PSI - Issue 78

Lorenzo Audisio et al. / Procedia Structural Integrity 78 (2026) 1277–1284

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As it is clear to note, R depends on several parameters, including the damping ratio ξ a of the NSE, its natural vibration period T a , and the vibration period T i associated with the i-th mode of the principal structure. The formulation assumed includes the coefficients a, b, c, d, and e, which are calibrated to effectively represent the dynamic behaviour of the NSE as a function of the ratio β =T a /T i and the damping ξ a . For completeness, Figure 3 shows the influence of these coefficients (a, b, c, d, e) on the behaviour of the dynamic amplification factor R as a function of the frequency ratio β =T a /T i , assuming a damping ratio ξ a of 5%. Each figure illustrates the variation of a single coefficient, keeping constant the remaining ones (a=2; b=2, c=1, d=2, c=0.5), and highlighting its effect on the shape and amplitude of the resonance peak. It is observed that the parameters affect differently the amplitude, width, and position of the peak, thus allowing the calibration of the FRS to reproduce the desired dynamic behaviour. 3.1. Calibration of Coefficients A series of parametric analyses are implemented in order to propose the optimal values of the coefficients of the amplification function R. Based on the obtained results, the following formulation for R is proposed: = [(2 ) 2 +(1−( ) ) 2 ] − (7) More in detail, parametric response spectrum analyses are performed on a two-degree-of-freedom model implemented in OpenSeesPy (Mazzoni et al., 2006; McKenna et al., 2000; Zhu et al., 2018). This model simulates the coupled dynamic interaction between the primary structure and the NSE. The parametric study is conducted varying both the structure vibration period and the mass ratio α, defined as the ratio of the NSE mass to the mass of the main structure.

Fig. 4. Variation of Optimal Coefficients c and e with Mass Ratio and Structural Period

Figure 4 illustrates the variation of the optimal values of the coefficients c and e in the element amplification function R, as a function of the mass ratio parameter α , defined as the ratio between the mass of the non-structural element (NSE) and that of the main structure. Each curve corresponds to a different fundamental vibration period of the structure. The results indicate that the coefficient c exhibits a decreasing trend with increasing α , showing a marked sensitivity to the structural period. Similarly, the coefficient e also decreases as α increases, although it displays less variability across different structural periods compared to c.