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Vibrations of nonhomogeneous orthotropic rectangular plates with bilinear thickness variation resting on Winkler foundation

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Abstract

Free transverse vibrations of nonhomogeneous orthotropic rectangular plates with bilinear thickness variation resting on Winkler foundation are presented here using two dimensional boundary characteristic orthogonal polynomials in the Rayleigh-Ritz method on the basis of classical plate theory. Gram-Schmidt process has been used to generate orthogonal polynomials. The nonhomogeneity of the plate is assumed to arise due to linear variations in elastic properties and density of the plate material with the in-plane coordinates. The two dimensional thickness variation is taken as the Cartesian product of linear variations along the two concurrent edges of the plate. Effect of nonhomogeneity parameters, aspect ratio and thickness variation together with foundation parameter on the natural frequencies has been illustrated for the first three modes of vibration for four different combinations of clamped, simply supported and free edges correct to four decimal places. Three dimensional mode shapes for specified plate for all the four boundary conditions have been plotted. A comparison of results in special cases with published one has been presented.

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Abbreviations

x,y,z :

Cartesian coordinate system

X,Y :

Non-dimensionalized variables

a,b :

Length and breadth of the plate

t :

Time

w(x,y,t):

Displacement at the point (x,y) at time t

\(\bar{W}(x,y)\) :

Maximum transverse displacement at the point (x,y)

E x ,E y :

Young’s moduli in proper directions

G xy :

Shear modulus

υ x ,υ y :

Poisson’s ratios in x and y directions

h(x,y):

Thickness

E 1, E 2, ρ 0,G 0 :

Material constants

α,β,γ 1,γ 2,δ 1,δ 2 :

Nonhomogeneity parameters

ψ 1,ψ 2 :

Thickness parameters

K f :

Foundation parameter

p 1,p 2,p 3,p 4 :

Non-negative integers

N :

Positive integer

V max ,T max  :

Maximum strain and kinetic energies of the plate

U f,max  :

Maximum strain energy stored in foundation

ω :

Circular frequency

j,k :

Integers used in displacement approximation

d k :

Unknowns used in displacement approximation

a jk :

Coefficients in (7)

φ k :

Orthogonal polynomials

\(\hat{\phi} _{k}\) :

Orthonormal polynomials

C:

Clamped edge

S:

Simply supported edge

F:

Free edge

\(\frac{a}{b}\) :

Aspect ratio

Ω:

Non-dimensionalized frequency parameter

δ jk :

Kronecker delta

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, 247 667, India

    Yajuvindra Kumar & R. Lal

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  1. Yajuvindra Kumar
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  2. R. Lal
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Correspondence to Yajuvindra Kumar.

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Kumar, Y., Lal, R. Vibrations of nonhomogeneous orthotropic rectangular plates with bilinear thickness variation resting on Winkler foundation. Meccanica 47, 893–915 (2012). https://doi.org/10.1007/s11012-011-9459-4

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