Nonlinear Vibration and Instability Analysis of a PVDF Cylindrical Shell Reinforced with BNNTs Conveying Viscose Fluid Using HDQ Method

Document Type: Research Paper


1 Faculty of Mechanical Engineering, University of Kashan

2 Faculty of Mechanical Engineering, University of Kashan--- Institute of Nanoscience & Nanotechnology, University of Kashan


Using harmonic differential quadrature (HDQ) method, nonlinear vibrations and instability of a smart composite cylindrical shell made from piezoelectric polymer of polyvinylidene fluoride (PVDF) reinforced with boron nitride nanotubes (BNNTs) are investigated while clamped at both ends and subjected to combined electro-thermo-mechanical loads and conveying a viscous-fluid. The mathematical modeling of the cylindrical shell and the resulting nonlinear coupling governing equations between mechanical and electrical fields are derived using Hamilton’s principle based on the first-order shear deformation theory (FSDT) in conjunction with the Donnell's non-linear shallow shell theory. The governing equations are discretized via HDQ method, and solved to obtain the resonant frequencies and critical flow velocities associated with divergence and flutter instabilities as well as re-stabilization of the system. Results indicate that the internal moving fluid plays an important role in the instability of the cylindrical shell. Application of a smart material such as PVDF improves significantly the stability and vibration of the system.


 [1] Païdoussis M.P., Denise J.P., 1972, Flutter of thin cylindrical shells conveying fluid, Journal of Sound and Vibration 20: 9-26.

[2] Amabili M., Garziera R., 2002, Vibrations of circular cylindrical shells with nonuniform constraints, elastic bed and added mass; Part II: shells containing or immersed in axial flow, Journal of Fluids and Structures 16: 31–51.

[3] Païdoussis M.P., 2004, Fluid–Structure Interactions: Slender Structures and Axial Flow, Elsevier Academic Press, London.

[4] Kotsilkova R., 2007, Thermoset Nanocomposites for Engineering Applications, Smithers, USA.

[5] Merharihybrid L., 2002, Nanocomposites for Nanotechnology, Springer Science, New York.

[6] Yu V., Christopher T., Bowen R., 2009, Electromechanical Properties in Composites Based on Ferroelectrics, Springer-Verlag, London.

[7] Ghorbanpour Arani A., Kolahchi R., Mosallaie Barzoki A.A., Loghman, A., 2011, Electro-thermomechanical behaviors of FGPM spheres using analytical method and ANSYS software, Applied Mathematical Modelling 36: 139–157.

[8] Ghorbanpour Arani A., Kolahchi R., Mosallaie Barzoki A.A., 2011, Effect of material inhomogeneity on electro-thermo-mechanical behaviors of functionally graded piezoelectric rotating cylinder, Applied Mathematical Modelling 35: 2771–2789.

[9] Bent A.A., Hagood N.W., Rodgers J.P., 1995, Anisotropic actuation with piezoelectric fiber composites, Journal of Material Sysistem and Structures 6: 338–349.

[10] Matsuna H., 2007, Vibration and buckling of cross-ply laminated composite circular cylindrical shells according to a global higher-order theory, International Journal of Mechanical Science 49: 1060-1075.

[11] Kadoli R., Ganesan N., 2003, Free vibration and buckling analysis of composite cylindrical shells conveying hot fluid, Composite Structures 60: 19–32.

[12] Messina A., Soldatos K.P., 1999, Vibration of completely free composite plates and cylindrical shell panels by a higher-order theory, International Journal of Mechanical Science 41: 891-918.

[13] Mosallaie Barzoki A.A., Ghorbanpour Arani A., Kolahchi R., Mozdianfard M.R., 2011, Electro-thermo-mechanical torsional buckling of a piezoelectric polymeric cylindrical shell reinforced by DWBNNTs with an elastic core, Applied Mathematical Modeling 36: 2983–2995.

[14] Bellman R.E., Casti J., 1971, Differential quadrature and long-term integration, Journal of Mathematics Analysis and Application 34: 235-238.

[15] Bellman R.E., Kashef B.G., Casti J., 1972, Differential quadrature: A technique for the rapid solution of nonlinear partial differential equations, Journal of Computational and Physics 10: 40-52.

[16] Farsa J., Kukreti A.R., Bert C.W., 1993, Fundamental frequency analysis of laminated rectangular plates by differential quadrature method, International Journal of Numerical Methods and Engineering 36: 2341–56.

[17] Jang S.K., Bert C.W., Striz A.G., 1989, Application of differential quadrature to static analysis of structural components, International Journal of Numerical Methods Engineering 28: 561–77.

[18] Sherbourne A.N., Pandey M.D., 1991, Differential quadrature method in the buckling analysis of beams and composite plates, Computers and Structures 40: 903–913.

[19] Striz A.G., Wang X., Bert C.W., 1995, Harmonic differential quadrature method and applications to analysis of structural components, Acta Mechanica 111: 85–94.

[20] Liew K.M., Teo T.M., Han J.B., 1999, Comparative accuracy of DQ and HDQ methods for three-dimensional vibration analysis of rectangular plates, International Journal of Numerical Methods and Engineering 45: 1831–1848.

[21] Civalek Ö., 2004, Application of differential quadrature (DQ) and harmonic differential quadrature (HDQ) for buckling analysis of thin isotropic plates and elastic columns, Engineering Structures 26: 171–186.

[22] Amabili M., 2008, Nonlinear Vibrations and Stability of Shells and Plates, Cambridge University Press, USA.

[23] Wang L., Ni Q., 2009, A reappraisal of the computational modelling of carbon nanotubes conveying viscous fluid, Mechanics Research Communications 36: 833–837.

[24] Liew K.M., Han J.B., Xiao Z.M., 1996, Differential Quadrature Method for Thick Symmetric Cross-Ply Laminates with First-Order Shear Flexibility, International Journal of Solids and Structures 33: 2647-2658.

[25] Shu C., Richards B.E., 1992, Application of generalized differential quadrature to solve two-dimensional incompressible Navier– Stokes equations, International Journal of Numerical Methods in Fluids 15: 791–798.

[26] Malekzadeh P., 2008, Nonlinear free vibration of tapered Mindlin plates with edges elastically restrained against rotation using DQM, Thin-Walled Structures 46: 11–26.

[27] Chen W., Shu C., He W., Zhong, T., 2000, The applications of special matrix products to differential quadrature solution of geometrically nonlinear bending of orthotropic rectangular plates, Computers and Structures 74: 65–76.

[28] Zhou X., 2012, Vibration and stability of ring-stiffened thin-walled cylindrical shells conveying fluid, Acta Mechanica Solida Sinica 25:168–176.