# Elasto-Plastic Short Exoskeleton to Improve the Dynamic and Seismic Performance of Frame Structures
## 1 Introduction  
In the last few years, multiple studies proposed the coupling with external mechanical systems as a means to reduce displacements and drifts of structures under external ground excitation. Examples of such external mechanical systems are rocking structures and rigid blocks. In [1], the authors investigated the effect of placing a structure on a rocking podium. Similarly, in [2], the authors analysed the effects of rocking isolation. Researchers also studied couplings between frames and rocking walls. In some cases [3, 4], frames and rocking walls have a rigid coupling; in other cases [5, 6, 7, 8], the connection between the frame and rocking wall was achieved by using visco-elastic devices. It was found that the rigid wall can work as an enhanced tuned mass damper for the frame.  
Other studies proposed to improve the performance of structures by coupling them with adjacent structures. For instance, in [9], the seismic performances of frame structures connected with adjacent towers were investigated. Instead, in [10, 11], improvement of the galloping motion under turbulent wind of two similar adjacent towers was obtained by a nonlinear viscous coupling. In [12], coupling between adjacent structures was used to increase the seismic performance of both structures.  
Two-degree-of-freedom models (2-DOF) were often used in the literature to study the conceptual aspects related to seismic protection methodologies. For example, the conceptual aspects of Base Isolation (BI) were studied in [13], and those of the Tuned Mass Damper (TMD) systems were studied in [14, 15, 16]. Such low-dimensional models were also used to analyse various modifications and combinations of the two techniques. In [17, 18],the reduction of the base displacement in a base isolated system through the use of a TMD was investigated. The same objective was lately pursued in [19; 20; 21] by means of a TMD and an inerter device.  
This paper investigates the dynamic and seismic behaviour of a structure rigidly coupled with a yielding, elasto-plastic system, representing a short exoskeleton. A two-degree-of-freedom model (2-DOF) is used to capture the main dynamic and seismic behaviour of the structure, whereas a single-degree-of freedom model (1-DOF) is used for the yielding system. A rigid link connects the bottom mass of the 2-DOF model of the structure to the yielding system.  
The yielding system has an elasto-plastic constitutive behaviour that is described by the Bouc-Wen model [22]. Such a model is chosen for its versatility, since the Bouc-Wen model is able to describe several different hysteretic behaviours by varying a few constitutive parameters, and it has already been used in combination with simplified 2-DOF models, as in [23]. Additionally, it is possible to find several generalisations of the original Bouc-Wen model that furtherly increase the range of hysteretic behaviours that such models can describe. Due to the rigid link between the structure and yielding system, the coupled system is described by a 2-DOF mechanical model. The nonlinear equations of motion are obtained by a direct approach and successively numerically integrated to analyse the behaviour of the coupled system. The coupling with the yielding system is considered beneficial for the structure when there is a reduction in its displacements or drifts.  
An extensive parametric analysis is performed to point out the role of the mechanical parameters describing the elasto-plastic constitutive behaviour of the yielding system. Both harmonic and seismic base excitations are considered. The results are summarised in behaviour maps plotted in the plane of the system's parameters. The maps provide the ratio between the maximum displacements of the coupled system and the stand-alone structure. A ratio less than unity highlights the effectiveness of the yielding system in improving the dynamics of the structure. Results reveal that such effectiveness can be achieved in large ranges of the parameters' values.