Dynamics of solid propellant motor composite casing under impact pressure

Abstract

A dynamic behavior of solid propellant motor composite casing under the action of an internal impact pressure is treated. This pressure describes the engine operation. The thin-walled casing consists of the cylindrical shell and two bottoms. These bottoms are truncated hemisphere. The casing is clamped along two edges of the bottoms. A shear, a rotary inertia and stress–strain relations for an orthotropic material are accounted. Semi analytical method is suggested to analyze the structure stress–strain state. The thin-walled casing dynamic is described by large dimension system of the ordinary differential equations.

Appendix: Parameters of the dynamical systems (26)

$$ \begin{aligned} K_{1} & = - \frac{{E_{xx} }}{{\rho \left( {1 - \mu_{x\varphi } \mu_{\varphi x} } \right)}}, \\ K_{2} & = \frac{{G_{x\varphi } }}{{\rho R^{2} }}, \, \\ K_{3} & = - \frac{1}{\rho R}\left( {\frac{{\mu_{x\varphi } E_{xx} }}{{1 - \mu_{x\varphi } \mu_{\varphi x} }} + G_{x\varphi } } \right), \, \\ K_{4} & = - \frac{{\mu_{x\varphi } E_{xx} }}{{\rho R\left( {1 - \mu_{x\varphi } \mu_{\varphi x} } \right)}}, \\ K_{5} & = - \frac{{G_{x\varphi } }}{\rho }, \\ K_{6} & = \frac{{E_{\varphi \varphi } }}{{\rho R^{2} \left( {1 - \mu_{x\varphi } \mu_{\varphi x} } \right)}}, \\ K_{7} & = \frac{{\kappa G_{\varphi z} }}{{\rho R^{2} }}, \\ K_{8}^{{}} & = - \frac{{E_{\varphi \varphi } }}{{\rho R^{2} \left( {1 - \mu_{x\varphi } \mu_{\varphi x} } \right)}}, \\ K_{81} & = \frac{1}{\rho R}\left( {\frac{{\mu_{\varphi x} E_{\varphi \varphi } }}{{R\left( {1 - \mu_{x\varphi } \mu_{\varphi x} } \right)}} + G_{x\varphi } } \right), \\ K_{82} & = \frac{1}{{\rho R^{2} }}\left( {\frac{{E_{\varphi \varphi } }}{{1 - \mu_{x\varphi } \mu_{\varphi x} }} + \kappa G_{\varphi z} } \right), \\ K_{9} & = - \frac{{\kappa G_{\varphi z} }}{\rho R},\quad K_{10} = - \frac{{\kappa G_{xz} }}{\rho },\quad K_{11} = \frac{{\kappa G_{\varphi z} }}{{\rho R^{2} }}, \\ K_{12} & = \frac{{\mu_{\varphi x} E_{\varphi \varphi } }}{{\rho R\left( {1 - \mu_{x\varphi } \mu_{\varphi x} } \right)}},\quad K_{13} = \frac{{12\kappa G_{xz} }}{{\rho h^{2} }},\quad K_{14} = \frac{{12\kappa G_{\varphi z} }}{{\rho h^{2} }},\quad K_{15} = - \frac{{12\kappa G_{\varphi z} }}{{\rho Rh^{2} }}. \\ \end{aligned} $$