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標題: Identification of multiple cracks in a beam by using local adaptive differential quadrature method
作者: 吳俊育
Jiun-Yu Wu
關鍵字: 多裂縫尤拉樑;轉動彈簧;轉換矩陣;區域微分值積法;有限單元法;牛頓法。;Multi-cracked Euler-Bernoulli beams;Rotational spring model;Transfer matrix;Local adaptive differential quadrature method;Finite element method;Newton-Raphson iteration method.
引用: 1. T. G. Chondros, and A. D. Dimarogonas, 'Identification of cracks in welded joints of complex structures', J. Sound Vibr. 1980; 69(4):531-538. 2 . A. D. Dimarogonas, S. A. Paipetis, and T. G. Chondros, 'Analytical Methods in Rotor Dynamics', Springer. 2013; 9 Second edition. 3. S. Christides, and A. D. S. Barr, 'One-dimensional theory of cracked Bernoulli-Euler beams', Int J Mech Sci. 1984; 26(11):639-648. 4. N. Papaeconomou, and A. Dimarogonas, 'Vibration of cracked beams', Comput Mech. 1989; 5:88-94. 5. P. F. Rizo, and N. Aspragathos, ' Identification of crack location and magnitude in a in cantilever beam from the vibrating modes', J. Sound Vibr. 1990; 138(3):381-388. 6. M.-H. H. Shen, and C. Pierre, 'Natural modes of Bernoulli-Euler beams with symmetric cracks', J. Sound Vibr. 1990; 138(1):115-134. 7. R. Y. Liang, F. K. Choy, and J. Hu, 'Detection of cracks in beam structures using measurements of natural frequencies', J Franklin Inst. 1991; 328(4):505-518. 8. J. Hu, and R. Y. Liang, 'An integrated approach to detection of cracks using vibration characteristics', J Franklin Inst. 1993; 330(5):841-853. 9. Y. Narkis, 'Identification of Crack Location in Vibrating Simply Supported Beams', J. Sound Vibr. 1994; 172(4):549-558. 10. H. P. Lin, and N. C. Perkins, 'Free Vibration of Complex Cable/Mass Systems: Theory and Experiment', J Sound Vibr. 1995; 179(1):131-149. 11. H. Dado. Mohammad, 'A comprehensive crack identification algorithm for beams under different end conditions', Appl Acoust. 1997; 51(4):381-398. 12. E. I. Shifrin, and R. Ruotolo, 'Natural frequencies of a beam with an arbitrary number of cracks', J Sound Vibr. 1999; 222(3):409-423. 13. Y. S. Lee, and M. J. Chung, 'A study on crack detection using eigenfrequency test data', Comput Struct, 2000; 77:327-342. 14. H. Tada, 'The Stress Analysis of Cracks Handbook', Hellertown, Pennsylvania. Del Research Corporation. 1985. 15. H. P. Lin, S. C. Chang, and J. D. Wu, 'Beam vibrations with an arbitrary number of cracks', J Sound Vibr. 2002; 258(5):987-999. 16. D. P. Patil, and S. K. Maiti, 'Detection of multiple cracks using frequency measurements', Eng Fract Mech. 2003; 70:1553-1572. 17. D. P. Patil, and S. K. Maiti, 'Experimental verification of a method of detection of multiple cracks in beams based on frequency measurements', J Sound Vibr. 2005; 281:493-451. 18. O Sadettin, ' Analysis of free and forced vibration of a cracked cantilever beam', NDT&E International. 2007; 40:443-450. 19. H. I. Yoon, I. S. Son, and S. J. Ahn, 'free vibration analysis of euler bernoulli beam with double cracks', J Mech Sci Technol. 2007; 21:476-485. 20. J. Lee, 'Identification of multiple cracks in a beam using natural frequencies', J Sound Vibr. 2009; 320:482-490. 21. B. S. Farook, and K. Bimlesh, 'Identification of crack location and crack size in a simply supported beam by measurement of natural frequencies', J. Vib. Control. 2013; 18(2):183-190. 22. C. S. Wang, and L. T. Lee, 'Modified and simplified sectional flexibility of a crack beam', J Appl Math. 2012; 16:1-17. 23. R. Bellman, and J. Casti, 'Differential quadrature and long term integration', J Math Anal Appl. 1971; 34: 235-238. 24. R. Bellman, and B. G. Kashef, ' Differential quadrature: a technique for the rapid solution of nonlinear partial differential qquations', J Comput Phys. 1972; 10:40-52. 25. C. Shu, W. Chen, H. Xue, and H. Du, 'Numerical study of grid distribution effect on accuracy of DQ analysis of beams and plates by error estimation of derivative approximation', Int J Numer Meth Engng. 2001; 51:159-179. 26. Y. Wang, Y. B. Zhao, and G. W. Wei, 'A note on the numerical solution of high-order differential equations'. J Comput Appl Math. 2003; 159:387–398. 27. M. H. Hsu, 'Vibration analysis of edge-cracked beam on elastic foundation with axial loading using the differential quadrature method', Comput Methods Appl Mech Engrg. 2005; 194:1-17. 28. M. A. De Rosa, N. M. Auciello, and M. Lippiello, 'Dynamic stability analysis and DQM for beams with variable cross-section', Mech Res Commun. 2008; 35: 187-192. 29. M. I. Char, F. P. Chang, and B. C. Tai, 'Inverse determination of thermal conductivity by differential quadrature method', Int Commun Heat Mass. 2008; 35:113-119. 30. B. C. Tai, M. I. Chang, 'Soret and Dufour effects on free convection flow of non-Newtonian fluids along a vertical plate embedded in a porous medium with thermal radiation', Int Commun Heat Mass. 2010; 37:480-483.
Multi-cracked Euler–Bernoulli beams is discussed in this article. The crack is considered as a rotational spring model which is related to the ratio of crack depth to the beam height (a/h). By using the transfer matrix method, the theoretical solution for natural frequencies of the cracked beam is obtained. The local adaptive differential quadrature method (La-DQM) and finite element method (FEM) were applied to evaluate the natural frequencies of cracked beams. The results of natural frequencies are in excellent agreement with literature results. By using Newton-Raphson iteration method, the crack locations and crack sizes are found for identification of multiple cracks in a beam. Numerical results were compared with results of other investigators and this method was found to be more accurate.

本文研究多裂縫尤拉樑(Euler-Bernoulli beam)的振動力學分析。在樑上的裂縫以轉動彈簧模擬,此轉動彈簧是與裂縫的深度有密切的關係,藉由基本假設引導出應力強度因子。應用轉換矩陣來連結裂縫的左右兩端樑,進一步推導出自然振動頻率。另外使用區域微分值積法(local adaptive differential quadrature method)與有限單元法(Finite element method)數值方法求解自然振動頻率。為了讓本文理論方法更完備,所獲得的結果與國外文獻的結果相互比較與討論。對於預測裂縫的位置與深度上,藉由上述三種方式(理論解、區域微分值積法、有限單元法)求解4個自然振動頻率,應用牛頓法進行預測裂縫樑的位置與深度。所獲得結果是比其它的文獻更精確。
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