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

(254)

The coefficients $A_1,B_2$ are given by Eq. (241). Substitution of Eq. (241) into Eqs. (253), (254) yields

$\begin{array}{l}{u}_r^A=-[\xi (-2\xi {\zeta }_S\sin {\zeta }_{S}d)\sin {\zeta }_{P}z+(-({\xi }^2-{\zeta }_S^2)\sin {\zeta }_{P}d)(-{\zeta }_S\sin {\zeta }_{S}z)]J_1\hfill \\ u_z^A=[-2\xi {\zeta }_S\sin {\zeta }_{S}d{\zeta }_P\cos {\zeta }_{P}z+\xi (-({\xi }^2-{\zeta }_S^2)\sin {\zeta }_{P}d)\cos {\zeta }_{S}z]J_0\hfill \end{array}$ (255)

(255)

$\begin{array}{l}{\sigma }_{zz}^A=\mu \{({\xi }^2-{\zeta }_S^2)(-2\xi {\zeta }_S\sin {\zeta }_{S}d)\sin {\zeta }_{P}z+[2\xi ({\xi }^2-{\zeta }_S^2)\sin {\zeta }_{P}d\left]{\zeta }_S\sin {\zeta }_{S}z\right\}{J}_0\hfill \\ {\sigma }_{rz}^A=-\mu \{2\xi (-2\xi {\zeta }_S\sin {\zeta }_{S}d){\zeta }_P\cos {\zeta }_{P}z+({\xi }^2-{\zeta }_S^2)[-({\xi }^2-{\zeta }_S^2)\sin {\zeta }_{P}d\left]\cos {\zeta }_{S}z\right\}{J}_1\hfill \end{array}$ (256)

(256)

Upon expansion and rearrangement, Eqs. (255), (256) become

$\begin{array}{l}{u}_r^A={\zeta }_S\hfill \end{array}$