DOI QR코드

DOI QR Code

Quadratic B-spline finite element method for a rotating non-uniform Rayleigh beam

  • Received : 2016.06.16
  • Accepted : 2016.11.23
  • Published : 2017.03.25

Abstract

The quadratic B-spline finite element method yields mass and stiffness matrices which are half the size of matrices obtained by the conventional finite element method. We solve the free vibration problem of a rotating Rayleigh beam using the quadratic B-spline finite element method. Rayleigh beam theory includes the rotary inertia effects in addition to the Euler-Bernoulli theory assumptions and presents a good mathematical model for rotating beams. Galerkin's approach is used to obtain the weak form which yields a system of symmetric matrices. Results obtained for the natural frequencies at different rotating speeds show an accurate match with the published results. A comparison with Euler-Bernoulli beam is done to decipher the variations in higher modes of the Rayleigh beam due to the slenderness ratio. The results are obtained for different values of non-uniform parameter ($\bar{n}$).

Keywords

References

  1. Auciello, N.M. (2013), "Dynamics analysis of rotating tapered beams using two general approaches", Proceedings of the Electronic International Interdisciplinary Conference, Zillina, September.
  2. Avcar, M. (2015), "Effects of rotary inertia, shear deformation and non-homogeneity on frequencies of beam", Struct. Eng. Mech., 55, 871-884. https://doi.org/10.12989/sem.2015.55.4.871
  3. Banerjee, J.R. and Jackson, D.R. (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
  4. Bauchau, O.A. and Hong, C.H. (1987), "Finite element approach to rotor blade modeling", J. Am. Helic. Soc., 32, 60-67. https://doi.org/10.4050/JAHS.32.60
  5. Bokaian, A. (1990), "Natural frequencies of beams under tensile axial loads", J. Sound Vib., 142, 481-498. https://doi.org/10.1016/0022-460X(90)90663-K
  6. Boor, C.D. (1972), "On calculating with B-splines", J. Approx. Theory, 6, 50-62. https://doi.org/10.1016/0021-9045(72)90080-9
  7. Bornemann, P.B. and Cirak, F. (2013), "A subdivision-based implementation of the hierarchical B-spline finite element method", Comput. Meth. Appl. Mech. Eng., 253, 584-598. https://doi.org/10.1016/j.cma.2012.06.023
  8. Dag, I. and Ozer, M.N. (2001), "Approximation of the RLW equation by the least square cubic B-spline finite element method" Appl. Math. Model., 25, 221-231. https://doi.org/10.1016/S0307-904X(00)00030-5
  9. Gardner, L.R.T., Gardner, G.A. and Dag, I. (1995), "A B-spline finite element method for regularized long wave equation", Commun. Numer. Meth. Eng., 11, 59-68. https://doi.org/10.1002/cnm.1640110109
  10. Giurgiutiu, V. and Stafford, R.O. (1977), "Semi-analytical methods for frequencies and mode shapes of rotor blades", Vertica, 1, 291-306.
  11. Gupta, A., Kiusalaas, J. and Saraph, M. (1991), "Cubic B-spline for finite element analysis of axisymmetric shells", Comput. Struct., 38, 463-468. https://doi.org/10.1016/0045-7949(91)90042-K
  12. Hoa, S.V. (1979), "Vibration of a rotating beam with tip mass", J. Sound Vib., 167, 369-381.
  13. Hodges, H.D. and Rutkowski, M.J. (1981), "Free-vibration analysis of rotating beams by a variable-order finite element method", AIAA J., 19, 1459-1466. https://doi.org/10.2514/3.60082
  14. Hughes, T.J.R., Cottrell, J.A. and Bazilevs, Y. (2005), "Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement", Comput. Meth. Appl. Mech. Eng., 194, 4135-4195. https://doi.org/10.1016/j.cma.2004.10.008
  15. Kagan, P., Fischer, A. and Bar-Yoseph, P.Z. (1998), "New B-spline finite element approach for geometric design and mechanical analysis", Int. J. Numer. Meth. Eng., 41, 435-458. https://doi.org/10.1002/(SICI)1097-0207(19980215)41:3<435::AID-NME292>3.0.CO;2-U
  16. Mao, Q. (2015), "AMDM for free vibration analysis of rotating tapered beams", Struct. Eng. Mech., 54, 419-432. https://doi.org/10.12989/sem.2015.54.3.419
  17. Mohammadi, R. (2014), "Sextic B-spline collocation method for solving Euler-Bernoulli beam models", Appl. Math. Comput., 241, 151-166.
  18. Mohammadnejad, M. (2015), "A new analytical approach for determination of flexural, axial and torsional natural frequencies of beams", Struct. Eng. Mech., 55, 655-674. https://doi.org/10.12989/sem.2015.55.3.655
  19. Nagaraj, V. T. and Shanthakumar, P. (1975), "Rotor blade vibration by the Galerkin finite element method", J. Sound Vib., 43, 575-577. https://doi.org/10.1016/0022-460X(75)90013-9
  20. Reddy, J.N. (2005), An Introduction to the Finite Element Method, Tata McGraw-Hill, New York, USA.
  21. Rostami, S., Shojaee, S. and Saffari, H. (2013), "An explicit time integration method for structural dynamics using cubic B-spline polynomial functions", Scientia Iranica, 20, 23-33.
  22. Sarkar, K., Ganguli, R. and Elishakoff, I. (2016), "Closed-form solutions for non-uniform axially loaded Rayleigh cantilever beams", Struct. Eng. Mech., 60, 455-470. https://doi.org/10.12989/sem.2016.60.3.455
  23. Shen, L., Liu, Z. and Wu, J.H. (2014), "B-spline finite element method based on node moving adaptive refinement strategy", Finite Elem. Anal. Des., 91, 84-94. https://doi.org/10.1016/j.finel.2014.07.007
  24. Shen, P.C. and Wang, J.G (1987), "Static analysis of cylindrical shells by using B-spline functions", Comput. Struct., 25, 809-816. https://doi.org/10.1016/0045-7949(87)90196-9
  25. Tang, A.Y., Li, X.F., Wu, J.X. and Lee, K.Y. (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
  26. Wang, G. and Wereley, N.M. (2004), "Free vibration analysis of rotating blades with uniform tapers", AIAA J., 42, 2429-2437. https://doi.org/10.2514/1.4302
  27. Zeid, I. (2007), Mastering CAD/CAM, Tata McGraw-Hill, New York, USA.

Cited by

  1. Transverse Vibration of Rotating Tapered Cantilever Beam with Hollow Circular Cross-Section vol.2018, pp.1875-9203, 2018, https://doi.org/10.1155/2018/1056397
  2. Quadratic B-spline finite element method for a rotating nonuniform Euler–Bernoulli beam pp.1550-2295, 2018, https://doi.org/10.1080/15502287.2018.1520757
  3. Finite element based stress and vibration analysis of axially functionally graded rotating beams vol.79, pp.1, 2017, https://doi.org/10.12989/sem.2021.79.1.023