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

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A system with fixed values of k c and k e can exhibit Mode 1a or Mode 1b for di ff erent values of m Re . Mode 1a requires a higher m Re than Mode 1b and its period is generally higher than that of Mode 1b. The variation between Mode 2a and Mode 2b also depends on the apparent mass m Re (for fixed values of k c and k e ). Mode 3 has the same shape for all values of the parameters within the investigated ranges. Finally, it is worth observing that the period of the first mode of the coupled structure is generally higher than the period of the first mode of the stand-alone structure. The periods of the second and third modes of the coupled structure are very close to the first and second modes of the stand-alone structure, respectively.

Mode 1

Mode 2

T = 0.116 s 2

T = 0.484 s 1

(a)

d 1 Stand-alone structure Coupled structure d 1 Mode 1a Mode 1b

Mode 2a

d >d 2 1

d 2

Mode 3

Mode 2b

d

d 2

(b)

Fig. 2. Spectral analysis of the structures coupled by the Kelvin-Voight device: a) Periods and modal shapes of the stand-alone frame structure; b) Modal shapes of the coupled structure.

To evaluate the dynamic influence of the coupling, the Mass Participating Coe ffi cients, MPC, of the first two modes of the coupled structure are obtained. Such MPC’s are arranged in contour maps in the ρ e − ρ c plane. Fig. 3 shows four maps arranged in matrix form, where the columns refer to the MPC’s of the first two modes of the coupled structure, whereas the rows refer to di ff erent values of ψ . All of the maps of Fig. 3 are obtained for c e = 0 (null damping of the connecting device). The maps also show labels indicating the shape of the mode. Such labels refer to the nomenclature adopted in Fig. 2 for the coupled structure. Thick line is the boundary between the two di ff erent modes. When no inerter is considered (first row of the figure) both MPC’s have significant values in all the parameter plane. When the inerter is considered, the MPC of Mode 1a is much higher than the MPC of Mode 2a, which becomes almost negligible, so that the dynamics of Mode 1a prevails on Mode 2a. In the region characterized by Mode 1b, the MPC’s of the first two modes are again both significant as in the case with no inerter.