Approximate analytical solution for Bernoulli-Euler beams under different boundary conditions with non-linear Winkler type foundation

• Autorzy: Mohammadpour A. , Rokni E. , Fooladi M. , Kimiaeifar A.

Warianty tytułu

PL

Przybliżone rozwiązanie analityczne dla belek Bernoulliego-Eulera opartych na nieliniowym podłożu winklerowskim przy różnych warunkach zamocowania

• Języki publikacji: EN

Abstrakty

EN

In this study, a powerful analytical method, known as Homotopy Analysis Method (HAM), is used to obtain an analytical solution to nonlinear ordinary deferential equations arising for Bernoulli-Euler beams with a non-linear foundation of the Winkler type. A comparison between the HAM solution and a solution obtained by a numerical method is made to show the accuracy of the method. It is shown that the present solution is valid for the whole domain of the solution and also for high nonlinear terms, where other methods such as the perturbation method fail to converge. The results clearly indicate that the convergence region can be controlled and adjusted by HAM. Finally, after validating the results, the effect of constant parameters on the deflection and slope for different boundary conditions is presented.

PL

W pracy przedstawiono zastosowanie bardzo wszechstronnej metody homotopii (HAM) do uzyskania analitycznego rozwiązania nieliniowych równań różniczkowych opisujących drgania belek Bernoulliego-Eulera spoczywających na nieliniowym podłożu winklerowskim. W celu zaprezentowania dokładności metody, uzyskane wyniki porównano z rezultatami symulacji numerycznych. Wykazano, że tak otrzymane rozwiązanie jest ważne w całej dziedzinie, także przy uwzględnieniu członów nieliniowych wyższego rzędu. Inne metody, m.in. analiza perturbacyjna, przestają być w takich przypadkach zbieżne. Przeprowadzone badania wyraźnie dowodzą, że obszar zbieżności może być monitorowany i dostosowywany w ramach metody HAM. Na zakończenie rozważań, po weryfikacji obliczeń, przedyskutowano wpływ stałych parametrów układu na ugięcie i kąt ugięcia belek przy równych warunkach zamocowania.

Słowa kluczowe

EN

Bernoulli-Euler beam; Winkler type foundation; HAM

• Wydawca: Polskie Towarzystwo Mechaniki Teoretycznej i Stosowanej
• Czasopismo: Journal of Theoretical and Applied Mechanics
• Rocznik: 2012
• Tom: Vol. 50 nr 2
• Strony: 339--355
• Opis fizyczny: Bibliogr. 25 poz., rys., tab.

Twórcy

autor

Mohammadpour A.

autor

Rokni E.

autor

Fooladi M.

autor

Kimiaeifar A.

• Aalborg University, Department of Mechanical and Manufacturing Engineering, Aalborg East, Denmark,

Bibliografia

• 1. Ames W.F., 1977, Numerical Methods for Partial Differential Equations, Section 1.6. Academic Press, New York, ISBN 0-12-056760-1
• 2. Bishop R.E.D., Johnson D.C., 1960, The Mechanics of Vibration, Cambridge University Press, Cambridge
• 3. Chowdhury M.S.H., Hashim I., Abdulaziz O., 2007, Comparison of homotopy analysis method and homotopy perturbation method for purely nonlinear fin-type problems, Communications in Nonlinear Science and Numerical Simulation
• 4. Courant R., Hilbert D., 1962, Methods of Mathematical Physics, Volume II, Wiley-Interscience
• 5. Hale J., 1969, Ordinary Differential Equations, Wiley, New York
• 6. Hildebrand F.B., 1968, Finite-Difference Equations and Simulations, Section 2.2, Prentice-Hall, Englewood Cliffs, New Jersey
• 7. Kantorovich L.V., Krylov V.I., 1964, Approximate Methods of Higher Analysis, Noordhoff, the Netherlands
• 8. Kimiaeifar A., 2010, An analytical approach to investigate the response and stability of Van der Pol-Mathieu-Duffing oscillators under different excitation functions, Journal of Mathematical Methods in the Applied Sciences, 33, 1571-1577
• 9. Kimiaeifar A., Bagheri G.H., Rahimpour M., Mehrabian M.A., 2009a, Analytical solution of two-dimensional stagnation flow towards a shrinking sheet by means of homotopy analysis method, Journal of Process Mechanical Engineering, 223, 3, 133-143
• 10. Kimiaeifar A., Saidi A.R., Bagheri G.H., Rahimpour M., Domairry D.G., 2009b, Analytical solution for Van der Pol-Duffing oscillators, Chaos, Solitons and Fractals, 42, 5, 2660-2666
• 11. Li Y.T., Wong R., 2008, Integral and series representations of the Dirac delta function, Commun. Pure Appl. Anal., 7, 2, 229-247, doi:10.3934/cpaa.2008.7.229
• 12. Liao S.J., 1992, The proposed homotopy analysis technique for the solution of nonlinear problems, Ph.D. thesis, Shanghai Jiao Tong University
• 13. Liao S.J., 1997, A kind of approximate solution technique which does not depend upon small parameters. II An application in fluid mechanics, International Journal of Non-Linear Mechanics, 32, 815-22
• 14. Liao S.J., 2003, Beyond Perturbation: Introduction to Homotopy Analysis Method, CRC Pr I Llc, ISBN 1-58488-407-X
• 15. Lyengar K.T.S.R., Anantharamu S., 1963, Finite beam-columns on elastic foundation, J. Engng Mech. Div. Proc. ASCE, 89, I39
• 16. Nadeem S., Hayat T., Abbasbandy S., Ali M., 2010, Effects of partial slip on a fourth-grade fluid next term with variable viscosity. An analytic solution, Nonlinear Analysis: Real World Applications, 11, 2, 856-868
• 17. Nayef A.H., 1985, Problems in Perturbation, John Wiley, New York
• 18. Ozis¸ T., Yıldırım A., 2007a, A note on He’s homotopy perturbation method for van der Pol oscillator with very strong nonlinearity, Chaos, Solitons and Fractals, 34, 3, 989-991
• 19. ¨ Ozis¸ T., Yıldırım A., 2007b, Determination of periodic solution for a u1/3 force by he’s modified Lindstedt-Poincar´e method, Journal of Sound and Vibration, 301, 1/2, 415-419
• 20. Sajid M., Hayat T., 2007, Comparison of HAM and HPM methods in nonlinear heat conduction and convection equations, Nonlinear Analysis: Real World Applications
• 21. Selvadurai A.P.S., 1979, Elastic Analysis of Soil-Foundation Interaction, Elsevier, Amesterdam
• 22. Sohouli A.R., Faomouri M., Kimiaeifar A., Domairry G., 2010, Application of homotopy analysis method for natural convection of Darcian fluid about a vertical full cone embedded in pours media prescribed surface heat flux, Communication in Nonlinear Science and Numerical Simulation, 15, 7, 1691-1699
• 23. Soldatos K.P., Selvadurai A.P.S., 1983, A perturbation-Galerkin for the analysis of beams resting on nonlinear elastic foundations, Proceeding of 9th Canadian Congress on Applied Mechanics, p. 241
• 24. Soldatos K.P., Selvadurai A.P.S., 1985, Flexure of a beams resting on hyperbolic elastic foundations, International Journal of Solids Structures, 21, 4, 373-388
• 25. Wang J., Chen J.K., Liao S.J., 2008, An explicit solution of the large deformation of a cantilever beam under point load at the free tip, Journal of Computational and Applied Mathematics, 212, 320-30