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Effect of thermal gradient on vibration of non-homogeneous visco-elastic elliptic plate of variable thickness

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Abstract

An analysis and numerical results are presented for free transverse vibrations of non-homogeneous visco-elastic elliptic plate whose temperature and thickness spatial variations both are parabolic along a line through plate centre. The variation in density is assumed as parabolic along a line through plate centre, which raise non-homogeneity of the plate materials and make problem interesting as introducing variation in non-homogeneity of the material mass density reduce the problem practical importance in comparison to homogenous plate. For visco-elastic, the basic elastic and viscous elements are combined. We have taken Kelvin model for visco-elasticity that is the combination of the elastic and viscous elements in parallel. Here the elastic element means the spring and the viscous element means the dashpot. The governing differential equation of motion has been solved by Galerkin’s technique. Deflection, time period and logarithmic decrement corresponding to the first two modes of vibrations of a clamped non-homogeneous visco-elastic elliptic plate for various values of taper constant, thermal constants, non-homogeneity constant and aspect ratio are obtained and shown graphically.

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Abbreviations

x,y :

coordinate in the plane of plate

M x , M y :

bending moments

M yx :

twisting moments

E :

Young’s modulus

G :

shear modulus

ν :

Poisson’s ratio

h :

thickness of plate

ρ :

mass density per unit length of plate material

D 1 :

=Eh 3/12(1−ν 2), flexural rigidity

\(\widetilde{D}\) :

visco elastic operator

t :

time

η :

visco elastic constant

w(x,y,t):

transverse deflection of plate at point

W(x,y):

deflection function

T(t):

time function

a, b :

length of semi major axis and semi minor axis of plate

α, α 1, α 2 :

temperature constants

β :

taper constant

α 3 :

non-homogeneity constant

τ :

temperature excess above a given reference

Λ:

logarithmic decrement

K :

time period

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Authors and Affiliations

  1. Department of Mathematics, M.S. College, Saharanpur, Uttar Pradesh, India

    A. K. Gupta

  2. Department of Mathematics, Kisan (P.G.) College, Simbhaoli, Dist. Ghaziabad, Uttar Pradesh, India

    Lalit Kumar

Authors
  1. A. K. Gupta
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  2. Lalit Kumar
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Correspondence to A. K. Gupta.

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Gupta, A.K., Kumar, L. Effect of thermal gradient on vibration of non-homogeneous visco-elastic elliptic plate of variable thickness. Meccanica 44, 507–518 (2009). https://doi.org/10.1007/s11012-008-9184-9

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  • DOI: https://doi.org/10.1007/s11012-008-9184-9

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