circular frequecy,, by (6) (6a) It ca readily be observed from Eq.(6) that the istataeous positio of the pedulum durig oscillatio is a circular fuctio of time ad is directly proportioal to the amplitude of motio,. From Eq.(6a), it is see that the period of oscillatio of the pedulum is a costat that is idepedet of the amplitude of motio. It follows that all amplitudes that are withi the limits of applicability of the goverig equatio yield the same period of oscillatio. Cosequetly, the period ad frequecy of oscillatio are ot affected by the iitial coditios. We will compare these results to those obtaied whe the pedulum assumes large amplitudes of oscillatio. 4. The solutio for ay agle: elliptic fuctios ad itegrals

5 Whe swigig agles may be large, Eq.() is trasformed ito Jacobi s elliptic itegral of the first kid by two successive itegratios ad a chage of variables. The exact solutio to Eq.() is () t Arcsisi s( t), (7) where s represets Jacobi s elliptic fuctio with the elliptic modulus suppressed 3-7. The elliptic fuctios of Jacobi are defied as iverses of Jacobi s elliptic itegral of the first kid. Thus, if oe writes u d 0 k si ( ), the, for example, s( u, k) si( ), c( u, k) cos( ) ad d( u, k) k si ( ). For the derivatio of Eq. (7), it is covetioal to trasform the origial differetial equatio ito a itegral as d t 0 si si The, by settig. (7a) si u, (7b) si ad usig this chage of variables i Eq. (7a), oe gets t where 0 du u k u (7c)

6 si si, (7d) ad k si. (7e) From Eq.(7c), the period of oscillatio is give by Eq.(8) as 4 Kk ( ), (8) where K deotes Jacobi s complete elliptic itegral of the first kid 3-7, which is defied as du Kk ( ). (8a) 0 u k u Expadig K ito a power series 4, oe gets k k.. 46 k k..., (9)... where 0. Rearragig Eq.(9), the ratio of the two periods is foud to be k k k k..., (9a) It ca readily be observed from Eq.(7) that the istataeous positio of the pedulum durig oscillatio is a oliear fuctio of the amplitude of motio,, ad a elliptic fuctio of time. From Eq.(8), it is see that the period of oscillatio of the pedulum depeds upo the amplitude of motio. Although the period varies with of the amplitude i a oliear way, oe ca see from Eq.(9) that it icreases mootoically with the amplitude. Cosequetly, the period ad frequecy of oscillatio are affected by the iitial coditios. We will compare results obtaied assumig small amplitudes of oscillatio to those obtaied assumig large amplitudes.

7 5. Comparig the solutios 5.a. Comparig the two periods It ca be see from Eq.(9) that,, the period obtaied from the oliear equatio, icreases with the amplitude; ad from Eq.(9a) that is always larger tha, or equal to 0, the period obtaied from the liearized equatio. This relatioship is illustrated graphically by the plot of 0 vs. k that is show i Fig.. t t k Fig.. Plot of the ratio of periods: 0 vs. k, Eq. (9a) We defie the error i the computatio of the period as the differece betwee the exact ad the approximate periods divided by the exact period, as show i Eq (0). 0 Error P (0)

8 Similarly, we defie the error i the computatio of the iitial swig agle as the differece betwee the iitial agle of swig ad the sie of divided by the sie of, as show i Eq.(). si Error si () The ratio betwee the errors foud i Eqs. () ad (0) is show i Eq.() ad plotted i Fig. (). Errorp Ratio () Error It ca be see from that figure that the error i the agle is always larger tha that i the period. Thus, sice the error that is made i usig the agle itself istead of its sie is much easier to compute, it ca be obtaied ad used as a upper boud o the error to be expected i the period. Ratio k Fig.. A ratio of the error i the period to that i the agle. 5.b. Comparig the swig agles To illustrate the differeces betwee the swig agles obtaied from the oliear ad liear equatios, six startig agular amplitudes have bee chose; ad, for each, a

9 solutio was obtaied usig the liearized equatio ad aother with the oliear equatio. The iitial agles used are 0 o, 30.3 o, 63 o, o,.3 o, ad 47 o ad they have bee idetified i Fig.b with dots. They correspod, respectively, to k = {0.0876, 0.648, 0.596, , 0.876, }. Plots of the correspodig variatios of the agular positios of the pedulum with time are show i Fig. 3, where the solid lies represet the liear solutio ad the dashed lies the oliear (exact) solutio. From Fig. 3, it ca be see that, as the iitial amplitude, that is give to the pedulum to iitiate its motio, icreases (from 0 o, to 30 o, 63 o, 88 o, o, ad 47 o ), so does, 0, the ratio betwee the exact period ad the approximate period 0, Eq.(9). The wideig differece betwee the periods prevets the two curves from beig i lock-step; this, i tur, icreases the discrepacy betwee the correspodig agular positios of the pedulum. For each iitial agle used i Fig. 3, differeces betwee the two solutios were computed usig Eq.(3) ad plotted i Fig. 4 i which the iitial agle is a parameter. = Arc sisi s( t) - si( t ) (3) It ca be see from Fig. 4 that these differeces vary cosiderably with both time ad amplitude ad ca chage algebraic sigs durig the motio.

10 Agle u Agle u Agle u Agle u Agle u Agle u Fig.3. Swig agle vs. time, for 0 o, 30 o, 60 o, 90 o, 0 o, ad 50 o. Exact values are for 0 o, 30.3 o, 63 o, o,.3 o, ad47 o. Solid lies (approximate solutio); dashed lies (exact solutio)

11 u u u u u u u u Fig. 4. eces i Eq.(3) are plotted over four cosecutive cycles, k is a parameter. (k = 0., 0.4, 0.36, 0.48, 0.60, 0.7, 0.84, ad 0.96, respectively).

12 6. Coclusios We preseted the solutio to the oscillatios of a pedulum that ecompasses large amplitudes of swig ad compared it to that which is valid oly for small amplitudes. This allowed for the determiatio of the limits of applicability of the liearized equatio. It was show that, i both cases, the agle of swig is a periodic fuctio of time but the mathematical ature of the fuctios ivolved chages with the amplitude of motio. For small iitial agular displacemets, the period of oscillatio is a costat that is idepedet of the iitial displacemet of the pedulum; ad the esuig agle of swig is represeted accurately by circular fuctios. For large amplitudes, however, the period of oscillatios is ot a costat, for it varies with the iitial amplitude give to the pedulum; it is represeted mathematically by Jacobi s complete elliptic itegral of the first kid; ad the correspodig agle of swig is expressed by meas of elliptic fuctios of Jacobi -8. As ca be see from Eq.9, the period of the liearized motio is always smaller tha, or equal to, that of the oliear motio. The approximatio si that is used to liearize the differetial equatio itroduces three kids of errors i the solutio: oe is i the magitude of the period of oscillatio; the secod oe is i the magitude of the swig agle; ad the third oe is i the phase of motio. These errors were determied exactly ad represeted graphically. Whe oe uses the approximate equatio, approximatio errors that are itroduced i the agle of swig affect the swig period of oscillatio. The error iduced i the period is fixed for a give motio of the pedulum (Eq. 0). So is the error i the agle itself (Eq. ). These two errors were compared i Eq.(). From the plot of their ratio that is show i Fig. (), it ca be see that the error i the period is always smaller tha that made i approximatig the si( ) by the agle. Therefore, the error i the agle ca be used as shortcut to the determiatio of a upper boud o the error to be expected i the period. Numerical experimetatio showed that the period of oscillatio of the pedulum ca be estimated reasoably well usig the liearized equatio up to agles of 30 o. Errors i the amplitude of swig ad i the phase vary with time ad the iitial amplitude. If the time elapsed is large, eve what might ordiarily be cosidered to be small agles of swig ca lead to large errors i the predicted positio of the pedulum. Ideed, umerical experimetatio showed that the liearized equatio represets the positio of the pedulum reasoably well oly up to about 0 o 8. Cosequetly, care must be take i determiig the amplitude ad the phase of swig of a pedulum from

13 the liearized equatio. As a geeral rule, therefore, it is iaccurate to use the magitude of the differece betwee si( ) ad the agle as a idicatio of the accuracy of the solutio obtaied from the liearized equatio. 7. Refereces. Rao, S.S., Mechaical Vibratios, 4th editio, Addiso-Wesley, Readig, Massachusetts, 004, Walker, Peter L., Elliptic Fuctios: A Costructive Approach, Joh Wiley ad Sos, Chichester, Eglad, Page, Leigh, Itroductio to Theoretical Physics, third editio Sixth Pritig, D. Va Nostrad Compay, Ic. Priceto, New Jersey, 965, Temme, Nico M., Special fuctios a itroductio to the classical fuctios of mathematical physics, Joh Wiley ad Sos, New York, 996, pp Whittaker, E. T. ad Watso, G. N. A Course i Moder Aalysis, 4th ed. Cambridge, Eglad: Cambridge Uiversity Press, Morse, P. M. ad Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, 953, Thomso, J.M.T. ad Stewart H.B., Noliear Dyamics ad Chaos: Geometric Methods for Egieers ad Scietists, Joh Wiley ad Sos, Chichester, Eglad, 986, A.B. Basset, A Treatise o Hydrodyamics, Vol., Dover Publicatios, New York, New York, 96,