Structural Health Monitoring with Piezoelectric Wafer Active Sensors, 2nd Edition

(82)

11.4.2.2 Antisymmetric Solution

Subtraction of the ${\sigma }_{yy}$ equations, i.e., Eq. (74)−Eq. (75), and addition of the ${\sigma }_{xy}$ equations, i.e., Eq. (76)+Eq. (77), yields

$\begin{array}{l}{A}_1({\xi }^2-{\eta }_S^2)\sin {\eta }_{P}d-B_{2}2\xi {\eta }_S\sin {\eta }_{S}d=0\hfill \\ A_{1}2\xi {\eta }_P\cos {\eta }_{P}d+B_2({\xi }^2-{\eta }_S^2)\cos {\eta }_{S}d=\frac{\tilde{\tau }}{2i\mu }\hfill \end{array}(\mathrm{antisymmetric}\mathrm{motion})$ (83)

(83)

In matrix notations, we write

$[\begin{array}{ll}({\xi }^2-{\eta }_S^2)\sin {\eta }_{P}d\hfill & -2\xi {\eta }_S\sin {\eta }_{S}d\hfill \\ 2\xi {\eta }_P\cos {\eta }_{P}d\hfill & ({\xi }^2-{\eta }_S^2)\cos {\eta }_{S}d\hfill \end{array}$