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Optimal design of minimum mass structures for a generalized Sturm-Liouville problem on an interval and a metric graph. (English) Zbl 1457.49027

Summary: We derive an optimal design of a structure that is described by a Sturm-Liouville problem with boundary conditions that contain the spectral parameter linearly. In terms of Mechanics, we determine necessary conditions for a minimum-mass design with the specified natural frequency for a rod of non-constant cross-section and density subject to the boundary conditions in which the frequency (squared) occurs linearly. By virtue of the generality in which the problem is considered other applications are possible. We also consider a similar optimization problem on a complete bipartite metric graph including the limiting case when the number of leafs is increasing indefinitely.

MSC:

49N99 Miscellaneous topics in calculus of variations and optimal control
49S05 Variational principles of physics
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
74P05 Compliance or weight optimization in solid mechanics
34B24 Sturm-Liouville theory
05C90 Applications of graph theory
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References:

[1] S. A. Avdonin, B. P. Belinskiy; On controllability of a rotating string, J. of Math. Analysis and Applications, 321 (1) (2006) 198-212. · Zbl 1112.93008
[2] B. P. Belinskiy, J. V. Matthews, J. W. Hiestand; Piecewise uniform optimal design of a bar with an attached mass, Electron. J. Diff. Equ., 133 (2015) 1-17. · Zbl 1327.80004
[3] B. P. Belinskiy, J. W. Hiestand, M. L. McCarthy; Optimal design of a bar with an attached mass for maximizing the heat transfer, Electron. J. Diff. Equ., 2012 (181) (2012), 1-13. · Zbl 1251.74003
[4] B. P. Belinskiy, J. P. Dauer; Eigenoscillations of mechanical systems with boundary conditions containing the frequency, Quarterly Appl. Math, 56 (3) (1998) 521-541. · Zbl 0958.34011
[5] A. Cardou; Piecewise uniform optimum design for axial vibration requirement, AIAA J., 11 (1973), 1760-1761.
[6] C. T. Fulton; Singular eigenvalue problems with eigenvalue parameter contained in the boundary conditions, Proc. Roy. Soc. Edinburg, 87A (1980), 1-34. · Zbl 0458.34013
[7] C. T. Fulton; Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions, Proc. Roy. Soc. Edinburg, 77A (1977), 293-308. · Zbl 0376.34008
[8] I. M. Gelfand, S. V. Fomin; Calculus of Variations, Mineola, New York: Dover, 1963. · Zbl 0127.05402
[9] D. Hinton, M. McCarthy; Bounds and optimization of the minimum eigenvalue for a vibrating system, Electron. J. Diff. Equ., 48 (2013), 1-22. · Zbl 1340.34334
[10] D. Hinton; An expansion theorem for an eigenvalue problem with eigenvalue parameter in the boundary condition, Quart. J. Math. Oxford, 2 (3) (1979), 3342. · Zbl 0427.34023
[11] H. Linden; Leighton’s bounds for Sturm-Liouville eigenvalues with eigenvalue parameter in the boundary conditions, J. of Math. Analysis and Applications, 156 (1991), 444-456. · Zbl 0738.34020
[12] J. E. Taylor; Minimum mass bar for axial vibration at specified natural frequency, AIAA Journal, 5 (10) (1967) 1911-1913.
[13] M. J. Turner; Design of minimum mass structures with specified natural frequencies, AIAA Journal, 5 (3) (1967), 406-412. · Zbl 0149.23202
[14] J. Walter; Regular eigenvalue problems with eigenvalue parameter in the boundary condition, Math. Z., 133 (1973), 301-312. · Zbl 0246.47058
[15] G. Q. Xu, N. E. Mastorakis; Differential Equation on Metric Graph, World Scientific Engineering Academy and Society Press, 2010.
[16] A. Zettl; Sturm-Liouville Theory. Mathematical Surveys and Monographs, v. 121, Rhode Island: American Mathematical Society, 2005. · Zbl 1103.34001
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