Sound Radiation Characteristics for Structural Damage Identification

This paper investigates the sound power radiated from a cracked plate. The sound power is obtained by using a lumped parameters model-based acoustics theory and vibration responses obtained by using the finite element method. It is shown that the vibration mode shapes and crack angle are closely related to the sound radiation characteristics, which can be applied to detect damages such as the cracks generated within a structure. Introduction Existence of structural damage within a structure may lead to the change in the sound radiation of the structure as well as the changes in dynamic characteristics such as the vibration response, natural frequency, mode shape, and modal damping. Therefore, the change in sound radiation of a structure can be used to detect, locate and quantify structural damages. Though a variety of structural damage identification methods (SDIM) have been developed based on the damage-induced changes of dynamic characteristics [1-3], there are very few SDIMs developed based on damage-induced sound radiation. The plates are the most commonly used structural elements along with beams and shells. When a cracked plate is excited by some external disturbances, it may radiate sound which is somewhat different from the sound radiated when it is undamaged. Thus, we may detect and identify the damage generated within a plate by comparing the damage-induced change in sound radiation characteristics of the plate. In the literature [4], the acoustic intensity has been well used to identify bearing defects. This paper considers a cantilevered rectangular flat plate containing a line-through crack and investigates the sound power with varying the angle (direction) of the line-through crack. The acoustic power is obtained by using the lumped parameters model-based acoustics theory by Fahnline and Koopmann [5] and the vibration response computed by the finite element analysis software ANSYS. It is observed that the vibration mode shapes and crack angle are closely related to the sound radiation characteristics, which can be in turn used to detect cracks within a plate. Sound Power Analysis Sound Power av Π . In the lumped parameter model of the sound radiation, the boundary surface of a vibrating structure is divided into small elements and it is assumed that each element vibrates as a piston. That is, the real volume velocity distribution of an element is approximated as its averaged value. As the primary quantity of interest is the time-averaged sound power radiated from a vibrating structure, the time-averaged sound power can be obtained as the surface integral of the multiplication of the pressure and normal velocity of the boundary surface of the structure. By using the Kirchhoff-Helmholz equation and by applying the assumption that each of the surface elements vibrates as a piston with the same elemental volume velocity distribution, the time-averaged sound power can be derived in the form as [5] Ru u H av 2 1 = Π (1) Advanced Materials Research Vols. 26-28 (2007) pp 7-10 online at http://www.scientific.net © (2007) Trans Tech Publications, Switzerland Online available since 2007/Oct/02 All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of the publisher: Trans Tech Publications Ltd, Switzerland, www.ttp.net. (ID: 130.203.133.33-16/04/08,10:41:38) Fig. 1. A cantilevered plate with a crack of orientation θ where R is the radiation resistance matrix and u = {uk; k = 1, 2, ..., N} is the vector of volume velocities of N surface elements. Volume Velocity u: The volume velocity can be computed by using the nodal displacements obtained by finite element analysis. Let the averaged normal velocity component of the k-th surface element be given by k k k k i n d n v ⋅ − = ⋅ ω (2) where vk is the velocity vector, dk is the displacement vector, nk is the unit directional vector normal to the k-th surface element, and ω is the circular frequency. The averaged volume velocity of the kth element of area Sk can be obtained from ∫∫ ⋅ − = ∫∫ ⋅ = k k S k k S k k k dS i dS u n d n v ω (3) Thus, we can experimentally measure the sound power by using the volume velocities which are determined by the surface displacements at each center of surface elements measured by, for instance, accelerometers. Radiation Resistance Matrix R: The radiation resistance is the real part of the radiation impedance. Thus, the radiation impedance matrix component Ri,j is defined by the real part of the ratio of the (averaged) pressure over the i-th element to the (averaged) volume velocity over the j-th element. Accordingly, the radiation resistance matrix is given by [5]