|
Unified Framework is a general formulation which yields ''n''th - order expressions giving mode shapes and natural frequencies for damaged elastic structures such as rods, beams, plates, and shells. The formulation is applicable to structures with any shape of damage or those having more than one area of damage. The formulation uses the geometric definition of the discontinuity at the damage location and perturbation to modes and natural frequencies of the undamaged structure to determine the mode shapes and natural frequencies of the damaged structure. The geometric discontinuity at the damage location manifests itself in terms of discontinuities in the cross-sectional properties, such as the depth of the structure, the cross-sectional area or the area moment of inertia. The change in cross-sectional properties in turn affects the stiffness and mass distribution. Considering the geometric discontinuity along with the perturbation of modes and natural frequencies, the initial homogeneous differential equation with nonconstant coefficients is changed to a series of non-homogeneous differential equations with constant coefficients. Solutions of this series of differential equations is obtained in this framework. This Framework is about using structural-dynamics based methods to address the existing challenges in the field of Structural Health Monitoring (SHM).〔Akash Dixit - Damage Modeling and Damage Detection For Structures Using A Perurbation Method (May 2012)〕 It makes no ad hoc assumptions regarding the physical behavior at the damage location such as adding fictitious springs or modeling changes in Young’s Modulus. == Introduction == Structural Health Monitoring (SHM) is a rapidly expanding field both in academia and research. This article is about the vibration based SHM techniques. Immense amount of literature is being generated in the field. Most of this literature is based on experimental observations and physically expected models. There are some mathematical models, that give, analytical theory to model the damage. Such mathematical models for structures with damage are useful in two ways, firstly; they allow understanding of the physics behind the problem, which helps in the explanation of experimental readings. Secondly, they allow prediction of response of the structure. These studies are also useful for the development of new experimental techniques. Examples of models based on expected physical behavior of damage are by Ismail et al. (1990),〔Ismail, F., Ibrahim, A., Martin, H.K., 1990. Identification of fatigue cracks from vibration testing. Journal of Sound and Vibration 140, 305–317.〕 who modeled the rectangular edge defect as a spring, by Ostachowicz and Krawczuk (1991),〔Ostachowicz, W., Krawczuk, M., 1991. Analysis of the effect of cracks on the natural frequencies of a cantilever beam. Journal of Sound and Vibrations 150,191–201.〕 who modeled the damage as an elastic hinge and by Thompson (1949),〔Thompson, W.T., 1949. Vibration of slender bars with discontinuities in stiffness. Journal of Applied Mechanics 16, 203–207.〕 who modeled the damage as a concentrated couple at the location of the damage. Other models based on expected physical behavior are by Joshi and Madhusudhan (1991),〔Joshi, A., Madhusudhan, B.S., 1991. A unified approach to free vibration of locally damaged beams having various homogeneous boundary conditions. Journal of Sound and Vibration 147, 475–488.〕 who modeled the damage as a zone with reduced Young’s modulus and by Ballo (1999),〔Ballo, I., 1999. Non-linear effects of vibration of a continuous transverse cracked slender shaft. Journal of Sound and Vibration 217 (2), 321–333.〕 who modeled it as spring with nonlinear stiffness. Krawczuk (2002)〔Krawczuk, M., 2002. Application of spectral beam finite element with a crack and iterative search technique for damage detection. Finite Elements in Analysis and Design 9–10, 991–1004.〕 used an extensional spring at the damage location, with its flexibility determined using the stress intensity factors ''KI''. Approximate methods to model the crack are by Chondros et al. (1998),〔Chondros, T., Dimarogonas, A., Yao, J., 1998. A continuous cracked beams vibration theory. Journal of Sound and Vibration 215 (1), 17–34.〕 who used a so-called crack function as an additional term in the axial displacement of Euler–Bernoulli beams. The crack functions were determined using stress intensity factors ''KI'', ''KII'' and ''KIII''. Christides and Barr (1984)〔Christides, S., Barr, A.D.S., 1984. One-dimensional theory of cracked Euler–Bernoulli beams. International Journal of Mechanical Sciences 26 (11–12), 339–348.〕 used the Rayleigh–Ritz method, Shen and Pierre (1990)〔Shen, M.H., Pierre, C., 1990. Natural modes of Euler–Bernoulli Beams with symmetric cracks. Journal of Sound and Vibration 138, 115–134.〕 used the Galerkin Method, and Qian et al. (1991)〔Qian, G.L., Gu, S.N., Jiang, J.S., 1991. The dynamic behavior and crack detection of a beam with a crack. Journal of Sound and Vibration 138, 233–243〕 used a Finite Element Model to predict the behavior of a beam with an edge crack. Law and Lu (2005)〔Law, S., Lu, Z.R., 2005. Crack identification in beam from dynamic responses. Journal of Sound and Vibration 285, 967–987.〕 used assumed modes and modeled the crack mathematically as a Dirac delta function. Wang and Qiao (2007)〔Wang, J., Qiao, P., 2007. Vibration of beams with arbitrary discontinuities and boundary conditions. Journal of Sound and Vibration 308 (1–2), 12–27.〕 approximated the modal displacements using Heaviside’s function, which meant that modal displacements were discontinuous at the crack location. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Unified Framework」の詳細全文を読む スポンサード リンク
|