Parametric optimization of complex systems using a multi-domain FRF-based substructuring method Doo-Ho Lee a, * , Woo-Seok Hwang b a Department of Mechanical Engineering, Dongeui University, San 24, Kaya-dong, Busanjin-ku, Busan 614-714, South Korea b School of Automotive, Industrial and Mechanical Engineering, Daegu University, Naeri, Kyungsan, Kyungbuk 712-714, South Korea Received 28 February 2003; accepted 12 June 2003 Abstract A general procedure for a parametric sensitivity analysis of a complex system has been presented using a multi- domain FRF-based substructuring formulation. The direct differentiation of reaction forces on the interface boundaries with respect to the design vector is used to obtain a design sensitivity formula of responses in any substructure. A simple example for a ladder structure is introduced to verify the formulation numerically. The proposed method is applied to an optimization problem of an engine mount system to minimize the interior sound over the concerned rpm range. The optimized interior noise in a passenger car shows that the proposed design sensitivity formulation is efficient and ac- curate. Ó 2003 Elsevier Ltd. All rights reserved. Keywords: Design sensitivity analysis; FRF-based substructuring method; Parametric optimization; Vibro-acoustic system 1. Introduction Complex dynamic systems such as automobiles, air- crafts and ships have a power generation system, which is the main source of noise and vibration. To make the passenger compartment more comfortable, engineers have introduced an isolation system between the engine and the body. A typical isolation system consists of springs and dampers, and their properties are deter- mined to minimize the transmissibility of the vibration. However, optimal parametric design of the isolation elements is very expensive since the whole complex system should be analyzed repeatedly to obtain the op- timal solution. The design of an engine mount system of a passenger car illustrates this problem well. Not only the characteristics of the trimmed body but also the stiffness of each engine mount and bushing strongly af- fect the interior noise. Therefore engineers usually select the whole body structure including the cabin cavity as the system boundary to be analyzed [13]. Generally the trimmed body of a car or the fuselage of an airplane have very complicated structure, since they are composed of thin shells with trims bolted or riveted to each other. Vibro-acoustic analysis of such structures is costly, especially in the design stage as nu- merous iterations are necessary. One of the most efficient methods under these circumstances is the substructuring method. In particular, the FRF-based substructuring method is widely used in the automobile, airplane and ship industries because the FRFs used in the method can be acquired from either tests or numerical analyses such as finite element analysis [2,3]. In addition to the evaluation of performance indexes itself, the performance variations with respect to design parameters are very important information because, from the designer’s viewpoint, they provide direction for design modifications. Design sensitivity is the rate of change of a design object with respect to the change of design variables. In most cases, the direction of design * Corresponding author. Tel.: +82-51-890-1658; fax: +82-51- 890-2232. E-mail address: dooho@dongeui.ac.kr (D.-H. Lee). 0045-7949/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0045-7949(03)00288-8 Computers and Structures 81 (2003) 2249–2257 www.elsevier.com/locate/compstruc