Free vibration and aeroelastic analyses of rectangular cantilever plates including correlation with experiment

Document Type : Full Length Article

Authors

1 Amirkabir University

2 Department of Mechanical Engineering, Amirkabir University of Technology, Tehran, Iran.

3 Department of Ocean Engineering, Amirkabir University of Technology, Tehran, Iran.

4 Department of Mechanical Engineering, Duke University Durham, North Carolina, USA.

10.22064/tava.2021.121930.1157

Abstract

In the present study, the free vibration and aeroelastic problems of rectangular cantilever plates with varying aspect ratio have been investigated. The classical plate theories based on the Kirchhoff hypothesis have been adopted to simulate the structural response of the plate. The Peter’s theory is selected to model the aerodynamic pressure on the plate due to the incompressible air flow. To discretize the partial deferential equations of the system, the ayleigh-Ritz method has been applied and by using Lagrange equations, the mass, damping, and stiffness matrices have been derived. Various numbers of mode shapes are used to show the convergence of the response of the system . The theoretical results including the natural frequencies and flutter speed have been evaluated by using the experimental data obtained from the ground vibration experiment carried out at Duke University. It has been shown that for
a relatively low aspect ratio rectangular cantilever plate, using some techniques in Rayleigh–Ritz method leads to an improvement of the results for both the natural frequencies and flutter speed. This technique ends up having two sets of decoupled equations and consequently, the number of equations that have to be solved simultaneously is divided by two. This could lead to a reduction of computational time significantly
.

Highlights

  • The free vibrations and aeroelastic analyses of a rectangular cantilever plate is presented
  • The classical plate theory based on the Kirchhoff hypothesis is the structural model.
  • Peters’ aerodynamic model is used to calculate the aerodynamic load.
  • Rayleigh-Ritz method has been adopted for the discretization of the problem.
  • Rayleigh-Ritz method improved the results for the natural frequencies and flutter speed.
  • The study includes a comparison with experimental data.

Keywords

Main Subjects


[1] A.W. Leissa, Vibration of plates, Scientific and Technical Information Division, National Aeronautics and …, 1969.
[2] H.-S. Shen, Y. Chen, J. Yang, Bending and vibration characteristics of a strengthened plate under various boundary conditions, Engineering structures, 25 (2003) 1157-1168.
[3] J. Seok, H.F. Tiersten, H.A. Scarton, Free vibrations of rectangular cantilever plates. Part 1: out-of-plane motion, Journal of sound and vibration, 271 (2004) 131-146.
[4] J. Seok, H.F. Tiersten, H.A. Scarton, Free vibrations of rectangular cantilever plates. Part 2: in-plane motion, Journal of sound and vibration, 271 (2004) 147-158.
[5] X. Wang, S. Xu, Free vibration analysis of beams and rectangular plates with free edges by the discrete singular convolution, Journal of Sound and Vibration, 329 (2010) 1780-1792.
[6] Q. Zhu, X. Wang, Free vibration analysis of thin isotropic and anisotropic rectangular plates by the discrete singular convolution algorithm, International Journal for Numerical Methods in Engineering, 86 (2011) 782-800.
[7] E. Carrera, F.A. Fazzolari, L. Demasi, Vibration analysis of anisotropic simply supported plates by using variable kinematic and Rayleigh-Ritz method, Journal of vibration and acoustics, 133 (2011).
[8] S.A. Eftekhari, A.A. Jafari, A mixed method for free and forced vibration of rectangular plates, Applied Mathematical Modelling, 36 (2012) 2814-2831.
[9] S.A. Eftekhari, A coupled ritz-finite element method for free vibration of rectangular thin and thick plates with general boundary conditions, Steel and Composite Structures, 28 (2018) 655-670.
[10] R. Li, P. Wang, Z. Yang, J. Yang, L. Tong, On new analytic free vibration solutions of rectangular thin cantilever plates in the symplectic space, Applied Mathematical Modelling, 53 (2018) 310-318.
[11] Y. Xing, Q. Sun, B.o. Liu, Z. Wang, The overall assessment of closed-form solution methods for free vibrations of rectangular thin plates, International Journal of Mechanical Sciences, 140 (2018) 455-470.
[12] M. Eisenberger, A. Deutsch, Solution of thin rectangular plate vibrations for all combinations of boundary conditions, Journal of Sound and Vibration, 452 (2019) 1-12.
[13] M. Goland, The flutter of a uniform cantilever wing, Journal of Applied Mechanics-Transactions of the Asme, 12 (1945) A197-A208.
[14] M.J. Patil, D.H. Hodges, C.E.S. Cesnik, Characterizing the effects of geometrical nonlinearities on aeroelastic behavior of high-aspect ratio wings, in:  NASA Conference Publication, NASA, 1999, pp. 501-510.
[15] M.J. Patil, D.H. Hodges, On the importance of aerodynamic and structural geometrical nonlinearities in aeroelastic behavior of high-aspect-ratio wings, Journal of Fluids and Structures, 19 (2004) 905-915.
[16] M.J. Patil, D.H. Hodges, C.E.S. Cesnik, Limit-cycle oscillations in high-aspect-ratio wings, Journal of fluids and structures, 15 (2001) 107-132.
[17] D. Tang, E.H. Dowell, Experimental and theoretical study on aeroelastic response of high-aspect-ratio wings, AIAA journal, 39 (2001) 1430-1441.
[18] D.M. Tang, E.H. Dowell, Effects of geometric structural nonlinearity on flutter and limit cycle oscillations of high-aspect-ratio wings, Journal of fluids and structures, 19 (2004) 291-306.
[19] D. Tang, E.H. Dowell, Experimental and theoretical study of gust response for high-aspect-ratio wing, AIAA journal, 40 (2002) 419-429.
[20] A.H. Modaress-Aval, F. Bakhtiari-Nejad, E.H. Dowell, D.A. Peters, H. Shahverdi, A comparative study of nonlinear aeroelastic models for high aspect ratio wings, Journal of Fluids and Structures, 85 (2019) 249-274.
[21] W. Zhao, M.P. Païdoussis, L. Tang, M. Liu, J. Jiang, Theoretical and experimental investigations of the dynamics of cantilevered flexible plates subjected to axial flow, Journal of Sound and Vibration, 331 (2012) 575-587.
[22] S.C. Gibbs, I. Wang, E. Dowell, Theory and experiment for flutter of a rectangular plate with a fixed leading edge in three-dimensional axial flow, Journal of Fluids and Structures, 34 (2012) 68-83.
[23] M. Colera, M. Pérez-Saborid, Numerical investigation of the effects of compressibility on the flutter of a cantilevered plate in an inviscid, subsonic, open flow, Journal of Sound and Vibration, 423 (2018) 442-458.
[24] A.H. Modaress-Aval, F. Bakhtiari-Nejad, E.H. Dowell, H. Shahverdi, H. Rostami, D.A. Peters, Aeroelastic analysis of cantilever plates using Peters’ aerodynamic model, and the influence of choosing beam or plate theories as the structural model, Journal of Fluids and Structures, 96 (2020) 103010.
[25] S.C. Gibbs, A. Sethna, I. Wang, D. Tang, E.H. Dowell, Aeroelastic stability of a cantilevered plate in yawed subsonic flow, Journal of Fluids and Structures, 49 (2014) 450-462.
[26] S.C. Gibbs IV, I. Wang, E.H. Dowell, Stability of rectangular plates in subsonic flow with various boundary conditions, Journal of Aircraft, 52 (2015) 439-451.
[27] T. Chen, M. Xu, D. Xie, X. An, Post-flutter response of a flexible cantilever plate in low subsonic flows, International Journal of Non-Linear Mechanics, 91 (2017) 113-127.
[28] E.H. Dowell, On asymptotic approximations to beam modal functions, Journal of Applied Mechanics, 51 (1984) 439.
[29] D.A. Peters, M.c.A. Hsieh, A. Torrero, A State‐Space Airloads Theory for Flexible Airfoils, Journal of the American Helicopter Society, 52 (2007) 329-342.