Fundamental Frequencies of Elliptical Plates using Static Deflections

Yıl 2022, Cilt: 33 Sayı: 1, 11569 - 11589, 01.01.2022

Murat Altekin

https://doi.org/10.18400/tekderg.817251

Öz

Fundamental frequencies of solid and annular elliptical plates were approximated using the static deflections by means of finite element method (FEM) without computing the eigenvalues. The problem was formulated within the framework of the first order shear deformation theory (FSDT). The effects of (i) the inner and outer boundary conditions, (ii) the size of the perforation, (iii) the aspect ratio, and (iv) the thickness of the plate on the performance of the method were examined via a large variety of numerical simulations. Convergence study was performed through h-refinement. Accuracy of the results was validated through comparison studies. The results reveal that the application of the Morley’s formula which does not require eigenvalue analysis approximates the fundamental frequency with finer mesh compared to the eigenvalue analysis. The method can be considered as a practical technique to approximate the fundamental frequency. However, the boundary conditions have dominant role on the accuracy of the solution particularly when the plate is perforated.

Anahtar Kelimeler

Vibration, Fundamental frequency, Static deflection, Plate, Finite Element Method

Kaynakça

• Liew, K.M., Kitipornchai, S., Lim, C.W., Free Vibration Analysis of Thick Superelliptical Plates, Journal of Engineering Mechanics, 124(2), 137-145, 1998. https://doi.org/10.1061/(ASCE)0733-9399(1998)124:2(137)
• Han, J.-B., Liew, K.M., Axisymmetric Free Vibration of Thick Annular Plates, International Journal of Mechanical Sciences, 41(9), 1089–1109, 1999. https://doi.org/10.1016/S0020-7403(98)00057-5
• Chakraverty, S., Bhat, R. B., Stiharu, I., Free Vibration of Annular Elliptic Plates Using Boundary Characteristic Orthogonal Polynomials as Shape Functions in the Rayleigh-Ritz Method, Journal of Sound and Vibration 241(3), 524-539, 2001. https://doi.org/10.1006/jsvi.2000.3243
• Zhong, S., Jin, G., Ye, T., Zhang, J., Xue, Y., Chen, M., Isogeometric Vibration Analysis of Multi-directional Functionally Gradient Circular, Elliptical and Sector Plates with Variable Thickness, Composite Structures, 250, 112470, 2020. https://doi.org/10.1016/j.compstruct.2020.112470
• Yuan, Y., Li, H., Parker, R.G., Guo, Y., Wang, D., Li, W., A Unified Semi-Analytical Method for Free In-plane and Out-of-plane Vibrations of Arbitrarily Shaped Plates with Clamped Edges, Journal of Sound and Vibration, 485, 115573, 2020. https://doi.org/10.1016/j.jsv.2020.115573
• Suganthi, G., Karthikeyan, S., Linear Modal Analysis of Ring Shaped - Elliptical Plate Using Finite Element Method, International Conference on New Trends in Mathematical Modelling with Applications, Uthangarai, India, July 29-30 2019. https://doi.org/10.1088/1742-6596/1597/1/012041
• Lim, C.W., Liew, K.M., Vibrations of Perforated Plates with Rounded Corners, Journal of Engineering Mechanics, 121(2), 203-213, 1995. https://doi.org/10.1061/(ASCE)0733-9399(1995)121:2(203)
• Karapınar, I.S., Determination of Buckling Temperatures for Elliptical FGM Plates with Variable Thicknesses, Uludağ University Journal of the Faculty of Engineering, 24(1), 75-88, 2019. https://doi.org/10.17482/uumfd.480661
• Okada, K., Maruyama, S., Nagai, K., Yamaguchi T., Analysis on Nonlinear Static Deflection and Natural Frequencies of a Thin Annular Plate with Initial Deformation, Proceedings of International Conference on Mechanical, Electrical and Medical Intelligent System 2017
• Sharma, P., Singh, R., Investigation on Modal Behaviour of FGM Annular Plate Under Hygrothermal Effect, IOP Conference Series: Materials Science and Engineering, 624, 012001
• Lin, C.-C., Tseng, C.-S., Free Vibration of Polar Orthotropic Laminated Circular and Annular Plates, Journal of Sound and Vibration, 209(5), 797-810, 1998. https://doi.org/10.1006/jsvi.1997.1293
• Zietlow, D.W., Griffin, D.C., Moorea, T.R., The Limitations on Applying Classical Thin Plate Theory to Thin Annular Plates Clamped on the Inner Boundary, AIP Advances 2, 042103, 2012. http://dx.doi.org/10.1063/1.4757928
• Bao, R.H., Chen, W.Q., Xu, R.Q., Free Vibration of Transversely Isotropic Circular Plates, AIAA Journal, 44(10), 2415-2418, 2006. https://doi.org/10.2514/1.18093
• Varghese, V., Dhakate, T., Khalsa, L., Thermoelastic Vibrations in a Thin Elliptic Annulus Plate with Elastic Supports, Theoretical and Applied Mechanics Letters, 8(1), 32-42, 2018. https://doi.org/10.1016/j.taml.2018.01.009
• Kang, S., Improved Non-dimensional Dynamic Influence Function Method for Vibration Analysis of Arbitrarily Shaped Plates with Simply Supported Edges, Advances in Mechanical Engineering, 10(2), 1-12, 2018. https://doi.org/10.1177/1687814018760082
• Liew, K.M., Wang, C.M., Xiang, Y., Kitipornchai, S., Vibration of Mindlin Plates, Elsevier Science Ltd., Oxford, UK, 1998, 131-132. https://doi.org/10.1016/B978-0-08-043341-7.X5000-6
• Leissa, A.W. Vibration of Plates, Columbus, Acoustical Society of America, 1993.
• Guenanou, A., Houmat, A., Hachemi, M., Chebout, R., Bachari, K., Free Vibration of Shear Deformable Symmetric VSCL Elliptical Plates by a Curved Rectangular P-element, Mechanics of Advanced Materials and Structures, 2020. https://doi.org/10.1080/15376494.2020.1770382
• Hasheminejad, S.M., Ghaheri, A., Exact Solution for Free Vibration Analysis of an Eccentric Elliptical Plate, Archive of Applied Mechanics, 84, 543–552, 2014. https://doi.org/10.1007/s00419-013-0816-8
• Powmya, A., Narasimhan, M.C., Free Vibration Analysis of Axisymmetric Laminated Composite Circular and Annular Plates Using Chebyshev Collocation, International Journal of Advanced Structural Engineering, 7, 129–141, 2015. https://doi.org/10.1007/s40091-015-0087-4
• Singh, A.V., A Numerical Free Vibration Analysis of Annular Elliptic Plates, Multidiscipline Modeling in Materials and Structures, 1(1), 53-62, 2005. https://doi.org/10.1163/1573611054455166
• Liew, K.M., Wang, J., Ng, T.Y., Tan, M.J., Free Vibration and Buckling Analysis of Shear-deformable Plates Based on FSDT Meshfree Method, Journal of Sound and Vibration, 276(3-5), 997-1017, 2004. https://doi.org/10.1016/j.jsv.2003.08.026
• Singh, B., Chakraverty, S., Transverse Vibration of Annular Circular and Elliptic Plates Using the Characteristic Orthogonal Polynomials in Two Dimensions, Journal of Sound and Vibration, 162(3), 537-546, 1993. https://doi.org/10.1006/jsvi.1993.1138
• Civalek, O., Gürses, M., Free Vibration of Annular Mindlin Plates with Free Inner Edge via Discrete Singular Convolution Method, The Arabian Journal for Science and Engineering, 34(1B), 81-90, 2009.
• Altekin, M., Free Transverse Vibration of Shear Deformable Super-Elliptical Plates, Wind and Structures, 24(4), 307-331, 2017. https://doi.org/10.12989/was.2017.24.4.307
• Merneedi, A., Nalluri, M.R., Vissakodeti, V.S.R., Free Vibration Analysis of an Elliptical Plate with Cut-out, Journal of Vibroengineering, 19(4), 2341-2353, 2017. https://doi.org/10.21595/jve.2016.17575
• Blevins, R.D. Formulas for Dynamics, Acoustics and Vibration, Wiley, United Kingdom, 2016. https://doi.org/10.1002/9781119038122
• Weiss, G., Approximated Fundamental Frequency for Thin Circular Plates Clamped or Pinned at the Edge, ASME International Mechanical Engineering Congress and Exposition (IMECE2007), Seattle, USA, November 11-15, 2007. https://doi.org/10.1115/IMECE2007-41018
• Jaroszewicz, J., Misiukiewicz, M., Puchalski, W., Limitations in Application of Basic Frequency Simplest Lower Estimators in Investigation of Natural Vibrations of Circular Plates with Variable Thickness and Clamped Edges, Journal of Theoretical and Applied Mechanics, 46(1), 109-121, 2008. [online] Available at: https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.608.4235&rep=rep1&type=pdf
• Jaroszewicz, J., Radziszewski, L., The General Formula for Calculation of Fundamental Frequency of Axisymmetric Vibrations of Circular Plates with Linearly Variable Thickness, Technical Sciences, 19(4), 401—410, 2016.[online] Available at: http://www.uwm.edu.pl/wnt/technicalsc/tech_19_4/jaroszew.pdf
• Altekin, M., Approximate Fundamental Frequency of a Circular Mindlin Plate by Morley’s Formula, International Conference on Numerical Analysis and Applied Mathematics (ICNAAM), Rhodes, Greece, Sept. 13-18, 2018. https://doi.org/10.1063/1.5114529
• Altekin, M., Approximate Fundamental Frequency of an Annular Plate by Using Static Deflections, Ulusal Mekanik Kongresi, Niğde, Turkey, September 02-06, 2019.
• Dhakate, T., Varghese, V., Khalsa, L., Integral Transform Approach for Solving Dynamic Thermal Vibrations in the Elliptical Disk, Journal of Thermal Stresses, 40(9), 1093-1110, 2017. https://doi.org/10.1080/01495739.2017.1285215
• Irie, T., Yamada, G., Aomura, S., Natural Frequencies of Mindlin Circular Plates, Journal of Applied Mechanics 47(3),652-655, 1980. https://doi.org/10.1115/1.3153748
• Chakraverty, S. Vibration of Plates, CRC Press, Boca Raton, USA, 2009.
• Demirhan, P.A., Taskin, V., Static Analysis of Simply Supported Functionally Graded Sandwich Plates by Using Four Variable Plate Theory, Teknik Dergi, 30(2), ‏ 8987-9007, 2019. https://dx.doi.org/10.18400/tekderg.396672
• Altunsaray, E., Bayer, I., Buckling Analysis of Symmetrically Laminated Rectangular Thin Plates under Biaxial Compression, Teknik Dergi, https://doi.org/10.18400/tekderg.606620
• Hughes, T.J.R., Taylor, R.L., Kanoknukulchai, W. A Simple and Efficient Finite Element for Plate Bending, International Journal for Numerical Methods in Engineering, 11(10), 1529–1543, 1977. https://doi.org/10.1002/nme.1620111005
• Krishnamoorthy, C.S. Finite Element Analysis: Theory and Programming (Second Edition), Tata McGraw-Hill Publishing Company Limited, 1994.
• Szilard, R. Theories and Applications of Plate Analysis, John Wiley & Sons Inc., Hoboken, USA, 2004.
• Han, J.-B., Liew, K.M., Analysis of Annular Reissner/Mindlin Plates Using Differential Quadrature Method, International Journal of Mechanical Sciences. 40(7), 651–661, 1998. https://doi.org/10.1016/S0020-7403(97)00087-8
• Golub, G.H., Van der Vorst, H.A., Eigenvalue Computation in the 20th Century, Journal of Computational and Applied Mathematics. 123, 35–65, 2000. https://doi.org/10.1016/S0377-0427(00)00413-1
• Civalek, Ö., Baltacıoglu, A.K., Free Vibration Analysis of Laminated and FGM Composite Annular Sector Plates, Composites Part B, 157, 182–194, 2019. https:// doi.org/10.1016/j.compositesb.2018.08.101
• Thai, H.-T., Kim, S.-E., Levy-Type Solution for Free Vibration Analysis of Orthotropic Plates Based on Two Variable Refined Plate Theory, Applied Mathematical Modelling, 36, 3870–3882, 2012. https://doi.org/10.1016/j.apm.2011.11.003
• Liu, W.H., Yeh, F.H., Non-linear Vibrations of Initially Imperfect, Orthotropic, Moderately Thick Plates with Edge Restraints, Journal of Sound and Vibration. 165(1), 101–122, 1993. https://doi.org/10.1006/jsvi.1993.1245
• Kolakowski, Z., Jankowski, J., Some Inconsistencies in the Nonlinear Buckling Plate Theories—FSDT, S-FSDT, HSDT, Materials, 14, 2154, 2021, 2154. https://doi.org/10.3390/ma14092154
• Thai, H.-T., Vo, T.P., A New Sinusoidal Shear Deformation Theory for Bending, Buckling, and Vibration of Functionally Graded Plates, Applied Mathematical Modelling, 37, 3269–3281, 2013. http://dx.doi.org/10.1016/j.apm.2012.08.008
• Szekrenyes, A., Analysis of Classical and First-Order Shear Deformable Cracked Orthotropic Plates, Journal of Composite Materials, 48(12), 1441–1457, 2014. https://doi.org/10.1177/0021998313487756
• Aksoylar, C., Omurtag, M.H., Mixed Finite Element Analysis of Composite Plates Under Blast Loading (Turkish), Teknik Dergi, 22(109), 5689-5711, 2011.
• Park, M., Choi, D.-H., A Simplified First-Order Shear Deformation Theory for Bending, Buckling and Free Vibration Analyses of Isotropic Plates on Elastic Foundations, KSCE Journal of Civil Engineering 22(4), 1235-1249, 2018. https://doi.org/10.1007/s12205-017-1517-6
• Oktem, A.S., Chaudhuri, R.A., Levy Type Analysis of Cross-Ply Plates Based on Higher-Order Theory, Composite Structures, 78, 243–253, 2007. https://doi.org/10.1016/j.compstruct.2005.09.012
• Zuo, H., Yang, Z., Chen, X., Xie, Y., Miao, H., Analysis of Laminated Composite Plates Using Wavelet Finite Element Method and Higher-Order Plate Theory, Composite Structures, 131, 248–258, 2015. http://dx.doi.org/10.1016/j.compstruct.2015.04.064
• Singh, S.K., Singh, I.V., Mishra, B.K., Bhardwaj, G., Singh, S.K., Analysis of Cracked Plate Using Higher-Order Shear Deformation Theory: Asymptotic Crack-tip Fields and XIGA Implementation, Computer Methods in Applied Mechanics and Engineering, 336, 594–639, 2018. https://doi.org/10.1016/j.cma.2018.03.009
• A. Bhar, A., Phoenix, S.S., Satsangi, S.K., Finite Element Analysis of Laminated Composite Stiffened Plates Using FSDT and HSDT: A Comparative Perspective, Composite Structures, 92, 312–321, 2010. https://doi.org/10.1016/j.compstruct.2009.08.002
• Zhou, D., Lo, S.H., Cheung, Y.K., Au, F.T.K., 3-D Vibration Analysis of Generalized Super Elliptical Plates Using Chebyshev-Ritz Method, International Journal of Solids Structures, 41(16-17), 4697-4712, 2004. https://doi.org/10.1016/j.ijsolstr.2004.02.045
• Nallim, L.G., Grossi, R.O., Natural Frequencies of Symmetrically Laminated Elliptical and Circular Plates, International Journal of Mechanical Sciences 50(7), 1153–1167, 2008. https://doi.org/10.1016/j.ijmecsci.2008.04.005
• Wang, Y., Ding, H., Xu, R., Three-Dimensional Analytical Solutions for the Axisymmetric Bending of Functionally Graded Annular Plates, Applied Mathematical Modelling, 40, 5393–5420, 2016. http://dx.doi.org/10.1016/j.apm.2015.11.051
• Yükseler, R.F., Exact Nonlocal Solutions of Circular Nanoplates Subjected to Uniformly Distributed Loads and Nonlocal Concentrated Forces, Journal of the Brazilian Society of Mechanical Sciences and Engineering, 42(1), 61, 2020. https://doi.org/10.1007/s40430-019-2144-6
• Altekin, M., Bending of Super-Elliptical Mindlin Plates by Finite Element Method, Teknik Dergi, 29(4), 8469-8496, 2018. https://doi.org/10.18400/tekderg.332384
• Civalek, Ö., Kiracioglu, O., Free Vibration Analysis of Timoshenko Beams by DSC Method, International Journal for Numerical Methods in Biomedical Engineering, 26(12), 1890-1898, 2010. https://doi.org/10.1002/cnm.1279