Issue 54

F. Brandão et alii, Frattura ed Integrità Strutturale, 54 (2020) 66-87; DOI: 10.3221/IGF-ESIS.54.05

In which M , C and K represent the (n+N TMD )x(n+N TMD ) mass, stiffness and damping matrices and n is the number of degrees of freedom. The damping matrix C is assumed to be proportional to the M and K matrices as: C = α M + β K . , u(t)   , u(t)   , u(t)  are the (n+N TMD )- dimensional acceleration, velocity and displacement vectors relative to ground, respectively. The matrix B is a (n+N TMD )xd matrix that contains the cosine directors of the angles formed between the base motion and the associated displacement direction with the considered degree of freedom and d is the number of considered ground motions (directions). , g u (t)   represents the base acceleration and is a d -dimensional vector.

Figure 1: n-degree-of-freedom system (n-DOF) structure equipped (a) with one TMD at the top floor and (b) with MTMD possibly located in all floors of the structure. In Fig. 1.(a), there is a single TMD attached at the top floor of the structure. The structure DOF is increased by one and the TMD contribution to M and K , is illustrated in Eqns. 2 and 3. For the C matrix, the procedure is analogous to the K .

m

1       2 0 0 m

0 0 0 0

       

       





M

(2)

m

0 0 0 0

0

 

n

m

0

TMD

   k k k

0 0

0 0

 

       

1

2

2

      



 k k k

2

2

3







 



K

(3)

 k k



k

0 0

0 0

 

n

TMD TMD



k

k

TMD

TMD

Fig. 1.(b) represents the MTMD possibly located on all floors of the structure, the TMD contribution to M and K is illustrated in Eqns. 4 and 5. Again, for the C matrix, the procedure is analogous to the K .

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