The maximization of fundamental frequency of structures under arbitrary initial stress states

• Published in: International journal for numerical methods in engineering Vol. 65; no. 4; pp. 445 - 460
• Main Authors: de Faria, Alfredo R., de Almeida, Sérgio Frascino M.
• Format: Journal Article
• Language: English
• Published: Chichester, UK John Wiley & Sons, Ltd 22.01.2006; Wiley
• Subjects: Assessments; Buckling; Computational techniques; Concavity; convex modelling; Exact sciences and technology; free vibration; Fundamental areas of phenomenology (including applications); Initial stresses; Mathematical methods in physics; Mathematical models; Maximization; Optimization; Physics; Resonant frequency; Solid mechanics; Strategy; Structural and continuum mechanics; Vibration, mechanical wave, dynamic stability (aeroelasticity, vibration control...); Normal variable; Prestress; Free vibration; Mechanical properties; Simple support; Modeling; Optimization; convex modelling; Natural frequency; Minimax problem; initial stresses; Minimax method; Rectangular plate; Buckling; Convex analysis
• ISSN: 0029-5981; 1097-0207
• DOI: 10.1002/nme.1438

A general technique is proposed to maximize the lowest natural frequency of structures subjected to an arbitrary state of initial stresses. The arbitrary states of initial stresses are represented by nondimensional loading parameters that describe an admissible loading space, i.e. every possible initial stress state lies within the admissible loading space. The key to the proposed optimization strategy is shown to be the concavity of the first natural frequency with respect to variations of the loading parameters within the admissible loading space. A rigorous demonstration is presented to show that, provided buckling has not occurred, all the possible initial stress states must not be considered. Instead, assessment of only a small number of initial stress states must be done in order to guarantee that the first natural frequency does not decrease for all the other initial stress states within the admissible loading space. A minimax optimization technique is used to maximize the lowest natural frequency of a simply supported rectangular plate where the thickness distribution is the design variable and normal and shear initial stress states are considered. Copyright © 2005 John Wiley & Sons, Ltd.