## 1 Introduction
The Component Mode Synthesis (CMS) method [13], first developed for the substructuring of a linear system formed by multiple substructures, is widely used for model order reduction of structures with friction contact [9; 10; 11; 12]. The CMS technique involves three steps: decomposition of the structure into non-overlapping components, reducing the size of each component by projective reduction, and finally, coupling the individual components' ROMs through their interfaces. This allows the retention of the contact DoFs in the reduced model while replacing the other DoFs with generalized coordinates. In this way, the resulting ROM can correctly simulate the different contact states such as full-stick, gross slip, microslip, and liftoff. Generally, the DoFs kept in the reduced coordinate vector and the DoFs projected on the reduced basis are called master and slave DoFs, respectively. CMS methods can be classified depending on the underlying reduction basis. Generally, two groups of modes are considered in CMS subspace: (i) component linear normal modes obtained by imposing specific boundary conditions at master DoFs; and (ii) static deflections due to applying either unit forces or unit displacements at master DoFs. The normal modes account for inherent component dynamics, while static deflections describe the interface behavior and are necessary for coupling the substructures. The Craig-Bampton method, as a fixed-interface-based CMS method [13], is the most widely used technique in which the static deformed shapes due to imposing unit displacement on a master DoF (constraint mode) and the normal modes of the body with fixed master DoFs are considered as reduction bases. Rubin, as a free-interface-based CMS method [14; 15; 16; 17], is the other widely used technique. The static deformed shapes due to a unit force applied at a master DoF (attachment modes) and the normal modes of the body with free master DoFs comprise its reduction basis. Component mode synthesis techniques have also been used for the substructuring of the systems with geometrically nonlinear substructures [18; 19]. [PERSON] et al. employed the Craig-Bampton method [18] and the accuracy of the assembled reduced models can be proved by comparing their nonlinear normal modes with those computed from the full-scale assembly. To better model the physical behavior induced by geometric nonlinearity, [PERSON] and his colleagues extended CMS methods with the addition of static modal derivatives (SMDs)to the reduction basis. The formulation was first introduced in [20; 21], then the extended Craig-Bampton method was more thoroughly studied in [22], in which SMDs of the constraint modes and fixed interface normal modes for each component were considered and modified as fixed-interface modes for the convenience of interface compatibility during the assembly procedure. The extension of Rubin was also studied in [23] and developed for a 3D flexible multibody system in [24], in which a nonlinear coupling between attachment modes and free-interface modes was considered. These studies concluded that extended CMS techniques can accurately predict the response, while the Rubin substructuring basis enhanced with SMDs provides a better representation of the geometric nonlinearities in the interfaces, since it comprises normal modes and SMDs featuring free motion of the interface. Some researchers have also considered model order reduction techniques of structures with both contact and geometric nonlinearities [1; 4; 25; 26]. In [1; 4; 25], Craig-Bampton is employed to keep the contact DoFs in generalized coordinates. [PERSON] et al. use only the linear modes [1] while [PERSON] et al. augmented reduction basis with SMDs [4; 25]. The comparison of linear Craig-Bampton and augmented Craig-Bapton in [4] shows slightly better results of the latter while it requires a larger reduction basis to get accurate results. To the authors' knowledge, Rubin ROM enhanced by SMDs has not yet been studied for structures with both types of nonlinearity.