The Variational Formulation of a Rod in Torsional Vibration for Crack Identification
In Chap. 10 the Hu-Washizu-Barr variational formulation is used for the development of the differential equation and boundary conditions for a cracked rod. Based on the general variational principle and independent assumptions about displacement, momentum
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The Variational Formulation of a Rod in Torsional Vibration for Crack Identification
Abstract In Chap. 10 the Hu-Washizu-Barr variational formulation is used for the development of the differential equation and boundary conditions for a cracked rod. Based on the general variational principle and independent assumptions about displacement, momentum, strain and stress fields of the cracked rod with one or more pairs of transverse symmetrically disposed open edge cracks along its length, the equations of motion in torsional vibration were derived. Crack is introduced as a stress disturbance function, and stress field is determined by fracture mechanics methods. Strain energy density theory has been used for an accurate evaluation of the stress disturbance function. The strain energy density criterion is based on local density of the energy field in the crack tip region, and no special assumptions on the direction in which the energy released by the separating crack surfaces is required.
10.1 Dynamic Behaviour of Cracked Shafts The assessment of the state of damage of a structural system depends on various factors, among which the identification of existing flaws, their location, type and severity and on damage tolerance. The latter is a measure of the capability of a damaged material or damaged structure to sustain load and/or maintain functional capability. Modern structures, especially in turbomachinery (Fig. 10.1), are designed on the basis of a damage tolerance philosophy, which allows for the presence of sub-critical cracks not growing to critical length between periodic inspections. The damage tolerance concept provides quantitative guidance for the balancing of cost of repair or replacement of damaged components against the possibility that continued service would lead to a catastrophic failure [1, 2]. Despite elements of uncertainty concerning environmental effects such as corrosion, moisture, and temperature effects, a predictive methodology taking into
A. D. Dimarogonas et al., Analytical Methods in Rotor Dynamics, Mechanisms and Machine Science 9, DOI: 10.1007/978-94-007-5905-3_10, Ó Springer Science+Business Media Dordrecht 2013
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The Variational Formulation of a Rod in Torsional Vibration
Fig. 10.1 Steam turbine stepped rotor disassembled for scheduled inspection and maintenance
account crack growth mechanisms is valuable in the evaluation procedures. Although, computational technology has provided vast possibilities, the predictive failure analyses tools available in the literature are impractical to implement with complex structural problems. Thus, the extension of current damage tolerance methods to complex structures is not easy [3, 4]. The development of crack detection methods in rotating shafts was initiated in about 1970 [5–8]. Theory and extensive laboratory and field experiments were used to develop methodologies for crack detection based on the second harmonic and the half critical speed sub-harmonic. Further, it was reported that on-line electronic instrument for monitoring
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