PSI - Issue 44

Sandro Carbonari et al. / Procedia Structural Integrity 44 (2023) 27–34 Sandro Carbonari et al. / Structural Integrity Procedia 00 (2022) 000–000

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2.2. The continuous-time state-space model The first-order continuous-time state-space model is obtained by introducing the state vector � � � �

(5)

that allows expressing the second-order equation (1) as � � � � �

(6)

where

� � � � � � � � � � � � By pre-multiplying equation (6) by �� , the following state equation can be obtained � � �

(7a,b,c)

(8)

where

� � � �� � �� � � � �� �

(9a,b)

in which � � � � � � (10a,b,c) and is the identity matrix of order 5. In modern control theory equation (8) is used together with the following observation equation to express system (1) in the form of a continuous-time state-space model: � � � (11) By assuming that the observation equation includes all the set of accelerations at the dofs , the following expressions hold for the output matrix � ∈ ℝ ���� and for the transmission matrix � ∈ ℝ ��� : � � � �� � �� � � �� (12a,b) It must be remarked that if the input is constituted by forces, the observed accelerations are absolute, as those registered in the real system through instrumentations during tests; on the contrary, if the input is represented by soil accelerations, Eq. (11) provides accelerations relative to the ground. Thus, in order to get the absolute accelerations if � � �� � , Eq (12b) should be substituted with � �� � � (13) It can be remarked that the eigenvalue problem associated to (6) is �� � �� � (14) where ����� contains the complex eigenvectors as columns and ����� is a diagonal matrix containing the 10 complex eigenvalues λ � ( i = 1…10). It can be proven that