Sensitivity Analysis and Shape Optimization of a Hole in a Vibrating Rectangular Plate for Eigenfrequency Maximization
Publication: Journal of Engineering Mechanics
Volume 138, Issue 6
Abstract
A hole inside a rectangular thin plate is optimized for maximization of an eigenfrequency of the structure. The shape of the hole is represented by an ellipse parametrization to guarantee a smooth curve along the boundary; and the hole’s center position, semiaxes, and rotation are treated as shape design variables. The Rayleigh-Ritz approach is applied to constitute the vibration frequency equation such that an exact procedure can be presented for the frequency sensitivity analysis. The hole shape optimization is performed by using a combined mathematical programming algorithm with different search directions for two sets of design variables in relation to the constraint of the hole’s size. In addition, a simple optimality criterion is provided for the optimum profile design of the internal hole of a given area. Several illustrative examples demonstrate the validity and effectiveness of the design sensitivity analysis and the proposed optimization method.
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Acknowledgments
This work was supported by the National Natural Science Foundation of China (50575181).
References
Bhat, R. B. (1985). “Natural frequencies of rectangular plates using characteristic orthogonal polynomials in Rayleigh-Ritz method.” J. Sound Vibrat., 102(4), 493–499.
Haftka, R. T., and Grandhi, R. V. (1986). “Structural shape optimization-A survey.” Comput. Methods Appl. Mech. Eng., 57(1), 91–106.CMMECC
Haftka, R. T., and Gürdal, Z. (1992). Elements of structural optimization, 3rd Ed., Kluwer Academic Publisher, Dordrecht, Netherlands.
Kim, C. S., and Dickinson, S. M. (1987). “The flexural vibration of rectangular plates with position supports.” J. Sound Vibrat., 117(2), 249–261.
Kim, N. H., Choi, K. K., and Botkin, M. E. (2002). “Numerical method for shape optimization using meshfree method.” Struct. Multidiscip. Optim.SMOTB4, 24(6), 418–429.
Kwak, M. K., and Han, S. (2007). “Free vibration analysis of rectangular plate with a hole by means of independent coordinate coupling method.” J. Sound Vibrat., 306(1-2), 12–30.
Laura, P. A. A., Romanelli, E., and Rossi, R. E. (1997). “Transverse vibrations of simply supported rectangular plates with rectangular cutouts.” J. Sound Vibrat., 202(2), 275–283.
Leissa, A. W. (1973). “The free vibration of rectangular plates.” J. Sound Vibrat., 31(3), 257–293.
Liew, K. M., Kitipornchai, S., Leungb, A. Y. T., and Lim, C. W. (2003). “Analysis of the free vibration of rectangular plates with central cut-outs using the discrete Ritz method.” Int. J. Mech. Sci., 45(5), 941–959.IMSCAW
Meske, R., Lauber, B., and Schnack, E. (2005). “A new optimality criteria method for shape optimization of natural frequency problems.” Struct. Multidiscip. Optim.SMOTB4, 31(4), 295–310.
Mundkur, G., Bhat, R. B., and Neriya, S. (1994). “Vibration of plates with cut-outs using boundary characteristic orthogonal polynomial functions in the Rayleigh-Ritz method.” J. Sound Vibrat., 176(1), 136–144.
Pedersen, N. L. (2004). “Optimization of holes in plates for control of eigenfrequencies.” Struct. Multidiscip. Optim.SMOTB4, 28(1), 1–10.
Pedersen, P. (2000). “On optimal shapes in materials and structures.” Struct. Multidiscip. Optim.SMOTB4, 19(3), 169–182.
Pedersen, P., and Pedersen, N. L. (2005). “An optimality criterion for shape optimization in eigenfrequency problems.” Struct. Multidiscip. Optim.SMOTB4, 29(6), 457–469.
Van Keulen, F., Haftka, R. T., and Kim, N. H. (2005). “Review of options for structural design sensitivity analysis. Part 1: Linear systems.” Comput. Methods Appl. Mech. Eng., 194(30-33), 3213–3243.CMMECC
Waisman, H., Chatzi, E., and Smyth, A. W. (2010). “Detection and quantification of flaws in structures by the extended finite element method and genetic algorithms.” Int. J. Numer. Methods Eng.IJNMBH, 82(3), 303–328.
Wang, C. M., and Wu, W. Q. (2002). “Optimal location of a cutout in rectangular Mindlin plates for maximum fundamental frequency of vibration.” Struct. Multidiscip. Optim.SMOTB4, 24(5), 400–404.
Wang, D., Yang, Z. C., and Yu, Z. G. (2010). “Minimum stiffness location of point support for control of fundamental natural frequency of rectangular plate by Rayleigh Ritz method.” J. Sound Vibrat., 329(14), 2792–2808.
Wu, Z. X. (2008). “Optimal hole shape for minimum stress concentration using parameterized geometry models.” Struct. Multidiscip. Optim.SMOTB4, 37(6), 625–634.
Zhang, W. H., Beckers, P., and Fleury, C. (1995). “A united parametric design approach to structural shape optimization.” Int. J. Numer. Methods Eng.IJNMBH, 38(13), 2283–2292.
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© 2012. American Society of Civil Engineers.
History
Received: Oct 21, 2010
Accepted: Dec 12, 2011
Published online: Dec 14, 2011
Published in print: Jun 1, 2012
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