PSI - Issue 78

Salvatore Dario Di Trapani et al. / Procedia Structural Integrity 78 (2026) 2118–2125

2120









These additional DOF are collected in the n × 1 vectors 1 2 () u();u ();...;u () n t t t t = u , respectively. In the close-up of the generic j th STLCD unit (with j =1,.. n ) shown in Figure 1, installed at the j th floor, the total liquid column length is denoted as , , 2 j v j h j L L L = + , where , v j L and , h j L represent the vertical and horizontal liquid segments, respectively. Based on these lengths, the horizontal-to-total length ratio is defined as , / j h j j L L  = . The horizontal and vertical cross-sectional areas of the j th container are denoted as , h j A and , v j A , respectively, with their area ratio denoted as , , / j h j v j r A A = . The j th unit has a total mass , t j M , constituted of the container mass, , c j m , and the liquid mass, , , , , , 2 l j h j v j v j h j m A L A L   = + . Dissipative effects due to the liquid motion are modeled through the head-loss coefficient j  , while the natural frequency of the liquid column is defined as , 2 / l j j g L  = . Regarding the j th spring-dashpot unit, its damping and stiffness properties are denoted by , c j c and , c j k , respectively, and can be expressed in canonical form as , c j  and , c j  . The governing equations for the M-STLCD-controlled multi-story structure are defined as follows: 1 2 () y();y ();...;y () n t t t t = y and

( ) t t x t + + =− Mx Cx Kx M ( ) t ( ) ( ) g

(1)

 

 

where ( ) s t x , ( ) t y and ( ) t u . The 3 n × 3 n mass, damping, and stiffness matrices, along with the 3 n × 1 array accounting for inertial contributions due to the ground excitation ( ) g x t , are respectively organized into submatrices that separately group the components associated with the main structure and those related to the M-STLCD, as follows:   1 ( ) ; nx t   1 ( ) ; nx t   1 nx ( ) t x x ( ) t s = y u is the 3 n × 1 vector collecting

     

     

     

     

 0 M M M M M M M    M 0   nxn   nxn   nxn 3 3 s nxn c c nx n =

  c nxn C C 0 −   s nxn

  nxn

  nxn

;  C

;

0 0

  C 0 c nxn

=

  nxn



  nxn

  nxn

, c l

3 3

nx n

0

C

  nxn

  nxn

  nxn

  nxn

  nxn

 

, c l

, c l

l

l nxn

(2 a-d)

     

     

     

     

M

τ τ

− K K 0

    1 s nxn nx

 

 

  nxn

s nxn

c nxn

;

 3 3 K 0 nx n = 

  K 0 c nxn

 3 1 nx M M = 

  nxn

  nxn

    1 nxn nx

c

0

0

K

M

τ

  nxn

  nxn

 

    1

, c l nxn nx

l nxn

With [1;1;...;1] = τ being a n × 1 vector of ones. The other submatrices in (2) are diagonal submatrices defined as:

1

M

A L  =

;

;

M

, h j A L (2

, h j L m + )

 =

+

 

 

, h j h j ,

, c l nxn

, v j

, c j

c nxn

r

j

M

(2 A L r L  = +

)

;

 

, h j

, v j

, j h j

l nxn

(3 a-g)

1 2 2

C

c =

;   l nxn C

, h j j j A r u t   ( ) j

=

 

, c j

c nxn

K

K

k =

, h j A g 

;

=

 

 

, c j

c nxn

l nxn