PSI - Issue 78

Enrico Bernardi et al. / Procedia Structural Integrity 78 (2026) 599–606

601

Dynamic system representing the structure can be modeled as a 2DOF system, similar to the approach commonly used in the analysis of Tuned Mass Dampers (TMDs). The 2DOF system can be defined in dimensionless terms, as summarize in Figure 1.

Fig. 1. Dynamic model of the structure with IIS.

To represent the stochastic nature of the earthquake, the Power Spectral Density function (PSD) of a stochastic Gaussian process, in which the mean value is zero ( S(ω) ), was considered. The seismic input was assumed as a ‘white noise’ signal, i.e. a signal of equal intensity at all frequencies. With this approach, the PSD no longer depends on the frequency of the signal ω , and S(ω) can be rewritten as S 0 . Based on these assumptions, the variance of the structural response (which also represents its intensity) in terms of relative displacement ( σ 2 ) and absolute acceleration ( σ a 2 ), can be described as:

 

σ

2 4  

2

G

( )

d  

=

Sub

S

(1)

0

−

 

σ

2

2

G

( )

d  

=

a

a

S

0

−

where G ( ρ ) and G a ( ρ ) are the Frequency Response Functions (FRFs), normalized to the dynamic characteristics of the primary structure, ω Sub is the frequency of the substructure (existing building in the case of retrofit) and ρ = ω / ω Sub . The multi-objective optimization approach presented aims to consider the overall structural performance by defining performance parameters that account for the response of the existing structure, the isolation system (by limiting the isolation drift), and the isolated superstructure (by limiting the acceleration of the superstructure). Performance indexes assumed in the optimization, which represent the normalized variance of the 2DOF system, relative to the performance of the SDOF system, are defined as follows:

 

 

 

2

2

2

G ( )

G ( ) G ( )  −

a,2 G ( )

d  

d  

d  

1

2

1

(2)

;

;

PI

PI

PI

=

=

=

−

−

−

1

2

3

 

 

 

2

2

2

G ( ) SUB

G ( ) SUB

a, G ( ) SUB

d  

d  

d  

−

−

−

where: