Cellular automata in nonlinear vibration problems of two-parameter elastic foundation

• Autorzy: Glabisz W.

Warianty tytułu

PL

Automaty komórkowe w nieliniowych zagadnieniach drgań dwuparametrowego podłoża sprężystego

• Języki publikacji: EN

Abstrakty

EN

Cellular automata procedure for the analysis of nonlinear viscously damped transverse vibrations of two-parameter elastic foundation was defined. Parameters were obtained by comparing the cellular automata defining rules with relations resulting from the discrete form of the mathematical description of the string modelling the foundation's surface layer. A series of numerical analysis of a foundation under arbitrary static or dynamic load, including moving forces and the simulation of the behaviour of rigid structures on a foundation were done. Numerical results demonstrate that cellular automata can constitute a simple and effective tool for the analysis of two-parameter elastic foundation complex problems which have not been analyzed in this way before.

PL

W artykule zdefiniowano procedurę automatów komórkowych, którą przystosowano do analizy nieliniowych, wiskotycznie tłumionych drgań poprzecznych dwuparametrowego podłoża sprężystego. Parametry automatów komórkowych otrzymano porównując reguły definiujące ewolucję CA ze związkami wprost wynikającymi z dyskretnej postaci matematycznego opisu drgań struny modelującej warstwę wierzchnią podłoża. Przeprowadzono szereg analiz numerycznych zachowania się podłoża sprężystego pod działaniem obciążeń statycznych i dynamicznych, obciążeń ruchomych i obciążeń sztywnymi blokami. Wykonane testy numeryczne pokazują, że automaty komórkowe mogą być prostym i skutecznym narzędziem analizy szeregu złożonych zagadnień zachowania się dwuparametrowego podłoża sprężystego dotychczas tym sposobem nie analizowanych.

Słowa kluczowe

EN

cellular automata; two-parameter elastic foundation; nonlinear vibration; dynamic loading; moving forces

PL

automat komórkowy; drgania poprzeczne; podłoże sprężyste

• Wydawca: Springer ; Wrocław University of Science and Technology
• Czasopismo: Archives of Civil and Mechanical Engineering
• Rocznik: 2011
• Tom: Vol. 11, no 2
• Strony: 285--299
• Opis fizyczny: Bibliogr. 50 poz., rys., wykr.

Twórcy

autor

Glabisz W.

• Wrocław University of Technology, Wybrzeże Wyspiańskiego 27 50-370 Wrocław, Poland

Bibliografia

• [1] Abdellaoui M., El Jai A., Shillor M.: Cellular automata model for contact problem, Mathematical and Computer Modelling, Vol. 36, 2002, pp. 1099-1114.
• [2] Ambrosi D.D., Di Gregorio S., Gabriele S., Gaudio R.: A cellular automata model for soil erosion by water, Phys. Chem. Earth (B), Vol. 26, 2001, pp. 33-39.
• [3] Balzter H., Braun P.W., Kohler W.: Cellular automata models for vegetation dynamics, Ecological Modelling, Vol. 107, 1998, pp. 113-125.
• [4] Bard G., Edelstein-Keshet L.: Cellular automata approaches to biological modelling, J. Theor. Biol., Vol. 160, 1993, pp. 97-133.
• [5] Barros F.J., Mendes M.T.: Forest fire modelling and simulation in DELTA environment, Simulation Practice and Theory, Vol. 5, 1997, pp. 185-197.
• [6] Bilbao S.: Conservative numerical methods for nonlinear strings, J. Acoust. Soc. Am., Vol. 118, 2005, pp. 3316-3327.
• [7] Boghosian B.M.: Lattice gases and cellular automata, Future Generation Computer Systems, Vol. 16, 1999, pp. 171-185.
• [8] Carrier G.F.: A note on the vibrating string, Q. J. Appl. Math., Vol. 7, 1949, pp. 97-101.
• [9] Carrier G.F.: On the nonlinear vibration problem of the elastic string, Q. J. Appl. Math., Vol. 3, 1945, pp. 157-165.
• [10] Celep Z., Demir F.: Circular rigid beam on a tensionless two-parameter elastic foundation, Z. Angew. Math. Mech., Vol. 85, 2005, pp. 431-439.
• [11] Celep Z.: Dynamic response of a beam on elastic foundation, Z. Angew. Math. Mech., Vol. 64, 1984, pp. 279-286.
• [12] Celep Z., Gençoglu M.: Forced vibrations of rigid circular plate on a tensionless Winkler edge support, Journal of Sound and Vibration, Vol. 263, 2003, pp. 945-953.
• [13] Celep Z.: In-plane vibrations of circular rings on a tensionless foundation, Journal of Sound and Vibration, Vol. 143, 1990, pp. 461-471.
• [14] Celep Z., Malaika A., Anu-Hussein M.: Forced vibrations of a beam on a tensionless elastic, Journal of Sound and Vibration, Vol. 128, 1989, pp. 235-246.
• [15] Chopard B., Masselot A.: Cellular automata and lattice Boltzmann methods: a new approach to computational fluid dynamics and particle transport, Future Generation Komputer Systems, Vol. 16, 1999, pp. 249-257.
• [16] De Rosa M.A., Maurizi M.J.: The influence of concentrated masses and Pasternak soil on the free vibrations of Euler beams - exact solution, Journal of Sound and Vibration, Vol. 212, 1998, pp. 573-581.
• [17] Del Rey A.M., Mateus J.P., Sanches G.R.: A secret sharing scheme based on cellular automata, Applied Mathematics and Computation, Vol. 170, 2005, pp. 1356-1364.
• [18] Doeschl A., Davison M., Rasmussen H., Reid G.: Assessing cellular automata based models using partial differential equations, Mathematical and Computer Modelling, Vol. 40, 2004, pp. 977-994.
• [19] Filipich C.P., Rosales M.B.: A further study about the behaviour of foundation piles and beams in a Winkler-Pasternak soil, Int. J. Mech. Sci., Vol. 44, 2002, pp. 21-36.
• [20] Filipich C.P., Rosales M.B.: A variant of Rayleigh's method applied to Timoshenko beams embedded in a Winkler-Pasternak medium, Journal of Sound and Vibration, Vol. 124, 1988, pp. 443-451.
• [21] Gardner M.: The fantastic combinations of John Conway's new solitaire game "Life", Sci. Am., Vol. 223, 1970, pp. 120-123.
• [22] Glabisz W.: Cellular automata in nonlinear string vibration, Archives of Civil and Mechanical Engineering, Vol. 10, 2010, pp. 27-41.
• [23] Guler K., Celep Z.: Static and dynamic responses of a circular plate on a tensionless elastic foundation, Journal of Sound and Vibration, Vol. 183, 1995, pp. 185-195.
• [24] Hogeweg P.: Cellular automata as a paradigm for ecological modelling, Applied Mathematics and Computation, Vol. 27, 1988, pp. 81-100.
• [25] Kansal A.R., Torqato S., Harsh IV G.R., Chiocca E.A., Deisboeck T.S.: Simulated brain tumour growth dynamics using a tree-dimensional cellular automata, J. Theor. Biol., Vol. 203, 2000, pp. 367-382.
• [26] Kari J.: Thory of cellular automata: a survey, Theoretical Computer Science, Vol. 334, 2005, pp. 3-33.
• [27] Kawamura S., Shirashige M., Iwatsubo T.: Simulation of the nonlinear vibration of a string using the cellular automation method, Applied Acoustics, Vol. 66, 2005, pp. 77-87.
• [28] Kawamura S., Yoshida T., Minamoto H., Hossain Z.: Simulation of the nonlinear vibration of a string using cellular automata based on the reflection rule, Applied Acoustics, Vol. 67, 2006, pp. 93-105.
• [29] Kerr A.D.: Elastic and viscoelastic foundation models, J. Appl. Mech., Vol. 31, 1964, pp. 491-498.
• [30] Kier L.B., Chao-Kun Cheng, Testa B.: Cellular automata models of biochemical phenomena, Future Generation Computer Systems, Vol. 16, 1999, pp. 273-289.
• [31] Kirchhoff G.: Vorlesungen uber Mathematische Physik: Mechanik, Leipzig, Druck und Verlag von B.G. Teubner, 1867.
• [32] Lafe O.: Data compression and encryption using cellular automata transforms, Engng Applic. Artif. Intell., Vol. 10, 1997, pp. 581-591.
• [33] Li-Quen Chen, Hu Ding: Two nonlinear models of a transversely vibrating string, Arch. Appl. Mech., Vol. 78, 2008, pp. 321-328.
• [34] Maerivoet S., De Moor B.: Cellular automata of road traffic, Physics Reports, Vol. 419, 2005, pp. 1-64.
• [35] Mitchell M., Crutchfield J., Das R.: Evolving cellular automata with genetic algorithms: a review of recent work, First International Conference on Evolutionary Computation and its Applications, 1996.
• [36] Molteno T.C., Tufillaro N.B.: An experimental investigation into the dynamics of a string, Am. J. Phys., Vol. 72, 2004, pp. 1157-1169.
• [37] Neumann J.V.: The theory of self-reproducing automata, Burks A.W. (ed). Urbana and London, Univ. of Illinois Press, 1966.
• [38] Pal Chaudhuri P., Chowdhury A.R., Nandi S., Chatterjee S.: Additive cellular automata - theory and applications, USA, IEEE Computer Society Press, Vol. 1, 1997.
• [39] Sarkar P.: A brief history of cellular automata, ACM Computing Systems, Vol. 32, 2000, pp. 80-107.
• [40] Schonfisch B., Lacoursiere C.: Migration in cellular automata, Physica D, Vol. 103, 1997, pp. 537-553.
• [41] Setoodeh S., Gurdal Z., Watson L.T.: Design of variable-stiffness composite layers rusing cellular automata, Comput. Methods Appl. Mech. Engrg., Vol. 195, 2006, pp. 836-851.
• [42] Toffoli T.: Cellular automata as an alternative to (rather than an approximation of) differentia equations in modelling physics, Physica 10D, 1984, pp. 117-127.
• [43] Toffoli T., Margolus N.: Invertible cellular automata: a review, Physica D, Vol. 66, 1994, pp. 1-24.
• [44] Tomassini M., Sipper M., Zolla M., Perrenoud M.: Generating high-quality random numbers in parallel by cellular automata, Future Generation Computer Systems, Vol. 16, 1999, pp. 291-305.
• [45] Townend J.: Modelling water release and absorption in soils using cellular automata, Journal of Hydrology, Vol. 220, 1999, pp. 104-112.
• [46] Vollmar T.: Cellular space and parallel algorithms: an introductory survey. Parallel computation-parallel mathematics, Feilmeier M. (ed), North Holland Publishing Co, 1977, pp. 49-58.
• [47] Weimar J.R.: Cellular automata for reaction-diffusion systems, Parallel Computing, Vol. 23, 1997, pp. 1699-1715.
• [48] White S.H., Del Rey A.M., Sanches G.R.: Modelling epidemics using cellular automata, Applied Mathematics and Computation, Vol. 186, 2007, pp. 193-202.
• [49] Wolfram S.: A new kind of science, Champaign, Wolfram Media Inc., 2002.
• [50] Zhang Y.: Tensionless contact of a finite beam resting on Reissner foundation, Int. J. Mech. Sci., Vol. 50, 2008, pp. 1035-1041.