Assignment of Geometrical and Physical Parameters for the Confinement of Vibrations in Flexible Structures
Publication: Journal of Aerospace Engineering
Volume 22, Issue 4
Abstract
A strategy for confinement of flexural vibrations in flexible structures by proper selection of their geometrical and physical parameters is proposed. We first show that the problem of vibration confinement can be formulated as an inverse eigenvalue problem (IEP) where the mode shapes and/or natural frequencies are assumed and the geometrical and physical properties are unknown functions of the space variables. It is required that the assumed modes form a complete and independent set of spatial functions that satisfy the boundary conditions and guarantee confinement within the desired spatial subdomain(s) of the structure. Using simple spatial functions, such as polynomials and exponentials, we determined approximate solutions of the geometrical and physical parameters by applying the orthogonality of the mode shapes with respect to the stiffness and mass density. The order of the selected polynomials or exponentials depends on the number of modes retained in the discretized model. Numerical simulations are presented on a beam and then on a plate to examine convergence of the solution to the IEP. We show that convergence is attained with few assumed mode shapes. The approximated parameters are finally substituted into the forward eigenvalue problem to confirm confinement at the desired locations.
Get full access to this article
View all available purchase options and get full access to this article.
References
Allaei, D. (1992). “Application of localized modes in vibration control.” Proc., 2nd Int. Congress on Recent Development of Air and Structure Borne Sound and Vibration, Auburn, Ala., 611–618.
Allaei, D. (1997). “Performance comparison between vibration control by confinement and conventional control techniques.” Proc., ASME 16th Biennial Conf. on Mechanical Vibration and Noise, Sacramento, Calif., 14–17.
Baccouch, M., Choura, S., El-Borgi, S., and Nayfeh, A. (2006). “On the selection of physical and geometrical properties for the confinement of vibrations in nonhomogeneous beams.” J. Aerosp. Eng., 19(3), 158–168.
Barcilon, V. (1976). “On the solution of the inverse problem with amplitude and natural frequency data. Part I.” Phys. Earth Planet. Inter., 13, P1–8.
Barcilon, V. (1979). “On the multiplicity of solutions of the inverse problems for a vibrating beam.” SIAM J. Appl. Math., 37, 605–613.
Choura, S., El-Borgi, S., and Nayfeh, A. (2005). “Axial vibration confinement in nonhomogeneous rods.” Shock Vib., 12(3), 177–195.
Choura, S., and Yigit, A. S. (1995). “Vibration confinement in flexible structures by distributed feedback.” Comput. Struct., 54(3), 531–540.
Chu, M. T. (1992). “Numerical methods for inverse singular numerical analysis.” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal., 29(3), 885–903.
Datta, B. N., Elhay, S., and Ram, Y. M. (1997). “Orthogonality and partial pole placement for the symmetric definite quadratic pencil.” Linear Algebr. Appl., 257, 29–48.
Elishakoff, I., and Candan, S. (2001). “Apparently first closed-form solution for vibrating inhomogenous beam.” Int. J. Solids Struct., 38, 3411–3441.
Gladwell, G. M. L. (1986). Inverse problems in vibration, Martinus Nijho, Dordrecht, The Netherlands.
Hodges, C. H. (1982). “Confinement of vibrations by structural irregularity.” J. Sound Vib., 82, 411–424.
Hu, Y. R., and Ng, A. (2005). “Active robust vibration control of flexible structures.” J. Sound Vib., 288, 43–56.
Karpeshina, Y. E. and McLaughlin, J. R. (1998). “Two methods of solution of the three-dimensional inverse nodal problem.” Journées, Equations aux dérivées partielles Séminaire, Ecole Polytechnique, U.M.R. 7640 du C.N.R.S., I.1–I.9.
Lai, E., and Ananthasuresh, G. K. (2002). “On the design of bars and beams for desired mode shapes.” J. Sound Vib., 254(2), 393–406.
Liu, T. X., Hua, H. X., and Zhang, Z. (2004). “Robust control of plate vibration via active constrained layer damping.” Thin-Walled Struct., 42, 427–448.
McCarthy, C. M. (1999). “Recovery of a density from the eigenvalues of a nonhomogeneous membrane.” Proc., 3rd Int. Conf. on Inverse Problems in Engineering: Theory and Practice, ASME, Port Ludlow, Wash.
McLaughlin, J. R. (2000). “Solving inverse problems with spectral data.” Surveys on solution methods for inverse problems, D. Colton, H. Engl, A. Louis, J. McLaughlin, and W. Rundell, eds., Springer, New York, 169–194.
McLaughlin, J. R., and Hald, O. H. (1995). “A formula for finding a potential from nodal lines.” Bull., New Ser., Am. Math. Soc., 32, 241–247.
Pierre, C., Tang, D. M., and Dowell, E. H. (1987). “Localized vibrations of disordered multispan beams: theory and experiment.” AIAA J., 25(9), 1249–1257.
Ram, Y. M. (1998). “Pole assignment for the vibrating rod.” Q. J. Mech. Appl. Math., 51, 461–492.
Ram, Y. M., and Elhay, S. (1998). “Constructing the shape of a rod from eigenvalues.” Commun. Numer. Methods Eng., 14, 597–608.
Singh, A. N. (2002a). “Nodal control of a vibrating beam.” Proc., Mechanical Engineering Graduate Student Conf., Louisiana State Univ., Baton Rouge, La.
Singh, K. V. (2002b). “Physical parameter estimation of vibrating structure from its spectral data: a new mathematical model.” Proc., Mechanical Engineering Graduate Student Conf., Louisiana State Univ., Baton Rouge, La.
Sivan, D. D., and Ram, Y. M. (1999). “Physical modifications to vibratory systems with assigned eigendata.” ASME Trans. J. Appl. Mech., 66, 427–432.
Yigit, A. S., and Choura, S. (1995). “Vibration confinement in flexible structures via alteration of mode shapes using feedback.” J. Sound Vib., 179, 553–567.
Information & Authors
Information
Published In
Copyright
© 2009 ASCE.
History
Received: Mar 12, 2008
Accepted: Sep 23, 2008
Published online: Sep 15, 2009
Published in print: Oct 2009
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.