Electron. J. Differential Equations, Vol. 2018 (2018), No. 119, pp. 1-18.

Optimal design of minimum mass structures for a generalized Sturm-Liouville problem on an interval and a metric graph

Boris P. Belinskiy, David H. Kotval

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.

Submitted December 4, 2017. Published May 17, 2018.
Math Subject Classifications: 34L15, 74P05, 49K15, 49S05, 49R05.
Key Words: Sturm-Liouville Problem; vibrating rod; calculus of variations; optimal design; boundary conditions with spectral parameter; complete bipartite graph.

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Boris P. Belinskiy
University of Tennessee at Chattanooga
Department of Mathematics
Dept 6956, 615 McCallie Ave.
Chattanooga TN 37403-2598, USA
email: boris-belinskiy@utc.edu
David H. Kotval
Middle Tennessee State University
Department of Mathematical Sciences
MTSU BOX 34, 1301 East Main Street
Murfreesboro TN 37132-0001, USA
email: dhk2e@mtmail.mtsu.edu

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