Semi-analytical approach for free vibration analysis of cracked beams resting on two-parameter elastic foundation with elastically restrained ends

Abstract

In present study, free vibration of cracked beams resting on two-parameter elastic foundation with elastically restrained ends is considered. Euler-Bernoulli beam hypothesis has been applied and translational and rotational elastic springs in each end considered as support. The crack is modeled as a mass-less rotational spring which divides beam into two segments. After governing the equations of motion, the differential transform method (DTM) has been served to determine dimensionless frequencies and normalized mode shapes. DTM is a semi-analytical approach based on Taylor expansion series that converts differential equations to recursive algebraic equations. The DTM results for the natural frequencies in special cases are in very good agreement with results reported by well-known references. Also, the DTM procedure yields rapid convergence beside high accuracy without any frequency missing. Comprehensive studies to analyze the effects of crack location, crack severity, parameters of elastic foundation and boundary conditions on dimensionless frequencies as well as effects of elastic boundary conditions on cracked beams mode shapes are carried out and some problems handled for first time in this paper. Since this paper deals with general problem, the derived formulation has capability for analyzing free vibration of cracked beam with every boundary condition.

Keywords

free vibration / cracked beam / elastic foundation / restrained ends / differential transform method

References

[1] AriaeiA, Ziaei-RadS, GhayourM. Vibration analysis of beams with open and breathing cracks subjected to moving masses. Journal of Sound and Vibration, 2009, 326(3-5): 709-724

[2] DimarogonasA D. Vibration of cracked structures: a state of the art review. Engineering Fracture Mechanics, 1996, 55(5): 831-857

[3] KirmsherP G. The effect of discontinuity on natural frequency of beams. Proceedings of the American Society of Testing and Materials, 1944, 44: 897-904

[4] ChristidesS, BarrA D. One dimensional theory of cracked Bernoulli-Euler beams. International Journal of Mechanical Sciences, 1984, 26(11-12): 639-648

[5] RizosP F, AspragathosN, DimarogonasA D. Identification of crack location and magnitude in a cantilever beam for the vibration mode. Journal of Sound and Vibration, 1990, 138(3): 381-388

[6] KhiemN T, LienT V. The dynamic stiffness matrix method in forced vibration analysis of multiple-cracked beam. Journal of Sound and Vibration, 2002, 254(3): 541-555

[7] HsuM H. Vibration analysis of edge-cracked beam on elastic foundation with axial loading using the differential quadrature method. Computer Methods in Applied Mechanics and Engineering, 2005, 194(1): 1-17

[8] YangJ, ChenY, XiangY, JiaX L. Free and forced vibration of cracked inhomogeneous beams under an axial force and a moving load. Journal of Sound and Vibration, 2008, 312(1-2): 166-181

[9] KhorramA, Bakhtiari-NejadF, RezaeianM. Comparison studies between two wavelet based crack detection methods of a beam subjected to a moving load. International Journal of Engineering Science, 2012, 51: 204-215

[10] ChenW Q, LuC F, BianZ G. A mixed method for bending and free vibration of beams resting on a Pasternak elastic foundation. Applied Mathematical Modelling, 2004, 28(10): 877-890

[11] CivalekO. Nonlinear analysis of thin rectangular plates on Winkler-Pasternak elastic foundations by DSC-HDQ methods. Applied Mathematical Modelling, 2007, 31(3): 606-624

[12] CivalekO, AcarM H. Discrete singular convolution method for the analysis of Mindlin plates on elastic foundations. International Journal of Pressure Vessels and Piping, 2007, 84(9): 527-535

[13] MirzabeigyA. Semi-analytical approach for free vibration analysis of variable cross-section beams resting on elastic foundation and under axial force. International Journal of Engineering, Transactions C: Aspects, 2014, 27(3): 455-463

[14] HoS H, ChenC K. Analysis of general elastically end restrained non-uniform beams using differential transform. Applied Mathematical Modelling, 1998, 22(4-5): 219-234

[15] MaoQ. Free vibration analysis of multiple-stepped beams by using Adomian decomposition method. Mathematical and Computer Modelling, 2011, 54(1-2): 756-764

[16] LeeJ. Free vibration analysis of beams with non-ideal clamped boundary conditions. Journal of Mechanical Science and Technology, 2013, 27(2): 297-303

[17] RaoL B, RaoC K. Fundamental buckling of circular plates with elastically restrained edges and resting on concentric rigid ring support. Frontiers of Mechanical Engineering, 2013, 8(3): 291-297

[18] ChenC K, HoS H. Application of differential transformation to eigenvalue problems. Applied Mathematics and Computation, 1996, 79(2-3): 173-188

[19] YalcinH S, ArikogluA, OzkolI. Free vibration analysis of circular plates by differential transformation method. Applied Mathematics and Computation, 2009, 212(2): 377-386

[20] BalkayaM, KayaM O, SaglamerA. Analysis of the vibration of an elastic beam supported on elastic soil using the differential transform method. Archive of Applied Mechanics, 2009, 79(2): 135-146

[21] DemirdagO, YesilceY. Solution of free vibration equation of elastically supported Timoshenko columns with a tip mass by differential transform method. Advances in Engineering Software, 2011, 42(10): 860-867

[22] ShariyatM, AlipourM M. Differential transform vibration and modal stress analyses of circular plates made of two-directional functionally graded materials resting on elastic foundations. Archive of Applied Mechanics, 2011, 81(9): 1289-1306

[23] MaoQ. Design of shaped piezoelectric modal sensors for cantilever beams with intermediate support by using differential transformation method. Applied Acoustics, 2012, 73(2): 144-149

[24] SuddoungK, CharoensukJ, WattanasakulpongN. Vibration response of stepped FGM beams with elastically end constraints using differential transformation method. Applied Acoustics, 2014, 77: 20-28

[25] ShahbaA, RajasekaranS. Free vibration and stability of tapered Euler–Bernoulli beams made of axially functionally graded materials. Applied Mathematical Modelling, 2012, 36(7): 3094-3111

[26] Jandaghi SemnaniS, AttarnejadR, Kazemi FirouzjaeiR. Free vibration analysis of variable thickness thin plates by two-dimensional differential transform method. Acta Mechanica, 2013, 224(8): 1643-1658

[27] WattanasakulpongN, ChaikittiratanaA. On the linear and nonlinear vibration responses of elastically end restrained beams using DTM. Mechanics Based Design of Structures and Machines, 2013,

[28] YaghoobiH, TorabiM. Analytical solution for settling of non-spherical particles in incompressible Newtonian media. Powder Technology, 2012, 221: 453-463

[29] NourazarS, MirzabeigyA. Approximate solution for nonlinear Duffing oscillator with damping effect using the modified differential transform method. Scientia Iranica, 2013, 20: 364-368

[30] CivalekO, KiraciogluO. Free vibration analysis of Timoshenko beams by DSC method. International Journal for Numerical Methods in Biomedical Engineering, 2010, 26: 1890-1898

[31] CivalekO. Application of differential quadrature (DQ) and harmonic differential quadrature (HDQ) for buckling analysis of thin isotropic plates and elastic columns. Engineering Structures, 2004, 26(2): 171-186

[32] CivalekO, UlkerM. Harmonic differential quadrature (HDQ) for axisymmetric bending analysis of thin isotropic circular plates. Structural Engineering & Mechanics, 2004, 17(1): 1-14

[33] MaoQ. Free vibration analysis of elastically connected multiple-beams by using the Adomian modified decomposition method. Journal of Sound and Vibration, 2012, 331(11): 2532-2542

[34] TorabiK, DastgerdiJ N, MarzbanS. Solution of free vibration equations of Euler-Bernoulli cracked beams by using differential transform method. Applied Mechanics and Materials, 2012, 110-116: 4532-4536

[35] TadaH, ParisP, IrwinG. The Stress Analysis of Cracks Handbook. Missouri: Research Corporation, 1985

[36] LaiH Y, HsuJ C, ChenC K. An innovative eigenvalue problem solver for free vibration of Euler-Bernoulli beam by using the Adomian decomposition method. Computers & Mathematics with Applications (Oxford, England), 2008, 56(12): 3204-3220

[37] NarkisY. Identification of crack location in vibrating simply supported beams. Journal of Sound and Vibration, 1994, 172(4): 549-558

RIGHTS & PERMISSIONS

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg