Issue 52

J. Akbari et alii, Frattura ed Integrità Strutturale, 52 (2020) 269-280; DOI: 10.3221/IGF-ESIS.52.21 272 Teager -Kayser Energy Operator (TEO) The free vibration response of the single degree of freedom (SDF) system with concentrated mass m and stiffness k is written as Eqn.(6)   x t =Acos( ω t+ θ ), (6) where, x(t) is the time variable position of the mass, A is the peak amplitude of the vibration, ω refers to the natural frequency of vibration and θ is the phase angle of the free vibration. For the mentioned SDF system, the total energy is computed as Eqn.(7) 2 2 2 2 1 1 1 E(t)= kx + mv E= m ω A 2 2 2 .  (7) This equation indicates that the energy of a system is dependent on the frequency and amplitude of the vibration. For a discrete signal, the free-response could be written as Eqn.(8)       n-1 n n+1 x =Acos Ω (n-1) + Φ x =Acos Ω n+ Φ x =Acos Ω (n+1) + Φ            (8) where  refers to the natural frequency of the system and  is the phase angle of the vibration. After processing and simplifying the above equation, the following equation could be obtained as Eqn.(9)   2 2 2 n n-1 n+1 A sin Ω = x x x  (9) Then, Teager- Kayser operator for a discrete signal n x is defined using  and could be present as Eqn.(10) 2 n n n-1 n+1 . [x ]= x x x   (10) P ROBLEM D EFINITION n this paper, the single and multiple damage detection for the beam-type structures with pined-pined and clamped- clamped boundary conditions were investigated. For the steel beams as depicted in Fig. 1 the geometrical and mechanical specifications are presented at Tab.1 Figure 1: The schematic figure of the studied beams (a), and the schematic locations of multiple damages (b). I