PSI - Issue 64

Amir Shamsaddinlou et al. / Procedia Structural Integrity 64 (2024) 360–367 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

362

3

{ ̇( ) = ( ) + ( ) ( ) = ( ) + ( )

(6)

where Z(t) is the state vector, and A and B are the system states. Also, R and Q are the system matrices defined based on the type of expected output Y . ( ) = [ ̇(( ))] (7) = [ 0 − −1 . − −1 . ] (8) = [ 0 −1 ] (9) It should be noted that if the external loading is the base acceleration, vector P(t) can be defined as: ( ) = {1} +1 ̈( ) (10) 3. Stability and Poles of System Consider a building structure as an open-loop system with a ground excitation input and a seismic response output. The transfer function of a system is defined as the ratio of the Laplace transform of the output to the Laplace transform of the input when the initial conditions are zero. So, the transfer function Ogata (1999) for a system is defined as follows: ( ) = (( )) (11) Also, this system can be represented in state space by the following equations: ̇ = A + B (12) = C + D (13)

where x is the state vector, u is the input, and y is the system's output. The Laplace transforms of state space equations are given by: ( ) − (0) = A ( ) + B ( ) ( ) = C ( ) + D ( ) The x(0) is set to be zero for zero initial conditions. Then, this obtained that: ( ) = ( I − A) −1 B ( ) By substituting equation (16) into equation (15), this is seen that: ( ) = [C( I − A) −1 B + D] ( )

(14) (15)

(16)

(17)

So: ( ) = (( )) = C( I − A) −1 B+D Hence ( ) can be written as: ( ) = | I (− )A|

(18)

(19) where Q(s) is a polynomial function of s . This is considered that | I−A| is equal to the characteristic polynomial of G(s). Therefore, the eigenvalues of A represent the poles of G(s). The transfer function can be written as: ( ) = (( )) = 0 + 1 −1 +⋯+ −1 + + 1 −1 +⋯+ −1 + (20) Therefore, based on the transfer function of a system, the roots and poles of a system can be determined. In general, the roots and poles of a system can be expressed as a complex number: = ± (21)