PSI - Issue 68

Gomez-Mancilla Julio C. et al. / Procedia Structural Integrity 68 (2025) 318–324 Gomez-Mancilla J / Structural Integrity Procedia 00 (2025) 000–000

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modes. Since 2014, the author has advised several Ph.D. theses on frequency split. Adequate, fast, efficient modeling and SFBI in an automated program capture the problem’s essence and allow quick applications. 2. Damaged Beam Model In 1-D beams, only the bending flexural modes split when a) in integral pristine structures, notch irregularities exist, and b) when damage occurs. Nor axial nor torsion modes split. This vibrating phenomenon has gone unnoticed or ignored. Fig. 1a shows beam finite elements with six degrees of freedom (dof) per node. In a crack plane, the weakest principal plane is along the ξ -direction, and orthogonal to it, the η -direction is the stiffer η-x plane. Fig. 1b illustrates six experiments focusing on the first flexural natural frequency whose juxtaposed frequency spectrums show a progressively larger distance between split frequencies as the crack deepens. Resorting to the J-Integral, J(A) , Rice (1961), Strain Energy Release Rate (SERR), the strain energy density, Papadopolous (2008), and to First Castigliano Theorem approaches allow computing a flexibility matrix for the cracked element, [ I g ]. Darpe et al. (2004) applied SERR to a rotor, circular cross-section Timoshenko beam, and developed a comprehensive analysis. Some mathematical expressions implemented in an automated code are presented, and shear factors for squared and orthogonal rectangular cross-sections are tested. They can be programmed in such a code. The current work scope comprehends only circular cross-section area as of typical rotors; & is the strain energy due to the crack presence, and J(A) function for plain stress considers and adds contributions from the six generalized forces and is expressed by, ! = ∫ ∫ ( ) % $ $ # " , (1) ( ) = &%' ! ( [(∑ )* + *,& ) - + (∑ ))* + *,& ) - + (∑ )))* + *,& ) - ] (2) ' , '' , ''' are the Stress Intensity Factors (SIF) for the open mode, sliding mode, and shearing mode, respectively, and = (1 + ) , where for each degree of freedom, six additional displacements ( & due to the crack pop-up, evaluated using the following expression, Considering the influence of the three SIF , such additional displacements are, where to save space just the axial additional displacement due to the crack, &! , is given, ( & = )* ! )+ " , = 1 6 , (3) &! = &%' ! ( 5 & .& + ( - − + ) .- + ( / + 0 ) ./ : (4) The method implements Eqs. (1-4) and many more. The analyses can be extended to other beam geometric cross sections; modifying the mathematical functions for finite geometric boundaries F(η,ξ) in ' , '' , ''' is required. The previous is the principal, though not unique, numerical adaptation to widen the application scope, that is, application to other beam cross-sectional areas such as ellipsoidal, squared, rectangular, and different beam cross-sections. Since such expressions are eighteen, due to space limitations, they are not presented. 2.1. Validation of bending mode frequencies split The numerical model implemented in a Matlab code is compared versus test data published in the literature for two crack depths, a c / D = 0, 0.26, 0.39, see Dirr et al. (1988), where the author added the integral case frequencies, and the ignored, missing ω ղ . Case details of Table 1: circular beam, slenderness L/D =22.5, Free-free boundary conditions, E =208 GPa , γ =7,800 Kg/m 3 , ν =0.3, L =0.90m, D =0.040m, crack location along beam span, L c /L =0.60; The ω ղ1 , ω ղ2 , ω ղ3 , were ignored in Dirr et al. (1988) but here are added. It shows a pretty accurate model; notice minimum discrepancy values (err), less than 1%.