PSI - Issue 44

A.Di Egidio et al. / Procedia Structural Integrity 44 (2023) 2136–2143 A. Di Egidio, S. Pagliaro, A. Contento / Structural Integrity Procedia 00 (2022) 000–000

2139

4

Table 1. Geometrical and mechanical characteristics of the reference 3-D frame structure

Storeys Connection Level Floor area Storey mass m s

Storey height

250 m 2

301.5x10 3 kg

4

1

3 m

The dynamic characteristics of the equivalent 2-DOF models that are shown in Table 2 are derived by applying the reducing order technique adopted in Fabrizio et al. (2019).

Table 2. Dynamic characteristics of the 2-DOF equivalent model Storeys k 1 [N / m]

k 2 [N / m]

m 1 [kg]

m 2 [kg]

4 6.610823x10 8

2.034100x10 8

301.5x10 3

904.5x10 3

The parameters that are varied in the analyses are the following:

• Ratio between the sti ff ness of the external structure and the sti ff ness of the substructure

k e k 1

(4)

ρ e =

• Ratio between the sti ff ness of the coupling device and the sti ff ness of the substructure

k c k 1

(5)

ρ c =

• Ratio between the apparent mass of the inerter device and the mass of the substructure

m Re m 1

(6)

ψ =

3.2. Performance indexes

The comparison between the displacement u 1 and the drift ∆ u = u 2 − u 1 of the coupled structure and those of the stand-alone frame structure provides an indication about the e ffi ciency of the coupling. The above-mentioned comparison is performed introducing two performance indexes

max | u 1 ( t ) | max | ˜ u 1 ( t ) |

max | u 2 ( t ) − u 1 ( t ) | max | ˜ u 2 ( t ) − ˜ u 1 ( t ) |

max | ∆ u ( t ) | max | ∆ ˜ u ( t ) |

(7)

, α 2 =

α 1 =

=

where the displacements ˜ u 1 and ˜ u 2 are those of the stand-alone frame structure. From Eq. 7 it follows that the lower are the values of α 1 and α 2 with respect to unity, the higher is the e ff ectiveness of the coupling. The aim of the parametric analysis is to evaluate α 1 and α 2 for each set of variable parameters and plot such values in a specific parameters plane, thus creating performance or gain maps.

4. Modal analysis of the coupled system

A preliminary investigation evaluates the e ff ects of coupling and inerter device on the spectral characteristics of the coupled structure. The analyses in this Section refer to the 4-storey frame structure whose characteristics are reported in Table 1 and Table 2. Figure. 2 shows the modal shapes for both stand-alone frame structure and coupled structure. Specifically, Fig. 2a shows the two periods and modal shapes of the 2-DOF model of the stand-alone frame structure. Fig. 2b shows the modal shapes of the coupled structure. Such modal shapes depend on the parameters characterizing the coupling be tween the frame and external structures, specifically k c , k e , and m Re . A fixed value of m e = 0 . 1 m 1 is always considered.