Joural of Soud ad Vibratio, Vol. 314, Nº 3-5, 775-78 (008) Harmoic balace approaches to the oliear oscillators i which the restorig force is iversely proportioal to the depedet variable A. Belédez, D. Médez, T. Belédez, A. Herádez ad M. L. Álvarez Departameto de Física, Igeiería de Sistemas y Teoría de la Señal. Uiversidad de Alicate. Apartado 99. E-03080 Alicate. SPAIN E-mail: a.beledez@ua.es Correspodig author: A. Belédez 1
Phoe: +34-96-5903651 Fax: +34-96-5903464
ABSTRACT The secod-order harmoic balace method is used to costruct three approximate frequecy-amplitude relatios for a coservative oliear sigular oscillator i which the restorig force is iversely proportioal to the depedet variable. Two procedures are used to solve the oliear differetial equatio approximately. I the first the differetial equatio is rewritte i a form that does ot cotai the y!1 expressio, while i the secod the differetial equatio is solved directly. The approximate frequecy obtaied usig the secod procedure is more accurate tha the frequecy obtaied with the first oe ad the discrepacy betwee the approximate frequecy ad the exact oe is lower tha 1.8%. Keywords: Noliear oscillator; Harmoic balace method; Approximate frequecy. 3
Mickes [1] has recetly aalyzed the oliear differetial equatio d y dt + 1 y = 0 (1) with iitial coditios y(0) = A,! " dy dt & % t = 0 = 0 () Mickes has also show that all the motios correspodig to Eq. (1) are periodic [1, 3]; the system will oscillate withi symmetric bouds [-A, A], ad the agular frequecy ad correspodig periodic solutio of the oliear oscillator are depedet o the amplitude A. Itegratio of Eq. (1) gives the first itegral 1! dy & " dt % + log y = log A (3) where the itegratio costat was evaluated usig the iitial coditios of Eq. (). From Eq. (3), the expressio for the exact period, T ex (A), for the oliear oscillator give by Eq. (1) takig ito accout the iitial coditios i Eq. () is A dy T ex (A) = 4! (4) 0 log(a / y) The trasformatio y = Ae t reduces this equatio to the form " T ex (A) = 4 A e!t dt = 4 A 0 = A (5) 4
From Eq. (5), the exact value for the agular frequecy is give by the expressio! ex (A) = " T ex (A) = " "A = " A = 1.533141 A (6) It is difficult to solve oliear differetial equatios ad, i geeral, it is ofte more difficult to get a aalytic approximatio tha a umerical oe for a give oliear oscillatory system [3, 4]. There are may approaches for approximatig solutios to oliear oscillatory systems. The most widely studied approximatio methods are the perturbatio methods [5]. The simplest ad perhaps oe of the most useful of these approximatio methods is the Lidstedt-Poicaré perturbatio method, whereby the solutio is aalytically expaded i the power series of a small parameter [3]. To overcome this limitatio, may ew perturbative techiques have bee developed. Modified Lidstedt-Poicaré techiques [6-8], homotopy perturbatio method [9-15] or liear delta expasio [16-18] are oly some examples of them. A recet detailed review of asymptotic methods for strogly oliear oscillators ca be foud i referece [4]. The harmoic balace method is aother procedure for determiig aalytical approximatios to the periodic solutios of differetial equatios by usig a trucated Fourier series represetatio [3, 19-5]. This method ca be applied to oliear oscillatory systems where the oliear terms are ot small ad o perturbatio parameter is required. The mai objective of this paper is to approximately solve Eq. (1) by applyig the harmoic balace method, ad to compare the approximate frequecy obtaied with the exact oe ad with aother approximate frequecy obtaied applyig the harmoic balace method to the same oscillatory system but rewritig Eq. (1) i a way suggested previously by Mickes [1]. The approximate frequecy derived here is more accurate ad closer to the exact solutio. The error i the resultig frequecy is reduced ad the maximum relative error is less tha 1.3% for all values of A. I order to approximately solve Eq. (1), Mickes has rewritte this equatio i a form that does ot cotai the y!1 expressio [1] 5
y d y dt +1= 0, y(0) = A,! dy & " dt % t = 0 = 0 (7) It is possible to solve Eq. (7) by applyig the harmoic balace method. Followig the lowest order harmoic balace method, a reasoable ad simple iitial approximatio satisfyig the coditios i Eq. (7) would be y 1 (t) = Acos!t (8) The substitutio of Eq. (8) ito Eq. (7) gives!a " cos "t +1= 0 (9) the expadig ad simplifyig the resultig expressio gives! 1 A " +1 + (higher! order harmoics) = 0 (10) ad the solutio for the agular frequecy,! M1 (A), is! M1 (A) = A = 1.41414 A (11) ad the percetage error of this approximate frequecy i relatio to the exact oe is percetage error =! ex "! M1! ex 100 =1.8% (1) Mickes [1] also used the secod-order harmoic balace approximatio 6
y (t) = A cos!t + B cos3!t (13) to the periodic solutio of Eq. (7). Substitutio of Eq. (13) ito Eq. (7), simplifyig the resultig expressio ad equatig the coefficiets of the lowest harmoics to zero gives two equatios ad takig ito accout that A = A + B, Mickes [1] obtaied A = 10A/9 ad B = -A/9, ad the secod-order approximate solutio (Eq. (13)) to Eq. (7) ca be writte as follows y M (t) = 10 9 Acos(! t) " 1 M 9 Acos(3! M t) (14) where the secod-order approximate frequecy,! M (A), is give by! M (A) = 16 10A = 1.779 A (15) ad the percetage error is percetage error =! ex "! M! ex 100 =1.55% (16) As we poited out previously, the mai objective of this paper is to solve Eq. (1) istead of Eq. (7) by applyig the harmoic balace method. Substitutio of Eq. (8) ito Eq. (1) gives!a" cos"t + 1 Acos"t = 0 (17) I order to apply the first order harmoic balace method to Eq. (17) we have to expad Eq. (17) ad to set the coefficiet of cos!t (the lowest order harmoic) equal to zero. To take this, firstly we do the followig Fourier series expasio 7
" 1 Acos!t = a +1 cos[( +1)!t] = a 1 cos!t + a 3 cos3!t + (18) =0 where the first term of this expasio ca be obtaied by meas of the followig equatio a 1 = 4! / 1 cos" d" =! 0 Acos" A (19) Substitutig Eq. (18) ito Eq. (17) ad takig ito accout Eq. (19) gives!" + & % A ' ( cos"t + (higher! order harmoics) = 0 (0) For the lowest order harmoic to be equal to zero, it is ecessary to set the coefficiet of cosωt equal to zero i Eq. (0), the! 1 ( A) = A = 1.414 A (1) Cosequetly, i this limit, the low-order harmoic balace method applied to Eq. (1) gives exactly the same results as the low-order harmoic balace method applied to Eq. (7). I order to obtai the ext level of harmoic balace, we express the periodic solutio to Eq. (1) with the assiged coditios i Eq. () i the form of [19-] y (t) = y 1 (t) +u(t) () where u(t) is the correctio part which is a periodic fuctio of t of period! /" ad 8
u(0) = 0,! " du dt & % t = 0 = 0 (3) Substitutig Eq. () ito Eq. (1) gives d y 1 dt + d u dt + 1 y 1 (t) +u(t) = 0 (4) Wu ad Lim [19-1] preseted a approach by combiig the harmoic balace method ad liearizatio of oliear oscillatio equatio with respect to displacemet icremet oly, u(t). This harmoic balace approach will be used to approximately solve Eq. (4). Liearizig the goverig Eq. (4) with respect to the correctio u(t) at y 1 (t) leads to ad d y 1 dt + d u dt + 1 y 1 (t)! u(t) y 1 (t) = 0 (5) u(0) = 0,! " du dt & % t = 0 = 0 (6) To obtai the secod approximatio to the solutio, u(t) i Eq. (), which must satisfy the iitial coditios i Eq. (6), we take ito accout the secod-order harmoic balace approximatio i Eq. (13) which ca be writte as follows y (t) = A cos!t + B cos3!t = Acos!t " B cos!t + B cos3!t = Acos!t + B (cos3!t " cos!t) where we have take ito accout that A = A + B. From Eqs. (8), () ad (7) we ca see that u(t) takes the form (7) 9
u(t) = B (cos3!t " cos!t) (8) where B is a costat to be determied. Substitutig Eqs. (), (8) ad (8) ito Eq. (5), expadig the expressio i a trigoometric series ad settig the coefficiets of the terms cos!t ad cos3!t equal to zero, respectively, leads to!(a! B )" + A (A + B ) = 0 (9) ad!a! 8B! 9B A " = 0 (30) A From Eqs. (9) ad (30) we ca obtai B ad ω as follows B WL = A (! 6) =!0.09354A (31) 14! WL (A) = A + 8B WL = 1.19373 "9B WL A A (3) With this value for B, Eq. (7) ca be writte y WL (t) = 0! 14 Acos(" WL t) +! 6 14 Acos(3" WL t) (33) The percetage error for the secod order approximate frequecy is 10
percetage error =! ex "! WL! ex 100 =.71% (34) which is higher tha the percetage error for the secod approximate frequecy obtaied by Mickes whe harmoic balace method is applied to Eq. (7). As we ca see substitutio of Eq. () ito Eq. (1) does ot give the same result as substitutio of Eq. (8) ito Eq. (7) ad applicatio of the secod-order harmoic balace method to Eq. (7) give a more accurate frequecy tha applicatio of Wu ad Lim s approach to Eq. (1). These questios have bee aalyzed i detail i refereces [3] ad [5] for other oscillators aalyzed by first-order harmoic balace method ad oe would wait to obtai better results whe the harmoic balace method is applied to Eq. (1) istead of to Eq. (7). This would be due to the fact that whe we substitute Eq. (13) ito Eq. (7) we obtai a equatio that icludes oly three eve powers of cosωt: 1 (cos 0 ωt), cos ωt ad cos 4 ωt ad the there are oly three cotributios to the coefficiet of the first term 1 (cos0ωt), from 1 (cos 0 ωt), cos ωt ad cos 4 ωt, ad two cotributios to the coefficiet of the secod harmoic cos(ωt), from cos ωt ad cos 4 ωt, Therefore, substitutig Eq. (13) ito Eq. (7) produces oly three terms, 1, cos(ωt) ad cos(4ωt). However, Eq. (17) icludes all odd powers of cosωt, which are cos +1 ωt with = 0, 1,,,, ad the there are ifiite cotributios to the coefficiet of the first harmoic cosωt, that is, 1 from cosωt, 3/4 from cos 3 ωt, 5/8 from cos 5 ωt,, "! + 1 % ' from & cos +1!t, ad so o. Therefore, substitutig Eq. (7) ito Eq. (1) produces the ifiite set of higher harmoics, cosωt, cos3ωt,, cos[(+1)ωt], ad so o, ad the secod-order agular frequecy i Eq. (3) would have to be more accurate tha the frequecy give i Eq. (15). But we obtaied the opposite result. The reaso is that we have ot applied the exact secod-harmoic balace method to Eq. (1), but a liearized approximatio to this method. I order to verify this affirmatio, we cosider a ew approach to obtai higherorder approximatios usig harmoic balace method. Istead of cosiderig the assumptio i Eq. (5), first we do the followig series expasio 11
1 y 1 (t) +u(t) = 1 y 1 (t)[1 + y!1 1 (t)u(t)] = u (t) (!1) (35) y +1 1 (t) " =0 ad substitutig Eq. (35) ito equatio (1) gives d y 1 dt + d " u dt + (!1) u (t) = 0 (36) y +1 1 (t) =0 To obtai the secod approximatio to the solutio, u(t) i Eq. (), which must satisfy the iitial coditios i Eq. (3), takes the form u(t) = B (cos3!t " cos!t) = 4B (cos 3!t " cos!t) (37) where c is a costat to be determied. Substitutig Eqs. (8), () ad (37) ito Eq. (36) gives 4 B!" (A! B )cos"t! 9" B cos3"t + (1! cos "t) = 0 (38) A +1 cos"t = 0 The formula that allows us to obtai (1! cos "t) is (1! cos "t) = ) k= 0 & (!1) k! % ((!cos "t) k = ) cos k "t (39) k' k!(! k)! k= 0 Substitutig Eq. (39) ito Eq. (38) gives 1
!" (A! B )cos"t + 9" B cos3"t + = 0 4!B (!1) k cos k!1 "t = 0 (40) A +1 k!(! k)! k= 0 It is possible to state the followig Fourier series expasio where cos k!1 (k ) (k "t = b j+1 cos[( j +1)"t] = b ) (k 1 cos"t + b ) 3 cos3"t + (41) j=0 (k ) b j+1 = 4!!/ cos k"1 cos[( j +1)]d = 0 (k "1)!%(k + 1 )!(k " j "1)!( j + k)! (4) ad where!( z) is the Euler gamma fuctio [6]. Substitutig Eqs. (41) ad (4) ito Eq. (40), we obtai!" (A! B )cos"t! 9" B cos3"t + + = 0 4!B A +1 k= 0 k (!1) k (k!1)!%(k + 1 ) cos[( j +1)"t] = 0 k!(! k)! &(k! j!1)!( j + k)! j = 0 (43) ad settig the coefficiets of the resultig items cos!t (j = 0) ad cos3!t (j = 1) equal to zero, respectively, yields!" (A! B ) + = 0 4!B (!1) k %(k + 1 ) = 0 (44) A +1 &(k!) (! k)! k= 0!9" B + = 0 4!B (!1) k (k!1)%(k + 1 ) = 0 (45) A +1 &k!(k +1)!(! k)! k= 0 13
which ca be writte as follows!" (A! B ) + A A A! 4B = 0 (46)!9" B + A + B! A! 4 AB B A! 4AB = 0 (47) I Eq. (47) the followig relatios have to be take ito accout (!1) k "(k + 1) = "(k + 1) (k!) (! k)! (!) (48) k =0! " = 0 4!B A +1 (k + 1 ) (!) = A A A % 4B (49) while i Eq. (48) the followig expressios have bee cosidered (!1) k (k!1)"(k + 1) =! "( + 1)!+(!1)!"( + 3) k!(! k)!(1 + k)! ( +1)!!(!1)! k =0 [ ] (50)! " = 0 (!1) 4!c A +1 [ ( + 1)!+(!1)!( + 3) ] =!A + B + A! 4AB (51) %( +1)!!(!1)! B A! 4AB These results have bee obtaied usig Mathematica. From Eqs. (46) ad (47) ad oce agai by usig Mathematica we ca obtai ω 14
ad B as follows B =! A " 159 + 5 106 50 ( ) 1/ 3 89! 6! ( 159 + 5 106 ) 1/ 3 % ' = 0.10158074A (5) ' &! (A) = A(A " B ) A = 1.37330058 A " 4AB A (53) With this value for B, Eq. () ca be writte y (t) =1.101581Acos(! t) " 0.101581Acos(3! t) (54) The percetage error for the secod order approximate frequecy is percetage error =! ex "!! ex 100 =1.75% (55) As we ca see, this error is lower tha the error obtaied by Mickes (1.55%) ad we ca coclude this is the percetage error obtaied whe the secod-order harmoic balace method is exactly applied to Eq. (1). The secod order harmoic balace method was used to obtai three approximate frequecies for a oliear sigular oscillator. The first approximate frequecy,! M, was obtaied by rewritig the oliear differetial equatio i a form that does ot cotai the y!1 term; while the secod ad the third oes,! WL ad!, were obtaied by solvig the oliear differetial equatio cotaiig the y!1 term. The secod-order approximate frequecy! WL is obtaied by usig the approach by Wu ad Lim [19-1]. This approach ca be described as a liearisatio of the harmoic balace method, ad works pretty well for the chose problems. Because the harmoic balace method does ot elimiate the 15
secular terms systematically, it is difficult to obtai secod ad higher order approximate solutios by the harmoic balace method. But this approach elimiates this difficulty ad may be applied to other oliear oscillators. We ca coclude that equatios (5) ad (53) are valid for the complete rage of oscillatio amplitude, icludig the limitig cases of amplitude approachig zero ad ifiity. Excellet agreemet of the approximate frequecies with the exact value was demostrated, ad discussed, ad the discrepacy betwee the third approximate frequecy, ω, ad the exact value ever exceeds 1.8%. The approximate frequecy, ω, derived here is the best frequecy that ca be obtaied usig the first-order harmoic balace method, ad the maximum relative error was reduced as compared with approximate frequecies! M ad! WL. Fially, we discussed the reaso why the accuracy of the approximate frequecy, ω, is better tha that of the frequecy! M obtaied by Mickes. This reaso is related to the umber of harmoics that applicatio of the secod-order harmoic balace method produces for each differetial equatio solved. Ackowledgemets This work was supported by the Miisterio de Educació y Ciecia, Spai, uder project FIS005-05881-C0-0, ad by the Geeralitat Valeciaa, Spai, uder project ACOMP/007/00. 16
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