Livija Cveticanin

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Structure element of a building connected with the ground in a line is usually modeled as a beam with the Winkler type support. The elastic property of the support is assumed to be linear or with cubic nonlinearity. Unfortunately, the experiments do not prove such an assumption. It is evident that the nonlinearity is with the order which is a real positive number which need not to be an integer. In this paper the generalization of the beam with Winkler support is done by introducing the nonlinearity of any non-integer order. The line structure, i.e., beam has transversal vibrations. The mathematical description of these vibrations is a nonlinear partial differential equation. To solve the equation, we suggest an analytic procedure. The solution is assumed as a product of a time and a displacement function. After averaging, the problem transforms into a second order nonlinear differential equation. The approximate solution has the form of a cosine (ca) Ateb function. Analyzing the obtained results the influence of support properties on the system behavior is considered. The attention is given to the influence of the Winkler-Pasternak foundation, too.

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Lenci, S., Tarantino, A.M., (1996), Chaotic dynamics of an elastic beam resting on a Winkler-type soil. Chaos, Solitons and Fractals, 7, 1601-1614.

Li, X.-F., Tang, G.-J., Lee, K.Y., (2012), Vibration of nonclassical shear beams with Winkler-Pasternak-type restraint. Acta Mechanica, 223, 953-966.

Burton, T.D., Hamdan, M.N., (1996), On the calculation of non-linear normal modes in continuos systems", Journal of Sound and Vibration, 197, 117-130.

Atanackovic, T.,M., Cveticanin, L., (1996) Dynamics of plane motion of an elastic rod. Journal of Applied Mechanics, Trans. ASME, 63, March, 392-398.

Shaw, S.W., Pierre, C., (1994), Normal modes of vibration for non-linear continuous systems. Journal of Sound and Vibration, 169, 319-347.

Cveticanin, L., (1997), Normal modes of vibration for continuous rotors with slow time variable mass. Mechanism and Machine Theory, 32(7), 881-891.

Cveticanin, L., (2009), Application of homotopy-perturbation to non-linear partial differential equations. Chaos, Solitons & Fractals, 40(1), 221-228.

Cveticanin, L., Pogany, T., (2012), Oscillator with a sum of noninteger-order nonlinearities. Journal of Applied Mathematics, 2012, Article ID 649050, 20 pages.


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