The reconstruction of a periodic structure from its dynamical behaviour

Authors

  • Raúl D. Jiménez Alarcón Universidad Católica del Norte

DOI:

https://doi.org/10.4067/10.4067/S0716-09172011000100009

Keywords:

Inverse eigenvalue problems, Periodic Structure.

Abstract

This work is related  to the inverse problem in vibration produced in a special type of mechanical structure known as periodic structure. This problem consist in determining the stiffness and mass parameter of the structure from the natural frequencies and vibrations modes. The problem concern with the inverse eigenvalue problem for a specially structured Jacobi matrix which contains the desired parameters. Necessary conditions to be applied to the data to obtain a real system are derived and a numerical procedure is develop. Some numerical examples are presented

Author Biography

Raúl D. Jiménez Alarcón, Universidad Católica del Norte

Departamento de Matemáticas.

References

[1] Bansal A. S. Free-wave Propagation Through Combinations of Periodic and Disordered Systems. Journal of The Acoustical Society of America, Vol.67, No. 2, pp. 377-389, (1980).

[2] Barcilon V. On the solution of inverse eigenvalue problems with high orders, Geophys. J. Royal Astronomical Society 39, pp. 287-298, (1974).

[3] Barcilon V. A two-dimensional inverse eigenvalue problem, Inverse Problems 6, No. 1, pp. 11-20, (1990).

[4] Bendiksen O. Mode localization phenomena in large space structures, AIAA Journal, Vol. 25, pp. 1241-1248, (1987).

[5] Boley D. L. and Golub G. H. A survey of matrix inverse eigenvalue problems, Inverse Problems, 3, No. 4, pp. 595-622, (1987).

[6] Brasil R. M. L. R. F. and Mazzilli C.E.N. Influence of loading on mode localization in periodic structures, Applied Mechanics Reviews, 48, No. 11, pp. s132-s137, (1995).

[7] Chu M. T and Golub G. H. Inverse Eigenvalue Problems, Oxford University Publications, (2005).

[8] de Boor C. and Golub G. H. The numerically stable reconstruction of a Jacobi Matrix from spectral data, Linear Algebra Appl.,21, No3, pp. 245-260, (1978).

[9] Gantmacher F. R. and Krein M. G. Oscillation Matrices and Small Vibrations of Mechanical Systems (1961 Translation by US Atomic Energy Commission, Washington DC), (1950).

[10] Gladwell G. M. L. Inverse Problems in Vibration, Martinus Nijhoff, Dordrecht, Netherlands, (1986).

[11] Gladwell G. M. L. The reconstruction of a tridiagonal system from its frequency response at an interior point, Inverse Problems, 4,, pp. 1013-1024, (1988).

[12] Hald O. Inverse eigenvalue problems for Jacobi matrices, Linear Algebra Appl., 14, pp. 63-85, (1976).

[13] Knobel R., and McLaughlin A reconstruction method for a two dimensional inverse eigenvalue problem, Z. Angew. Math. Phys. 45, No. 5, pp. 794-826, (1994).

[14] McLaughlin J. R. and Hald H. A formula for finding a potential from nodal lines, Bull. Amerc. Math. Soc., New Series, 32, pp. 241-247, (1995).

[15] Jiménez R., et als. The reconstruction of a specially structured Jacobi matrix with an application to damage detection in rods, Computer and Mathematics with Applications, (49), pp. 1815-1823, (2005).

[16] Jiménez R., et als. An Inverse Eigenvalue Procedure for Damage Detection in Rods, Computer and Mathematics with Applications (47), pp. 643-657, (2004).

[17] Jiménez R. Aplicacoes de Métodos Inversos em Autovalores à Deteccao de Danos, Ph.D. thesis, IME-USP, Universidade de S˜ao Paulo, (2002).

[18] Jiménez R., Kuhl N. and Castro L. On detection of structural failure using an inverse technique on vibration, Proceedings of the third international conference on nonlinear dynamics, chaos, control and their applications in engineering sciences, Iconne 2000, Campos de Jordao, Brasil 5, pp. 11-22, (2000).

[19] Ram Y. M., Gladwell G. M. L. Constructing a finite element model of a vibratory rod from eigendata, J. Sound Vibration, 169, pp. 229-237, (1994).

Published

2011-05-25

How to Cite

[1]
R. D. Jiménez Alarcón, “The reconstruction of a periodic structure from its dynamical behaviour”, Proyecciones (Antofagasta, On line), vol. 30, no. 1, pp. 91-109, May 2011.

Issue

Section

Artículos