Unified Framework
Unified framework is a general formulation which yields nth - 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). 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.
Structural health monitoring (SHM) is a rapidly expanding field both in academia and research. Most of the literature on SHM 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. They allow understanding of the physics behind the problem, which helps in the explanation of experimental readings, and 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), who modeled the rectangular edge defect as a spring, by Ostachowicz and Krawczuk (1991), who modeled the damage as an elastic hinge and by Thompson (1949), 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), who modeled the damage as a zone with reduced Young’s modulus and by Ballo (1999), who modeled it as spring with nonlinear stiffness. Krawczuk (2002) 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), 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) used the Rayleigh–Ritz method, Shen and Pierre (1990) used the Galerkin Method, and Qian et al. (1991) used a finite element method to predict the behavior of a beam with an edge crack. Law and Lu (2005) used assumed modes and modeled the crack mathematically as a Dirac delta function. Wang and Qiao (2007) approximated the modal displacements using Heaviside’s function, which meant that modal displacements were discontinuous at the crack location.
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