Phát triển phương pháp runge kutta nystrom tính toán dao động của cơ hệ có phần từ đàn phớt cấp phân số

B GIO DC V O TO TRNG I HC BCH KHOA H NI DNG VN LC PHT TRIN PHNG PHP RUNGE-KUTTA-NYSTRM TNH TON DAO NG CA C H Cể PHN T N PHT CP PHN S Chuyờn ngnh : C IN T LUN VN THC S KHOA HC C IN T NGI HNG DN KHOA HC GS.TSKH NGUYN VN KHANG H Ni - 2016 MC LC LI CAM OAN DANH MC CC Kí HIU V CC T VIT TT DANH MC CC BNG .5 DANH MC CC HèNH V V TH M U CHNG O HM CP PHN S 1.1 KHI NIM O HM V TCH PHN 1.2 BIU THC HP NHT GIA O HM V TCH PHN 10 1.2.1 o hm cp n 10 1.2.2 Tớch phõn nhiu lp ca mt hm s 11 1.2.3 S hp nht gia toỏn t o hm cp n v tớch phõn n lp 12 1.3 NH NGHA O HM CP PHN S 13 1.3.1 énh ngha o hm cp phõn s Grỹnwald-Letnikov 13 1.3.2 énh ngha o hm cp phõn s Riemann Liouville 13 1.3.3 énh ngha o hm cp phõn s Caputo 14 1.3.4 Mt s nh ngha o hm cp phõn s khỏc 15 1.3.5 S tng ng ca nh ngha Riemann-Liouville v Grỹwald-Letnikov 17 1.4 CC TNH CHT CA O HM CP PHN S 18 1.4.1 Tớnh cht tuyn tớnh 18 1.4.2 Quy tc Leibniz 18 1.4.3 Tớnh cht bin i thang bc 18 1.4.4 o hm cp phõn s ca mt chui 19 1.4.5 Tớnh cht hp thnh 19 1.5 MT S V D V O HM CP PHN S 20 1.5.1 o hm cp phõn s ca mt hng s 20 1.5.2 o hm cp phõn s ca hm f (t ) 1.5.3 o hm cp phõn s ca f (t ) t a 21 (t a ) p 21 1.5.4 o hm v tớch phõn cp phõn s ca hm f (t ) (1 t ) p 21 1.5.5 o hm ca hm bc nhy n v v hm Delta-Dirac 22 1.5.6 o hm cp phõn s ca hm f (t ) e at 23 1.6 PHẫP BIN I FOURIER V LAPLACE CA O HM CP PHN S 23 1.6.1 Phộp bin i Laplace 23 1.6.2 S tn ti ca phộp bin i Laplace 24 1.6.3 Tớnh cht phộp bin i Laplace 24 1.6.4 Phộp bin i Laplace ca o hm cp phõn s 24 1.6.5 Phộp bin i Fourier ca o hm cp phõn s 25 CHNG PHNG PHP S TNH TON DAO NG CA C H Cể O HM CP PHN S 27 2.1 HAI THUT TON XC NH GN NG O HM CP PHN S 27 2.1.1 Thut toỏn s dng o hm cp mt 27 2.1.2 Thut toỏn s dng o hm cp hai 32 2.2 PHNG PHP S TNH TON DAO NG C H CHA O HM CP PHN S 35 2.2.1 Phng phỏp sai phõn 35 2.2.2 Phng phỏp Newmark 35 2.2.3 Phng phỏp Runge-Kutta 37 2.2.4 Phỏt trin phng phỏp Runge-Kutta-Nystrửm tớnh toỏn dao ng c h cú o hm cp phõn s 39 2.2.5 So sỏnh chớnh xỏc v thi gian tớnh gia cỏc phng phỏp 40 CHNG TNH TON DAO NG CA H Cể PHN T N NHT CP PHN S BNG PHNG PHP RUNGE-KUTTA-NYSTRM 44 3.1 DAO NG CA H CHU KCH NG VA P 44 3.1.1 Mụ hỡnh dao ng ca h chu kớch ng va p 44 3.1.2 Tớnh toỏn dao ng ca h chu kớch ng va p 45 3.2 DAO NG PHI TUYN CA H Cể PHN T O HM CP PHN S 49 3.2.1 Dao ng ca h Duffing 49 3.2.2 Dao ng ca h Vander Pol 59 KT QU V BN LUN 67 TI LIU THAM KHO 68 PH LC 73 LI CAM OAN Tờn tụi l DNG VN LC, hc viờn cao hc lp 14BCT.KH, khúa CH2014B, chuyờn ngnh C in t Sau thi gian hc tp, nghiờn cu ti trng i Hc Bỏch Khoa H Ni, c s giỳp hng dn ca thy NGUYN VN KHANG, tụi ó hon thnh lun tt nghip thc s Tụi xin cam oan cỏc ni dung c trỡnh by lun ny l kt qu nghiờn cu ca bn thõn tụi, khụng cú s chộp hay copy ca bt c tỏc gi no Tụi xin t chu trỏch nhim v li cam oan ca mỡnh H Ni, Ngy 28 thỏng 03 nm 2016 Tỏc gi DNG VN LC DANH MC CC Kí HIU V CC T VIT TT Ký hiu Ni dung, ý ngha p a Dt R p a t C p a t G p a t Ký hiu o hm cp phõn s D Ký hiu o hm cp phõn s theo Riemann Liouville D Ký hiu o hm cp phõn s theo Caputo D s Ký hiu o hm cp phõn s Grunwald-Letnikov p, q m c k f t RKN Hm Gramma Hm Bờta Khi lng cn nht cng Ngoi lc Phng phỏp Runge-Kutta-Nystrửm DANH MC CC BNG Tờn Ni dung Bng 2.1 Bng 2.2 So sỏnh nghim chớnh xỏc v kt qu tớnh toỏn ca cỏc phng phỏp So sỏnh sai s % tng i ca cỏc phng phỏp Trang 41 41 DANH MC CC HèNH V V TH Ni dung Tờn Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh 1.1 1.2 2.1 2.2 2.3 2.4 2.5 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18 3.19 3.20 3.21 3.22 3.23 3.24 3.25 3.26 3.27 3.28 3.29 3.30 3.31 3.32 3.33 3.34 3.35 ng cong Jordan trn tng khỳc cú hai im mỳt ng cong kớn C D thuc mt phng phc Xp x tớch phõn bng cụng thc hỡnh thang Nghim chớnh xỏc (vớ d 2.1) Kt qu gii v thi gian tớnh (vớ d 2.2) Kt qu gii, v thi gian tớnh (vớ d 2.3) Kt qu gii, v thi gian tớnh (vớ d 2.4) H dao ng chu kớch ng va p Mụ hỡnh dao ng sau va chm Kt qu so sỏnh mụ hỡnh lý thuyt IIa v thc nghim vi h=30mm Kt qu so sỏnh mụ hỡnh lý thuyt IIa v thc nghim vi h=60mm Kt qu so sỏnh mụ hỡnh lý thuyt IIa v thc nghim vi h=100mm Kt qu so sỏnh mụ hỡnh lý thuyt IIb v thc nghim vi h=30mm Kt qu so sỏnh mụ hỡnh lý thuyt IIb v thc nghim vi h=60mm Kt qu so sỏnh mụ hỡnh lý thuyt IIb v thc nghim vi h=100mm Kt qu so sỏnh mụ hỡnh lý thuyt IIIc v thc nghim vi h=30mm Kt qu so sỏnh mụ hỡnh lý thuyt IIIc v thc nghim vi h=60mm Kt qu so sỏnh mụ hỡnh lý thuyt IIIc v thc nghim vi h=100mm Kt qu so sỏnh mụ hỡnh lý thuyt IVc v thc nghim vi h=30mm Kt qu so sỏnh mụ hỡnh lý thuyt IVc v thc nghim vi h=60mm Kt qu so sỏnh mụ hỡnh lý thuyt IVc v thc nghim vi h=100mm th dch chuyn theo thi gian (vớ d 3.1) th gia tc theo thi gian (vớ d 3.1) th pha o hm cp phõn s theo dch chuyn (vớ d 3.1) th pha o hm cp phõn s theo gia tc (vớ d 3.1) th dch chuyn theo thi gian (vớ d 3.2) th gia tc theo thi gian (vớ d 3.2) th pha o hm cp phõn s theo dch chuyn (vớ d 3.2) th pha o hm cp phõn s theo gia tc (vớ d 3.2) th dch chuyn theo thi gian (vớ d 3.3) th gia tc theo thi gian (vớ d 3.3) th pha o hm cp phõn s theo dch chuyn (vớ d 3.3) th pha o hm cp phõn s theo gia tc (vớ d 3.4) th dch chuyn theo thi gian (vớ d 3.4) th o hm cp phõn s theo thi gian (vớ d 3.4) th pha o hm cp phõn s theo dch chuyn (vớ d 3.4) th dch chuyn theo thi gian (vớ d 3.5) th dch chuyn theo thi gian (vớ d 3.5) th dch chuyn theo thi gian (vớ d 3.5) th dch chuyn theo thi gian (vớ d 3.5) th pha o hm cp phõn s theo dch chuyn (vớ d 3.5) th pha o hm cp phõn s theo dch chuyn (vớ d 3.5) Trang 16 17 33 41 42 42 43 44 44 45 46 46 46 47 47 47 48 48 48 49 49 50 50 51 51 51 52 52 52 53 53 53 54 54 54 55 55 56 56 56 57 57 Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh Hỡnh 3.36 3.37 3.38 3.39 3.40 3.41 3.42 3.43 3.44 3.45 3.46 3.47 3.48 3.49 3.50 3.51 3.52 3.53 3.54 3.55 3.56 3.57 3.58 3.59 3.60 th dch chuyn theo thi gian (vớ d 3.6) th gia tc theo thi gian (vớ d 3.6) th pha tc theo dch chuyn (vớ d 3.6) th pha o hm cp phõn s theo dch chuyn (vớ d 3.6) th pha o hm cp phõn s theo gia tc (vớ d 3.6) th dch chuyn theo thi gian (vớ d 3.7) th gia tc theo thi gian (vớ d 3.7) th pha o hm cp phõn s theo dch chuyn (vớ d 3.7) th pha o hm cp phõn s theo gia tc (vớ d 3.7) th dch chuyn theo thi gian (vớ d 3.8) th gia tc theo thi gian (vớ d 3.8) th pha o hm cp phõn s theo dch chuyn (vớ d 3.8) th pha o hm cp phõn s theo gia tc (vớ d 3.8) th dch chuyn theo thi gian (vớ d 3.9) th o hm cp phõn s theo thi gian (vớ d 3.9) th pha o hm cp phõn s theo dch chuyn (vớ d 3.9) th dch chuyn theo thi gian (vớ d 3.10) th o hm cp phõn s theo thi gian (vớ d 3.10) th pha o hm cp phõn s theo dch chuyn (vớ d 3.10) th dch chuyn theo thi gian (vớ d 3.11) th o hm cp phõn s theo thi gian (vớ d 3.11) th pha o hm cp phõn s theo dch chuyn (vớ d 3.11) th dch chuyn theo thi gian (vớ d 3.12) th o hm cp phõn s theo thi gian (vớ d 3.12) th pha o hm cp phõn s theo dch chuyn (vớ d 3.12) 57 58 58 58 59 60 60 60 61 61 61 62 62 62 63 63 64 64 64 65 65 65 66 66 66 M U Vi thp k gn õy, nhiu ng dng ca o hm cp phõn s cỏc lnh vc vt lý, húa hc, c khớ, giao thụng ti, xõy dng, ti chớnh v cỏc ngnh khoa hc khỏc ó c quan tõm nghiờn cu Oldham v Spanier ng dng o hm cp phõn s vo quỏ trỡnh khuych tỏn; Kempfle mụ t c h tt dn; Bagley v Torvik, Caputo nghiờn cu v tớnh cht ca cỏc vt liu mi Cỏc nh c hc cng bt u nghiờn cu vic ỏp dng o hm cp phõn s vo cỏc h dao ng, ng lc hc nh h n nht v nht Nutting (1921, 1943) l mt nhng nh nghiờn cu u tiờn ngh rng hin tng chựng ng sut cú th c mụ hỡnh thụng qua thi gian bc phõn s Shimizu (1995) nghiờn cu dao ng v c tớnh xung ca b dao ng vi mụ hỡnh Kelvin Voigt phõn s ca vt liu n nht da trờn gel silicone v chng minh mt s tớnh cht khỏc bit gia kh nng gim chn ca vt liu ny so vi vt liu o hm cp nguyờn Zhang v Shimizu (1999) nghiờn cu mt vi phng din quan trng v trng thỏi tt dn ca b dao ng n nht c mụ hỡnh bi quy lut kt cu Kelvin Voigt phõn s H ó tho lun s nh hng ca iu kin u ti trng thỏi tt dn Ta ó bit quan h gia lc v bin dng ca cỏc b gim chn n nht cú dng f t Dpx t , p Trong ú f t l lc tỏc dng, x t l dch chuyn, D p l h s cn khụng tuyn tớnh, dp l toỏn t o hm cp phõn s dt p Thc t rng i vi nhng bin dng ln, ỏp ng phi tuyn xut hin Mt s mụ hỡnh c xut gii thớch s ỏp ng phi tuyn Mt mụ hỡnh cú th c a l mt lũ xo phi tuyn c thờm vo v phi ca phng trỡnh trờn Mt mụ hỡnh khỏc c a bi Zhimizu v Nasuno yờu cu i vi mt s vt liu n nht, tớnh phi tuyn xut phỏt t o hm cp phõn s ca bin dng nộn Lun ny trỡnh by cỏc phng phỏp s gii phng trỡnh vi phõn cha o hm cp phõn s, ú phng phỏp Runge-Kutta-Nystrửm c phỏt trin tớnh toỏn dao ng ca c h cú thnh phn o hm cp phõn s Lun s dng cỏc phng phỏp s ny tớnh toỏn mt vi mụ hỡnh dao ng phi tuyn ca cỏc h n nht cú cha o hm cp phõn s Qua õy em xin gi li cm n chõn thnh n thy GS.TSKH Nguyn Vn Khang, thy ó tn tỡnh giỳp em sut thi gian thc hin lun ny CHNG O HM CP PHN S 1.1 KHI NIM O HM V TCH PHN Chỳng ta s dng n v N l nhng s nguyờn dng, v p l s bt k Cho mt hm s f t Ta ký hiu o hm cp 1, cp 2, cp n ,ca hm f t d2 f t df t nh sau dn f t , , , , dt dt dt n Ngoi ta cng cú cỏc ký hiu o hm tng t df t d f t dn f t , , , , n dt dt dt o hm ca hm f t theo df d t a df t dt , (1.1) (1.2) a bng o hm theo t ca nú t d2 f t d2 f t d t a dt dn f t , , d t a dn f t n dt n , (1.3) Do tớch phõn l s nghch o ca o hm nờn ta vit t d 1f t f t0 dt0 dt Cỏc tớch phõn nhiu lp c ký hiu d d n dt dt1 tn dtn n t1 t t f t f t dt (1.4) (1.5) t2 dtn f t0 dt0 t1 dt1 f t0 dt0 0 (1.6) Khi gii hn di khỏc 0, cỏc tớch phõn s c vit t d 1f t d n d t a a tn t f t dtn n (1.7) f t0 dt0 d t a dtn a t2 t1 dt1 f t0 dt0 a a (1.8) a Lu ý phng trỡnh sau ỳng vi o hm nhng khụng ỳng vi tớch phõn dn f t d t a dn f t n dt n (1.9) Tc l x 0.2, x 0.2 (3.35) Kt qu mụ phng Hỡnh 3.52 th dch chuyn theo thi gian Hỡnh 3.53 th o hm cp phõn s theo thi gian Hỡnh 3.54 th pha o hm cp phõn s theo dch chuyn Vớ d 3.11: Kho sỏt dao ng Vander Pol 64 D0.95 x y y x 0.5x y 0.5 y (3.36) iu kin u: x 0, y (3.37) Kt qu mụ phng Hỡnh 3.55 th dch chuyn theo thi gian Hỡnh 3.56 th o hm cp phõn s theo thi gian Hỡnh 3.57 th pha o hm cp phõn s theo dch chuyn 65 Vớ d 3.12: Kho sỏt dao ng Vander Pol D0.95 x y y 2cos t cos3 t x x2 y y (3.38) iu kin u: x 0, y (3.39) Kt qu mụ phng Hỡnh 3.58 th dch chuyn theo thi gian Hỡnh 3.59 th o hm cp phõn s theo thi gian Hỡnh 3.60 th pha o hm cp phõn s theo dch chuyn 66 KT QU V BN LUN Vi s phỏt trin khoa hc k thut, ngy cng cú nhiu cỏc vt liu mi i (nh vt liu silicone, vt liu cao su), nhng mụ hỡnh n nht c in vi o hm cp nguyờn khụng th hin c y tớnh cht ca vt liu Do ú gii quyt ny, o hm cp phõn s c ỏp dng nghiờn cu thi gian gn õy Lun gm chng vi ni dung nh sau: - Chng mt: Trỡnh by cỏc c bn ca o hm cp phõn s da trờn ti liu [1, 5, 7, 30] - Chng hai: Trỡnh by v cỏc phng phỏp s khỏc xp x thnh phn o hm cp phõn s, cng nh cỏc phng phỏp s tớnh toỏn dao ng ca c h cha thnh phn o hm cp phõn s da trờn nhng ti liu [3, 20, 28, 29, 30, 31, 32, 33] - Chng ba: Trỡnh by v ng dng cỏc thut toỏn tớnh toỏn dao ng ca cỏc h c cú thnh phn o hm cp phõn s da trờn cỏc ti liu [33-58] Lun a cỏc phng phỏp s gii quyt cỏc bi toỏn dao ng phi tuyn cp phõn s v ng dng tớnh toỏn dao ng ca múng mỏy, dao ng hn n ca h Duffing v Vander Pol Hng nghiờn cu sp ti l m rng tớnh toỏn dao ng cho nhiu mụ hỡnh phc hn, h dao ng cú tr, nghiờn cu v s n nh ca cỏc phng phỏp s, cng nh x dng cỏc kt qu thc nghim tỡm cỏc quy lut phi tuyn cp phõn s mi Hy vng nhng kt qu ca lun ny gúp phn nghiờn cu cỏc bi toỏn lnh vc dao ng phi tuyn cp phõn s Tuy nhiờn kh nng v trỡnh cũn hn ch cng nh thi gian cú hn nờn lun ca em chc chn cũn nhiu thiu sút, em rt mong s nhn c nhng ý kin úng gúp phờ bỡnh v b sung ca cỏc thy cụ giỏo b mụn em cú th hon thin v b sung thờm kin thc Mt ln na em xin chõn thnh cm n Thy GS.TSKH Nguyn Vn Khang ó tn tỡnh giỳp em thi gian nghiờn cu v hon thnh lun ny 67 TI LIU THAM KHO [1] Nguyn Vn Khang, Bi ging ng lc hc h cú o hm cp phõn s, Trng i hc Bỏch Khoa, (2009) [2] Nguyn Vn Khang, Dao ng k thut, NXB Khoa hc k thut, H Ni, (2004) [3] Nguyen Van Khang, N Shimizu, M Fukunaga, Duong Van Lac, Bui Thi Thuy, Calculation of responses of a nonlinear fractional derivative model of impulse motion for viscoelastic materials using Runge-Kutta-Nystrửm method, Tuyn hi ngh c hc ton quc, H Ni (2014) [4] M Fukunaga and N Shimizu Comparison of fractional derivative models for nite deformation with experiments of impulse response Journal of Vibration and Control July 10, (2013) [5] M Fukunaga, N Shimizu, H Nasuno, A nonlinear fractional derivative model of impulse motion for viscoelastic materials, Physica Scripta T136, 014010 (6pp), (2009) [6] K B Oldham, J Spanier, The Fractional Calculus, Academic Press, Boston, New York (1974) [7] I Podluny, Fractional Differential Equations, Academic Press, Boston, New York (1999) [8] R.L Bagley, P.J Torvik, A theoretical basis for the application of fractional calculus to viscoelasticity, Journal of Rheology, pp 201-210, (1983) [9] R.L Bagley, P.J Torvik, Fractional calculus in the transient analysis of viscoelastically damped structures, AIAA Journal, pp 918-925, (1985) [10] K Diethelm, An Algorithm for the Numerical Solution of Differential Equations of Fractional Order, IMA J Numer Anal, Vol.5, pp 1-6, (1997) [11] K Diethelm, Fractional Differential Equations, Vorlesungskrifit der TU Braunschweig, (2003) [12] L E Suarez, A Shokooh, Response of Systems with Damping Materials Modeled using Fractional Calculus, ASME J Appl Mech, Vol.48, No.11, pp 9, (1995) [13] L E Suarez, A Shokooh, An Eigenvector Expansion Method for the Solution of Motion Containing Fractional Derivatives, ASME J Appl Mech, Vol.64, pp 629-635, (1957) 68 [14] L Gaul, P Klein, S Kemple, Damping description involving fractional operators, Mechanical systems and signal processing 5(2), pp 8188, (1991) [15] Dumitru Baleanu et all , Fractional Calculus: Models and Numerical Methods, World scientific publishing, Singapore, (2012) [16] M Fukunaga, N Shimiz, Analysis of Impulse Response of Gel by Nonlinear Fractional Derivative Models, Proceedings of the ASME 2009 International [17] Design Engineering Technical Conferences, San Diego, California USA, (2009) Neville J Ford and A Charles Simpson, Numerical and Analytical Treatment of Differential Equations of Fractional Order, Numerical Analysis Report 387, Manchester Centre for Computational Mathematics, Manchester, (2003) [18] N Gil-Negrete, J Vinolas, L Kari, A Nonlinear Rubber Material Model Combing Fractional Order Viscoelasticity and Amplitude Dependent Effects, ASME J Appl Mech, Vol.76, pp 110091-110099, (2009) [19] N Shimizu, Dynamic Characteristics of a Viscoelastic Oscillator, Trans Jps Soc Mech Eng., Vol.61, No.583, C, p 166-170, (1995) [20] N Shimizu, W Zhang, Fractional Calculus Approach to Dynamic Problems of Viscoelastic Materials, International Journal of JSME, Series C, Vol.42, No.4, pp [21] 825-837, (1999) N Shimizu, H Nasuno, Modeling and Analysis of Nonlinear Viscoelastic Systems by means of Fractional Calculus Numerical Integration Algorithms, International Conference on Material Theory and Nonlinear Dynamics, Hanoi, (2007) [22] P G Nutting, A New General Law of Deformation, J of the Frankin Inst, 191, pp 679-685, (1921) [23] P.G Nutting, A General Stress-Strain-Time Formula, J of the Frankin Inst, 235, pp 513-524, (1943) [24] R C Koller, Applications of Fractional Calculus to the Theory of Viscoelasticity, ASME J Appl Mech., Vol.51, pp 299-307, (1984) [25] S Saha Ray, B P Poddar, R K Bera, Analytical Solution of a Dynamic System Containing Fractional Derivative of Order One-Half by Adomian Decomposition Method, ASME J Appl Mech., Vol.72, pp 290-295, (2005) [26] S Sakakibara, Properties of Vibration with Fractional Derivative Damping of Order 1/2 , International Journal of JSME, Series C, Vol.40, pp 393-399, (1997) 69 [27] W Zhang, N Shimizu, Damping Properties of the Viscoelastic Material Described by Fractional Kelvin-Voigt Model, International Journal of JSME, Vol.42, No.1, pp 1-9, (1999) [28] Zhang W., Shimizu N Numerical algorithm for dynamic problems involving fractional operators Int J of the Japan Society of Mechanical Engineers, Series C 41(3): 364-370, (1998) [29] Lixia Yuan and Om P Agrawal A numerical scheme for dynamic systems containing fractional derivatives J Vib Acoust., 124:321324, (2002) [30] T.M Atanackovic, B Stankovic On a numerical scheme for solving differential equations of fractional Oder Mechanics Research Communications, Volume 35, Issue 7, pp 429438, (2008) [31] Chan ghee KOH, James M Kelly, Application of Fractional derivative to seismic analysis of base-isolated models, Earthquake Engineering & Structural Dynamics Volume 19, Issue 2, pp 229241, (1990) [32] E.Sousa How to approximate the fractional derivative of order < International Journal of Bifurcation and Chaos, Vol 22, No 04, (2012) [33] HuaqingLi, Xiaofeng Liaoa, Saleem Ullaha,Li Xiao, Analytical proof on the existence of chaos in a generalized Duffing-type oscillator with fractional-order deflection, Nonlinear Analysis: Real World Application, pp 27242733, (2012) [34] Xilin Fu, Shasha Zheng, Chatter dynamic analysis for Van der Pol Equation with impulsive effect via the theory of ow switchability, Commun Nonlinear Sci Numer Simulat pp 30233035, (2014) [35] A.Y.T.Leung, Zhongjin Guo, H.X.Yang, Fractional derivative and time delay damper characteristics in Dufngvan der Pol oscillators, Commun Nonlinear Sci Numer Simulat pp 29002915, (2013) [36] Yongjun Shen, Shaopu Yang, Haijun Xing, Guosheng Gao, Primary resonance of Dufng oscillator with fractional-order derivative, Commun Nonlinear Sci Numer Simulat pp 30923100, (2012) [37] Li-Guo Yuan, Qi-Gui Yang, Parameter identication and synchronization of fractional-order chaotic systems, Commun Nonlinear Sci Numer Simulate, pp 305 316, (2012) [38] Xin Gao, Juebang Yu, Chaos in the fractional order periodically forced, Chaos, Solitons and Fractals pp 109711041, (2005) [39] J Yang, Y.P Xiong n, J.T Xing, Nonlinear power flow analysis of the Duffing oscillator, Mechanical Systems and Signal Processing 45, pp 563578, (2014) 70 [40] Giuseppe Quaranta, Giorgio Monti, Giuseppe Carlo Marano, Parameters identication of Van der PolDufng oscillators via particle swarm optimization and differential evolution, Mechanical Systems and Signal Processing 24, pp 20762095, (2010) [41] Neville J Forda, M Manuela Rodrigues, Jingyu Xiao, Yubin Yana, Numerical analysis [42] of a two-parameter fractional telegraph equation, Journal of Computational and Applied Mathematics 249, pp 95106, (2013) M.L Morgadoa, N.J Fordb, P.M Limac, Analysis and numerical methods for fractional differential equations with delay, Journal of Computational and Applied Mathematics 252, pp 159168, (2013) [43] Mohammad Rostami, Mohammad Haeri, Undamped oscillations in fractionalorder Duffing oscillator, Signal Processing 107, pp 361367, (2015) [44] Saùd Abbas, Mouffak Benchohra, Fractional order integral equations of two independent variables, Applied Mathematics and Computation 227, pp 755761, (2014) [45] Nguyen Van Loi, Valeri Obukhovskii, On controllability of Dufng equation, Applied Mathematics and Computation 219, pp 1046810474, (2013) [46] Xiuyun Zhang, Some results of linear fractional order time-delay system, pplied Mathematics and Computation 197, pp 407411, (2008) [47] M Siewe Siewea, C Tchawouab, P Woafoc, Melnikov chaos in a periodically driven RayleighDufng oscillator, Mechanics Research Communications 37, pp 363368, (2010) [48] H Jafaria, C.M Khaliquea, M Nazarib, An algorithm for the numerical solution of nonlinear fractional-order Van der Pol oscillator equation, Mathematical and Computer Modelling 55, pp 17821786, (2012) [49] Yongjun Shen n, Shaopu Yang, Haijun Xing, Huaixiang Ma, Primary resonance of Dufng oscillator with two kinds of fractional-order derivatives, International Journal of Non-Linear Mechanics 47, pp 975983, (2012) [50] X.L.Yang, Z.K Sun, Research on parametric resonance in a stochastic van der Pol oscillator under multiple time delayed feedback control, International Journal of Non-Linear Mechanics 45, pp 621627, (2010) [51] ZhongjinGuo, A.Y.T.Leung, H.X.Yang, Oscillatory region and asymptotic solution of fractional van der Pol oscillator via residue harmonic balance technique, Applied Mathematical Modelling 35, pp 39183925, (2011) [52] Ivo Petrỏ, Fractional-Order Nonlinear System: Modeling, Analysis and Simulation, Springer Publishing, (2011) 71 [53] Junyi Cao, Chengbin Ma, Hang Xie, Zhuangde Jiang, Nonlinear Dynamics of Dufng System With Fractional Order Damping, Journal of Computational and Nonlinear Dynamics, Vol.5, (2010) [54] Syta, Litak, Lenci, Scheffler, Chaotic vibrations of the duffing system with fractional damping, Chaos, (2014) [55] R Magana, A.R Hermosillo, M Perez, Fractional chaotics oscillators, 15 WCEE, [56] (2012) Bựi Th Thỳy, Gúp phn nghiờn cu dao ng phi tuyn ca c h cú o hm cp phõn s, Lun thc s khoa hc, trng H khoa hc t nhiờn i hc quc gia H Ni, (2010) [57] Dng Vn Lc, Tớnh toỏn dao ng ca múng mỏy trờn nn n nht cp phõn s, ỏn tt nghip k s, trng i hc Bỏch khoa H Ni, (2014) 72 PH LC HM GAMMA 4.1 nh ngha hm Gamma hay tớch phõn Euler loi Vi s>0 ta cú nh ngha hm Gamma (tớch phõn Euler loi 2) e x x s 1dx, s nh ngha trờn cú c bng phộp i bin u log t nh ngha ban u ca Euler s s log x dx 4.2 Cụng thc c bn th nht ca hm Gamma s Vi s s s , s dx ta cú: 2 x2 e , Mt khỏc theo cụng thc trờn ta cú n 2 n n 2 1.3.5 2n 2n 4.3 Cụng thc c bn th ca hm Gamma v hm Bờta Hm Bờta (tớch phõn Euler loi 1) cú dng xp p, q 1 x q dx, p 0, q Bng phộp i bin x cos dng tớch phõn suy rng ta cú hm p, q cos p sin q dx, p 0, q t x sin ta cng bin i c hm xq q, p 1 x p dx vi p 0, q 0 v dng p, q q, p 73 Nh vy hm i xng v cú th c tớnh toỏn thụng qua hm nh tớnh cht quan trng sau nh lý: p q p q p, q q, p S dng tớnh cht va chng minh ta cú s s s,1 s ts s,1 s s t dt Thay t x x suy s s xs dx x sin s , s nh ngha: n k n n n k! n k n! , n k !k ! k n Cỏc h qu n n n n n n 1, n k n, n n k n k k n, , n k n Quan h gia hm Gamma v h s ca nh thc: n k n! n k !k ! n k n k n! n k n k HM MITTAG LEFFLER 5.1 Hm Mittag Leffler mt tham s Hm Mittag Leffler mt tham s c nh ngha k zk , k 1 z2 z3 z Dng khai trin l dóy vụ hn E z z , Hm ny ó c gii thiu bi Mittag-Leffler nm 1903 74 Vi mt s giỏ tr ca ta cú cỏc hm Mittag Leffler x x x ex , x cosh x 3x e x cos x , x 2e x 33 x cos cosh e x erf x 12 , x x , , , x 5.2 Hm Mittag-Leffler hai tham s Hm Mittag-Leffler hai tham s úng vai trũ rt quan trng phộp tớnh khụng nguyờn Kiu hm ny c a bi R.P.Agarwal v Erdelyi vo nm 1953-1954 Hm hai tham s c nh ngha E , zk k z k E zk k z ,1 k 0, z l hm Mittag-Leffler mt tham s E Nhng ng nht thc sau c suy t nh ngha k zk k k zk k E1,1 z E1,2 z k zk k! zk k k E1,3 z k z k ez , k 1! z k k k 2! zk z2 zk k 1! k zk k 2! ez , z ez z z2 Phng trỡnh trờn cú dng tng quỏt E1,m z m zm ez k zk k! Nhng hm lng giỏc v hyperbolic cng l nhng s biu hin ca hm Mittag-Leffler hai tham s 75 E2,1 z k z 2k 2k k z 2k 2k 2 E2,2 z z 2k cosh z , 2k ! k zk z 2k 2k ! sinh z , z 5.3 Hm sai s Hm sai s c nh ngha z erf z e t dt , v c biu din bi dóy n erf z n z 2n 2n n ! z3 z z5 10 z7 42 z9 216 5.4 Hm bự sai s erfc z erf z z 2 e t dt e t dt z Khai trin dóy ca hm bự sai s e z erfc z e z z2 1 1.3.5 2n n 2z2 n z2 1 2n ! n n! z n 2n n 2.5 Phộp bin i Laplace ca hm Mittag - Leffler Ta cú: f t fL s f t t k s fL s k , E at k! k , s a t dk E dt k Trong ú: E Vi k k , , t toỏn t l vi phõn ca hm Mittag-Leffler v vi k phõn ca hm Mittag-Leffler toỏn t l tớch 76 Khi 1, k E at ,1 E s s a s at ,1 s , a 5.6 Cỏc hm phỏt trin t hm Mittag - Leffler Ta cú: E zk k z , k 0, 5.7 Hm Agarwal Hm Mittag-Leffler c khỏi quỏt bi Agarwal vo nm 1953 Hm ny quan tõm c bit ti h cp phõn s phộp bin i Laplace ca nú c a bi Agarwal Hm c nh ngha m E t t , L E , , m m s s t 5.8 Hm Erdelyi Erdelyi (1954) ó nghiờn cu dng tng quỏt ca hm Mittag-Leffler E , t m tm m , , ú s m ca t l s nguyờn 5.9 Hm R v hm G suy rng a Rq , a, c, t n n n 1q t c n 1q Rq , a, t c Trong ú, t l bin c lp v c l gii hn di ca o hm v tớch phõn cp phõn s i vi t c s quan tõm ca ta ti hm ny s l bỡnh thng Phộp bin i Laplace ca hm R Rq , a,0, t Rq , a, c, t s , sq a e cs s sq a 77 R1,0 a,0, t aR2,0 R2,1 e at , a ,0, t a ,0, t a t a 2t 3! a 2t 2! a 4t 5! a 4t 5! sin at , cos at 78 ... đƣợc phát triển để tính toán dao động hệ có thành phần đạo hàm cấp phân số Luận văn sử dụng phƣơng pháp số để tính toán vài mô hình dao động phi tuyến hệ đàn nhớt có chứa đạo hàm cấp phân số Qua... CHƢƠNG TÍNH TOÁN DAO ĐỘNG CỦA HỆ CÓ PHẦN TỬ ĐÀN NHỚT CẤP PHÂN SỐ BẰNG PHƢƠNG PHÁP RUNGE-KUTTA-NYSTRӦM 44 3.1 DAO ĐỘNG CỦA HỆ CHỊU KÍCH ĐỘNG VA ĐẬP 44 3.1.1 Mô hình dao động hệ chịu kích động. .. 44 3.1.2 Tính toán dao động hệ chịu kích động va đập 45 3.2 DAO ĐỘNG PHI TUYẾN CỦA HỆ CÓ PHẦN TỬ ĐẠO HÀM CẤP PHÂN SỐ 49 3.2.1 Dao động hệ Duffing 49 3.2.2 Dao động hệ Vander