DISTRIBUTION OF SUB AND SUPER HARMONIC SOLUTION OF MATHIEU EQUATION WITHIN STABLE ZONES

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1 Fifth ASME Interntionl Conference on Multibody Systems, Nonliner Dynmics nd Control Symposium on Dynmics nd Control of Time-Vrying nd Time-Dely Systems nd Structures September 2-2, 05, Long Bech, Cliforni, SA DETC05/VIB-5391 DISTRIBTION OF SB AND SPER HARMONIC SOLTION OF MATHIE EQATION WITHIN STABLE ZONES G. Nkhie Jzr, Mhinflh, M., M. Rstgr Agh, N. Mhmoudin, Dept. of Mech. Eng. North Dkot Stte niversity, Frgo, , SA KEYWORDS: Mthieu eqution dynmics, Sub nd super-hrmonics oscilltion, Prmetric vibrtions. Abstrct The third stble region of the Mthieu stbility chrt, surrounded by one π-trnsition nd one 2π-trnsition curve is investigted. It is known tht the solution of Mthieu eqution is either periodic or qusi-periodic when its prmeters re within stble regions. Periodic responses occur when they re on splitting curve. Splitting curves re within stble regions nd re corresponding to coexisting of periodic curves where n instbility tongue closes. Distributions of sub nd superhrmonics, s well s qusi-periodic solutions re nlyzed using power spectrl density method. In Figure 1, ech stble zone is bounded by two trnsition curves strting off the horizontl xes, from integer roots 2 of eqution n = 0, n=0, 1, 2, 3,. We nme the stble zones s first, for the zone bounded between trnsient curves strted from =0, 1; second, for the zone bounded between trnsient curves strted from =1,, nd so on. The third stble zone, which is bounded between curves strting from = nd =9, will be exmined in this pper s n exmple. 1. INTRODCTION Stbility digrm for the ngulr Mthieu eqution dx x+ x 2b x cos( 2t) = 0 x = (1) dt is depicted in Figure 1 [1]. The stbility digrm is developed by plotting reltionship between constnt prmeters nd b for π nd 2π-periodic solutions of Mthieu Eqution [2]. The prmeters nd b re clled control prmeters simply becuse they control dynmic behvior of the eqution. More specificlly, depending on the vlue of nd b, response of the eqution cn be periodic, qusi-periodic, or unstble. The stbility chrt is mde by plotting the curves clled trnsition curves which indicte the boundry of stble nd unstble regions [3-10]. ce 0 se 1 ce1 se 2 ce 2 se 3 S ce 3 se S S ce se 5 nstble Stble Figure 1. Mthieu stbility digrm bsed on the numericl vlues, generted by McLchln (197). spectrl density for time response of the eqution is utilized to determine the generl behvior of the ce 5 se 6 1 Copyright 05 by ASME

2 system. This investigtion lso indictes how we my find the splitting (or coexistence) curves within ech stble zone. In ddition, pplying energy-rte method provides geometricl illustrtion of the stbility within ech region. 2. THE THIRD STABLE ZONE Applying Energy-Rte method [11], n illustrtion of three dimensionl view of 2π-Mthieu stbility surfce in the third stble zone is shown in Figure 2. Contours of Energy-Rte of the 2π-stbility surfce re plotted in Figure 3. Γ b A B E Figure 2. The third stbility zone of 2π-stbility surfce of Mthieu eqution 10 6 b 2 0 A E D F w 1 π-periodic ridge π-trnsition curve B F C D 2π-trnsition curve Figure 3. Contours of Energy-Rtes in the third stbility zone for 2π-stbility surfce of Mthieu eqution w 2 w 1 C Time integrl of n eqution of motion over period for periodic solution must be zero. Since Mthieu eqution hs periodic solution on trnsition curves, its time integrl would be zero on these curves, while it is equl to positive number for point in unstble region nd negtive number for point within stble region. Evluting time integrl of Mthieu eqution, Γ, over 2π for domin of prmetric plne cn determine the stble nd unstble regions, s well s splitting lines. Skew symmetry chrcteristic of Mthieu eqution implies the 2T-periodic trnsition curves will lso pper when Energy-Rte is pplied on T-periodic function [11]. Therefore, the ridge in Figure 2 indictes π-periodic splitting curves [9]. As result, zero level line of the left wll of the stble vlley in Figure 2 is 1 to 1 or π- periodic trnsition, while the zero level line of the right wll of the third stble vlley is 2 to 1 or 2π-periodic trnsition curve. It is seen in Figure 2, tht there re two wells w 1, nd w 2 in the stble vlley t points ( 1 =.192, b 1 =3.0) nd ( 2 =.19, b 2 =7.0). The dividing ridge touches the zero level plne nd indictes π-periodic curve. In order to investigte the behvior of Mthieu eqution in the third stble region, we will wlk long pth from point B on the π-trnsition wll to point C on 2πtrnsition wll pssing over w 1. We will lso nlyze the behvior of the eqution on the π nd 2π-trnsition curves s well s the π-periodic ridge. The two nd three dimensionl illustrtion of the pths of investigtion re shown in Figures 2 nd 3. Tble 1. Chrcteristics of point 0 to point b We study the Mthieu equtions on 5 different points on the line b=3.0, corresponding to the pth B-C. Point 0 is on the π-trnsition curve, while points 1 is n rbitrry investigting points between the π-trnsition curve nd the π-periodic ridge t locl minimum. Point 2 is on the 2 Copyright 05 by ASME

3 π-periodic ridge. Point 3 is corresponding to the well w 1, nd point is n investigting point on the 2π-trnsition curve of Mthieu stbility digrm. The informtion nd chrcteristics of points 0 to re summrized in Tble ANALYSIS Phse plne, time response, nd power spectrl density of Mthieu eqution for prmeters t points 0 to re plotted in Figure () to Figure (e), pplying initil conditions x(0) = 1,x(0) = 0. The powers re clculted bsed on fst Fourier trnsform lgorithm over π time response []. Time history responses re plotted for the first 10π time units, while phse plne responses re showing for π time units. In ddition, the first four Poincre points re lso indicted on the phse plne responses for shooting time T p =π. Figure () indictes tht point 0 hs π-periodic response bsed on time response, Phse plne, nd repeting Poincre points. In ddition, power density designtes strong π-periodic spike s well s (π/2)- periodic nd (π/3)-periodic sub-hrmonics. It lso represents existence of very long period superhrmonic. The power of π-periodic response is much higher thn sub-hrmonics, indicting tht π-periodic response is the dominnt period, but it is n symmetric oscilltion. of the system. Point 2 t =7.5 corresponds to the πperiodic ridge. =6.17 b=3.0 Γ= Figure (b). Phse plne, time response, nd power of Mthieu eqution of point 1 on pth BC. =7.5 b=3.0 Γ=-1.63E-5 Figure (c). Phse plne, time response, nd power of Mthieu eqution of point 2 on pth BC. ẋ x PSD =6.1 b=3.0 Γ= =.192 b=3.0 Γ= x t Figure (). Phse plne, time response, nd power of Mthieu eqution of point 0 on pth BC. In Figure (b), is 6.17 corresponding to the locl minimum of Energy-Rte t point 1. The response of the system t this point is qusi-periodic response with n overll period very close to 10π s indicted in phse plot Figure (d). Phse plne, time response, nd power of Mthieu eqution of point 3 on pth BC. The π-periodic response is obvious by investigting the phse plot, Poincre points, nd time response of the system in Figure (c). At this point, the spikes of the power spectrl pper exctly t points 0.5, 1.5, 2.5, 3.5,.5, 5.5, 6.5, which re corresponding to (π), (π/3), (π/5), (π/7), (π/9), (π/11), nd (π/13) periods. The power of (π), nd (π/5) re stronger thn the other sub- 3 Copyright 05 by ASME

4 hrmonics, while (π/11), nd (π/13) sub-hrmonics re the wekest. Figure (d) illustrtes the behvior t the well w 1, nd Figure (e) depicts response of the system t point corresponding to 2π-trnsition curve =9.22 b=3.0 Γ= Figure (e). Phse plne, time response, nd power of Mthieu eqution of point on pth BC. Figure 5(b). Top view of power spectrl density surfce of Mthieu eqution in third stble region, for b=3.0. Investigting powers in Figures () to (e) presumes tht by moving from π-trnsition curve to 2π-trnsition curve, the period must grdully chnge from the period set {(π), (π/2), (π/3), (π/), } to the set {(2π), (2π/3), (2π/5), (2π/7), }. Figure 5() depicts the power spectrl density of Mthieu eqution when the investigting point moves on line BC slowly. The resolution of the grph is points per unit. It clerly shows the vrition of super nd sub hrmonics of Mthieu eqution in third stble region, for b=3.0, nd 6.1<<9.22. It lso indictes the power of sub-hrmonics re much lower thn the power of hrmonics round (π), nd (π/2). Projection of the power surfce mkes two dimensionl grph indicting the pek vlues of power on (, ) plne, s shown in Figure 5(b) Figure 5(). 3-dimensionl illustrtion of power spectrl density of Mthieu eqution in third stble region, for b=3.0. Response of Mthieu eqution for b=0 must be simple hrmonic oscilltion with unique principl period equl to T = 2 π /, while its response is periodic or qusiperiodic for b 0. In order to investigte distribution of sub nd super hrmonics of Mthieu eqution on trnsition curves nd on periodic ridge, we illustrte power spectrl density of the eqution on the lines AB, DC nd EF of Figure 2 or 3. Line AB is on the π-periodic ce 2 (cosine elliptic of second order), line DC is on the 2πperiodic se 3 (sine elliptic of third order), nd line EF is on the π-periodic ridge. A 3-dimensionl illustrtion of power spectrl density of Mthieu eqution on the π-trnsition curve, between points A nd B is depicted in Figure 6. Obviously, the response of Mthieu eqution t point A must be pure π- periodic response with no sub or super hrmonics. But when b 0, sub nd super hrmonics pper, lthough nd b re set on trnsition line. A unique nd lone π- periodic response t point A is obvious in Figure 6. By moving from A towrd B, very long super-hrmonic, nd (π/2), (π/3), (π/),, sub-hrmonics re ppering nd growing. The power of π-periodic response is much higher thn the other hrmonics while b is sufficiently smll. The power of π-periodic response hs negtive rte, while the long super-hrmonic nd sub-hrmonics hve positive rtes, indicting tht prticiption of super nd sub-hrmonics re more t higher vlues of b. Copyright 05 by ASME

5 B c 2 A Figure 6. Three-dimensionl illustrtion of power spectrl density of Mthieu eqution on the π-trnsition curve ce 2, between points A nd B of Figure 3. C s 3 D Figure 7. Three -dimensionl illustrtion of power spectrl density of Mthieu eqution on the π-trnsition curve se 3, between points C nd D of Figure 3. F π-periodic ridge E Figure 1. Three -dimensionl illustrtion of power spectrl density of Mthieu eqution on the π-periodic ridge, between points E nd F of Figure 3. A power spectrl density on 2π-trnsition curve, between points D nd C is illustrted in Figure 7. Once gin, the eqution hs pure hrmonic response with period T=2π/3 t point D, but when we move towrd point C, sub nd super-hrmonics re dded to the response, s shown in Figure 7. spectrl in Figure 7 lso indictes tht sensitivity of Mthieu eqution to vrition of is getting higher t high vlues of b. In fct, detecting the trnsition curves of Mthieu eqution t high vlues of both nd b, re not esy becuse of incresing sensitivity. The first point E, of the π-periodic ridge EF is t =6.25 providing simple hrmonic motion with period T=π/5. By moving up, power spectrl grph shown in Figure, show ppernce of super nd sub hrmonics t (π), (π/3), (π/5), (π/7), (π/9), (π/11), nd (π/13). By incresing b, the power of π-hrmonic is growing fster thn sub-hrmonics, while the power of (π/5)-hrmonic hs decresing trend. CONCLSION The third stble region of the Mthieu stbility digrm hs been investigted s n exmple of the stble regions. It is shown tht the 2π-periodic Energy-Rte surfce shows one π-periodic ridge s coexistence splitting curve. The existence of super nd sub-hrmonics in qusiperiodic response of Mthieu eqution hs been nlyzed on trnsition curves nd on the π-splitting curve of the third stble region, using power spectrl density. The Mthieu eqution reduces to simple hrmonic oscilltor with period T = = n, n N, when b=0. For nonzero vlues of b, the response of Mthieu eqution is qusi-periodic. When the vlues of nd b re such tht the corresponding power plot indictes spike t = m/ k, { k,m} N, then the point (, b) is locted on periodic curve strting from 2 nk + m =, { k,m} N, k 2, m < k. The k periodic curve is trnsition curve, if k mod m, otherwise it is splitting curve. The splitting curves hve periods T=2kπ/m. 5 Copyright 05 by ASME

6 REFERENCES 1. McLchln, N. W., (197), Theory nd Appliction of Mthieu Functions, Clrendon, Oxford. P., Reprinted by Dover, New York, Hyshi, C., (196), Nonliner Oscilltions in Physicl Systems, Princeton. 3. Alhrgn, F. A., (1996), A Complete Method for the Computtions of Mthieu Chrcteristic Numbers of Integer Orders, SIAM Review, 3(2), Whittker, E. T., nd Wtson, G. N., (1927), A Course of Modern Anlysis, Cmbridge niv. Press, th ed. 5. Vn der Pol, B.,, nd Strutt, M. J. O., (192), On the Stbility of the solution of Mthieu s Eqution, The Philosophicl Mgzine, vol. 5, p Hill, G. W., (), On the prt of the moon's motion which is function of the men motions of the sun nd the moon, Act Mth.,, Esmilzdeh E. nd Nkhie Jzr G., "Periodic Solution of Mthieu-Duffing Type Eqution", (1997), Interntionl Journl of Nonliner Mechnics, 32(5), p ,.. Cesri, L., (196), Asymptotic Behvior nd Stbility Problems in Ordinry Differentil Equtions, Second Edition, Acdemic Press, New York, SA. 9. Ng, L., nd Rnd, R., (02), Nonliner Effects on Coexistence Phenomenon in Prmetric Excittion, Proceeding of IMECE 02, New Orlens, Louisin, SA, November Ng, L., R., Rnd, nd O Neil, M., (03), Slow Pssge Through Resonnce in Mthieu s Eqution, Journl of Vibrtion nd Control, 9, Nkhie Jzr, G., (0), "Stbility Chrt of Prmetric Vibrting Systems, sing Energy Method", Interntionl Journl of Nonliner Mechnics, 39(), Brighm, E. O., (197), The Fst Fourier Trnsform, Prentice-Hll Inc., New Jersey, 6 Copyright 05 by ASME