Flutter analysis of rotating beams with elastic restraints

doi:10.1007/s10483-022-2850-6

Abstract

Abstract: The aeroelastic stability of rotating beams with elastic restraints is investigated. The coupled bending-torsional Euler-Bernoulli beam and Timoshenko beam models are adopted for the structural modeling. The Greenberg aerodynamic model is used to describe the unsteady aerodynamic forces. The additional centrifugal stiffness effect and elastic boundary conditions are considered in the form of potential energy. A modified Fourier series method is used to assume the displacement field function and solve the governing equation. The convergence and accuracy of the method are verified by comparison of numerical results. Then, the flutter analysis of the rotating beam structure is carried out, and the critical rotational velocity of the flutter is predicted. The results show that the elastic boundary reduces the critical flutter velocity of the rotating beam, and the elastic range of torsional spring is larger than the elastic range of linear spring.

Key words: rotating beam, modified Fourier series method, elastic restraint, flutter

2010 MSC Number:

74F10
74H15

Lüsen WANG, Zhu SU, Lifeng WANG. Flutter analysis of rotating beams with elastic restraints. Applied Mathematics and Mechanics (English Edition), 2022, 43(5): 761-776.

TrendMD

References

[1] FRIEDMANN, P. P. Rotary-wing aeroelasticity:current status and future trends. AIAA Journal, 42, 1953-1972(2004)
[2] ZHANG, P. and HUANG, S. Review of aeroelasticity for wind turbine:current status, research focus and future perspectives. Frontiers in Energy, 5, 419-434(2011)
[3] ZHU, T. L. The vibrations of pre-twisted rotating Timoshenko beams by the Rayleigh-Ritz method. Computational Mechanics, 47, 395-408(2011)
[4] RAFIEE, M., NITZSCHE, F., and LABROSSE, M. Dynamics, vibration and control of rotating composite beams and blades:a critical review. Thin-Walled Structures, 119, 795-819(2017)
[5] ADAIR, D. and JAEGER, M. A power series solution for rotating nonuniform Euler-Bernoulli cantilever beams. Journal of Vibration and Control, 24, 3855-3864(2018)
[6] LIU, G. Z., GUO, X. M., and ZHU, L. Dynamic analysis of wind turbine tower structures in complex ocean environment. Applied Mathematics and Mechanics (English Edition), 41(7), 999-1010(2020) https://doi.org/10.1007/s10483-020-2624-8
[7] RONG, Y. F., SUN, Q., and LIANG, K. Modified unified co-rotational framework with beam, shell and brick elements for geometrically nonlinear analysis. Acta Mechanica Sinica (2021)
[8] GUO, H. L., OUYANG, X., ZUR, K. K., and WU, X. T. Meshless numerical approach to flutter analysis of rotating pre-twisted nanocomposite blades subjected to supersonic airflow. Engineering Analysis with Boundary Elements, 132, 1-11(2021)
[9] MENG, H., JIN, D. Y., LI, L., and LIU, Y. Q. Analytical and numerical study on centrifugal stiffening effect for large rotating wind turbine blade based on NREL 5 MW and WindPACT 1.5 MW models. Renewable Energy, 183, 321-329(2022)
[10] YOO, H. H., RYAN, R. R., and SCOTT, R. A. Dynamics of flexible beams undergoing overall motions. Journal of Sound and Vibration, 181, 261-278(1995)
[11] YOO, H. H. and SHIN, S. H. Vibration analysis of rotating cantilever beams. Journal of Sound and Vibration, 212, 807-828(1998)
[12] CHUANG, J. and YOO, H. H. Dynamic analysis of a rotating cantilever beam by using the finite element method. Journal of Sound and Vibration, 249, 147-164(2002)
[13] BANERJEE, J. R. and SOBEY, A. J. Energy expressions for rotating tapered Timoshenko beams. Journal of Sound and Vibration, 254, 818-822(2002)
[14] BANERJEE, J. R. and KENNEDY, D. Dynamic stiffness method for inplane free vibration of rotating beams including Coriolis effects. Journal of Sound and Vibration, 333, 7299-7312(2014)
[15] BANERJEE, J. R. and JACKSON, D. R. Free vibration of a rotating tapered Rayleigh beam:a dynamic stiffness method of solution. Computers and Structure, 124, 11-20(2013)
[16] YANG, X. D., LI, Z., ZHANG, W., YANG, T. Z., and LIM, C. W. On the gyroscopic and centrifugal effects in the free vibration of rotating beams. Journal of Vibration and Control, 25, 219-227(2019)
[17] YANG, X. D., WANG, S. W., ZHANG, W., YANG, T. Z., and LIM, C. W. Model formulation and modal analysis of a rotating elastic uniform Timoshenko beam with setting angle. European Journal of Mechanics-A/Solids, 72, 209-222(2018)
[18] LIM, I. and LEE, I. Aeroelastic analysis of bearingless rotors with a composite flexbeam. Composite Structures, 88, 570-578(2009)
[19] CULP, J. D. and MURTHY, V. R. Free vibration analysis of branched blades by the integrating matrix method. Journal of Sound and Vibration, 155, 303-315(1992)
[20] HODGES, D. H. A theoretical technique for analyzing aeroelastic stability of bearingless rotors. AIAA Journal, 17, 400-407(1979)
[21] LIN, B. C., QIN, Y., LI, Y. H., and YANG, J. The deflection of rotating composite tapered beams with an elastically restrained root in hygrothermal environment. Zeitschrift für Naturforschung A, 74, 849-859(2019)
[22] LIN, S. PD control of a rotating smart beam with an elastic root. Journal of Sound and Vibration, 312, 109-124(2008)
[23] CHEN, Q. and DU, J. A Fourier series solution for the transverse vibration of rotating beams with elastic boundary supports. Applied Acoustics, 155, 1-15(2019)
[24] ZENG, J., MA, H., YU, K., XU, Z. T., and WEN, B. C. Coupled flapwise-chordwise-axialtorsional dynamic responses of rotating pre-twisted and inclined cantilever beams subject to the base excitation. Applied Mathematics and Mechanics (English Edition), 40(8), 1053-1082(2019) https://doi.org/10.1007/s10483-019-2506-6
[25] HODGES, D. H. and ORMISTON, R. A. Stability of elastic bending and torsion of uniform cantilevered rotor blades in hover. 14th Structures, Structural Dynamics, and Materials Conference, AIAA, Williamsburg (1974)
[26] HANSEN, M. H. Aeroelastic instability problems for wind turbines. Wind Energy, 10, 551-577(2007)
[27] LEE, J., YEE, K., YOO, S. J., LEE, I., SHIN, S. J., and KIM, D. K. Aeroelastic analysis of a hingeless rotor using a dynamic wake model. Journal of Aircraft, 48, 1817-1822(2011)
[28] WANG, D., CHEN, Y., WIERCIGROCH, M., and CAO, Q. J. Bifurcation and dynamic response analysis of rotating blade excited by upstream vortices. Applied Mathematics and Mechanics (English Edition), 37(9), 1251-1274(2016) https://doi.org/10.1007/s10483-016-2128-6
[29] SICARD, J. and SIROHI, J. Aeroelastic stability of a flexible ribbon rotor blade. Journal of Fluids and Structures, 67, 106-123(2016)
[30] AMOOZGAR, M. R., SHAHVERDI, H., and NOBARI, A. S. Aeroelastic stability of hingeless rotor blades in hover using fully intrinsic equations. AIAA Journal, 55, 2450-2460(2017)
[31] SAYED, M., KLEIN, L., LUTZ, T., and KRAMER, E. Bifurcation and dynamic behavior analysis of a rotating cantilever plate in subsonic airflow. Renewable Energy, 140, 304-318(2019)
[32] MA, L., YAO, M. H., ZHANG, W., and CAO, D. X. Bifurcation and dynamic behavior analysis of a rotating cantilever plate in subsonic airflow. Applied Mathematics and Mechanics (English Edition), 41(12), 1861-1880(2020) https://doi.org/10.1007/s10483-020-2668-8
[33] SABALE, A. K. and GOPAL, N. K. V. Nonlinear aeroelastic analysis of large wind turbines under turbulent wind conditions. AIAA Journal, 57, 4416-4432(2019)
[34] WANG, S. W., HAN, J. L., CHEN, Q. L., YUN, H. W., and CHEN, X. M. New method for analyzing the flutter stability of hingeless blades with advanced geometric configureurations in hovering. International Journal of Aerospace Engineering, 2020, 1-16(2020)
[35] REN, J., HUANG, H., WANG, D. X., DONG, X., and CAO, B. C. An efficient coupled-mode flutter analysis method for turbomachinery. Aerospace Science and Technology, 106, 106215(2020)
[36] LI, W. L. Free vibrations of beams with general boundary conditions. Journal of Sound and Vibration, 237, 709-725(2000)
[37] JIN, G. Y., SU, Z., SHI, S. X., YE, T. G., and GAO, S. Y. Three-dimensional exact solution for the free vibration of arbitrarily thick functionally graded rectangular plates with general boundary conditions. Composite Structures, 108, 565-577(2014)
[38] SU, Z., JIN, G. Y., SHI, S. X., YE, T. G., and JIA, X. Z. A unified solution for vibration analysis of functionally graded cylindrical, conical shells and annular plates with general boundary conditions. International Journal of Mechanical Sciences, 80, 62-80(2014)
[39] WANG, Q. S., SHI, D. Y., LIANG, Q., and SHI, X. J. A unified solution for vibration analysis of functionally graded circular, annular and sector plates with general boundary conditions. Composites Part B:Engineering, 88, 264-294(2016)
[40] ZHONG, R., WANG, Q. S., TANG, J. Y., SHUAI, C. J., and QIN, B. Vibration analysis of functionally graded carbon nanotube reinforced composites (FG-CNTRC) circular, annular and sector plates. Composites Part B:Engineering, 194, 49-67(2018)
[41] YANG, X. D., WANG, S. W., ZHANG, W., QIN, Z. H., and YANG, T. Z. Dynamic analysis of a rotating tapered cantilever Timoshenko beam based on the power series method. Applied Mathematics and Mechanics (English Edition), 38(10), 1425-1438(2017) https://doi.org/10.1007/s10483-017-2249-6
[42] GREENBERG, J. M. Airfoil in sinusoidal motion in a pulsating stream. National Advisory Committee for Aeronautics, Technical Notes, No. 1326(1947)
[43] HODGES, D. H. and RUTKOWSKI, M. J. Free-vibration analysis of rotating beams by a variableorder finite-element method. AIAA Journal, 19, 1459-1466(1981)
[44] DU, H., LIM, M. K., and LIEW, K. M. A power series solution for vibration of a rotating Timoshenko beam. Journal of Sound and Vibration, 175, 505-523(1994)