MODAL ANALYSIS OF MULTISTEP TIMOSHENKO BEAM WITH A NUMBER OF CRACKS
Keywords:Timoshenko beam theory, multi-stepped beam, multi-cracked beam, natural frequencies, transfer matrix method
AbstractModal analysis of cracked multistep Timoshenko beam is accomplished by the Transfer Matrix Method (TMM) based on a closed-form solution for Timoshenko uniform beam element. Using the solution allows significantly simplifying application of the conventional TMM for multistep beam with multiple cracks. Such simplified transfer matrix method is employed for investigating effect of beam slenderness and stepped change in cross section on sensitivity of natural frequencies to cracks. It is demonstrated that the transfer matrix method based on the Timoshenko beam theory is usefully applicable for beam of arbitrary slenderness while the Euler-Bernoulli beam theory is appropriate only for slender one. Moreover, stepwise change in cross-section leads to a jump in natural frequency variation due to crack at the steps. Both the theoretical development and numerical computation accomplished for the cracked multistep beam have been validated by an experimental study
Sato H. Free vibration of beams with abrupt changes of cross-section. Journal of Sound and Vibration 89 (1983) 59-64.
Jang S. K. and Bert C. W. Free vibration of stepped beams: exact and numerical solutions. Journal of Sound and Vibration 130 (1989) 342-346.
Maurizi M.J. and Belles P.M. Natural frequencies of one-span beams with stepwise variable cross-section. Journal of Sound and Vibration 168 (1) (1993) 184-188.
Stanton S.C. and Mann B.P. On the dynamic response of beams with multiple geometric or material discontinuities. Mechanical Systems and Signal Processing 24 (2010) 1409-1419.
Mao Q. Free vibration analysis of multiple-stepped beams by using Adomian decomposition method. Mechanical and Computer Modelling 54 (2012) 756-764.
Cunha-Vaz J. and Lima-Junior J. J. Vibration analysis of Euler-Bernoulli beams in multiple steps and different shapes of cross section. Journal of Vibration and Control (2014) DOI: 10.1177/1077546314528366.
Koplow M.A., Bhattachayya A. and Mann B.P. Closed form solutions for the dynamic response of Euler-Bernoulli beams with step changes in cross section. Journal of Sound and Vibration 295 (206) 214-225.
Farghary S.H. Vibration and stability analysis of Timoshenko beams with discontinuities in cross-section. Journal of Sound and Vibration 174(5) (1994) 591-605.
Maguleswaran S. Vibration of a Euler-Bernoulli beam on elastic end supports and with up to three step changes in cross-section. Intern. J. Mech. Sciences 44 (2002) 2541-2555.
Wang X. W. and Wang Y. L. Free vibration analysis of multiple-stepped beams by the differential quadrature element method. Applied Mathematics and Computation 219 (2013) 5802-5810.
Sarigul (Aydin) A.S. and Aksu G. A finite difference method for free vibration analysis of stepped Timoshenko beams and shafts. Mechanism and Machine Theory 21 (1986) 1-12.
Subramanian G. and Balasubramanian T.S. Beneficial effects of steps on the free vibration characteristics of beams. Journal of Sound and Vibration 118(3) (1987) 555-560.
Kukla S. and Zamojska I. Frequency analysis of axially loaded stepped beams by Green’s function method. Journal of Sound and Vibration 300 (2007) 1341-1041.
Jaworski J. W. and Dowell E. H. Free vibration of a cantilevered beam with multiple steps: Comparison of several theoretical methods with experiment. Journal of Sound and Vibration 312 (2008) 713-725.
Nandwana B. P. and Maiti S. K. Detection of the location and size of a crack in stepped cantilever beams based on measurements of natural frequencies, Journal of Sound and Vibration 203(3) (1997) 435-446.
Kukla S. Free vibrations and stability of stepped columns with cracks. Journal of Sound and Vibration 319 (2009) 1301-1311.
Zhang W., Wang Z. and Ma H. Crack identification in stepped cantilever beam combining wavelet analysis with transform matrix. Acta Mechanica Solida Sinica 22(4) (2009) 360-368.
Maghsoodi A., Ghadami A. and Mirdamadi H. R. Multiple crack damage detection in multi-step beams by a novel local flexibility-based damage index. Journal of Sound and Vibration 332 (2013) 294-305.
Skrinar M (2013) Computational analysis of multi-stepped beams and beams with linearly-varying heights implementing closed-form finite element formulation for multi-cracked beam elements. International Journal of Solids and Structures 50: 2527-2541.
Li Q.S. Vibratory characteristics of multi-step beams with an arbitrary number of cracks and concentrated masses. Applied Acoustics 62 (2001) 691-706.
Attar M. A transfer matrix method for free vibration analysis and crack identification of stepped beams with multiple edge cracks and different boundary conditions. International Journal of Mechanical Sciences 57 (2012) 19-33.
Lele S.P. and Maiti S.K. Modeling of transverse vibration of short beams for crack detection and measurement of crack extension. Journal of Sound and Vibration 257(3) (2002) 559-583.
Swamidas A.S.J., Yang X. and Seshadri R. Identification of cracking in beam structures using Timoshenko and Euler formulations. ASCE J. Eng. Mech. 130 (11) (2004) 1297-1338.
Loya JA, Rubio L and Fernandez-Sàez J. Natural frequencies for bending vibrations of Timoshenko cracked beams. Journal of Sound and Vibration 290 (2006) 640-653.
Li Q.S. Vibratory characteristics of Timoshenko beams with arbitrary number of cracks. ASCE J. Eng. Mech. 129 (2009) 1355-1359.
Aydin K. Influence of Crack and Slenderness Ratio on the Eigenfrequencies of Euler-Bernoulli and Timoshenko Beams. Mechanics of Advanced Materials and Structures, 20 (5) (2012) 339-352.
Tsai T. C. and Wang Y. Z. Vibration analysis and diagnosis of a cracked shaft. Journal of Sound and Vibration 192(3) (1996) 607-620.
Vu Thi An Ninh, Luu Quynh Huong, Tran Thanh Hai, Nguyen Tien Khiem. The transfer matrix method for modal analysis of cracked multistep beam. Vietnam Journal of Science and Technology 55 (5) (2017) 598-611.
Khiem N.T. and Hung D. T. A closed form solution for free vibration of multiple cracked Timoshenko beam and application. Vietnam Journal of Mechanics 39 (4) (2017) 315-328.
Karnovsky I.A. and Lebed O.I. Formulas for Structural Dynamics: Tables, Graphs and Solutions. McGraw-Hill, Inc. NY 2000, 638p.
Chondros T. G., Dimarogonas A. D. and Yao J. Longitudinal vibration of a continous cracked bar. Engineering Fracture Mechanics 61 (1998) 593-606.
Authors who publish with Vietnam Journal of Science and Technology agree with the following terms:
- The manuscript is not under consideration for publication elsewhere. When a manuscript is accepted for publication, the author agrees to automatic transfer of the copyright to the editorial office.
- The manuscript should not be published elsewhere in any language without the consent of the copyright holders. Authors have the right to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal’s published version of their work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are encouraged to post their work online (e.g., in institutional repositories or on their websites) prior to or during the submission process, as it can lead to productive exchanges or/and greater number of citation to the to-be-published work (See The Effect of Open Access).