PSI - Issue 78

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

602

2    i

1

G ( ) ( SUB 

2 1) SUB +

−

=− − +

(3)

2   =−

a, G ( ) SUB

G ( ) 1 SUB  +

while G 1 and G 2 are the first and second components of the vector G , respectively, and G a,2 the second component of the vector G a . Specifically, PI 1 represents the dimensionless variance of the drift in the existing structure, PI 2 represents the dimensionless variance of the isolation drift, and PI 3 represents the dimensionless variance of the acceleration in the isolated superstructure. Once the mass ratio μ is defined, the performance indices ( PIs ) depend only on the frequency ratio ν and the damping ratio ζ IS . Therefore, two multi-objective optimizations were performed with the goal of optimizing these latter parameters ( ν and ζ IS ) to minimize the structural response. The two optimizations investigated ( OPT1 and OPT2 ) simultaneously optimize two objective functions ( OFs ). OPT1 minimizes the response of the existing structure ( OF 1 ) and the drift in the isolation system ( OF 2 ), and OPT2 minimizes OF 1 and the superstructure’s acceleration ( OF 3 ). Specifically, the objective functions ( OF 1 , OF 2 , OF 3 ) and optimizations ( OPT1 and OPT2 ) are illustrate in Figure 2.

Fig. 2. Performed multi-objective optimizations.

Multi-objective optimizations were performed using the ‘Non - dominated Sorting Genetic Algorithm’ (NSGA -II) (Deb et al. 2002), in which a population size of 80 individuals was adopted, 100 generations were performed, and the crossover and mutation probabilities were set to 0.9 and 0.1, respectively. 3. Proposed design model Figure 3 presents the main optimization results in both OPT1 and OPT2 , expressed in terms of Pareto Fronts. The curves indicate that the minimum value of OF 1 ( OP L ) is approximately 0.4, for both optimizations and all values of μ . Conversely, the OP U point is where the gradient of the curve equals 1, representing the “optimal compromise” point. This is the threshold beyond which a reduction in OF 2 or OF 3 results in a greater increase in OF 1 . To provide a model for the calculation of optimal isolation parameters, equations has been calibrated based on the optimization results. To allow the performance level to be chosen, the performance parameter β has been defined (as illustrated in Figure 4), which indicates the normalized distance from OP L to OP U in the Pareto Fronts, and assumes values from 0 ( OP L ) to 1 ( OP U ). In particular, β should be defined based on the desired level of performance to be achieved. For instance, if the objective is to optimize the substructure performance, β should be assumed equal to 0. Conversely, if the aim is to achieve an optimal solution in terms of isolation drift or superstructure acceleration, β should be chose equal to 1 ( previously defined as “optimal compromise” point ). Values of β between 0 and 1 indicate an intermediate level of performance.