## 1 Introduction  
In recent years, numerous researchers have explored various methods through which to enhance the performance of frame structures under seismic excitation. Methods involving the coupling of structures to a specific device have garnered particular interest. Such couplings can be achieved in series or in parallel. Base isolation [1] is among those requiring a series coupling with the frame structure. This technique involves connecting the structure with an oscillating base, which can be positioned either at the base of the structure or at different intermediate levels ([2; 3; 4; 5; 6; 7; 8]). Among the series connections of external devices to a structure that is to be protected, one extensively studied method is coupling with devices acting as tuned mass dampers (TMD) [9]. Typically, TMDs necessitate a small mass to be adequately effective. However, under seismic excitation, the use of a small mass may compromise the effectiveness of the TMD. This challenge can be effectively addressed by coupling the TMD with an inerter device, thus augmenting its inertial forces, as demonstrated in [10]. The resulting device, known as the tuned mass damper inerter (TMDI), has been extensively investigated in various studies [11; 12; 13; 14; 15; 16]. Some of these studies specifically focused on enhancing the efficiency of base isolation [17]. Various other coupling devices connected in parallel have been studied in the scientific literature. For example, external exoskeletons rigidly connected to the frame structure [18; 19; 20], hysteretic mass damper inerters (HMDI) [21; 22], and rocking walls [23; 24; 25; 26] are among the instances that have been explored. Another more recent device is the vibro-impacting nonlinear energy sink (VI-NES). This comprises a mass that can move along a path with limited clearance. During this motion, the mass impacts on the limits of the clearance, thereby dissipating the mechanical energy. The VI-NES has been studied as a device capable of improving the seismic response of frame structures in [27; 28; 29; 30]. Typically, the VI-NES is positioned on a frame structure story and fixed to it. Consequently, the impacting mass cannot be very high as it directly affects the structural elements of the story. This paper proposes a protection mechanism based on a coupling with an auxiliary structure that is equipped with a vibro-impacting nonlinear energy sink--one that is rigidly connected to the first story of the frame structure. The frame structure to be protected is modeled by a low-dimensional 2-DOF model, which is obtained by means of an established dynamical equivalence criterion [4]. Due to the rigid connection between the frame and external structures, the resulting low-dimensional model has three degrees of freedom: two that describe the motion of a frame structure that is rigidly connected to an external structure, and one that captures the motion of the vibro-impacting mass. An ideal model assuming instantaneous impact was adopted for the vibro-impacting mass. This established model was used to conduct an parametric analysis, where the several parameters that characterize the external structure, a vibro-impacting mass, and the frame structure to be protected were varied. Three earthquake records were utilized to excite the structure. The seismic effectiveness of the coupling with the external structure was assessed by introducing two gain coefficients, which were defined as the ratio of the displacements of the coupled and stand-alone uncoupled frame structures. The results of the parametric analyses were presented in gain maps, and they represent the contour plot of the gain coefficients in a designated parameter plane. It should be stressed that, since this is considered a preliminary investigation (which is essential before implementing the findings in practical engineering applications), only a limited number of earthquake records were used in the analyses.