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Prediction of Vibrational Behavior of Composite Cylindrical Shells under Various Boundary Conditions

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Abstract

In this paper, a unified analytical approach is applied to investigate the vibrational behavior of composite cylindrical shells. Theoretical formulation is established based on Sanders’ thin shell theory. The modal forms are assumed to have the axial dependency in the form of Fourier series whose derivatives are legitimized using Stoke's transformation. The Influence of some commonly used boundary conditions and the effect of variations in shell geometrical parameters on the shell frequencies are studied. The results obtained for a number of particular cases show good agreement with those available in the open literature. The simplicity and the capability of the present method are also discussed.

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Authors and Affiliations

  1. Department of Mechanical Engineering, University of Guilan, P.O. Box 3756, Rasht, Iran

    Milad Hemmatnezhad, Reza Ansari & Mansour Darvizeh

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  1. Milad Hemmatnezhad
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  2. Reza Ansari
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  3. Mansour Darvizeh
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Correspondence to Reza Ansari.

Appendices

Appendix A

$$ \begin{array}{*{20}{c}} {u\left( {x,\theta ,t} \right) = \left( {{A_{0n}} + \sum\limits_{m = 1}^\infty {{A_{mn}}} \cos \frac{{m\pi x}}{L}} \right)\cos n\theta \sin \omega t,\quad 0 \leqslant x \leqslant L,} \hfill \\ {{u_{,x}} = - \left( {\frac{\pi }{L}} \right)\sum\limits_{m = 1}^\infty {m{A_{mn}}} \sin \frac{{m\pi x}}{L}\cos n\theta \sin \omega t,\quad 0 < x < L,} \hfill \\ {\quad \quad \quad {u_{,x}}\left( {0,\theta } \right) = \left( {\frac{{{\pi ^2}}}{{2L}}} \right){{\overline u }_0}\cos n\theta ,\quad {u_{,x}}\left( {L,\theta } \right) = \left( {\frac{{{\pi ^2}}}{{2L}}} \right){{\overline u }_L}\cos n\theta ,} \hfill \\ {{u_{,xx}} = {{\left( {\frac{\pi }{L}} \right)}^2}\left[ {\frac{{{{\overline u }_0} + {{\overline u }_L}}}{2} + \sum\limits_{m = 1}^\infty {\left\{ {{{\overline u }_0} + {{\overline u }_L}{{\left( { - 1} \right)}^m} - {m^2}{A_{mn}}} \right\}\cos \frac{{m\pi x}}{L}} } \right]\cos n\theta \sin \omega t,\quad 0 \leqslant x \leqslant L,} \hfill \\ {v\left( {x,\theta ,t} \right) = \sum\limits_{m = 1}^\infty {{B_{mn}}\sin \left( {m\pi x/L} \right)\sin n\theta \sin \omega t,\quad 0 < x < L,} } \hfill \\ {\quad \quad \quad \quad v\left( {0,\theta } \right) = - \left( {\frac{\pi }{2}} \right){v_0}\sin n\theta ,\quad v\left( {L,\theta } \right) = - \left( {\frac{\pi }{2}} \right){v_L}\sin n\theta ,} \hfill \\ {{v_{,x}} = \left( {\frac{\pi }{L}} \right)\left[ {\frac{{{v_0} + {v_L}}}{2} + \sum\limits_{m = 1}^\infty {\left\{ {{v_0} + {v_L}{{\left( { - 1} \right)}^m} + m{B_{mn}}} \right\}\cos \frac{{m\pi x}}{L}} } \right]\sin n\theta \sin \omega t,\quad 0 \leqslant x \leqslant L,} \hfill \\ {{v_{,xx}} = - {{\left( {\frac{\pi }{L}} \right)}^2}\left[ {\sum\limits_{m = 1}^\infty {\left\{ {{v_0}m + {v_L}m{{\left( { - 1} \right)}^m} + {m^2}{B_{mn}}} \right\}\sin \frac{{m\pi x}}{L}} } \right]\sin n\theta \sin \omega t,\quad 0 < x < L,} \hfill \\ {\quad \quad \quad \quad {v_{,xx}}\left( {0,\theta } \right) = - \left( {\frac{{{\pi ^3}}}{{2{L^2}}}} \right){{\overline{\overline v} }_0}\sin n\theta ,\quad {v_{,xx}}\left( {L,\theta } \right) = - \left( {\frac{{{\pi ^3}}}{{2{L^2}}}} \right){{\overline{\overline v} }_L}\sin n\theta ,} \hfill \\ {w\left( {x,\theta ,t} \right) = \sum\limits_{m = 1}^\infty {{C_{mn}}\sin \left( {m\pi x/L} \right)\cos n\theta \sin \omega t,\quad 0 < x < L,} } \hfill \\ {\quad \quad \quad \quad w\left( {0,\theta } \right) = - \left( {\frac{\pi }{2}} \right){w_0}\cos n\theta ,\quad v\left( {L,\theta } \right) = - \left( {\frac{\pi }{2}} \right){w_L}\cos n\theta ,} \hfill \\ {{w_{,x}} = \left( {\frac{\pi }{L}} \right)\left[ {\frac{{{w_0} + {w_L}}}{2} + \sum\limits_{m = 1}^\infty {\left\{ {{w_0} + {w_L}{{\left( { - 1} \right)}^m} + m{C_{mn}}} \right\}\cos \frac{{m\pi x}}{L}} } \right]\cos n\theta \sin \omega t,\quad 0 \leqslant x \leqslant L,} \hfill \\ {{w_{,xx}} = - {{\left( {\frac{\pi }{L}} \right)}^2}\left[ {\sum\limits_{m = 1}^\infty {\left\{ {{w_0}m + {w_L}m{{\left( { - 1} \right)}^m} + {m^2}{C_{mn}}} \right\}\sin \frac{{m\pi x}}{L}} } \right]\cos n\theta \sin \omega t,\quad 0 < x < L,} \hfill \\ {\quad \quad \quad \quad {w_{,xx}}\left( {0,\theta } \right) = - \left( {\frac{{{\pi ^3}}}{{2{L^2}}}} \right){{\overline{\overline w} }_0}\cos n\theta ,\quad {w_{,xx}}\left( {L,\theta } \right) = - \left( {\frac{{{\pi ^3}}}{{2{L^2}}}} \right){{\overline{\overline w} }_L}\cos n\theta ,} \hfill \\ {{w_{,xxx}} = {{\left( {\frac{\pi }{L}} \right)}^3}\left[ {\frac{{{{\overline{\overline w} }_0} + {{\overline{\overline w} }_L}}}{2} + \sum\limits_{m = 1}^\infty {\left\{ {{{\overline{\overline w} }_0} + {{\overline{\overline w} }_L}{{\left( { - 1} \right)}^m} - {m^2}{w_0} - {m^2}{w_L}{{\left( { - 1} \right)}^m} + {m^3}{C_{mn}}} \right\}\cos \frac{{m\pi x}}{L}} } \right]} \hfill \\ {\quad \quad \quad \quad \quad \quad \quad \quad \quad \times \cos n\theta \sin \omega t,\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad 0 \leqslant x \leqslant L,} \hfill \\ {{w_{,xxxx}} = {{\left( {\frac{\pi }{L}} \right)}^4}\left[ {\sum\limits_{m = 1}^\infty {\left\{ { - m{{\overline{\overline w} }_0} - m{{\overline{\overline w} }_L}{{\left( { - 1} \right)}^m} + {m^3}{w_0} + {m^3}{w_L}{{\left( { - 1} \right)}^m} + {m^4}{C_{mn}}} \right\}\sin \frac{{m\pi x}}{L}} } \right]} \hfill \\ {\quad \quad \quad \quad \quad \quad \quad \quad \quad \times \cos n\theta \sin \omega t,\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad 0 < x < L,} \hfill \\ {{w_{,xxxx}}\left( {0,\theta } \right) = {{\overline{\overline {\overline{\overline w} }} }_0}\cos n\theta ,\quad \quad \quad \quad \quad \quad {w_{,xxxx}}\left( {L,\theta } \right) = {{\overline{\overline {\overline{\overline w} }} }_L}\cos n\theta } \hfill \\ \end{array} $$

The successive derivatives with respect to θ and t are simply achieved. For example, the successive derivatives of u(x, θ, t) with respect to θ are as follows

$$ \begin{gathered} {u_{,\theta }} = - n\left( {{A_{0n}} + \sum\limits_{m = 1}^\infty {{A_{mn}}\cos \frac{{m\pi x}}{L}} } \right)\sin n\theta \sin \omega t, \hfill \\ {u_{,\theta \theta }} = - {n^2}\left( {{A_{0n}} + \sum\limits_{m = 1}^\infty {{A_{mn}}\cos \frac{{m\pi x}}{L}} } \right)\cos n\theta \sin \omega t \hfill \\ \end{gathered} $$

Appendix B

$$ \begin{array}{*{20}{c}} {{K_{11}} = R{A_{11}}{{\left( {\frac{{m\pi }}{L}} \right)}^2} + {A_{66}}\frac{{{n^2}}}{R}} \hfill \\ {{K_{12}} = - \left( {{A_{12}} + {A_{66}} + \frac{{{B_{12}}}}{R} + \frac{{{B_{66}}}}{{2R}}} \right)\left( {\frac{{mn\pi }}{L}} \right)} \hfill \\ {{K_{13}} = - {A_{12}}\left( {\frac{{m\pi }}{L}} \right) - R{B_{11}}{{\left( {\frac{{m\pi }}{L}} \right)}^3} + \left( {{B_{12}} - {B_{66}}} \right)\left( {\frac{{m\pi {n^2}}}{{RL}}} \right)} \hfill \\ {{K_{22}} = \left( {R{A_{66}} + \frac{{3{B_{66}}}}{2} + \frac{{{D_{66}}}}{{2R}}} \right){{\left( {\frac{{m\pi }}{L}} \right)}^2} + \left( {R{A_{22}} + 2{B_{22}} + \frac{{{D_{22}}}}{R}} \right){{\left( {\frac{n}{R}} \right)}^2}} \hfill \\ {{K_{33}} = {{\left( {\frac{{m\pi }}{L}} \right)}^2}{{\left( {2{B_{12}} + R{D_{11}}(\frac{{m\pi }}{L}} \right)}^2} + \left. {2\frac{{{D_{12}} + {D_{66}}}}{R}{n^2}} \right)} \hfill \\ { + 2{B_{22}}{{\left( {\frac{n}{R}} \right)}^2} + \frac{{{A_{22}}}}{R} + {D_{22}}\frac{{{n^4}}}{{{R^3}}}} \hfill \\ {{K_{01}} = {A_{66}}\frac{{{n^2}}}{R}} \hfill \\ \end{array} $$

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Hemmatnezhad, M., Ansari, R. & Darvizeh, M. Prediction of Vibrational Behavior of Composite Cylindrical Shells under Various Boundary Conditions. Appl Compos Mater 17, 225–241 (2010). https://doi.org/10.1007/s10443-009-9108-4

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