## 1 Introduction  
Multibody systems with lightweight and slender links exhibit excellent performance in terms of high attainable speed and low energy consumption. However, flexibility of the links makes such systems prone to vibration, and consequently, prone to fatigue and poor in-position accuracy. Such issues can be greatly mitigated if the system is designed or modified with appropriate eigenfrequencies through modification of physical parameters [1].  
From the mathematical point of view, a problem in which a system is designed for specific eigenfrequencies can be regarded as an inverse eigenvalue problem [2], in contrast to the direct eigenvalue problem, in which the eigenfrequencies of a vibrating system are computed. For linear time-invariant systems, the latter has been extensively studied, and the state-of-the-art can be considered very well developed [3]. The task of computing or measuring the eigenfrequencies of a system is often referred to as modal analysis. The inverse problem, which aims at finding a feasible system model from prescribed eigenfrequencies, is ill-posed and therefore intrinsically more challenging. The significance of the inverse problem in engineering (not limited to vibrations) has stimulated fertile research over the past few decades, e.g., [4, 5, 6, 7].  
For a flexible-link multibody system (FLMS), linear models are satisfactory only in the proximity of an equilibrium configuration. If large displacements and/or rotations of the links are considered, a nonlinear model is essential in order to account for the dependence on the multibody system configuration. Some of the FLMS modeling techniques available in the literature represent the system dynamics by means of a minimum set of second-orderordinary differential equations, which allow for a straightforward application of modal analysis [8]. One such flexible multibody formulation is the equivalent rigid-link system (ERLS) [9; 10].  
The iterative nature of design optimization is inherently afflicted with high computational effort. Approximation-based design optimization, also known as surrogate-based design optimization, replaces a computationally expensive physical model (e.g., finite-element analysis or multibody system simulation) with a computationally cheap mathematical approximation. Methods of mathematical approximation include polynomial regression (response surface methodology, e.g., [11]), radial basis functions, Gaussian process inference (Kriging, ref. [12]), amongst others, cf. [13]. The former method will be addressed in this work. The use of approximation-based design optimization has been proposed for aerospace structures [14; 15] and nonlinear structural mechanics [16], including crash [17] and multibody dynamics [8].  
The typical approach to the modal analysis of FLMSs consists of the interpolation of several static modal analyses, which span large displacements of the multibody system [18]. A parametric representation as an approximation of the eigenvalues (and eigenvectors) of second-order mechanical systems is proposed by [PERSON] et al. [19], which is adapted to ERLS and validated by [PERSON] et al. in [20; 21]. This parameteric approximation is used in the optimization here.  
The present paper addresses eigenfrequency assignment to flexible-link multibody systems, which is still an open problem. The novel method introduced in Section 2 relies on the aforementioned parametric approximation of the eigenfrequencies, which is briefly summarized in Sections 2.1 and 2.2. The formulation of the assignment problem as a quadratic program is described in Sections 2.3 and 2.4. For validation, a test-case is presented in Section 3 and the obtained results are discussed in Section 4.