(2) 4 48 Irregular vbratons n mult-mass dscrete-contnuous systems torsonally deformed Abstract In the paper rregular vbratons of dscrete-contnuous systems consstng of an arbtrary number rgd bodes connected by shafts torsonally deformed are studed. In the systems a local nonlnearty descrbed by the polynomal of the thrd degree s ntroduced. It s assumed that the characterstc of the local nonlnearty s of a hard type. Governng equatons are solved usng the wave approach leadng to equatons wth a retarded argument. Exemplary numercal calculatons are done for the three-mass system. The possblty of occurrng of rregular vbratons s dscussed on the bass of the Poncaré maps, bfurcatons dagrams and the exponents of Lyapunov. Keywords nonlnear dynamcs, rregular vbratons, dscrete-contnuous systems, torsonal systems, wave approach A. Pelorz and D. Sado 2 Kelce Unversty of Technology, Poland, e-mal: apelorz@tu.kelce.pl 2 Warsaw Unversty of Technology, Poland, e-mal: dsa@smr.pw.edu.pl INTRODUCTION The paper deals wth nonlnear vbratons of dscrete-contnuous mechancal systems torsonally deformed wth a local nonlnearty havng the characterstc of a hard type. The systems consst of shafts wth crcular cross-sectons connected by rgd bodes. Local nonlneartes, justfed by many engneerng solutons, are descrbed by the polynomal of the thrd degree. Regular vbratons n nonlnear mult-mass dscrete-contnuous torsonal systems for the hard characterstc case are dscussed n [5]. Irregular nonlnear vbratons ncludng chaos are studed manly n dscrete systems, [,2,4,2-5]. In the present paper an attempt to study rregular nonlnear vbratons n a dscrete-contnuous system s undertaken by the generalzaton of the approach used n [2-4] for dscrete systems. Governng equatons for mult-mass dscrete-contnuous systems torsonally deformed are derved n [5,6] ncludng the local nonlneartes havng the characterstc of a hard as well as of a soft type. In the studes a wave approach leadng to solvng equatons wth retarded argument s used, [5-7]. In [8,9] rregular nonlnear vbratons n dscrete-contnuous systems torsonally deformed wth local nonlneartes havng a soft type characterstc are dscussed. Here smlar consderatons are presented for systems havng the hardenng characterstcs of the local nonlneartes. Numercal results are presented for the three-mass system. The possblty of occurrng of rregular vbratons Latn Amercan Journal of Solds and Structures (2) 4 48
42 A. Pelorz and D. Sado / Irregular vbratons n mult-mass dscrete-contnuous systems torsonally deformed s dscussed on the bass of bfurcaton dagrams and the Poncaré maps. Exemplary dagrams of the exponent of Lyapunov are also gven. 2 ASSUMPTIONS, GOVERNING EQUATIONS The dscrete-contnuous system dscussed n the paper s shown n Fg..The -th shaft n a mult-mass system, =,2,...,N, s characterzed by length l, densty, shear modulus G and polar moment of nerta I are, [5,6]. The mass moment of nerta of rgd bodes, =,2, N+, J. The frst rgd body J s loaded by the harmonc moment M( t) M sn pt, where M and p are the ampltude and frequency of the external moment, correspondngly. A local nonlnear dscrete element, descrbed by the polynomal of the thrd degree, wth a hardenng characterstc s located n the cross-secton x =. Equvalent external and nternal dampng, havng coeffcents d and D, are takng nto account n approprate cross-sectons. It s assumed that dsplacements and veloctes of the shaft cross-sectons are equal to zero at tme nstant t =. Fgure Mult-mass system torsonally deformed. The determnaton of angular dsplacements of shaft cross-sectons, n approprate nondmensonless quanttes gven [5,6], s reduced to solvng N equatons, tt, xx,,2,..., N () wth the followng nonlnear boundary condtons, tt r, xt, x, t M sn pt K ( D ) d k k for x, ( x, t) ( x, t) for x l,,2,..., N, k k K B E ( D ) K B E ( D ) E d, tt r, xt, x r, xt, x, t for x l,,2,..., N, k k N, tt r N N N N, xt N, x N N N, t k k N K B E ( D ) E d for x l, (2) Latn Amercan Journal of Solds and Structures (2) 4 48
A. Pelorz and D. Sado / Irregular vbratons n mult-mass dscrete-contnuous systems torsonally deformed 4 and wth zero ntal condtons. Comma denotes partal dfferentaton. The solutons of equatons () are sought n the form k k ( x, t) f ( t x) g ( t x 2 l ),,2,..., N () Substtutng () nto the boundary condtons (2) we obtan the followng set of ordnary nonlnear dfferental equatons wth a retarded argument for unknown functons f and g r g ( z) r g ( z) r f ( z 2 l ) r f ( z 2 l ), N, N N,2 N N, N N N,4 N N g ( z) f ( z 2 l ) g ( z 2 l ) f ( z 2 l ),,2,..., N, r f ( z) M( z) r g ( z) r f ( z) r g ( z) M ( z ) 2 4 r f ( z) r f ( z) r g ( z) r g ( z) r f ( z) r f ( z), 2,,..., N, 2 4 5 6 sp (4) where r K D, r2 K D, r r Kr d, 4 r r r K d, r KrE( BD B D ), 2 r r E [ K ( B B ) d ], r KrE( BD B D ), 4 r r 5 2KrB ED, 6 r r E [ K ( B B ) d ], r 2 K B E, 2,,..., N, r N, K r B N E N D N, rn,2 EN ( KrBN d N ), r N, K r B N E N D N, rn,4 EN ( KrBN d N ). (5) Nonlnear equatons (4) are solved numercally by means of the Runge-Kutta method. In the case of the local nonlneartes havng the hard characterstc, k, such equatons can be solved numercally wth zero or nonzero ntal condtons. It should be ponted out that ordnary dfferental equatons wth shfts n the arguments of unknown functons have an attenton n the lterature, eg., n []. NUMERICAL ANALYSIS The am of the numercal analyss s to study the possblty of occurrence of rregular vbratons n dscrete-contnuous systems consdered. Ths s done on the bass of the bfurcaton dagrams and the Poncaré maps for the three-mass torsonal system, characterzed by the followng basc parameters: N 2, l l 2, B B 2, E2 E.8, K 5, k 5, r Latn Amercan Journal of Solds and Structures (2) 4 48
44 A. Pelorz and D. Sado / Irregular vbratons n mult-mass dscrete-contnuous systems torsonally deformed k 5, [5,]. The three frst natural frequency for lnear the system are 89, 2.26 and.76. 24. A k = 2. A M =. 2 5.5 6. 8. 2. 5 8. 4..25 4.. p p.4.8.2.4.8.2 Fgure 2 The effect of parameters k and M on dsplacement ampltudes. Cases when solutons are harmonc vbratons wth the perod equal to the perod of the external loadng are presented n Fg. 2. They are gven for large dampng havng coeffcents equal d d D. and show the effect of the parameter k wth M and the effect the ampltude of the external moment wth k 5. Nonlnear effects are observed n the frst three resonant regons, smlarly to other results gven n [5]. Especally, t s seen that n the thrd resonant regon nonlnear effects have the form of ampltude jumps. Two ampltude jumps are observed. They correspond to zero and nonzero ntal condtons, respectvely. From dagrams n Fg. 2 t follows that dstances between jumps ncrease wth the ncrease of the parameter k representng the local nonlnearty and wth the ncrease of M. It appears that dstances of jumps ncrease also wth the decrease of dampng, [5]. Further numercal results are presented n Fgs. - 7. They concern the ampltude of the external moment equal M and k 5, however small dampng,.e., all dampng coeffcent are equal d. In Fg. bfurcaton dagrams are shown for the angular dsplacement as well as for the angular velocty n the cross-secton x. From these dagrams t follows that rregular vbratons can be expected for the frequency of the external moment p <.2. In bfurcatons dagrams perods of the solutons are taken nto account. Latn Amercan Journal of Solds and Structures (2) 4 48
A. Pelorz and D. Sado / Irregular vbratons n mult-mass dscrete-contnuous systems torsonally deformed 45 5 5 5-5 -5 - -5 p - p.4.8.2.6.4.8.2.6 Fgure Bfurcaton dagrams for angular dsplacements and veloctes wth d. In Fg. 4 the Poncaré maps are presented for selected frequency p of the external moment, namely equal to p =.2,.48,.78,.8 wth dampng coeffcents d d D and the ampltude of the external moment M. One can see that the Poncaré maps have varous shapes, n the dependence of the frequency p of the external moment Mt (). In Fg. 4 strange atractors also are notced. For ths reason n the case of p =.78 and p =.8 maxmal exponents of Lyapunov are checked. From Fg. 5 t follows that maxmal exponents of Lyapunov are postve, so motons n these cases are chaotc. From the bfurcaton dagrams as well as from detaled calculatons wth p t was found that rregular vbratons n the studed three-mass system can occur for frequency p.5. j 4. 8. p=.2 p=.48 2. 4. -2. -4. -4. -8. -8. -4. 4. 8. -2-2 Latn Amercan Journal of Solds and Structures (2) 4 48
46 A. Pelorz and D. Sado / Irregular vbratons n mult-mass dscrete-contnuous systems torsonally deformed.. p=.78 p=.8.5.5 -.5 -.5 -. -4.5 -. -.5.5. 4.5-6. -4. -2. 2. 4. Fgure 4 Poncaré maps for p =.2,.48,.78,.8. -..8 max.8 max.7 p=.78.6 p=.8.7.4.6.6.2.5 n n 2 4 6 8 2 4 6 8 Fgure 5 Maxmal Lyapunov exponents for angular dsplacements wth p =.78,.8. The numercal results presented n Fgs. - 5 concern approprate solutons n the crosssectons x. Elastc elements n dscrete-contnuous systems have fnte length. The wave approach appled n the papers allows to determne smultaneously solutons n requred crosssectons of shafts. In Fg. 6 the Poncaré maps for the frequency p =.8 and cross-sectons x =,.5,.,.5, 2. are presented wth dampng coeffcents equal to d.. From these dagrams t follows that the maxmal angular veloctes decrease wth the ncrease of x. Latn Amercan Journal of Solds and Structures (2) 4 48
A. Pelorz and D. Sado / Irregular vbratons n mult-mass dscrete-contnuous systems torsonally deformed.5 47 x=. x=.5.5..5 2. -.5 -. -.5-2. -.. 2. Fgure 6 Poncaré maps for p =.8 n cross-sectons x =,.5,.,.5, 2. for d.. x= x=2..5.5 -.5 -.5 -. -. -6. -.... 6. -6. -. Fgure 7 Poncaré maps for p =.8 n cross-sectons x =, 2. for d The Poncaré maps for p =.8 and cross-sectons x =, 2. wth d. 6.., shown n Fg. 7, nform that dagrams have qute dfferent shapes n the each consdered cross-secton. The above numercal results concern the three-mass torsonal system wth the local nonlnearty havng the characterstc of a hard type. Smlar consderatons were carred out n [] n the case of a two-mass system. The possblty of occurrence of rregular vbratons were also done on the bass of the bfurcaton dagrams and the Poncaré maps. 4 CONCLUSIONS From the consderatons n the paper t follows that n dscrete-contnuous systems torsonally deformed wth a local nonlnearty havng a hardenng characterstc and loaded by the external Latn Amercan Journal of Solds and Structures (2) 4 48
48 A. Pelorz and D. Sado / Irregular vbratons n mult-mass dscrete-contnuous systems torsonally deformed moment harmoncally changng n tme, regular and rregular vbratons may appear. Dfferent knds of rregular vbratons ncludng chaotc vbratons can be found n the lmted range of the change of the parameters representng the system and the external moment. Presented numercal calculatons concern the three-mass system, however governng equatons allow us to wden consderatons to other dscrete-contnuous systems. Exemplary dagrams show bfurcaton dagrams and varety of Poncaré maps. Chaotc motons notced for certan frequences of the external moment are justfed by the postve values of the maxmal exponents of Lyapunov. Acknowledgment The results contaned n the paper were presented durng the conference DYNAMICAL SYSTEMS - THEORY AND APPLICATIONS held n 5-8 December 2, Łódź, Poland. References [] Awrejcewcz, J. (99) Bfurcaton and Chaos n Smple Dynamcal Systems. Sngapore, World Scentfc. [2] Awrejcewcz J. and Olejnk P. (2) Stck-slp dynamcs of a two-degree-of-freedom system, Internatonal Journal of Bfurcaton and Chaos (4), do:.42/s28274696, 84-86. [] Cherepennkov V.B., (28) Polynomal quassolutons of a boundary value problem for lnear dfferentaldfference equatons, Functonal Dfferental Equatons, 5(4), 4-57. [4] Moon F.C. (987) Chaotc Vbratons. New York, John Wley and Sons Inc. [5] Pelorz A. (995) Dynamc analyss of a nonlnear dscrete-contnuous torsonal system by means of wave method, ZAMM, 75, 69-698. [6] Pelorz A. (999) Non-lnear vbratons of a dscrete-contnuous torsonal system wth non-lneartes havng characterstc of a soft type, Journal of Sound and Vbraton, 225(2), 75-89. [7] Pelorz A. (27) Nonlnear equatons wth a retarded argument n dscrete-contnuous systems, Mathematcal Problems n Engneerng, Vol. 27, Artcle ID 284, do:.55/27/284, -. [8] Pelorz A. and Sado D. (29) Note on regular and rregular nonlnear vbratons n dscrete-contnuous systems, Proc. of th Conference on DYNAMICAL SYSTEMS - THEORY and APPLICATIONS, 25-. [9] Pelorz A. and Sado D. (2) On regular and rregular nonlnear vbratons n torsonal dscretecontnuous systems, Internatonal Journal of Bfurcaton and Chaos, 2(), do:.42/s2827486, 7-82. [] Pelorz A. and Sado D. (2) Regular and rregular vbratons n nonlnear dscrete-contnuous systems torsonally deformed, Vbraton Problems ICOVP 2: The th Internatonal Conference on Vbraton Problems 2, Prague, Czech Republc, Sempember 5-8, Sprnger Proceedngs n Physcs 9, do:.7/978-94- 7-269-5_9, 9-44. [] Pelorz A. and Sado D. (2) On rregular vbratons n a three-mass dscrete-contnuous system torsonally deformed, th Conference on DYNAMICAL SYSTEMS THEORY AND APPLICATIONS, Łódź, December 5-8, 79-84. [2] Sado D. and Gajos K. (2) Note on chaos n three degree of freedom dynamcal system wth double pendulum, Meccanca, 8, 79-729. [] Sado D. and Kot M. (27) Chaotc vbraton of an autoparametrcal system wth a non-deal source of power, Journal of Theoretcal and Appled Mechancs, 45, 9-. [4] Sado D. (2) Regular and chaotc vbratons n selected systems wth pendulum (n Polsh)., Wydawnctwa Naukowo-Technczne, Warszawa. [5] Szemplńska-Stupncka W. (2) Chaos, Bfurcatons and Fractals Around Us, World Scentfc, London. Latn Amercan Journal of Solds and Structures (2) 4 48