Wave Propagation in Piezoelectric Circular Curved Rods

Abstract

Dispersion equations of wave propagation in piezoelectric circular curved rods in an orthogonal curvilinear coordinate system are established in this paper, in which the displacements and electrical potential fields are described using Bessel functions. Characteristics of dispersion relations, distributions of displacements and electrical potential over the cross section are calculated. In the numerical examples, effects of various parameters on wave dispersions, distributions of displacement and electric potential of the first several modes over the cross section are investigated, in which influence of the changing of curvature radius of circular rods on distributions of displacement and electrical potential is compared, and also influence of whether considering the piezoelectric parameters in curved rods on dispersion is discussed.

Appendix

The coefficient $a_{ij}$ in the operator matrix in Eq. (7) is defined as follows:

$$ \begin{aligned} a_{11} & = - \frac{{c_{12} \cos \theta }}{{rR\left( {1 - \frac{r\cos \theta }{R}} \right)}} - \frac{{c_{13} \cos^{2} \theta }}{{R^{2} \left( {1 - \frac{r\cos \theta }{R}} \right)^{3} }} + \frac{{rc_{13} \cos^{3} \theta }}{{R^{3} \left( {1 - \frac{r\cos \theta }{R}} \right)^{3} }} - \frac{{c_{13} \cos \theta }}{{rR\left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} + \frac{{2c_{13} \cos^{2} \theta }}{{R^{2} \left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} - \frac{{c_{22} }}{{r^{2} }} \\ & \quad + \frac{{2c_{23} \cos \theta }}{{rR\left( {1 - \frac{r\cos \theta }{R}} \right)}} - \frac{{c_{33} \cos^{2} \theta }}{{R^{2} \left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} + \frac{{c_{55} }}{{\left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }}\frac{{\partial^{2} }}{{\partial s^{2} }} + \frac{{c_{66} \sin \theta }}{{rR\left( {1 - \frac{r\cos \theta }{R}} \right)}}\frac{\partial }{\partial \theta } + \left[ {\frac{{c_{66} }}{{r^{2} \left( {1 - \frac{r\cos \theta }{R}} \right)}} - \frac{{c_{66} \cos \theta }}{{rR\left( {1 - \frac{r\cos \theta }{R}} \right)}}} \right]\frac{{\partial^{2} }}{{\partial \theta^{2} }} \\ & \quad + \left[ {\frac{{c_{11} }}{{r\left| {1 - \frac{r\cos \theta }{R}} \right|}} - \frac{{2c_{11} \cos \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)}} - \frac{{c_{12} }}{r} + \frac{{c_{12} }}{{r\left( {1 - \frac{r\cos \theta }{R}} \right)}} - \frac{{c_{12} \cos \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)}} - \frac{{c_{13} \cos \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} + \frac{{rc_{13} \cos^{2} \theta }}{{R^{2} \left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} + \frac{{c_{13} \cos \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)}}} \right]\frac{\partial }{\partial r} \\ & \quad + \left[ {\frac{{c_{11} }}{{1 - \frac{r\cos \theta }{R}}} - \frac{{c_{11} r\cos \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)}}} \right]\frac{{\partial^{2} }}{{\partial r^{2} }} - \rho \frac{{\partial^{2} }}{{\partial t^{2} }} \\ \end{aligned} $$

$$ \begin{aligned} a_{12} & = \frac{{c_{13} \cos \theta \sin \theta }}{{R^{2} \left( {1 - \frac{r\cos \theta }{R}} \right)^{3} }} - \frac{{rc_{13} \cos^{2} \theta \sin \theta }}{{R^{3} \left( {1 - \frac{r\cos \theta }{R}} \right)^{3} }} + \frac{{c_{13} \sin \theta }}{{rR\left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} - \frac{{2c_{13} \cos \theta \sin \theta }}{{R^{2} \left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} - \frac{{c_{23} \sin \theta }}{{rR\left( {1 - \frac{r\cos \theta }{R}} \right)}} + \frac{{c_{33} \cos \theta \sin \theta }}{{R^{2} \left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} \\ & \quad - \frac{{c_{66} \sin \theta }}{{rR\left( {1 - \frac{r\cos \theta }{R}} \right)}} - \left[ {\frac{{c_{12} \cos \theta }}{{rR\left( {1 - \frac{r\cos \theta }{R}} \right)}} + \frac{{c_{22} }}{{r^{2} }} - \frac{{c_{23} \cos \theta }}{{rR\left( {1 - \frac{r\cos \theta }{R}} \right)}} + \frac{{c_{66} }}{{r^{2} \left( {1 - \frac{r\cos \theta }{R}} \right)}} - \frac{{c_{66} \cos \theta }}{{rR\left( {1 - \frac{r\cos \theta }{R}} \right)}}} \right]\frac{\partial }{\partial \theta } \\ & \quad + \left[ {\frac{{c_{13} \sin \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} - \frac{{c_{13} r\cos \theta \sin \theta }}{{R^{2} \left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} + \frac{{c_{66} \sin \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)}}} \right]\frac{\partial }{\partial r} + \left[ {\frac{{c_{12} }}{{r\left( {1 - \frac{r\cos \theta }{R}} \right)}} - \frac{{c_{12} \cos \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)}} + \frac{{c_{66} }}{{r\left( {1 - \frac{r\cos \theta }{R}} \right)}} - \frac{{c_{66} \cos \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)}}} \right]\frac{{\partial^{2} }}{\partial r\partial \theta } \\ \end{aligned} $$

$$ \begin{aligned} a_{13} =\,& \left[ {\frac{{c_{13} \cos \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)^{3} }} - \frac{{c_{13} r\cos^{2} \theta }}{{R^{2} \left( {1 - \frac{r\cos \theta }{R}} \right)^{3} }} + \frac{{c_{13} }}{{r\left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} - \frac{{2c_{13} \cos \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} - \frac{{c_{23} }}{{r\left( {1 - \frac{r\cos \theta }{R}} \right)}}} \right. \hfill \\ &\left. { + \frac{{c_{33} \cos \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} + \frac{{c_{55} \cos \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }}} \right]\frac{\partial }{\partial s} + \left[ {\frac{{c_{13} }}{{\left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} - \frac{{c_{13} r\cos \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} + \frac{{c_{55} }}{{1 - \frac{r\cos \theta }{R}}}} \right]\frac{{\partial^{2} }}{\partial r\partial s} \hfill \\ \end{aligned} $$

$$ \begin{aligned} a_{14} & = \left[ {\frac{{e_{31} \cos \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)^{3} }} - \frac{{e_{31} r\cos^{2} \theta }}{{R^{2} \left( {1 - \frac{r\cos \theta }{R}} \right)^{3} }} + \frac{{e_{31} }}{{r\left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} - \frac{{2e_{31} \cos \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} - \frac{{e_{32} }}{{r\left( {1 - \frac{r\cos \theta }{R}} \right)}} + \frac{{e_{33} \cos \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }}} \right]\frac{\partial }{\partial s} \\ & \quad + \left[ {\frac{{e_{15} }}{{1 - \frac{r\cos \theta }{R}}} + \frac{{e_{31} }}{{\left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} - \frac{{e_{31} r\cos \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }}} \right]\frac{{\partial^{2} }}{\partial r\partial s} \\ \end{aligned} $$

$$ \begin{aligned} a_{21} & = \frac{{c_{22} \sin \theta }}{{rR\left( {1 - \frac{r\cos \theta }{R}} \right)}} + \frac{{c_{23} \sin \theta \cos \theta }}{{R^{2} \left( {1 - \frac{r\cos \theta }{R}} \right)^{3} }} - \frac{{c_{23} r\sin \theta \cos^{2} \theta }}{{R^{3} \left( {1 - \frac{r\cos \theta }{R}} \right)^{3} }} + \frac{{c_{23} \sin \theta }}{{rR\left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} - \frac{{2c_{23} \cos \theta \sin \theta }}{{R^{2} \left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} - \frac{{c_{23} \sin \theta }}{{rR\left( {1 - \frac{r\cos \theta }{R}} \right)}} + \frac{{c_{33} \sin \theta \cos \theta }}{{R^{2} \left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} \\ & \quad + \left[ {\frac{{c_{22} }}{{r^{2} \left( {1 - \frac{r\cos \theta }{R}} \right)}} - \frac{{c_{22} \cos \theta }}{{rR\left( {1 - \frac{r\cos \theta }{R}} \right)}} - \frac{{c_{23} \cos \theta }}{{rR\left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} + \frac{{c_{23} \cos^{2} \theta }}{{R^{2} \left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} + \frac{{c_{66} }}{{r^{2} }} - \frac{{c_{66} \cos \theta }}{{rR\left( {1 - \frac{r\cos \theta }{R}} \right)}}} \right]\frac{\partial }{\partial \theta } \\ & \quad + \left[ {\frac{{c_{12} \sin \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)}} - \frac{{c_{13} \sin \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)}}} \right]\frac{\partial }{\partial r} + \left[ {\frac{{c_{12} }}{{r\left( {1 - \frac{r\cos \theta }{R}} \right)}} - \frac{{c_{12} \cos \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)}} + \frac{{c_{66} }}{{r\left( {1 - \frac{r\cos \theta }{R}} \right)}} - \frac{{c_{66} \cos \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)}}} \right]\frac{{\partial^{2} }}{\partial r\partial \theta } \\ \end{aligned} $$

$$ \begin{aligned} a_{22} & = \frac{{c_{23} \cos \theta }}{{rR\left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} - \frac{{c_{23} \cos^{2} \theta }}{{R^{2} \left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} - \frac{{c_{23} \sin^{2} \theta }}{{R^{2} \left( {1 - \frac{r\cos \theta }{R}} \right)^{3} }} + \frac{{c_{23} r\cos \theta \sin^{2} \theta }}{{R^{3} \left( {1 - \frac{r\cos \theta }{R}} \right)^{3} }} + \frac{{c_{23} \sin^{2} \theta }}{{R^{2} \left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} - \frac{{c_{33} \sin^{2} \theta }}{{R^{2} \left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} \\ & \quad - \frac{{c_{66} }}{{r^{2} }} + \frac{{c_{66} \cos \theta }}{{rR\left( {1 - \frac{r\cos \theta }{R}} \right)}} + \frac{{c_{44} }}{{\left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }}\frac{{\partial^{2} }}{{\partial s^{2} }} + \left[ {\frac{{c_{22} \sin \theta }}{{rR\left( {1 - \frac{r\cos \theta }{R}} \right)}} + \frac{{c_{23} \sin \theta }}{{rR\left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} - \frac{{c_{23} \cos \theta \sin \theta }}{{R^{2} \left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} - \frac{{c_{23} \sin \theta }}{{rR\left( {1 - \frac{r\cos \theta }{R}} \right)}}} \right]\frac{\partial }{\partial \theta } \\ & \quad + \left[ {\frac{{c_{22} }}{{r^{2} \left( {1 - \frac{r\cos \theta }{R}} \right)}} - \frac{{c_{22} \cos \theta }}{{rR\left( {1 - \frac{r\cos \theta }{R}} \right)}}} \right]\frac{{\partial^{2} }}{{\partial \theta^{2} }} + \left[ {\frac{{c_{66} }}{r} - \frac{{c_{66} \cos \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)}}} \right]\frac{\partial }{\partial r} + \left[ {\frac{{c_{66} }}{{1 - \frac{r\cos \theta }{R}}} - \frac{{c_{66} r\cos \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)}}} \right]\frac{{\partial^{2} }}{{\partial r^{2} }} - \rho \frac{{\partial^{2} }}{{\partial t^{2} }} \\ \end{aligned} $$

$$ \begin{aligned} a_{23} & = \left[ { - \frac{{c_{23} \sin \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)^{3} }} + \frac{{c_{23} r\sin \theta \cos \theta }}{{R^{2} \left( {1 - \frac{r\cos \theta }{R}} \right)^{3} }} + \frac{{c_{23} \sin \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} - \frac{{c_{33} \sin \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} - \frac{{c_{44} \sin \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }}} \right]\frac{\partial }{\partial s} \\ & \quad + \left[ {\frac{{c_{23} }}{{r\left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} - \frac{{c_{23} \cos \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} + \frac{{c_{44} }}{{r\left( {1 - \frac{r\cos \theta }{R}} \right)}}} \right]\frac{{\partial^{2} }}{\partial \theta \partial s} \\ \end{aligned} $$

$$ \begin{aligned} a_{24} & = \left[ { - \frac{{e_{32} \sin \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)^{3} }} + \frac{{e_{32} r\sin \theta \cos \theta }}{{R^{2} \left( {1 - \frac{r\cos \theta }{R}} \right)^{3} }} + \frac{{e_{32} \sin \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} - \frac{{e_{33} \sin \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }}} \right]\frac{\partial }{\partial s} \\ & \quad + \left[ {\frac{{e_{24} }}{{r\left( {1 - \frac{r\cos \theta }{R}} \right)}} + \frac{{e_{32} }}{{r\left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} - \frac{{e_{32} \cos \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }}} \right]\frac{{\partial^{2} }}{\partial \theta \partial s} \\ \end{aligned} $$

$$ \begin{aligned} a_{31} & = \left[ {\frac{{c_{23} }}{{r\left( {1 - \frac{r\cos \theta }{R}} \right)}} - \frac{{c_{33} \cos \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} + \frac{{c_{55} \cos \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)^{3} }} - \frac{{c_{55} r\cos^{2} \theta }}{{R^{2} \left( {1 - \frac{r\cos \theta }{R}} \right)^{3} }} + \frac{{c_{55} }}{{r\left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} - \frac{{3c_{55} \cos \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }}} \right]\frac{\partial }{\partial s} \\ & \quad + \left[ {\frac{{c_{13} }}{{1 - \frac{r\cos \theta }{R}}} + \frac{{c_{55} }}{{\left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} - \frac{{c_{55} r\cos \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }}} \right]\frac{{\partial^{2} }}{\partial r\partial s} \\ \end{aligned} $$

$$ \begin{aligned} a_{32} & = \left[ {\frac{{c_{33} \sin \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} - \frac{{c_{44} \sin \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)^{3} }} + \frac{{c_{44} r\cos \theta \sin \theta }}{{R^{2} \left( {1 - \frac{r\cos \theta }{R}} \right)^{3} }} + \frac{{2c_{44} \sin \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }}} \right]\frac{\partial }{\partial s} \\ & \quad + \left[ {\frac{{c_{23} }}{{r\left( {1 - \frac{r\cos \theta }{R}} \right)}} + \frac{{c_{44} }}{{r\left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} - \frac{{c_{44} \cos \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }}} \right]\frac{{\partial^{2} }}{\partial \theta \partial s} \\ \end{aligned} $$

$$ \begin{aligned} a_{33} & = - \frac{{c_{44} \cos \theta }}{{rR\left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} + \frac{{c_{44} \cos^{2} \theta }}{{R^{2} \left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} + \frac{{c_{44} \sin^{2} \theta }}{{R^{2} \left( {1 - \frac{r\cos \theta }{R}} \right)^{3} }} - \frac{{c_{44} r\cos \theta \sin^{2} \theta }}{{R^{3} \left( {1 - \frac{r\cos \theta }{R}} \right)^{3} }} - \frac{{2c_{44} \sin^{2} \theta }}{{R^{2} \left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} + \frac{{c_{55} \cos^{2} \theta }}{{R^{2} \left( {1 - \frac{r\cos \theta }{R}} \right)^{3} }} \\ & \quad - \frac{{c_{55} r\cos^{3} \theta }}{{R^{3} \left( {1 - \frac{r\cos \theta }{R}} \right)^{3} }} + \frac{{c_{55} \cos \theta }}{{rR\left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} - \frac{{3c_{55} \cos^{2} \theta }}{{R^{2} \left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} + \frac{{c_{33} }}{{\left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }}\frac{{\partial^{2} }}{{\partial s^{2} }} - \left[ {\frac{{c_{44} \sin \theta }}{{rR\left( {1 - \frac{r\cos \theta }{R}} \right))^{2} }} - \frac{{c_{44} \sin \theta \cos \theta }}{{R^{2} \left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }}} \right. \\ & \quad \left. { + \frac{{2c_{44} \sin \theta }}{{rR\left( {1 - \frac{r\cos \theta }{R}} \right)}}} \right]\frac{\partial }{\partial \theta } + \left[ {\frac{{c_{44} }}{{r^{2} \left( {1 - \frac{r\cos \theta }{R}} \right)}} - \frac{{c_{44} \cos \theta }}{{rR\left( {1 - \frac{r\cos \theta }{R}} \right)}}} \right]\frac{{\partial^{2} }}{{\partial \theta^{2} }} + \left[ {\frac{{c_{55} \cos \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} - \frac{{c_{55} r\cos^{2} \theta }}{{R^{2} \left( {1 - \frac{r\cos \theta }{R}} \right))^{2} }} + \frac{{c_{55} }}{{r(\left( {1 - \frac{r\cos \theta }{R}} \right)}}} \right. \\ & \quad \left. { - \frac{{3c_{55} \cos \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)}}} \right]\frac{\partial }{\partial r} + \left[ {\frac{{c_{55} }}{{1 - \frac{r\cos \theta }{R}}} - \frac{{c_{55} r\cos \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)}}} \right]\frac{{\partial^{2} }}{{\partial r^{2} }} - \rho \frac{{\partial^{2} }}{{\partial t^{2} }} \\ \end{aligned} $$

$$ \begin{aligned} a_{34} & = \frac{{e_{33} }}{{\left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }}\frac{{\partial^{2} }}{{\partial s^{2} }} + \frac{{2e_{24} \sin \theta }}{{rR\left( {1 - \frac{r\cos \theta }{R}} \right)}}\frac{\partial }{\partial \theta } + \left[ {\frac{{e_{24} }}{{r^{2} \left( {1 - \frac{r\cos \theta }{R}} \right)}} - \frac{{e_{24} \cos \theta }}{{rR\left( {1 - \frac{r\cos \theta }{R}} \right)}}} \right]\frac{{\partial^{2} }}{{\partial \theta^{2} }} + \left[ {\frac{{e_{15} }}{{r\left( {1 - \frac{r\cos \theta }{R}} \right)}} - \frac{{3e_{15} \cos \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)}}} \right]\frac{\partial }{\partial r} \\ & \quad + \left[ {\frac{{e_{15} }}{{1 - \frac{r\cos \theta }{R}}} - \frac{{e_{15} r\cos \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)}}} \right]\frac{{\partial^{2} }}{{\partial r^{2} }} \\ \end{aligned} $$

$$ \begin{aligned} a_{41} & = \left[ {\frac{{e_{15} \cos \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)^{3} }} - \frac{{e_{15} r\cos^{2} \theta }}{{R^{2} \left( {1 - \frac{r\cos \theta }{R}} \right)^{3} }} - \frac{{e_{15} \cos \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} + \frac{{e_{15} }}{{r\left( {1 - \frac{r\cos \theta }{R}} \right)}} + \frac{{e_{32} }}{{r\left( {1 - \frac{r\cos \theta }{R}} \right)}} - \frac{{e_{33} \cos \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }}} \right]\frac{\partial }{\partial s} \\ & \quad + \left[ {\frac{{e_{15} }}{{\left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} - \frac{{e_{15} r\cos \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} + \frac{{e_{31} }}{{1 - \frac{r\cos \theta }{R}}}} \right]\frac{{\partial^{2} }}{\partial r\partial s} \\ \end{aligned} $$

$$ \begin{aligned} a_{42} & = \left[ { - \frac{{e_{24} \sin \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)^{3} }} + \frac{{e_{24} r\cos \theta \sin \theta }}{{R^{2} \left( {1 - \frac{r\cos \theta }{R}} \right)^{3} }} + \frac{{e_{24} \sin \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} + \frac{{e_{33} \sin \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }}} \right]\frac{\partial }{\partial s} \\ & \quad + \left[ {\frac{{e_{24} }}{{r\left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} - \frac{{e_{24} \cos \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} + \frac{{e_{32} }}{{r\left( {1 - \frac{r\cos \theta }{R}} \right)}}} \right]\frac{{\partial^{2} }}{\partial \theta \partial s} \\ \end{aligned} $$

$$ \begin{aligned} a_{43} = \frac{{e_{15} \cos^{2} \theta }}{{R^{2} \left( {1 - \frac{r\cos \theta }{R}} \right)^{3} }} - \frac{{e_{15} r\cos^{3} \theta }}{{R^{3} \left( {1 - \frac{r\cos \theta }{R}} \right)^{3} }} - \frac{{e_{15} \cos^{2} \theta }}{{R^{2} \left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} + \frac{{e_{15} \cos \theta }}{{rR\left( {1 - \frac{r\cos \theta }{R}} \right)}} - \frac{{e_{24} \cos \theta }}{{rR\left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} + \frac{{e_{24} \cos^{2} \theta }}{{R^{2} \left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} \\ \quad + \frac{{e_{24} \sin^{2} \theta }}{{R^{2} \left( {1 - \frac{r\cos \theta }{R}} \right)^{3} }} - \frac{{e_{24} r\cos \theta \sin \theta }}{{R^{3} \left( {1 - \frac{r\cos \theta }{R}} \right)^{3} }} - \frac{{e_{24} \sin^{2} \theta }}{{R^{2} \left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} + \frac{{e_{33} }}{{\left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }}\frac{{\partial^{2} }}{{\partial s^{2} }} + \left[\frac{{e_{24} \cos \theta \sin \theta }}{{R^{2} \left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} + \frac{{e_{24} \sin \theta }}{{rR\left( {1 - \frac{r\cos \theta }{R}} \right)}} \right.\\ \quad \left.- \frac{{e_{24} \sin \theta }}{{rR(\left( {1 - \frac{r\cos \theta }{R}} \right))^{2} }}\right]\frac{\partial }{\partial \theta } + \left[ {\frac{{e_{24} }}{{r^{2} \left( {1 - \frac{r\cos \theta }{R}} \right)}} - \frac{{e_{24} \cos \theta }}{{rR\left( {1 - \frac{r\cos \theta }{R}} \right)}}} \right]\frac{{\partial^{2} }}{{\partial \theta^{2} }} + \left[ {\frac{{e_{15} }}{r} + \frac{{e_{15} \cos \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }} - \frac{{e_{15} r\cos^{2} \theta }}{{R^{2} \left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }}} \right. \\ \quad \left. { - \frac{{e_{15} \cos \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)}}} \right]\frac{\partial }{\partial r} + \left[ {\frac{{e_{15} }}{{1 - \frac{r\cos \theta }{R}}} - \frac{{e_{15} r\cos \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)}}} \right]\frac{{\partial^{2} }}{{\partial r^{2} }} \\ \end{aligned} $$

$$ \begin{aligned} a_{44} & = - \frac{{\varSigma_{33} }}{{\left( {1 - \frac{r\cos \theta }{R}} \right)^{2} }}\frac{{\partial^{2} }}{{\partial s^{2} }} - \frac{{\varSigma_{22} \sin \theta }}{{rR\left( {1 - \frac{r\cos \theta }{R}} \right)}}\frac{\partial }{\partial \theta } + \left[ {\frac{{\varSigma_{22} \cos \theta }}{{rR\left( {1 - \frac{r\cos \theta }{R}} \right)}} - \frac{{\varSigma_{22} }}{{r^{2} \left( {1 - \frac{r\cos \theta }{R}} \right)}}} \right]\frac{{\partial^{2} }}{{\partial \theta^{2} }} \\ & \quad + \left[ {\frac{{\varSigma_{11} \cos \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)}} - \frac{{\varSigma_{11} }}{r}} \right]\frac{\partial }{\partial r} + \left[ {\frac{{\varSigma_{11} r\cos \theta }}{{R\left( {1 - \frac{r\cos \theta }{R}} \right)}} - \frac{{\varSigma_{11} }}{{1 - \frac{r\cos \theta }{R}}}} \right]\frac{{\partial^{2} }}{{\partial r^{2} }}. \\ \end{aligned} $$