Variational Formulation of Consistent: Continuous Cracked Structural Members
In Chap. 9, modeling and formulation of the governing dynamic equations for cracked Euler-Bernoulli beams in flexural vibration are studied. The results of three independent evaluations of the lowest natural frequency of lateral vibrations of beams with s
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Variational Formulation of Consistent: Continuous Cracked Structural Members
Abstract In Chap. 9, modeling and formulation of the governing dynamic equations for cracked Euler-Bernoulli beams in flexural vibration are studied. The results of three independent evaluations of the lowest natural frequency of lateral vibrations of beams with single-edge cracks and various end conditions are investigated: continuous cracked beam vibration theory, lumped crack flexibility model vibration analysis, a finite element method, and experimental results. For the case of torsional vibration of a shaft with a peripheral crack, the Hu-WashizuBarr variational formulation is adopted for obtaining the differential equation of motion, with plausible assumptions about displacements, momentum, strain and stress fields, along with the associated boundary conditions. For the experimental procedure crack propagation and formation of stationary cracks is achieved by a vibration technique. Continuous cracked beam theory agrees better with experimental results than lumped crack theory.
9.1 Variational Formulation of Cracked Beams and Rods Cracks on elastic structural elements introduce considerable local flexibility due to strain energy concentration around the crack tip [1, 2]. Great efforts have been expended to develop models capable of simulating the vibrational characteristics of cracked beams, however, the said models usually assume ideal geometry and material properties [3–20]. Variational approaches seem promising in such cases, as the one used by Chondros et al. [21–29] to develop a continuous vibration model for the lateral vibration of cracked Euler-Bernoulli beams with open single-edge or double-edge cracks, and torsional vibration of cracked rods. The Hu-Washizu-Barr
A. D. Dimarogonas et al., Analytical Methods in Rotor Dynamics, Mechanisms and Machine Science 9, DOI: 10.1007/978-94-007-5905-3_9, Ó Springer Science+Business Media Dordrecht 2013
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9 Variational Formulation of Consistent
[30, 31] variational formulation was used to develop the differential equation and boundary conditions for flexural and torsional vibration of structural members with flaws considered as one-dimensional continua. Cracks are modelled as continuous flexibilities, and displacement field, found with fracture mechanics methods is investigated. This consistent continuous cracked beam vibration theory is useful in predicting changes in flexural vibration of prismatic members with single- or double open-edge surface cracks. The variational formulation along with fracture mechanics methods, provide analytical solutions, for the development of the differential equation and the boundary conditions of the cracked beam. Analytical methods are always convenient, since they deliver accurate results, are more efficient and provide deep physical insight into the problem. In this chapter the problem of cracked beams with a single- or double-edge surface crack and various end-conditions is investigated. The cracked beam model satisfies the Euler-Bernoul
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