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Vibration of Nonuniform Beams Under Moving Point Loads: An Approximate Analytical Solution in Time Domain

Machine Design Section, Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai 600 036, India

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C. Sujatha

Machine Design Section, Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai 600 036, India

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, and

K. Shankar

Machine Design Section, Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai 600 036, India

E-mail Address: skris@iitm.ac.in

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https://doi.org/10.1142/S0219455417500353Cited by:6 (Source: Crossref)

Abstract

The forced-free responses of nonuniform beams under moving point loads are analyzed in this paper. Simple approximate analytical formulae for the forced responses of undamped nonuniform beams, derived using the fundamental mode by the Rayleigh–Ritz (R–R) method, are presented. The responses of both simply supported and clamped–clamped beams are analyzed. The responses are also determined by the finite element method (FEM) in which nonuniform elements are used for fast convergence. It is found that the present method yields results that are very close to those obtained by the FEM. As this method does not require time integration, it is faster and computationally more efficient. Though the single-mode analysis of forced vibration of uniform beams under moving loads has been done by several researchers, its application to nonuniform beams has not been reported.

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References

1. Fryba L. , Vibration of Solids and Structures Under Moving Loads (Thomas Telford, Prague, Czech Republic, 1999). Crossref, Google Scholar
2. Yang B. and Noh K. , Exact transient vibration of nonuniform bars, shafts and strings governed by wave equations, Int. J. Struct. Stabil. Dyn. 12 (4) (2012). Link, Web of Science, Google Scholar
3. R. H. Jessen, Optimizing the dynamic shape of a vibrating beam, M. S. thesis, University of Wyoming, Wyoming (1963). Google Scholar
4. Laura P. A. A., Greco B. V. D., Utjes J. C. and Carnicer R. , Numerical experiments on free and forced vibrations of beams of non-unform cross-section, J. Sound Vib. 120 (3) (1988) 587–596. Crossref, Web of Science, Google Scholar
5. Geannakakes G. N. , Vibration analysis of arbitrarily shaped plates using beam characteristic orthogonal polynomials in the semi-analytical finite strip method, J. Sound Vib. 137 (2) (1990) 283–303. Crossref, Web of Science, Google Scholar
6. Auciello A. M. , On the transverse vibration of nonuniform beams with axial loads and elastically restrained ends, Int. J. Mech. Sci. 43 (2001) 193–208. Crossref, Web of Science, Google Scholar
7. Chehil D. S. and Jateganonkar R. , Determination of natural frequencies of a beam with varying section properties, J. Sound Vib. 115 (3) (1987) 423–436. Crossref, Web of Science, Google Scholar
8. Jateganonkar R. and Chehil D. S. , Natural frequencies of a beam with varying section properties, J. Sound Vib. 133 (2) (1989) 303–322. Crossref, Web of Science, Google Scholar
9. Aristizabal-Ochoa J. D. , Statics, stability and vibration of nonprismatic beams and columns, J. Sound Vib. 162 (3) (1993) 441–455. Crossref, Web of Science, Google Scholar
10. Qiao H., Li Q. S. and Li G. Q. , Vibratory characteristics of flexural nonuniform Euler-Bernoulli beams carrying an arbitrary number of spring-mass systems, Int. J. Mech. Sci. 44 (2002) 725–743. Crossref, Web of Science, Google Scholar
11. Martinez-Castro A. E., Museros P. and Castillo-Linares A. , Semi-analytical solution in the time domain for nonuniform multi-span Bernoulli–Euler beams traversed by moving loads, J. Sound Vib. 294 (2006) 278–297. Crossref, Web of Science, Google Scholar
12. Tufekci E. and Yigit O. O. , In-plane vibration of circular arches with varying cross-sections, Int. J. Struct. Stabil. Dyn. 13 (1) (2013). Link, Web of Science, Google Scholar
13. Hou Y. C., Tseng C. H. and Ling S. F. , A new high-order nonuniform Timoshenko beam finite element on variable two-parameter foundations for vibration analysis, J. Sound Vib. 191 (1) (1996) 91–106. Crossref, Web of Science, Google Scholar
14. Gutierrez R. H. and Laura P. A. A. , Vibrations of a beam of nonuniform cross-section traversed by a time varying concentrated force, J. Sound Vib. 207 (3) (1997) 419–425. Crossref, Web of Science, Google Scholar
15. Lenci S., Clementi F. and Mazzilli C. E. N. , Simple formulas for the natural frequencies of nonuniform cables and beams, Int. J. Mech. Sci. 7 (2013) 155–163. Crossref, Web of Science, Google Scholar
16. Mehmet C. E., Aydogdu M. and Taskin V. , Vibration of variable cross-section beam, Mech. Res. Commun. 34 (2007) 78–84. Crossref, Web of Science, Google Scholar
17. Zheng D. Y., Cheung Y. K., Au F. T. K. and Cheng Y. S. , Vibration of multi span nonuniform beams under moving loads by using modified beam vibration functions, J. Sound Vib. 212 (3) (1998) 455–467. Crossref, Web of Science, Google Scholar
18. Cheung Y. K., Au F. T. K., Zheng D. Y. and Cheng Y. S. , Vibration of multi-span nonuniform bridges under moving vehicles and trains by using modified beam vibration functions, J. Sound Vib. 228 (3) (1999) 611–628. Crossref, Web of Science, Google Scholar
19. Dugush Y. A. and Eisenberger M. , Vibration of nonuniform continuous beams under moving loads, J. Sound Vib. 254 (5) (2002) 911–926. Crossref, Web of Science, Google Scholar
20. Omolofe B., Oni S. T. and Tolorunshagba J. M. , On the transverse motions of nonprismatic deep beam under the action of variable magnitude moving loads, Lat. Am. J. Solids Struct. 6 (2009) 153–167. Web of Science, Google Scholar
21. Datta A. K. and Sil S. N. , An analysis of free undamped vibration of beams of varying cross-section, Comput. Struct. 59 (3) (1996) 479–483. Crossref, Web of Science, Google Scholar
22. Oni S. T. and Awodola T. O. , Dynamic response under a moving load of an elastically supported nonprismatic Bernoulli–Euler beam on variable elastic foundation, Lat. Am. J. Solids Struct. 7 (2010) 3–20. Crossref, Web of Science, Google Scholar
23. Ahmadi M. and Nikkhoo A. , Utilization of characteristic polynomials in vibration analysis of nonuniform beams under a moving mass excitation, Appl. Math. Model. Available at http://dx.doi.org/10.1016/j.apm.2013.10.011. Google Scholar
24. Sudheesh Kumar C. P., Sujatha C. and Shankar K. , Vibration of simply supported beams under a single moving load: A detailed study of cancellation phenomenon, Int. J. Mech. Sci. 99 (2015) 40–47. Crossref, Web of Science, Google Scholar
25. Biggs J. M. , Introduction to Structural Dynamics (McGraw-Hill, New York, 1964). Google Scholar
26. Pesterev A. V., Yang B., Bergman L. A. and Tan C. A. , Revisiting the moving force problem, J. Sound Vib. 261 (2003) 75–91. Crossref, Web of Science, Google Scholar
27. Yang Y. B., Yau J. D. and Hsu L. C. , Vibration of simple beams due to trains moving at high speeds, Eng. Struct. 19 (1997) 936–944. Crossref, Web of Science, Google Scholar
28. Puand J. and Liu P. , Numerical calculation of dynamic response for multi-span nonuniform beam subjected to moving mass with friction, Eng. 2 (2010) 367–377. Crossref, Google Scholar