Evaluation of cantilever plates in the spinning situation: time histories and modal characteristics

Document Type : Full Length Article

Authors

1 PhD, Department of Ocean Engineering, Amirkabir University of Technology, Tehran, Iran

2 Professor, Department of Mechanical Engineering, University of Maryland at Baltimore County, USA & Department of Mechanical Engineering, Amirkabir University of Technology, Tehran, Iran

3 PhD Student, Department of Mechanical Engineering, Amirkabir University of Technology, Tehran, Iran

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

10.22064/tava.2020.121868.1155

Abstract

A study on the dynamics of cantilever orthotropic plates under spinning conditions is presented in this article. The governing equations of motions are containing the centrifugal and Coriolis effects. Two approximation methods, the extended Galerkin method, and extended Kantorovich method, are utilized for the investigation of the mathematical model. The verification of the obtained results is conducted by comparing two methods that show good agreement. This
investigation is concentrated on the time histories and the natural frequencies of the system. First, using time responses, the effects of different types and numbers of admissible functions used in the approximate solution are discussed.
Next, the results are obtained to explore the impact of dimensionless parameters like material, hub radius ratio, stagger angle, etc. on the modal characteristics of the spinning structures. The results of the simulations exhibit the importance of the proper choice of both type and number for trial functions. Furthermore, the selection of orthogonal functions can be vital to guarantee the convergence speed of an approximate solution. Further discussion on the modal characteristic reveals that in different stiffness ratios of the plate, the centrifugal stiffening rate caused by spinning motion is affected by rotational speed. Moreover, this stiffening rate is depended on setting angle and hub radius ratio. Finally, the last part of the paper is devoted to the forced  response analysis of the rotating plate

Highlights

  • Approximate solutions are presented for the evaluation of the rectangular cantilever plates under spinning conditions.
  •  The effects of types and numbers of the admissible functions as well as their orthogonal conditions on the time responses are investigated.
  •  The Campbell diagram of the system is plotted.
  •  Parametric studies are conducted to obtain the effect of different parameters on the modal characteristics

Keywords

Main Subjects


[1] M.A. Dokainish, S. Rawtani, Vibration analysis of rotating cantilever plates, International Journal for Numerical Methods in Engineering, 3 (1971) 233-248.
[2] J.T.-S. Wang, D. Shaw, O. Mahrenholtz, Vibration of rotating rectangular plates, Journal of Sound and vibration, 112 (1987) 455-468.
[3] J.S. Rao, K. Gupta, Free vibrations of rotating small aspect ratio pretwisted blades, Mechanism and Machine Theory, 22 (1987) 159-167.
[4] J. Sun, L. Kari, I.L. Arteaga, A dynamic rotating blade model at an arbitrary stagger angle based on classical plate theory and the Hamilton's principle, Journal of Sound and Vibration, 332 (2013) 1355-1371.
[5] S.K. Sinha, K.E. Turner, Natural frequencies of a pre-twisted blade in a centrifugal force field, Journal of Sound and Vibration, 330 (2011) 2655-2681.
[6] H.H. Yoo, S.K. Kim, Free vibration analysis of rotating cantilever plates, AIAA journal, 40 (2002) 2188-2196.
[7] H.H. Yoo, S.K. Kim, D.J. Inman, Modal analysis of rotating composite cantilever plates, Journal of sound and vibration, 258 (2002) 233-246.
[8] L. Li, D.G. Zhang, Free vibration analysis of rotating functionally graded rectangular plates, Composite Structures, 136 (2016) 493-504.
[9] S.K. Sinha, R.P. Zylka, Vibration analysis of composite airfoil blade using orthotropic thin shell bending theory, International Journal of Mechanical Sciences, 121 (2017) 90-105.
[10] M. Yao, Y. Niu, Y. Hao, Nonlinear dynamic responses of rotating pretwisted cylindrical shells, Nonlinear Dynamics, 95 (2019) 151-174.
[11] Y. Chen, D. Zhang, L. Li, Dynamics analysis of a rotating plate with a setting angle by using the absolute nodal coordinate formulation, European Journal of Mechanics-A/Solids, 74 (2019) 257-271.
[12] X.J. Gu, Y.X. Hao, W. Zhang, L.T. Liu, J. Chen, Free vibration of rotating cantilever pre-twisted panel with initial exponential function type geometric imperfection, Applied Mathematical Modelling, 68 (2019) 327-352.
[13] J. Fang, H. Wang, X. Zhang, On size-dependent dynamic behavior of rotating functionally graded Kirchhoff microplates, International Journal of Mechanical Sciences, 152 (2019) 34-50.
[14] H. Rostami, A.R. Ranji, F. Bakhtiari-Nejad, Free in-plane vibration analysis of rotating rectangular orthotropic cantilever plates, International Journal of Mechanical Sciences, 115 (2016) 438-456.
[15] H. Rostami, A.R. Ranji, F. Bakhtiari-Nejad, Vibration characteristics of rotating orthotropic cantilever plates using analytical approaches: a comprehensive parametric study, Archive of Applied Mechanics, 88 (2018) 481-502.
[16] H. Rostami, F. Bakhtiari-Nejad, A.R. Ranji, Vibration of the rotating rectangular orthotropic Mindlin plates with an arbitrary stagger angle, Journal of Vibration and Control, 25 (2019) 1194-1209.
[17] R. Xiang, Z.-Z. Pan, H. Ouyang, L.-W. Zhang, A study of the vibration and lay-up optimization of rotating cross-ply laminated nanocomposite blades, Composite Structures, 235 (2020) 111775.
[18] A.R. Ranji, H.R. Hoseynabadi, A semi-analytical technique for bending analysis of cylindrical panels with general loading and boundary conditions, Journal of mechanical science and technology, 26 (2012) 1711-1718.
[19] A.R. Ranji, H.R. Hoseynabadi, A semi-analytical solution for forced vibrations response of rectangular orthotropic plates with various boundary conditions, Journal of Mechanical Science and Technology, 24 (2010) 357-364.
[20] B. Tian, Y. Zhong, R. Li, Analytic bending solutions of rectangular cantilever thin plates, Archives of Civil and Mechanical Engineering, 11 (2011) 1043-1052.
[21] X. Wang, Y.L. Wang, R.B. Chen, Static and free vibrational analysis of rectangular plates by the differential quadrature element method, Communications in Numerical Methods in Engineering, 14 (1998) 1133-1141.
[22] L. Xiao-song, Y. Wen-bo, Solution of bending of cantilever rectangular plates under uniform surface-load by the method of two-direction trigonometric series, Applied Mathematics and Mechanics, 6 (1985) 789-799.