Research Papers

An Improved Fourier–Ritz Method for Analyzing In-Plane Free Vibration of Sectorial Plates

[+] Author and Article Information
Siyuan Bao

School of Civil Engineering,
University of Science and Technology of Suzhou,
Suzhou 215011, China;
School of Mechanical and
Aerospace Engineering,
Oklahoma State University,
Stillwater, OK 74078
e-mail: sy.bgwl.bao@hotmail.com

Shuodao Wang, Bo Wang

School of Mechanical and
Aerospace Engineering,
Oklahoma State University,
Stillwater, OK 74078

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received March 10, 2017; final manuscript received June 7, 2017; published online July 7, 2017. Assoc. Editor: George Kardomateas.

J. Appl. Mech 84(9), 091001 (Jul 07, 2017) (10 pages) Paper No: JAM-17-1136; doi: 10.1115/1.4037030 History: Received March 10, 2017; Revised June 07, 2017

A modified Fourier–Ritz approach is developed in this study to analyze the free in-plane vibration of orthotropic annular sector plates with general boundary conditions. In this approach, two auxiliary sine functions are added to the standard Fourier cosine series to obtain a robust function set. The introduction of a logarithmic radial variable simplifies the expressions of total energy and the Lagrangian function. The improved Fourier expansion based on the new variable eliminates all the potential discontinuities of the original displacement function and its derivatives in the entire domain and effectively improves the convergence of the results. The radial and circumferential displacements are formulated with the modified Fourier series expansion, and the arbitrary boundary conditions are simulated by the artificial boundary spring technique. The number of terms in the truncated Fourier series and the appropriate value of the boundary spring retraining stiffness are discussed. The developed Ritz procedure is used to obtain accurate solution with adequately smooth displacement field in the entire solution domain. Numerical examples involving plates with various boundary conditions demonstrate the robustness, precision, and versatility of this method. The method developed here is found to be computationally economic compared with the previous method that does not adopt the logarithmic radial variable.

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Leissa, A. W. , McGee, O. G. , and Huang, C. S. , 1993, “ Vibrations of Circular Plates Having V-Notches or Sharp Radial Cracks,” J. Sound Vib., 161(2), pp. 227–239. [CrossRef]
Huang, C. S. , Leissa, A. W. , and McGee, O. G. , 1993, “ Exact Analytical Solutions for the Vibrations of Sectorial Plates With Simply Supported Radial Edges,” ASME J. Appl. Mech., 60(2), pp. 478–483. [CrossRef]
Jomehzadeh, E. , and Saidi, A. R. , 2009, “ Analytical Solution for Free Vibration of Transversely Isotropic Sector Plates Using a Boundary Layer Function,” Thin Walled Struct., 47(1), pp. 82–88. [CrossRef]
Cheung, Y. K. , and Kwok, W. L. , 1975, “ Dynamic Analysis of Circular and Sector Thick, Layered Plates,” J. Sound Vib., 42(2), pp. 147–158. [CrossRef]
Liew, K. M. , Xiang, Y. , and Kitipornchai, S. , 1995, “ Research on Thick Plate Vibration: A Literature Survey,” J. Sound Vib., 180(1), pp. 163–176. [CrossRef]
Onoe, M. , 1956, “ Contour Vibrations of Isotropic Circular Plates,” J. Acoust. Soc. Am., 28(6), pp. 1158–1162. [CrossRef]
Holland, R. , 1966, “ Numerical Studies of Elastic-Disk Contour Modes Lacking Axial Symmetry,” J. Acoust. Soc. Am., 40(5), pp. 1051–1057. [CrossRef]
Irie, T. , Yamada, G. , and Muramoto, Y. , 1984, “ Natural Frequencies of In-Plane Vibration of Annular Plates,” J. Sound Vib., 97(1), pp. 171–175. [CrossRef]
Leung, A. Y. T. , Zhu, B. , Zheng, J. , and Yang, H. , 2004, “ Analytic Trapezoidal Fourier p-Element for Vibrating Plane Problems,” J. Sound Vib., 271(1–2), pp. 67–81. [CrossRef]
Lyon, R. H. , 1986, “ In-Plane Contribution to Structural Noise Transmission,” Noise Control Eng. J., 26(1), pp. 22–27. [CrossRef]
Farag, N. , and Pan, J. , 2003, “ Modal Characteristics of In-Plane Vibration of Circular Plates Clamped at the Outer Edge,” J. Acoust. Soc. Am., 113(4), pp. 1935–1946. [CrossRef] [PubMed]
Chen, S. S. , and Liu, T. M. , 1975, “ Extensional Vibration of Thin Plates of Various Shapes,” J. Acoust. Soc. Am., 58(4), pp. 828–831. [CrossRef]
Bashmal, S. , Bhat, R. , and Rakheja, S. , 2009, “ In-Plane Free Vibration of Circular Annular Disks,” J. Sound Vib., 322(1), pp. 216–226. [CrossRef]
Park, C. I. , 2008, “ Frequency Equation for the In-Plane Vibration of a Clamped Circular Plate,” J. Sound Vib., 313(1–2), pp. 325–333. [CrossRef]
Seok, J. , and Tiersten, H. , 2004, “ Free Vibrations of Annular Sector Cantilever Plates—Part 2: In-Plane Motion,” J. Sound Vib., 271(3), pp. 773–787. [CrossRef]
Ravari, M. K. , and Forouzan, M. , 2011, “ Frequency Equations for the In-Plane Vibration of Orthotropic Circular Annular Plate,” Arch. Appl. Mech., 81(9), pp. 1307–1322. [CrossRef]
Vladimir, N. , Hadžić, N. , Senjanović, I. , and Xing, Y. , 2014, “ Potential Theory of In-Plane Vibrations of Rectangular and Circular Plates,” Int. J. Eng. Modell., 27(3–4), pp. 69–84.
Kim, C.-B. , Cho, H. S. , and Beom, H. G. , 2012, “ Exact Solutions of In-Plane Natural Vibration of a Circular Plate With Outer Edge Restrained Elastically,” J. Sound Vib., 331(9), pp. 2173–2189. [CrossRef]
Singh, A. , and Muhammad, T. , 2004, “ Free In-Plane Vibration of Isotropic Non-Rectangular Plates,” J. Sound Vib., 273(1), pp. 219–231. [CrossRef]
Wang, Q. , Shi, D. , Liang, Q. , and Fazl e Ahad , 2016, “ A Unified Solution for Free In-Plane Vibration of Orthotropic Circular, Annular and Sector Plates With General Boundary Conditions,” Appl. Math. Modell., 40(21–22), pp. 9228–9253. [CrossRef]
Li, W. L. , 2000, “ Free Vibrations of Beams With General Boundary Conditions,” J. Sound Vib., 237(4), pp. 709–725. [CrossRef]
Li, W. L. , 2002, “ Comparison of Fourier Sine and Cosine Series Expansions for Beams With Arbitrary Boundary Conditions,” J. Sound Vib., 255(1), pp. 185–194. [CrossRef]
Shi, X. , Li, W. , and Shi, D. , 2014, “ Free In-Plane Vibrations of Annular Sector Plates With Elastic Boundary Supports,” Meetings on Acoustics Acoustical Society of America (ASA), Indianapolis, IN, Oct. 27–31.
Zhang, K. , Shi, D. , Teng, X. , Zhao, Y. , and Liang, Q. , 2015, “ A Series Solution for the In-Plane Vibration of Sector Plates With Arbitrary Inclusion Angles and Boundary Conditions,” J. Vibroeng., 17(2), pp. 870–882.
Du, J. , Li, W. L. , Jin, G. , Yang, T. , and Liu, Z. , 2007, “ An Analytical Method for the In-Plane Vibration Analysis of Rectangular Plates With Elastically Restrained Edges,” J. Sound Vib., 306(3–5), pp. 908–927. [CrossRef]
Jin, G. , Ye, T. , Chen, Y. , Su, Z. , and Yan, Y. , 2013, “ An Exact Solution for the Free Vibration Analysis of Laminated Composite Cylindrical Shells With General Elastic Boundary Conditions,” Compos. Struct., 106, pp. 114–127. [CrossRef]
Chen, Y. , Jin, G. , and Liu, Z. , 2014, “ Flexural and In-Plane Vibration Analysis of Elastically Restrained Thin Rectangular Plate With Cutout Using Chebyshev–Lagrangian Method,” Int. J. Mech. Sci., 89, pp. 264–278. [CrossRef]
Wang, Q. , Shi, D. , Liang, Q. , and Shi, X. , 2016, “ A Unified Solution for Vibration Analysis of Functionally Graded Circular, Annular and Sector Plates With General Boundary Conditions,” Composites, Part B, 88, pp. 264–294. [CrossRef]
Wang, Q. , Shi, D. , and Shi, X. , 2016, “ A Modified Solution for the Free Vibration Analysis of Moderately Thick Orthotropic Rectangular Plates With General Boundary Conditions, Internal Line Supports and Resting on Elastic Foundation,” Meccanica, 51(8), pp. 1985–2017. [CrossRef]
Yao, W. , Zhong, W. , and Lim, C. W. , 2009, Symplectic Elasticity, World Scientific, Singapore.
Kim, K. , and Yoo, C. H. , 2010, “ Analytical Solution to Flexural Responses of Annular Sector Thin-Plates,” Thin-Walled Struct., 48(12), pp. 879–887. [CrossRef]
Dozio, L. , 2010, “ Free In-Plane Vibration Analysis of Rectangular Plates With Arbitrary Elastic Boundaries,” Mech. Res. Commun., 37(7), pp. 627–635. [CrossRef]
Gorman, D. J. , 2006, “ Exact Solutions for the Free In-Plane Vibration of Rectangular Plates With Two Opposite Edges Simply Supported,” J. Sound Vib., 294(1–2), pp. 131–161. [CrossRef]
Budiansky, B. , and Hu, P. C. , 1946, “ The Lagrangian Multiplier Method of Finding Upper and Lower Limits to Critical Stresses of Clamped Plates,” NASA Langley Research Center, Hampton, VA, Technical Report No. 848. https://ntrs.nasa.gov/search.jsp?R=19960017539
Ilanko, S. , Monterrubio, L. , and Mochida, Y. , 2014, The Rayleigh-Ritz Method for Structural Analysis, Wiley, Hoboken, NJ.
Monterrubio, L. E. , and Ilanko, S. , 2015, “ Proof of Convergence for a Set of Admissible Functions for the Rayleigh–Ritz Analysis of Beams and Plates and Shells of Rectangular Planform,” Comput. Struct., 147, pp. 236–243. [CrossRef]


Grahic Jump Location
Fig. 1

An orthotropic annular sector plate with arbitrary in-plane elastic edge supports

Grahic Jump Location
Fig. 3

The first three mode shapes of the annular sector plate with different boundary conditions: CCCC—(a) first, (b) second, and (c) third, FFFF—(d) first, (e) second, and (f) third, and S2S2S2S2—(g) first, (h) second, and (i) third

Grahic Jump Location
Fig. 2

Derivation of the frequency parameters versus the elastic boundary restraint parameters



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