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Abstract of the Doctoral Thesis mgr inż. Błażej Witkowski

Title: “Synchronization of Coupied Spherical Per i Aa”

supen isor: prof. dr hab. inż. Tomasz Kapitaniak additonal supervisor: dr inż. Łukasz Borkot .

Faculty of Mechanical Engineering Division of Dynamics

The doctoral dissertation contains such issues as synchronization of -.amic systems. non-linear dynamics and modelling. The sini of this study is to describe the syn ■' ,-onizji.on cf the system composed of two spherical rotating pendula, coupied by a rigid beam . The bcar.; is attached :o the ends of two weightless and inextensible strings. At both ends of the beam n\ o Łientical. mathematical pendula are suspended. The thesis was formulated as follows: the change of the initial conditions and the type of an excitation of the system of the coupied spherical pendula results in qualitative changes in their dynamics and the type of me synchronization between thein.

The different mathematical models of the system were introduced:

1.    the    beam defmed in    the Carte ^:an coordinates, using the Lagrange mu..., ...rs.

2.    the    beam defined in    the Cartesian and spherical coordinates, using the ¹ -.grange multipiiers;

3.    the    beam defmed in    the spherical coordinates.

Based on the Lagrange equatio . of the second type. coupied ordinary . o et e.uia cqualions of the second order were derived. The bchaviour of the mathematical modę: • • ere c >mparec! w:vh the behaviour of the real one. The ‘test mathematical model was chosen - le oeam described in the spherical coordinates.

The system was simplified to a zero-length beam to determine the non-lii car norma! modes. Then smali amplitudes ofthe motion of m: pendula were assumed. Three normt ■ des w e.e obtaiaed analytically:

1.    the phase shift between the pendula is n, they rotate in the same direc’    mm, is m rest;

2.    the beam and the pendula rota e in-phase in the same direction:

3.    the beam and the pendula rotate in the same direction but the phase sin ii nc.wcei' che pendula and the beam is n.

Using the Newton-Raphson algorithm the obtained Solutions were correct- >. i - cer ampli',.des.

The viscous damping was modelied. so that each of the nodes was e.    . . ;d fhe onstant

and harmonie rorce was applLa co me pendula, which is perpendicular . _r .. . u m. the XY piane.

The numerical analyses ofthe initial conditions and excitations of :.c s siem were dotie. The foliowi ng obsewations were madę:

1.    the type of synchronization strongly depends on the initial conditions . penduia;

2.    the chaeged ofthe beam's s - ■ ensioifs length maj cause periodic or . o . - . f or..

The reai system consisting of two coupied spherical pendula was . ws t..c exeitation, an aireraft toy with electric motors was used. The results of the . ... c. ica! analyses were comparcd with the results ot che rea! system, which proved to be the s :e.