Skip to main content
SpringerLink
Log in

Thermal effect on vibration of non-homogenous visco-elastic rectangular plate of linearly varying thickness

  • Published:
Meccanica Aims and scope Submit manuscript

Abstract

An analysis for vibration of non-homogenous visco-elastic rectangular plate of linearly varying thickness subjected to thermal gradient has been discussed in the present investigation. 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 at different points for the first two modes of vibration are calculated for various values of thermal gradients, non homogeneity constant, taper constant and aspect ratio for non-homogenous visco-elastic rectangular plate which is clamped on two parallel edges and simply supported on remaining two edges. Comparison studies have been carried out with homogeneous visco-elastic rectangular plate to establish the accuracy and versatility.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

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 :

Flexural rigidity

\(\tilde{D}\) :

Visco elastic operator

t :

Time

η :

Visco elastic constant

w(x,y,t):

Transverse deflection of plate at point

a, b :

Length and breath of the 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

References

  1. Tomar JS, Gupta AK (1983) Thermal effect on frequencies of an orthotropic rectangular plate of linearly varying thickness. J Sound Vib 90(3):325–331

    Article  MATH  ADS  Google Scholar 

  2. Fanconneau G, Marangoni RD (1970) Effect of thermal gradient on the natural frequencies of a rectangular plate. Int J Mech Sci 12(2):113–122

    Article  Google Scholar 

  3. Rao CK, Satyanarayana B (1975) Effect of thermal gradient of frequencies of tapered rectangular plates. AIAA J 131(8):1123–1126

    Article  ADS  Google Scholar 

  4. Tomar JS, Gupta AK (1984) Vibration of an orthotropic elliptic plate of non-uniform thickness and temperature. J Sound Vibr 96(1):29–35

    Article  MATH  ADS  Google Scholar 

  5. Tomar JS, Gupta AK (1985) Thermal effect on axisymmetric vibrations of an orthotropic circular plate of parabolically varying thickness. Ind J Pure Appl Math 16(5):537–545

    MATH  Google Scholar 

  6. Bambill DV, Rossit CA, Laura PAA, Rossi RE (2000) Transverse vibrations of an orthotropic rectangular plate of linearly varying thickness and with a free edge. J Sound Vib 235(3):530–538

    Article  ADS  Google Scholar 

  7. Li SR, Zhou YH (2001) Shooting method for non linear vibration and thermal buckling of heated orthotropic circular plates. J Sound Vib 248(2):379–386

    Article  ADS  Google Scholar 

  8. Hoff NJ (1958) High temperature effect in aircraft structures. Pergamon, New York

    Google Scholar 

  9. Pronsato ME, Laura PAA, Juan A (1999) Transverse vibrations of rectangular membrane with discontinuously varying density. J Sound Vib 222(2):341–344

    Article  ADS  Google Scholar 

  10. Bhatnagar NS, Gupta AK (1988) Vibration analysis of viscoelastic circular plate subjected to thermal gradient. Model Simul Control B 15(1):17–31

    Google Scholar 

  11. Gupta AK, Khanna A (2007) Vibration of visco-elastic rectangular plate with linearly thickness variations in both directions. J Sound Vib 301(3–5):450–457

    Article  ADS  Google Scholar 

  12. Pradeep V, Ganesan N (2007) Buckling & vibration of rectangular composite visco-elastic sandwich plates under thermal loads. Compos Struct 77(4):419–429

    Article  Google Scholar 

  13. Li S, Batra RC, Ma L (2007) Vibration of thermally post-buckled orthotropic circular plate. J Therm Stresses 30(1):43–57

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, M.S. College, Saharanpur, UP, India

    A. K. Gupta

  2. Department of Mathematics, Kisan (PG) College, Simbhaoli, Ghaziabad, UP, India

    Lalit Kumar

Authors
  1. A. K. Gupta
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lalit Kumar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. K. Gupta.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gupta, A.K., Kumar, L. Thermal effect on vibration of non-homogenous visco-elastic rectangular plate of linearly varying thickness. Meccanica 43, 47–54 (2008). https://doi.org/10.1007/s11012-007-9093-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11012-007-9093-3

Keywords

Search

Navigation