Determination of Economical and Stable Rotating Tapered Sandwich Beam Experiencing Parametric Vibration and Temperature Gradient

  • Pusparaj DashEmail author
  • Dipesh Kumar Nayak
Original Contribution


In this research article, the optimum profile has been generated for a tapered rotating sandwich beam, subjected to parametric vibration at steady-state temperature. For this, two groups of constant-width elastic layers are considered, each having three different profiles as linearly tapered, parabolic and one with constant thickness. The groups are designed in such a way that they are of same weight and length, but differ with respect to other dimensions. The different possible forms of tapered sandwich beams have been considered. Out of 21 possible tapered sandwich beams of uniform kind, the optimum one has been selected. Thereafter, considering the optimum one, the study of static and dynamic stability with respect to various non-dimensional parameters has been done with constant temperature gradient for uniform beam and variable temperature gradient for tapered beam. For this, the sequential steps followed here are Hamilton’s principle, Galerkin’s method and Hsu’s method. The different equivalent elastic profiles selected have been described in ‘Appendix’.


Viscoelastic core Parametric instability Constant temperature gradient Variable temperature gradient Equal weight Rotational parameter Optimum tapered rotating sandwich beam profile 

List of symbols

\(A_{i,x} (i = 1,2,3)\)

Cross-sectional area of ith layer at ‘x’, i = 1 for upper layer

\(A_{i} (i = 1,2,3)\)

Cross-sectional area at \(x = 0\)


Beam width

\(E_{i,x} (i = 1,2,3)\)

Young’s modulus of ith layer at ‘x

\(E(\xi )\)

Variation of Young’s modulus of beam


\({{E_{3,x} } \mathord{\left/ {\vphantom {{E_{3,x} } {E_{1,x} }}} \right. \kern-0pt} {E_{1,x} }}\)


\(g(1 + j\eta )\), complex shear parameter


Shear parameter


Hub radius




Shear strain of core

\(\left( {h_{i} } \right)_{x} \left( {i = 1,2,3} \right)\)

ith layer’s thickness at ‘x


\({{\left( {h_{3} } \right)_{x} } \mathord{\left/ {\vphantom {{\left( {h_{3} } \right)_{x} } {\left( {h_{1} } \right)}}} \right. \kern-0pt} {\left( {h_{1} } \right)}}_{x}\)


\({{\left( {h_{2} } \right)_{x} } \mathord{\left/ {\vphantom {{\left( {h_{2} } \right)_{x} } {\left( {h_{1} } \right)}}} \right. \kern-0pt} {\left( {h_{1} } \right)}}_{x}\)

\(I_{i,x} \left( {i = 1,2,3} \right)\)

Second moment of inertia about relevant axis at ‘x


Beam length


\(l/(h_{1} )_{0}\)


Non-dimensional mass per unit length of beam




Non-dimensional time

\(U_{i} (x,t)\left( {i = 1,3} \right)\)

Axial displacement of the ith layer of beam


\({{\partial U_{i} } \mathord{\left/ {\vphantom {{\partial U_{i} } {\partial x}}} \right. \kern-0pt} {\partial x}} \left( {i = 1,3} \right)\)

\(u_{1}^{{\prime }}\)

\({{\partial u_{1} } \mathord{\left/ {\vphantom {{\partial u_{1} } {\partial x}}} \right. \kern-0pt} {\partial x}}\)

\(\overline{{u_{1} }}^{{\prime \prime }}\)

\(\partial^{2} \overline{{u_{1} }} {{} \mathord{\left/ {\vphantom {{} {\partial \overline{x}^{2} }}} \right. \kern-0pt} {\partial \overline{x}^{2} }}\)


Lateral deflection of beam at ‘x’ and time ‘t

\(w^{{\prime \prime }}\)

\({{\partial^{2} w} \mathord{\left/ {\vphantom {{\partial^{2} w} {\partial x^{2} }}} \right. \kern-0pt} {\partial x^{2} }}\)


\({{\partial^{2} \overline{w} } \mathord{\left/ {\vphantom {{\partial^{2} \overline{w} } {\partial \overline{t}^{2} }}} \right. \kern-0pt} {\partial \overline{t}^{2} }}\)

\(\overline{w}^{{\prime \prime }}\)

\({{\partial^{2} \overline{w} } \mathord{\left/ {\vphantom {{\partial^{2} \overline{w} } {\partial \overline{x}^{2} }}} \right. \kern-0pt} {\partial \overline{x}^{2} }}\)


Density of ith layer

\(\overline{\omega }\)

Non-dimensional forcing frequency


Reference temperature


Coefficient of thermal expansion of beam material

\(T\left( \xi \right)\)

Distribution of Young’s modulus

\(\lambda_{0} ,\lambda_{1}\)

Rotation parameter


\(\frac{{E_{1,x} A_{1,x} }}{{E_{3,x} A_{3,x} }}\)


  1. 1.
    R.C. Kar, T. Sujata, Parametric instability of a non-uniform beam with thermal gradient resting on a Pasternak foundation. Comput. Struct. 29(4), 591–599 (1988)CrossRefGoogle Scholar
  2. 2.
    R.C. Kar, T. Sujata, Parametric instability of Timoshenko beam with thermal gradient resting on a variable Pasternak foundation. Comput. Struct. 36(4), 659–665 (1990)CrossRefGoogle Scholar
  3. 3.
    R.C. Kar, T. Sujata, Dynamic stability of a rotating beam with various boundary conditions. Comput. Struct. 40(3), 753–773 (1991)CrossRefGoogle Scholar
  4. 4.
    K. Ray, R.C. Kar, Parametric instability of a sandwich beam under various boundary conditions. Comput. Struct. 55(5), 857–870 (1995)CrossRefGoogle Scholar
  5. 5.
    P.R. Dash, B.B. Maharathi, R. Mallick, B.B. Pani, K. Ray, Parametric instability of an asymmetric, rotating sandwich beam. J. Aerosp. Sci. Technol. 60(4), 292 (2008)Google Scholar
  6. 6.
    R. Ghosh, P. Dash, S. Dharmavaram, V.V.S. Pavankumar, K. Ray, Parametric instability of a tapered, asymmetric, sandwich beam under various boundary conditions. Adv. Vib. Eng. 8(2), 71–89 (2008)Google Scholar
  7. 7.
    B. Nayak, S.K. Dwivedy, K.S.R.K. Murthy, Dynamic stability of a rotating sandwich beam with magnetorheological elastomer core. Eur. J. Mech. A. Solids 47, 143–155 (2014)CrossRefGoogle Scholar
  8. 8.
    M. Pradhan, P.R. Dash, Stability of an asymmetric tapered sandwich beam resting on a variable Pasternak foundation subjected to a pulsating axial load with thermal gradient. Compos. Struct. 140, 816–834 (2016)CrossRefGoogle Scholar
  9. 9.
    M. Pradhan, M.K. Mishra, P.R. Dash, Stability analysis of an asymmetric tapered sandwich beam with thermal gradient. Proc. Eng. 144, 908–916 (2016)CrossRefGoogle Scholar
  10. 10.
    M. Pradhan, P.R. Dash, P.K. Pradhan, Static and dynamic stability analysis of an asymmetric sandwich beam resting on a variable Pasternak foundation subjected to thermal gradient. Meccanica 51(3), 725–739 (2016)MathSciNetCrossRefGoogle Scholar
  11. 11.
    R. Parida, P. Dash, Dynamic stability analysis of a circularly tapered rotating beam subjected to axial pulsating load and thermal gradient under various boundary conditions. Int. J. Acoust. Vib. 21(2), 139–144 (2016)Google Scholar
  12. 12.
    E.M. Kerwin Jr., Damping of flexural waves by a constrained viscoelastic layer. J. Acoust. Soc. Am. 31(7), 952–962 (1959)CrossRefGoogle Scholar
  13. 13.
    H. Saito, K. Otomi, Parametric response of viscoelastically supported beams. J. Sound Vib. 63(2), 169–178 (1979)CrossRefGoogle Scholar
  14. 14.
    H. Leipholz, Stability theory, 2nd edn. (Wiley, Chichestar, 1987)CrossRefGoogle Scholar

Copyright information

© The Institution of Engineers (India) 2018

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringVeer Surendra Sai University of TechnologySambalpurIndia

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