Abstract
This paper addresses the free transverse vibrations of thin simply supported nonhomogeneous isotropic rectangular plates of bilinearly varying thickness with elastically restrained edges against rotation. The Gram-Schmidt process has been used to generate two-dimensional boundary characteristic orthogonal polynomials, which have been used in the Rayleigh-Ritz method to study the problem. The lowest three frequencies have been computed for various values of nonhomogeneous parameters, thickness parameters, aspect ratio and flexibility parameters. A comparison of the results with those available in the literature has been made. Three-dimensional mode shapes for the specified plate have been presented.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Ashour AS (2004), “Vibration of Variable Thickness Plates with Edges Elastically Restrained Against Translation and Rotation,” Thin-Walled Structures, 42: 1–24.
Cheung YK and Zhou D (1999a), “Eigenfrequencies of Tapered Rectangular Plates with Line Supports,” International Journal of Solids and Structures, 36: 143–166.
Cheung YK and Zhou D (1999b), “The Free Vibrations of Tapered Rectangular Plates Using a New Set of Beam Functions with the Rayleigh-Ritz Method,” Journal of Sound and Vibration, 223(5): 703–722.
Cheung YK and Zhou D (2000), “Vibrations of Rectangular Plates with Elastic Intermediate Linesupports and Edge Constraints,” Thin-walled Structures, 37: 305–331.
Cheung YK and Zhou D (2003), “Vibrations of Tapered Mindlin Plates in Terms of Static Timoshenko Beam Functions,” Journal of Sound and Vibration, 260: 693–709.
Grossi RO and Bhat RB (1995), “Natural Frequencies of Edge Restrained Tapered Rectangular Plates,” Journal of Sound and Vibration, 185: 335–343.
Huang M, Ma XQ, Sakiyama T, Matsuda H and Morita C (2007), “Free Vibration Analysis of Rectangular Plates with Variable Thickness and Point Support,” Journal of Sound and Vibration, 300(3–5): 435–452.
Hung KC, Lim MK and Liew KM (1993), “Boundary Beam Characteristics Orthogonal Polynomials in Energy Approach for Vibration of Symmetric Laminates-II: Elastically Restrained Boundaries,” Composite Structures, 26: 185–209.
Kobayashi H and Sonoda K (1991), “Vibration and Buckling of Tapered Rectangular Plates with Two Opposite Edges Simply Supported and the Other Two Edges Elastically Restrained Against Rotation,” Journal of Sound and Vibration, 146: 323–337.
Kumar Y and Lal R (2011a), “Buckling and Vibration of Orthotropic Nonhomogeneous Rectangular Plates with Bilinear Thickness Variation,” Journal of Applied Mechanics, ASME, 78: 1–11.
Kumar Y and Lal R (2011b), “Vibrations of Nonhomogeneous Orthotropic Rectangular Plates with Bilinear Thickness Variation Resting on Winkler Foundation,” Meccanica, DOI: 10.1007/s11012-011-9459-4.
Lal R and Kumar Y (2011a), “Rayleigh-Ritz Method in the Study of Transverse Vibration of Nonhomogeneous Orthotropic Rectangular Plates of Uniform Thickness Resting on Winkler Foundation,” Advances in Vibration Engineering, 10: 207–220.
Lal R and Kumar Y (2011b), “Boundary Characteristic Orthogonal Polynomials in the Study of Transverse Vibrations of Nonhomogeneous Rectangular Plates with Bilinear Thickness Variation,” Shock and Vibration, DOI:10.3233/SAV-2011-0635.
Lal R and Kumar Y (2011c), “Characteristic Orthogonal Polynomials in the Study of Transverse Vibrations of Nonhomogeneous Rectangular Orthotropic Plates of Bilinearly Varying Thickness,” Meccanica, DOI:10.1007/s11012-011-9430-4.
Lal R, Kumar Y and Gupta US (2010), “Transverse Vibrations of Nonhomogeneous Rectangular Plates of Uniform Thickness Using Boundary Characteristic Orthogonal Polynomials,” International Journal of Applied Mathematics and Mechanics, 6(14): 93–109.
Laura PAA and Grossi R (1979), “Transverse Vibrations of Rectangular Plates with Thickness Varying in Two Directions and with Edges Elastically Restrained Against Rotation,” Journal of Sound and Vibration, 63(4): 499–505.
Leissa AW (1969), Vibration of Plates, NASA SP-160 Washington, D.C. U.S. Government Printing Office.
Leissa AW (1978), “Recent Research in Plate Vibrations 1973–1976: Complicating Effects,” The Shock and Vibration Digest, 10(12): 21–35.
Leissa AW (1981), “Plate Vibration Research, 1976–1980: Complicating Effects,” The Shock and Vibration Digest, 13(10): 19–36.
Leissa AW (1987a), “Recent Studies in Plate Vibrations, 1981–1985 Part I: Classical Theory,” The Shock and Vibration Digest, 19(2): 11–18.
Leissa AW (1987b), “Recent Studies in Plate Vibrations, 1981–1985 Part II: Complicating Effects,” The Shock and Vibration Digest, 19(3): 10–24.
Li WL (2004), “Vibration Analysis of Rectangular Plates with General Elastic Boundary Supports,” Journal of Sound and Vibration, 273: 619–635.
Li WL, Zhang XF, Du JT and Liu ZG (2009), “An Exact Series Solution for the Transverse Vibration of Rectangular Plates with General Elastic Boundary Supports,” Journal of Sound and Vibration, 321: 254–269.
Malekzadeh P and Shahpari SA (2005), “Free Vibration Analysis of Variable Thickness Thin and Moderately Thick Plates with Elastically Restrained Edges by DQM,” Thin Walled Structures, 43(7): 1037–1050.
Sakiyama T and Huang M (1998), “Free Vibration Analysis of Rectangular Plates with Variable Thickness,” Journal of Sound and Vibration, 216(3): 379–397.
Singh B and Chakraverty S (1994), “Flexural Vibration of Skew Plates Using Boundary Characteristic Orthogonal Polynomials in Two Variables,” Journal of Sound and Vibration, 173(2): 157–178.
Singh B and Saxena V (1996), “Transverse Vibration of a Rectangular Plate with Bidirectional Thickness Variation,” Journal of Sound and Vibration, 198(1): 51–65.
Zhang X and Li WL (2009), “Vibrations of Rectangular Plates with Arbitrary Non-uniform Elastic Edge Restraints,” Journal of Sound and Vibration, 326: 221–234.
Zhou D (1996), “An Approximate Solution of Eigen-Frequencies of Transverse Vibration of Rectangular Plates with Elastical Restraints,” Applied Mathematics and Mechanics, 17(5): 451–456.
Zhou D (2001), “Vibrations of Mindlin Rectangular Plates with Elastically Restrained Edges Using Static Timoshenko Beam Functions with the Rayleigh-Ritz Method,” International Journal of Solids and Structures, 38: 5565–5580.
Zhou D (2002), “Vibrations of Point-supported Rectangular Plates with Variable Thickness Using a Set of Static Tapered Beam Functions,” International Journal of Mechanical Sciences, 44: 149–164.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kumar, Y. Free vibrations of simply supported nonhomogeneous isotropic rectangular plates of bilinearly varying thickness and elastically restrained edges against rotation using Rayleigh-Ritz method. Earthq. Eng. Eng. Vib. 11, 273–280 (2012). https://doi.org/10.1007/s11803-012-0117-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11803-012-0117-1