11 th Intenational Confeence on Vibation Poblems Z. Dimitovová et al. (eds.) Lisbon, Potugal, 9-12 Septembe 213 BIFURCATION ANALYSIS FOR A QUASI-LINEAR CENTRIFUGAL PENDULUM ABSORBER Eugen B. Keme* 1, Mikhail V. Zakzhevsky 2, Igo T. Schukin 3, Alexey V. Klokov 2 1 LuK GmbH & Co. OHG, Gemany eugen.keme@schaeffle.com 2 Institute of Mechanics of Riga Technical Univesity, Latvia mzak@latnet.lv alex_klokov@inbox.lv 3 Daugavpils banch of Riga Technical Univesity, Latvia igo@df.tu.lv Keywods: Centifugal Pendulum Absobe, Complete Bifucation Analysis, Dynamical Stability. Abstact. A vibational pendulum absobe with tapezoidal suspension is a modification of the classical pendulum absobe, which allows not only its tanslational motion but some complex 2D-motion of the pendulum. It povides not only highe pefomances of the absobe, but also some new inteesting featues. About them it is a possibility fo patial compensation of non-lineaity. It allows avoiding the paasite supe-hamonical espond of the pendulum. The bifucation analysis of such a pendulum make possible to estimate effectiveness of a new type pendulum absobe. This pape is a continuation of pevious investigations [1,2] devoted to a pendulum absobe poblem. Main methods used hee ae methods based on the bifucation theoy [3-11] of nonlinea dynamical systems. Among them ae the method of complete bifucation goups with paamete (obit) continuation, concepts of peiodic skeletons with passpots of stable and unstable obits, unstable peiodic infinitiums, detemined chaotic egions and ae attactos. Fo numeical ealization of the bifucation theoy fo pendulum absobe poblem, oiginal softwae is used.
1 INTRODUCTION The diect global bifucation analysis of pendulum-like systems is pesented by the main featues and components of a new so-called bifucation theoy of nonlinea dynamics and chaos and its applications. The descibed pat of bifucation theoy uses dynamical peiodic systems, descibed by a model of ODE equations. Ou appoach is based on ideas of Poincaé, Andonov and othe scientists' esults concening stuctual stability and bifucations of diffeent dynamical nonlinea systems and thei topological popeties [3-8]. Fo illustation of the advantages of the new bifucation theoy we use in this pape the pendulum vibation absobe model [1,2] with soft impact (Figue 1,a). We have found impotant unknown egula o chaotic attactos and new bifucation goups with ae attactos. The method of complete bifucation goups gives obust stability to the pendulum vibation absobe by changing the system paametes, that allows saving chaacteistics of the vibation absobe system and excluding othe amazing egimes. Recent effots in nonlinea dynamics of pendulum systems show, that the behaviou of the diven damped pendula may be complex and sometimes with the unexpected phenomena such as stable hilltop vibations, complex subhamonical and quasi-peiodical vibations, diffeent otations and othe [9-11]. Ou aim is doing complete bifucation analysis fo impotant paamete of pendulum vibation absobe systems by using the fundamental concepts of the bifucation theoy: a peiodic skeleton, complete bifucation goup nt, subgoup of unstable peiodic infinitium (UPI), concepts of ae attactos, complex potubeances, typical bifucation topological goups and the topological stuctue of chaotic attactos and chaotic tansients [3-8]. All esults wee obtained numeically, using softwae NLO [3] and SPRING [5], ceated in Riga Technical Univesity. 2 MODEL OF THE PENDULUM ABSORBER The studied nonlinea dynamical model of the pendulum vibation absobe [1,2] with soft impact (see Figue 1,a) is epesented. System has tilinea dissipation and estoing foces (Figue 1,b). The equation of motion fo studied pendulum system is such: 2 2 1 f f h cost 1 d 1 2hsint sin, 1 whee angle of the pendulum elative to basic disk (flywheel); angula velocity of the pendulum elative to basic disk, whee d / dt ; b 1, b 2 linea dissipation coefficients, accodingly to linea subegions of visco-elastic chaacteistic; c stiffness coefficient of linea elastic chaacteistic at soft impact in coodinate = ±15º: f b2 1 b2 if b if ; if f 1 (1) c if if, (2) c if atio of moments of inetia of the pendulum and disk (<1); h moment fom the moto (oscillating component) (h 1.5); excitation ode ( = 2 o 3, espectively fo 4-cylinde and 6-cylinde engine); d eo in setting up the pendulum vibation absobe at antiesonance (-.1 d.1). The vaiable paamete of the pendulum system Eq.1 is moment fom the moto h =...1.5 and excitation ode =...3.5. The base paametes fo analysis ae: =.1, b 1 =.2, b 2 = 5, c = 5, d =, h =.5. 2
3 CONSTRUCTION OF PERIODIC SKELETON FOR BASE PARAMETERS OF THE PENDULUM ABSORBER Befoe the constuction of complete bifucation diagams, peiodic skeleton fo base paametes of the pendulum vibation absobe was build. (a) (b) Figue 1: (a) A flywheel with pendulum vibation absobe with soft impact; (b) elastic dissipation foces chaacteistics and dependence of dissipation coefficient on state of system. Fom the gid of 2 2 = 4 initial conditions inside the ectangle (-/12; -1/ /12; 1) we have obtained peiodic skeleton [2] consisted fom 17 egimes (one unstable egime; one unstable P2 egime; fou P4 egimes two of them ae stable; two unstable P5 egimes; two unstable P6 egimes; one unstable P7 egime; one unstable P8 egime; thee unstable P9 egimes; one unstable 1 egime and one unstable) fo h =.5 and = 3 with egime scanning fo peiod fom 1 till 16 by using the Newton-Kantoovich method. By using the Poincaé map founded egimes can be specified by data of the solution (obit) ode, by the numbe of loops in the pojection of the phase tajectoies, the coodinates of the fixed point and stability chaacteistics. Fo example, 1 (1/1), Fixed point (-.185917, -.52712), 1 = -.781, 2 = -4.5831. This is the main passpot of peiodic egime (at a given point of the paamete space of the system). The examples of extended passpot of peiodic egimes with phase pojections and time histoies ae shown in [2,11]. 4 COMPLETE BIFURCATION ANALYSIS OF PENDULUM VIBRATION ABSORBER HEADINGS The constuction of bifucation diagams was done by the Runge-Kutta 4-th ode with constant integation step: t = T /2 k, whee the peiod of extenal foce is T = 2/ and coefficient k = 7. The esults of complete bifucation analysis of the pendulum vibation absobe model ae epesented in bifucation diagams of the fixed peiodic points of the coodinates p, p and Am vesus paamete h (Figue 2). Thee ae two 1T bifucations goups: 1T (with depth - P2) with complex potubeances and 1T isle. The pendulum system has many RA of diffeent kinds. In this figue stable solutions ae plotted by solid lines and unstable - by thin lines (eddish online). The banches of bifucation diagam ae not completed, because of poblem of high instability. 3
p (a) =.1, b 1 =.2, b 2 = 5, c = 5, = 3, d =, h = va. 1T bifucation goups P2 P2 isle stable egimes unstable egimes.28 Am.27 (b) Fagment P2 isle h.265.26.255.25 =.1, b 1 =.2, b 2 = 5, c = 5, = 3, d =, h = va. h Figue 2: Pendulum vibation absobe with soft impact (see Eq.1 and Figue 1,a). Bifucation diagams of the fixed peiodic points of the coodinates p, p and Am vesus paamete h. Thee ae two 1T bifucations goups: 1T (with depth -P2) with complex potubeances and 1T isle. The pendulum system has many ae attactos of diffeent kinds. Paametes: =.1, b 1 =.2, b 2 = 5, c = 5, = 3, d =, h = va. (a) h = 1.238831 (b) h = 1.238831 isle NT = 3T NT = 3T Figue 3: Coexistence of ae attacto and isle fo h = 1.238831. Phase pojections with NT = 3T. 4
(a) b 1 =.2, b 2 = 5, = 3, h = 1.47 NT = (5-1)T ChA Cm 4 1Q 1T (b) h = 1.47 NT = 3T (c) h = 1.47 NT = 3T = -.12, = -1.26 = -.12, = -1.26 t Figue 4: Chaotic attacto in pendulum absobe fo h = 1.47: () Poincaé map fom a contou Cm 4 1Q (5-1)T; (b)-(c) phase pojection and time histoies with fom initial conditions = -.12, = -1.26;. The example of coexistence of ae attacto () with peiod-1 and isle attacto fo coss-section h = 1.238831 of bifucation diagam (Figue 2) on phases pojections is shown in Figue 3. Sometimes oscillation amplitudes of ae attactos ae tenfold bigge than oscillating amplitudes of stable egimes [2,1,11]. Rae attacto has highe velocity of oscillations than isle. In ou case oscillation amplitudes ae limited by two elastic detents, which ealise soft impacts. The example of globally stable chaotic attacto fo coss-section h = 1.47 of bifucation diagam (Figue 2) obtained using the contou mapping Cm 4 1Q (5-1)T fom a contou (-.4,-3/.4,2), phase pojection and time histoies with NT = 3T fom initial conditions = -.12, = -1.26, is shown in Figue 4. 5 CONCLUSIONS The pendulum systems ae widely used in the engineeing, but thei qualitative behavio hasn t been investigated enough. Theefoe in this wok the new nonlinea effects in pendulum vibation absobe, which can be used in dynamics of the machines and mechanisms, wee shown. MCBG gives obust stability to the studied vibation absobe system by changing the system paametes, which allows saving chaacteistics of the vibation absobe system and excluding othe amazing egimes, which can lead to small beakages of machine and mechanisms and may be to global technical catastophes, because they ae unexpected and usually have lage amplitudes. This wok is peliminay, because it will be continued fo the nonlinea dynamical systems with two o seveal degees of feedom. 5
ACKNOWLEDGMENT This wok has been suppoted by the Euopean Social Fund within the poject Suppot fo the implementation of doctoal studies at Riga Technical Univesity 21-213 and by the Latvian Council of Science unde the gants 9.1587 (21-212). REFERENCES [1] Kause T., Keme E., Movlazada P., Theoy and Simulation of Centifugal Pendulum Absobe with Tapezoidal Suspension, Poceedings of the 1th Intenational Confeence on Vibation Poblems, Pague, Czech Republic (211) 322-327. [2] Keme E., Zakzhevsky M., Klokov, Complete Bifucation Analysis with Rae Attactos of a Pendulum Vibation Absobe, The 4th IEEE Intenational Confeence on Nonlinea Science and Complexity NSC 212, Budapest, Hungay (212) 25-21. [3] Zakzhevsky M., Ivanov Y. and Folov V. NLO: Univesal Softwae fo Global Analysis of Nonlinea Dynamics and Chaos. // In Poceeding of the 2nd ENOC, Pague, Septembe 9-13 1996, pp. 261-264. [4] Zakzhevsky M., New concepts of nonlinea dynamics: complete bifucation goups, potubeances, unstable peiodic infinitiums and ae attactos. // Jounal of Viboengineeing - JVE 1, Issue 4, 12 (28), 421-441. [5] Schukin I., Development of the methods and algoithms of simulation of nonlinea dynamics poblems. Bifucations, chaos and ae attactos. // PhD Thesis), Riga - Daugavpils, Latvia (25), 25 p. [6] Zakzhevsky M., Global Nonlinea Dynamics Based on the Method of Complete Bifucation Goups and Rae Attactos. // Poceedings of the ASME 29 (IDETC/CIE 29), CD, San Diego, USA (29), 8 p. [7] Zakzhevsky M., How to Find Rae Peiodic and Rae Chaotic Attactos? Abstacts of the 8th AIMS Intenational Confeence, http://www.math.tudesden.de/koksch/aims/, Desden, Gemany (21), 9 p. [8] Zakzhevsky M. Bifucation Theoy of Nonlinea Dynamics and Chaos. Peiodic Skeletons and Rae Attactos. // Poceedings of the 2nd Intenational Symposium RA 11 on Rae Attactos and Rae Phenomena in Nonlinea Dynamics, May 16-2, 211, Riga - Jumala, Latvia, pp. 1-1. [9] Klokov A., Zakzhevsky M. Nonlinea Dynamics and Rae Attactos of the Pendulum with the Vetical Vibating Point of Suspension // Poceedings of the 2nd Intenational Symposium RA 11 on Rae Attactos and Rae Phenomena in Nonlinea Dynamics, May 16-2, 211, Riga - Jumala, Latvia, pp. 16-2. [1] Klokov A., Zakzhevsky M., Paametically excited pendulum systems with seveal equilibium positions. Bifucation analysis and ae attactos. // Intenational Jounal of Bifucation and Chaos, - Vol. 21, No. 1 (211) pp. 2825-2836. [11] Zakzhevsky M.V. and Klokov A.V., How to find ae attactos in nonlinea dynamical systems by the method of complete bifucation goups. The collection of the pendulum diven systems. Riga, 213 31 p. 6