Engineering Transactions, 64, 2, pp. 197–211, 2016

Isogeometric Approximation for Dynamics of Infinite String Using Difference Equation Method

Jerzy RAKOWSKI
Poznan University of Techlology Institute of Structural Engineering ul. Piotrowo 5 60-965 Poznan Poland
Poland

Przemyslaw WIELENTEJCZYK
Poznan University of Techlology Institute of Structural Engineering ul. Piotrowo 5 60-965 Poznan Poland
Poland

An efficient method of vibration investigation of an infinite string using the isogeometric analysis (IGA) with B-spline basis functions is considered. The research objective is to compare IGA, finite element method (FEM) and exact formulation approaches. In the IGA approximation a system is divided into a set of regularly distributed coordinates assembled in a uniform knot vector. Transverse displacements are described by linear, quadratic, cubic and quartic B-spline basis functions. The geometrical and mass matrices are found for all types of approximations. The equilibrium conditions for an arbitrary interior element are expressed in the form of one difference equation equivalent to the infinite set of equations obtained by numerical IGA formulation for this dynamic problem. Assuming the wavy nature of a vibration propagation phenomenon the dispersive equations are obtained. The ranges of vibration frequencies for the dispersive and reactive cases are determined. The influences of the adopted discretization, mass distribution and initial axial force effects on the wave propagation phenomenon are examined.
Keywords: FEM; FDM; NURBS; IGA
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References

Brillouin L., Wave propagation in periodic structures, Dover Publications, 1953.

Belytschko T., Mindle W.L., Flexural wave propagation behaviour of lumped mass approximations, Computer & Structures, 12(6): 805–812, 1980.

Cottrell J.A., Reali A., Bazilevs Y., Hughes T.J.R, Isogeometric analysis of structural vibrations, Computer Methods in Applied Mechanics and Engineering, 195(41–43): 5257–5296, 2006.

Cottrell J.A., Hughes T.J.R., Bazilevs Y., Isogeometric analysis: toward integration of CAD and FEA, John Wiley and Sons, 2009.

Hughes T.J.R., The finite element method: linear static and dynamic finite element analysis, Dover Publications,, Mineola, NY, 2000.

Hughes T.J.R., Cottrell J.A., Bazilevs Y., Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Computer Methods in Applied Mechanics and Engineering, 194(39–41): 4135–4195, 2005.

Hughes T.J.R., Reali A., Sangalli G., Duality and unified analysis of discrete approximations in structural dynamics and wave propagation: comparison of p-method finite elements with k-method NURBS, Computer Methods in Applied Mechanics and Engineering, 197(49–50): 4104–4124, 2008.

Ihlenburg F., Babuška I., Dispersion analysis and error estimation of Galerkin finite element methods for the Helmholtz equation, International Journal for Numerical Methods in Engineering, 38(22): 3745–3774, 1995.

Mead D.J., Yaman Y., The harmonic response of uniform beams on multiple linear supports: a flexural wave analysis, Journal of Sound and Vibration, 141(3): 465–484, 1990.

Mead D.J., Yaman Y., The response of infinite periodic beams to point harmonic forces: A flexural wave analysis, Journal of Sound and Vibration, 144(3): 507–530, 1991.

Orris R.H., Petyt M., A finite element study of harmonic wave propagation in periodic structures, Journal of Sound and Vibration, 33(2), 223–236, 1974.

Rakowski J., A new methodology of evaluation of Co bending finite elements, Computer Methods in Applied Mechanics and Engineering, 91(1–3), 1327–1338, 1991.

Rakowski J., Vibrations of infinite one-dimensional periodic systems by finite element method, Proceedings of the Second European Conference on Structural Dynamics EURODYN’93, pp. 557–562, Trondheim, Norway, 1993.

Rakowski J., Wielentejczyk P., Vibrations of infinite periodic beams by finite element method, Zeitschrift für Angewandte Mathematik und Mechanik, 76(1), 411–412, 1996.

Rakowski J., Wielentejczyk P., Dynamic analysis of infinite discrete structures, Foundations of Civil and Environmental Engineering, 3: 91–106, 2002.

Rakowski J., Wielentejczyk P., Application of the difference equation method to the vibrations analysis of infinite Rayleigh beams by the isogeometric approach, Archives of Civil and Mechanical Engineering, 15(4), 1108–1117, 2015.




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