JOURNAL OF SOUND AND VIBRATION Journal of Sound and Vibration 292 (2006) 390–401 Regulating the vibratory motion of beams using shape optimization J.T. Katsikadelis à , G.C. Tsiatas School of Civil Engineering, National Technical University of Athens, Zografou Campus, GR-15780 Athens, Greece Received 27 October 2004; received in revised form 19 July 2005; accepted 2 August 2005 Available online 20 October 2005 Abstract In this paper, shape optimization is used to regulate the vibrations of an Euler–Bernoulli beam having constant material volume. This is achieved by varying appropriately the beam cross-section and thus its stiffness and mass properties along its length, so that the beam vibrates with its minimum, maximum or a prescribed eigenfrequency as well as with the minimum or maximum difference between two successive eigenfrequencies. The problem is reduced to a nonlinear optimization problem under equality and inequality constraints as well as specified lower and upper bounds. The evaluation of the objective function requires the solution of the free vibration problem of a beam with variable mass and stiffness properties. This problem is solved using the analog equation method (AEM) for hyperbolic differential equations with variable coefficients. Besides its accuracy, this method overcomes the shortcoming of a FEM solution, which would require resizing of the elements and re-computation of their stiffness and mass properties during the optimization process. Certain example problems are presented, which illustrate the method and demonstrate its efficiency. r 2005 Elsevier Ltd. All rights reserved. 1. Introduction Shape optimization is a subject which has attracted the interest of the researches for many years. It refers to the optimal design of the shape of structural components and is of great importance in structural and mechanical engineering. The problem consists in finding the best shape of a structural component under certain loading, in order to have minimum weight, or uniformly distributed equivalent stresses or even to control the deflections of the structural components. A shape optimization procedure is an iterative process in which repeated improvements are carried out over successive designs until the optimal design is acceptable. There is an extensive literature on the subject, especially, in the area of structural dynamics [1] on various mathematical and computational techniques which can be applied, including finite element model updating [2] and the use Volterra series [3]. In this paper, we consider the problem of determining the optimum shape of an Euler–Bernoulli beam with a given material volume to regulate its vibratory motion. Namely, we look for the variation law of the cross- section along the beam length so that the beam vibrates with the minimum, the maximum or a prescribed ARTICLE IN PRESS www.elsevier.com/locate/jsvi 0022-460X/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsv.2005.08.002 à Corresponding author. Tel.: +30 2107721654; fax: +30 2107721655. E-mail addresses: jkats@central.ntua.gr (J.T. Katsikadelis), gtsiatas@central.ntua.gr (G.C. Tsiatas).