## 1 Introduction  
It is no longer sufficient to design a technical system that fulfills its function; it has become important to achieve the highest performance possible with the smallest quantity of material, i.e., optimization. Many studies into structural optimization in the past few decades have responded to this need. Although the optimization of structures can be realized in either the time or frequency domain, frequency domain analysis techniques are widely employed in engineering structures on account of their advantages over their time domain counterparts. For instance, it is more challenging to obtain the time domain response of a system analytically than that of the frequency domain, especially in the case of random loading. In addition, the difficulty increases as the order of the system increases. Furthermore, the sensitivity of a system to parameter variations can be measured or evaluated more conveniently in the frequency domain [1]. The frequency domain provides great convenience, enabling analyses to be performed over a wide frequency range under random loading. Consequently, the optimum design of structures, especially those exposed to harmonic and random loading, is preferably studied in the frequency domain. The frequency domain approach was first predominantly deployed in control theory and has since been used in the analysis of mechanical systems due to its convenience. In chronological order, the relevant texts include [2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18]. For the purpose of optimizing a mechanical system in the frequency domain, various problem formulations have been proposed in the literature. These can be categorized according to their objective functions as follows: response (vibration) control, eigenfrequency maximization, and weight minimization. Although vibration control can be accomplished using an active control approach, deploying a passive control approach during the design phase of mechanical systems is more economical [19]. Vibration control is mainly achieved by reducing the structural response by either directly defining response amplitudes as objective functions [20] or by using transfer function norms (i.e., \(\mathbf{H}_{2}\) and \(\mathbf{H}_{\infty}\) norms) as objective functions [6; 17; 21]. Additionally, the root mean square (RMS) or weighted RMS of structural responses such as acceleration and displacement responses are other objective functions that have been used in several studies [4; 15; 22]. Another approach to vibration control is to formulate an objective function by utilizing different compliance expressions for the purpose of optimizing the frequency response. In [23], an attempt was made to minimize the absolute value (modulus) of the mean compliance and its integration, while in [9], dynamic compliance was used to optimize the dynamic response of the structure. In another study, the maximization of the imaginary part of the complex dynamic compliance of damping materials was investigated in order to reduce the resonance peak response [13]. Moreover, research has utilized both mean compliance formulations [24] and new approaches to compliance minimizations for topology optimization [25]. Additionally, a considerable amount of study has been conducted on the multi-objective optimization of structures over the past few years. This is of especial importance given that mechanical systems must be designed in a manner that considers multiple phenomena [2; 22; 26; 27]. The objective functions, which constitute the cost function, can conflict with each other, as in the cases of weight and displacement [28; 29; 30] or weight and fatigue/damage [16]. Cost functions with multiple objectives, such as weight, damage, and response amplitude, are used for the optimization problems within this study. Three main methods are reported for solving these multi-objective optimization problems: a priori, a posteriori, and interactive preference methods [31].