# Despite of the extensive studies of vibration

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Despite of the extensive studies of vibration analysis on damaged structures, only few effective and practical techniques are found for very small damage identification. This paper, therefore, focuses on the study of the FRF curvature energy damage index method for damage detection purposes. The results indicate that the present method is quite sensitive to assessing damage in addition to identifying location of damages. 2 Formulations 2.1 Frequency response function (FRF) The general mathematical representation of a single degree of freedom (SDOF) system is expressed by m ¨ x ( t ) + c ˙ t + kx ( t ) = F ( t ) . (1) Assuming that the forcing is harmonic of the form F ( t ) = F O e iΩt . In more general case the Receptance matrix for MDOF systems with viscous damping, can be expressed as [ H ( Ω )] = [[ K ] - Ω 2 [ M ] + [ c ]] - 1 . (2) Similarly without viscous damping the above equation can written as [ H ( Ω )] = [[ K ] - Ω 2 [ M ]] - 1 . (3) The Receptance matrix is symmetric for linear systems and therefore H rz ( Ω ) = ¯ X r F z = H zr = ¯ X S F z , (4) where ¯ X k and F k are, respectively the Fourier transform of the displacement and applied force time histories at the k th degree of freedom. The functions H rz ( Ω ) can be arranged in matrix form. This leads to a Receptance matrix defined as WJMS email for contribution
World Journal of Modelling and Simulation, Vol. 8 (2012) No. 2, pp. 147 - 153 149 [ H ( Ω )] = H 11 H 12 · · · H 1 n H 21 H 22 · · · H 2 n . . . . . . . . . . . . H nl H n 2 · · · H nn (5) 2.2 The frequency response function (FRF) curvature method This method presented by Pandey et al. [ 9 ] have found that in place of using a displacement mode shape, strain or curvature shapes (surface strain in a beam is proportional to curvature) are more effective at identi- fying the location of damage. Sampaio et al. [ 8 ] showed in place of using displacement mode shapes. Strain or curvature shapes (surface strain in a beam is proportional to curvature) are more effective at identifying the location of damage]. This paper is extended to plate like structures by using FRF curvature data rather than mode shape data. The FRF-curvature for any frequency is defined by H i,j ( Ω ) = - H i - 2 ,j ( Ω ) + 16 H i - 1 ,j ( Ω ) - 30 H i,j ( Ω ) + 16 H i +1 ,j ( Ω ) - H i +2 ,j ( Ω ) 2 h 2 , (6) where H i,j ( Ω ) : FRF curvature measured at location i due to a force input at position j . h : The distance between two consecutive measurement points. 2.3 The FRF curvature energy damage index In this section a new damage index based on the concept of FRF curvature energy is proposed. The damage indices are based on the variation of the FRF curvature energy at the element of the structure for a given excitation frequency. In a plate structure the FRF curvature energy can be defined as η ( Ω ) = L 0 [ H ( x ; Ω )] 2 d x, (7) where L is Span of the plate and H ( Ω ) is FRF curvature for a frequency Ω . x and y are the horizontal and vertical directions of the plate. In above equation showed only x direction.

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• Fall '19
• World Journal of Modelling and Simulation, D. Reddy

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