Minimization of the dynamic response of composite laminated doubly curved shells using design and control optimization M.E. Fares * , Y.G. Youssif, A.E. Alamir Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt Abstract A design control optimization approach is used to determine optimal levels of ply thickness, fiber orientation angle and closed- loop control force for composite laminated doubly curved shells. The optimization objective is the minimization of the dynamic response of a shell subject to constraints on the thickness and control energy. A higher-order shell theory is used to formulate the control objective for various cases of boundary conditions. The dynamic response is expressed as the sum of the total elastic energy of the shell and a penalty functional of a closed-loop control force. Comparative examples are presented for symmetric (or anti- symmetric) spherical and cylindrical shells with various cases of boundary conditions. The advantages of the present control op- timization over some design and control approaches are examined. The effect of number of layers, aspect ratio and orthotropy ratio on the control process is demonstrated. The discrepancy between optimal results obtained using the classical, first-order and higher- order shell theories is studied. Ó 2003 Elsevier Science Ltd. All rights reserved. Keywords: Structures design and control; Closed-loop control force; Optimal layer thickness; Fiber orientation angle; Higher-order shell theory 1. Introduction An important area of application of fiber composite structures occurs in the field of aerospace engineering and, in particular, in the construction of large space structures. Material tailoring and active control are effective means of improving the performance of these structures because of the adaptability of composite materials to a given design situation. For aerospace structures, weight considerations invariably lead to highly flexible structures with low natural damping. However, serviceability and safety requirements restrict the allowable limits of the dynamic response to external disturbances to specified values. So, optimization is a necessary part of the design process for these structures. Design optimization of composite laminated structures is concerned with the best use of the tailoring capabilities of fiber-reinforced laminated beams, plates and shells to minimize (or maximize) a given design objective. The vibration damping involves the damping out of the excessive vibrations by means of active structural control. These two subjects were treated in literature separately [1–4], while, in more recent studies, they were treated as an integrated approach for simultaneous design and control of these structures using unified formulation [5–8]. Most recent studies on these subjects may be found in the works [9–14]. Many studies indicate that transverse shear deformation can have significant effect on the global response and, consequently, on the dynamic response of laminated plates and shells made of advanced composite material [15,16]. As a result, the classical theories of laminated plates and shells underpredict the optimum values of the design variables. In addition, the boundary conditions at the edges play an important role in decreasing (or increasing) the dynamic re- sponse of laminated composites [17]. However, most of studies related to the design and control optimization of laminated composite plates and shells were carried out based on the classical theories of plates and shells for special cases of boundary conditions, and there exist few papers formulated based on shear deformation theories for various cases of boundary conditions [18,19]. * Corresponding author. Tel.: +20-50-346-781; fax: +20-50-346-254. E-mail address: sinfac@mum.mans.eun.eg (M.E. Fares). 0263-8223/03/$ - see front matter Ó 2003 Elsevier Science Ltd. All rights reserved. PII:S0263-8223(02)00241-6 Composite Structures 59 (2003) 369–383 www.elsevier.com/locate/compstruct