Structural Engineering and Mechanics

  • 제60권3호
  • /
  • Pages.455-470
  • /
  • 2016
  • /
  • 1225-4568(pISSN)
  • /
  • 1598-6217(eISSN)
테크노프레스 (Techno-Press)
권호 표지
DOI QR코드

DOI QR Code

Closed-form solutions for non-uniform axially loaded Rayleigh cantilever beams

  • Sarkar, Korak (Department of Aerospace Engineering, Indian Institute of Science) ;
  • Ganguli, Ranjan (Department of Aerospace Engineering, Indian Institute of Science) ;
  • Elishakoff, Isaac (Department of Mechanical Engineering, Florida Atlantic University)
  • 투고 : 2015.08.24
  • 심사 : 2016.09.06
  • 발행 : 2016.11.10
⟨ 이전 논문 다음 논문 ⟩

초록

In this paper, we investigate the free vibration of axially loaded non-uniform Rayleigh cantilever beams. The Rayleigh beams account for the rotary inertia effect which is ignored in Euler-Bernoulli beam theory. Using an inverse problem approach we show, that for certain polynomial variations of the mass per unit length and the flexural stiffness, there exists a fundamental closed form solution to the fourth order governing differential equation for Rayleigh beams. The derived property variation can serve as test functions for numerical methods. For the rotating beam case, the results have been compared with those derived using the Euler-Bernoulli beam theory.

키워드

참고문헌

  1. Abedinnasab, M.H., Zohoor, H. and Yoon, Y.J. (2012), "Exact formulations of non-linear planar and spatial Euler-Bernoulli beams with finite strains", P. I. Mech. Eng. C - J. Mech. Eng. Sci., 226(5), 1225-1236. https://doi.org/10.1177/0954406211420206
  2. Auciello, N. and Lippiello, M. (2013), "Vibration analysis of rotating non-uniform Rayleigh beams using "CDM" method", Global Virtual Conference.
  3. Aydin, K. (2013), "Influence of Crack and Slenderness Ratio on the Eigenfrequencies of Euler-Bernoulli and Timoshenko Beams", Mech. Adv. Mater. Struct., 20(5), 339-352. https://doi.org/10.1080/15376494.2011.627635
  4. Bagdatli, S.M. and Uslu, B. (2015), "Free vibration analysis of axially moving beam under non-ideal conditions", Struct. Eng. Mech., 54(3), 597-605. https://doi.org/10.12989/sem.2015.54.3.597
  5. Baghani, M., Mohammadi, H. and Naghdabadi, R. (2014), "An analytical solution for shape-memorypolymer Euler-Bernoulli beams under bending", Int. J. Mech. Sci., 84, 84-90. https://doi.org/10.1016/j.ijmecsci.2014.04.009
  6. Bambill, D., Rossit, C., Rossi, R., Felix, D. and Ratazzi, A. (2013), "Transverse free vibration of non uniform rotating Timoshenko beams with elastically clamped boundary conditions", Meccanica, 48(6), 1289-1311. https://doi.org/10.1007/s11012-012-9668-5
  7. Banerjee, J. R. and Jackson, D. (2013), "Free vibration of a rotating tapered Rayleigh beam: a dynamic stiffness method of solution", Comput. Struct., 124, 11-20. https://doi.org/10.1016/j.compstruc.2012.11.010
  8. Calim, F.F. (2016), "Dynamic response of curved Timoshenko beams resting on viscoelastic foundation", Struct. Eng. Mech., 59(4), 761-774. https://doi.org/10.12989/sem.2016.59.4.761
  9. Chang, J.R., Lin, W.J., Huang, C.J. and Choi, S.T. (2010), "Vibration and stability of an axially moving Rayleigh beam", Appl. Math. Model., 34(6), 1482-1497. https://doi.org/10.1016/j.apm.2009.08.022
  10. Datta, P. and Ganguli, R. (1990), "Vibration characteristics of a rotating blade with localized damage including the effects of shear deformation and rotary inertia", Comput. Struct., 36(6), 1129-1133. https://doi.org/10.1016/0045-7949(90)90221-M
  11. Ebrahimi, F. and Jafari, A. (2016), "Thermo-mechanical vibration analysis of temperature dependent porous FG beams based on Timoshenko beam theory", Struct. Eng. Mech., 59(2), 343-371. https://doi.org/10.12989/sem.2016.59.2.343
  12. Elishakoff, I. (2004), Eigenvalues Of Inhomogeneous Structures: Unusual Closed-form Solutions: CRC Press, Boca Raton, Florida, USA.
  13. Elishakoff, I. and Guede, Z. (2004), "Analytical polynomial solutions for vibrating axially graded beams", Mech. Adv. Mater. Struct., 11(6), 517-533. https://doi.org/10.1080/15376490490452669
  14. Gunda, J.B. and Ganguli, R. (2008), "Stiff-string basis functions for vibration analysis of high speed rotating beams", J. Appl. Mech., 75(2), 024502. https://doi.org/10.1115/1.2775497
  15. Ke, L.L., Yang, J., Kitipornchai, S. and Xiang, Y. (2009), "Flexural vibration and elastic buckling of a cracked Timoshenko beam made of functionally graded materials", Mech. Adv. Mater. Struct., 16(6), 488-502. https://doi.org/10.1080/15376490902781175
  16. Li, X., Tang, A. and Xi, L. (2013), "Vibration of a Rayleigh cantilever beam with axial force and tip mass", J. Constr. Steel Res., 80, 15-22. https://doi.org/10.1016/j.jcsr.2012.09.015
  17. Liu, Z., Yin, Y., Wang, F., Zhao, Y. and Cai, L. (2013), "Study on modified differential transform method for free vibration analysis of uniform Euler-Bernoulli beam", Struct. Eng. Mech., 48(5), 697-709. https://doi.org/10.12989/sem.2013.48.5.697
  18. Lou, P., Dai, G. and Zeng, Q. (2006), "Finite-element analysis for a Timoshenko beam subjected to a moving mass", P. I. Mech. Eng. C - J. Mech. Eng. Sci., 220(5), 669-678. https://doi.org/10.1243/09544062JMES119
  19. Ma'en, S.S. and Butcher, E.A. (2012), "Free vibration analysis of non-rotating and rotating Timoshenko beams with damaged boundaries using the Chebyshev collocation method", Int. J. Mech. Sci., 60(1), 1-11. https://doi.org/10.1016/j.ijmecsci.2012.03.008
  20. Maiz, S., Bambill, D.V., Rossit, C.A. and Laura, P. (2007), "Transverse vibration of Bernoulli-Euler beams carrying point masses and taking into account their rotatory inertia: exact solution", J. Sound Vib., 303(3), 895-908. https://doi.org/10.1016/j.jsv.2006.12.028
  21. Mao, Q. (2015), "AMDM for free vibration analysis of rotating tapered beams", Struct. Eng. Mech., 54(3), 419-432. https://doi.org/10.12989/sem.2015.54.3.419
  22. Pai, P.F., Qian, X. and Du, X. (2013), "Modeling and dynamic characteristics of spinning Rayleigh beams", Int. J. Mech. Sci., 68, 291-303. https://doi.org/10.1016/j.ijmecsci.2013.01.029
  23. Sarkar, K. (2012), "Closed-Form Solutions for Rotating and Non-Rotating Beams: An Inverse Problem Approach", M.Sc (Engg) Thesis.
  24. Sarkar, K. and Ganguli, R. (2013), "Closed-form solutions for non-uniform Euler-Bernoulli free-free beams", J. Sound Vib., 332(23), 6078-6092. https://doi.org/10.1016/j.jsv.2013.06.008
  25. Sarkar, K. and Ganguli, R. (2014a), "Analytical test functions for free vibration analysis of rotating nonhomogeneous Timoshenko beams", Meccanica, 49(6), 1469-1477. https://doi.org/10.1007/s11012-014-9927-8
  26. Sarkar, K. and Ganguli, R. (2014b), "Modal tailoring and closed-form solutions for rotating non-uniform Euler-Bernoulli beams", Int. J. Mech. Sci., 88, 208-220. https://doi.org/10.1016/j.ijmecsci.2014.08.003
  27. Sarkar, K. and Ganguli, R. (2014c), "Tailoring the second mode of Euler-Bernoulli beams: an analytical approach", Struct. Eng. Mech., 51(5), 773-792. https://doi.org/10.12989/sem.2014.51.5.773
  28. Shahba, A., Attarnejad, R. and Hajilar, S. (2011), "Free vibration and stability of axially functionally graded tapered Euler-Bernoulli beams", Shock Vib., 18(5), 683-696. https://doi.org/10.1155/2011/591716
  29. Stojanovic, V. and Kozic, P. (2012), "Forced transverse vibration of Rayleigh and Timoshenko double-beam system with effect of compressive axial load", Int. J. Mech. Sci., 60(1), 59-71. https://doi.org/10.1016/j.ijmecsci.2012.04.009
  30. Tang, A.Y., Li, X.F., Wu, J.X. and Lee, K. (2015), "Flapwise bending vibration of rotating tapered Rayleigh cantilever beams", J. Constr. Steel Res., 112, 1-9. https://doi.org/10.1016/j.jcsr.2015.04.010
  31. Tang, A.Y., Wu, J.X., Li, X.F. and Lee, K. (2014), "Exact frequency equations of free vibration of exponentially non-uniform functionally graded Timoshenko beams", Int. J. Mech. Sci., 89, 1-11. https://doi.org/10.1016/j.ijmecsci.2014.08.017
  32. Xi, L.Y., Li, X.F. and Tang, G.J. (2013), "Free vibration of standing and hanging gravity-loaded Rayleigh cantilevers", Int. J. Mech. Sci., 66, 233-238. https://doi.org/10.1016/j.ijmecsci.2012.11.013
  33. Yesilce, Y. (2015), "Differential transform method and numerical assembly technique for free vibration analysis of the axial-loaded Timoshenko multiple-step beam carrying a number of intermediate lumped masses and rotary inertias", Struct. Eng. Mech., 53(3), 537-573. https://doi.org/10.12989/sem.2015.53.3.537
  34. Zahrai, S.M., Mortezagholi, M.H. and Mirsalehi, M. (2016), "Effect of higher order terms of Maclaurin expansion in nonlinear analysis of the Bernoulli beam by single finite element", Struct. Eng. Mech., 58(6), 949-966. https://doi.org/10.12989/sem.2016.58.6.949

피인용 문헌

  1. Isospectrals of non-uniform Rayleigh beams with respect to their uniform counterparts vol.5, pp.2, 2018, https://doi.org/10.1098/rsos.171717
  2. Quadratic B-spline finite element method for a rotating non-uniform Rayleigh beam vol.61, pp.6, 2016, https://doi.org/10.12989/sem.2017.61.6.765
  3. A transfer matrix method for in-plane bending vibrations of tapered beams with axial force and multiple edge cracks vol.66, pp.1, 2018, https://doi.org/10.12989/sem.2018.66.1.125
  4. Free vibration analysis of cracked functionally graded non-uniform beams vol.7, pp.1, 2016, https://doi.org/10.1088/2053-1591/ab6ad1
  5. The Artificial Intelligence and Design of Multibody Systems with Predicted Dynamic Behavior vol.14, pp.None, 2016, https://doi.org/10.46300/9106.2020.14.124