PSI - Issue 78

Predaricka Deastra et al. / Procedia Structural Integrity 78 (2026) 2038–2045

2040

u str

m

u IViBa

u f,IViBa

u f,I

k IViBa

b

m IViBa

k

u f

u f,IViBa

c IViBa

u f,I

k f,I

(b)

c

k f

IViBa

k SSSI

u f,IViBa

c f,I

k f, IViBa

u f,I

k IViBa

m f

b

m f, IViBa

c f

c f, IViBa

c SSSI

c IViBa

u IViBa

u g (a)

(c)

Fig. 1. (a) Analytical model of a single-degree-of-freedom structure equipped with an IViBa. (b) TMDI. (c) TID.

damping c f , IViBa . The soil at the inerter’s ground connection is represented by sti ff ness k f , I and damping coe ffi cient c f , I . Two alternative IViBa configurations are illustrated in Fig. 1(b) and (c). The first configuration corresponds to the conventional ViBa, which consists of a spring and a dashpot arranged in parallel—with sti ff ness k IViBa and damping coe ffi cient c IViBa —supporting a secondary mass m IViBa . An inerter with inertance b is connected to the secondary mass, forming a TMDI configuration. The application of the TMDI as an IViBa was first investigated in Cacciola et al. (2020). The second configuration adopts the TID model, which can be regarded as a limiting case of the TMDI, where the secondary mass is negligible ( m IViBa = 0), o ff ering a potentially more compact and lightweight alternative. The equation of motion of the system shown in Fig. 1(a) in absolute coordinate can be written as

M ¨ u ( t ) + C ˙ u ( t ) + Ku ( t ) = ¯ C ˙ u g ( t ) + ¯ K u g ( t )

(1)

Considering the TMDI configuration, the M , C , K , ¯ C and ¯ K matices are given as follows:

M =    

    ; C =

    c f + c SSSI + c − c − c c − c SSSI 0 0 0 0

   

m f 0 0

0 0 0 0 0 0

− c SSSI

0 0 0 0

0 m

0

0

0 0 m f , IViBa

0 c SSSI + c f , IViBa + c IViBa − c IViBa 0

b + m IViBa − b − b b

c IViBa 0

0 0 0 0 0 0

− c IViBa

0

0

c f , I

K =    

   

    ; ¯ K =

   

    ; ¯ C =

   

k f + k SSSI + k − k

− k SSSI

0 0 0 0

− k f 0 − k f , IViBa 0 − k f , I

− c f 0 − c f , IViBa 0 − c f , I

− k

k

0

0 k SSSI + k f , IViBa + k IViBa − k IViBa 0

− k SSSI

0 0

0 0

− k IViBa

k IViBa 0

0

0

k f , I