×

zbMATH — the first resource for mathematics

Natural frequencies of a functionally graded cracked beam using the differential quadrature method. (English) Zbl 1422.74060
Summary: The present work is concerned with the free vibration analysis of an elastically supported cracked beam. The beam is made of a functionally graded material and rested on a Winkler-Pasternak foundation. The line spring model is employed to formulate the problem. The method of differential quadrature is applied to solve it. The obtained results agreed with the previous similar ones. Further, a parametric study is introduced to investigate the effects of the geometric and elastic characteristics of the problem on the values of natural frequencies and mode shape functions.

MSC:
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74H45 Vibrations in dynamical problems in solid mechanics
74A45 Theories of fracture and damage
74S30 Other numerical methods in solid mechanics (MSC2010)
74A40 Random materials and composite materials
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Doebling, S.W.; Farrar, C.R.; Prime, M.B., Summary review of vibration-based damage identification methods, Shock vib. digest, 30, 91-105, (1998)
[2] Dimarogonas, A.D., Vibration of cracked structures: a state of the art review, Eng. fract. mech., 55, 831-857, (1996)
[3] Li, Q.S., Vibratory characteristics of multi-step beams with an arbitrary number of cracks and concentrated masses, Appl. acoust., 62, 691-706, (2001)
[4] Shifrin, E.I.; Ruotolo, R., Natural frequencies of a beam with an arbitrary number of cracks, J. sound vib., 222, 409-423, (1999)
[5] Binici, B., Vibration of beams with multiple open cracks subjected to axial force, J. sound vib., 227, 277-295, (2005)
[6] Mengcheng, C., An approximate analysis of frequency response of a cracked hinged – hinged beam, ACTA mech. solida sinca, 7, 367-375, (1994)
[7] Yokoyama, T.; Chen, M.C., Vibration analysis of edge-cracked beams using a line-spring model, Eng. fract. mech., 59, 403-409, (1998)
[8] Zheng, D.Y.; Fan, S.C., Natural frequency changes of a cracked Timoshenko beam by modified Fourier series, J. sound vib., 297, 297-317, (2001)
[9] Matbuly, M.S.; Nassar, M., Exact series solution for elastically supported cracked beams, J. eng. appl. sci., 50, 419-431, (2003)
[10] Hsu, M.H., Vibration analysis of edge-cracked beam on elastic foundation with axial loading using the differential quadrature method, Comput. methods appl. mech. eng., 194, 1-17, (2005) · Zbl 1112.74392
[11] Alsabbagh, A.Y.; Abuzeid, O.M.; Dado, M.H., Simplified stress correction factor to study the dynamic behavior of a cracked beam, Appl. math. modell., 33, 127-139, (2009) · Zbl 1167.74572
[12] Zhu, H.; Sankar, B.V., A combined Fourier series - Galerkin method for the analysis of functionally graded beams, ASME J. appl. mech., 71, 421-424, (2004) · Zbl 1111.74747
[13] Sankar, B.V.; Tzeng, J.T., Thermal stresses in functionally graded beams, Aiaa, 40, 1228-1232, (2002)
[14] Chakraborty, A.; Gopalakrishnan, S.; Reddy, J.N., A new beam finite element for the analysis of functionally graded materials, Int. J. mech. sci., 45, 519-539, (2003) · Zbl 1035.74053
[15] Xiang, H.J.; Yang, J., Free and forced vibration of a laminated FGM Timoshenko beam of variable thickness under heat conduction, Compos. part B, 39, 292-303, (2008)
[16] Carpinteri, A.; Pugno, M.Paggi.N., An analytical approach for fracture and fatigue in functionally graded materials, Int. J. fract., 141, 535-547, (2006) · Zbl 1197.74095
[17] Upadhyay, A.K.; Simha, K.R.Y., Equivalent homogeneous variable depth beams for cracked FGM beams; compliance approach, Int. J. fract., 144, 209-213, (2007)
[18] Li, X.-F., A unified approach for analyzing static and dynamic behaviors of functionally graded Timoshenko and euler – bernoulli beams, J. sound vib., 318, 1210-1229, (2008)
[19] Jagan, U.; Chauhan, P.S.; Parameswaran, V., Energy release rate for interlaminar cracks in graded laminates, Compos. sci. technol., 68, 1480-1488, (2008)
[20] Chen, W.Q.; Jung, Jin Pyo; Lee, Kang Yong, Static and dynamic behavior of simply-supported cross-ply laminated piezoelectric cylindrical panels with imperfect bonding, Compos. struct., 74, 265-276, (2006)
[21] Dag, S.; Yildirim, B.; Sarikaya, D., Mixed-mode fracture analysis of orthotropic functionally graded materials under mechanical and thermal loads, Int. J. solids struct., 44, 7816-7840, (2007) · Zbl 1167.74546
[22] Kapuria, S.; Bhattacharyya, M.; Kumar, A.N., Bending and free vibration response of layered functionally graded beams: a theoretical model and its experimental validation, Compos. struct., 82, 390-402, (2008)
[23] Yang, J.; Chen, Y.; Xiang, Y.; Jia, X.L., Free and forced vibration of cracked inhomogeneous beams under an axial force and a moving load, J. sound vib., 312, 166-181, (2008)
[24] Nie, G.J.; Zhong, Z., Vibration analysis of functionally graded annular sectorial plates with simply supported radial edges, Compos. struct., 84, 167-176, (2008)
[25] Broek, D., Elementary engineering fracture mechanics, (1986), Matinus Nijhoff
[26] Erdogan, F.; Wu, B.H., The surface crack problem for a plate with functionally graded properties, ASME J. appl. mech., 64, 448-456, (1997) · Zbl 0900.73615
[27] Yang, J.; Chen, Y., Free vibration and buckling analysis of functionally graded beams with edge cracks, Compos. struct., 83, 48-60, (2008)
[28] Talookolaei, R.A.J.; Ahmadian, M.T., Free vibration analysis of a cross-ply laminated composite beam on Pasternak foundation, J. comput. sci., 3, 51-56, (2007)
[29] Shu, C.; Du, H., Implementation of clamped and simply supported boundary conditions in the GDQ free vibration analysis of beams and plates, Int. J. solids struct., 34, 819-835, (1997) · Zbl 0944.74645
[30] Shu, C., Differential quadrature and its application in engineering, (2000), Springer · Zbl 0944.65107
[31] Chen, C.N., Discrete element analysis methods of generic differential quadrature, Lecture notes in applied and computational mechanics, vol. 25, (2006), Springer
[32] Quan, J.R.; Chang, C.T., New insights in solving distributed system equations by the quadrature methods, Comput. chem. eng., 13, 779-788, (1989)
[33] Bert, C.W.; Malik, M., Free vibration analysis of tapered rectangular plates by differential quadrature method, a semi-analytical approach, J. sound vib., 190, 41-63, (1996)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.