Study on Vibration Isolation Performance of Double-Layer Wave Impeding Block Based on Wave Impedance Ratio

Abstract

Based on the theory of single-phase elastic media and unsaturated porous media, the vibration isolation effect of double-layer wave impeding block (WIB) in the unsaturated soil foundation is investigated. Using the Fourier transform and Helmholtz vector decomposition, the calculation formula of the dynamic response of unsaturated ground subjected to a strip harmonic load on the ground surface is established. By analyzing the wave impedance ratio at the interface between the double-layer WIB and the unsaturated soil foundation on the vibration isolation effect of the double-layer WIB, the corresponding interlayer wave impedance ratio of the double-layer WIB with the best vibration isolation effect is selected. On this basis, the influence of load frequency, saturation, thickness, and embedded depth on the vibration isolation performance of the double-layer WIB is analyzed. The results show that the best vibration isolation effect of the double-layer WIB can be obtained by designing the wave impedance ratio at the intersection between the layers of the double-layer WIB. At the same thickness, the vibration isolation effect of the double-layer WIB is better than that of the homogeneous WIB. The double-layer WIB enhances the frequency width of homogeneous WIB vibration damping, which has a better vibration isolation effect on low-frequency, medium-frequency and high-frequency vibration (5 Hz < f < 70 Hz). When the thickness of double-layer WIB exceeds the critical thickness, the vibration isolation effect decreases with the increase in thickness. Soil saturation has a significant effect on the vibration isolation effect of the double-layer WIB, and the double-layer WIB can achieve a better vibration isolation effect at high saturation.

Appendices

Appendix A

$\beta_{1} = - B_{1} a_{12} a_{23} + B_{1} a_{13} a_{22} + B_{2} a_{11} a_{23} - B_{2} a_{13} a_{21} - B_{3} a_{11} a_{22} + B_{3} a_{12} a_{21} - \mu_{p} a_{12} a_{23} + \mu_{p} a_{13} a_{22}$,$\begin{gathered} \beta_{2} = - B_{1} a_{12} b_{33} + B_{1} a_{13} b_{32} + B_{1} a_{22} b_{23} - B_{1} a_{23} b_{22} + B_{2} a_{11} b_{33} - B_{2} a_{13} b_{31} - B_{2} a_{21} b_{23} + B_{2} a_{23} b_{21} - \hfill \\ B_{3} a_{11} b_{32} + B_{3} a_{12} b_{31} + B_{3} a_{21} b_{22} - B_{3} a_{22} b_{21} - a_{11} a_{22} b_{13} + a_{11} a_{23} b_{12} + a_{12} a_{21} b_{13} - a_{12} a_{23} b_{11} - a_{13} a_{21} b_{12} \hfill \\ + a_{13} a_{22} b_{11} - \mu_{p} a_{12} b_{33} + \mu_{p} a_{13} b_{32} + \mu_{p} a_{22} b_{23} - \mu_{p} a_{23} b_{22} \hfill \\ \end{gathered}$,$\begin{gathered} \beta_{3} = - B_{1} b_{22} b_{33} + B_{1} b_{23} b_{32} + B_{2} b_{21} b_{33} - B_{2} b_{23} b_{31} - B_{3} b_{21} b_{32} + B_{3} b_{22} b_{31} + a_{11} b_{12} b_{33} - a_{11} b_{13} b_{32} - a_{12} b_{11} b_{33} \hfill \\ + a_{12} b_{13} b_{31} + a_{13} b_{11} b_{32} - a_{13} b_{12} b_{31} - a_{21} b_{12} b_{23} + a_{21} b_{13} b_{22} + a_{22} b_{11} b_{23} - a_{22} b_{13} b_{21} - a_{23} b_{11} b_{22} + a_{23} b_{12} b_{21} - \hfill \\ \mu_{p} b_{22} b_{33} + \mu_{p} b_{23} b_{32} \hfill \\ \end{gathered}$,$\beta_{4} = - b_{11} b_{22} b_{33} + b_{11} b_{23} b_{32} + b_{12} b_{21} b_{33} - b_{12} b_{23} b_{31} - b_{13} b_{21} b_{32} + b_{13} b_{22} b_{31}$,$\beta_{5} = - \mu_{p} d_{22} d_{33}$,$\beta_{6} = - d_{11} d_{22} d_{33} + d_{21} d_{12} d_{33} + d_{13} d_{22} d_{31}$.

Appendix B

The expression of the system of Eqs. (35) is:

$$ \begin{gathered} T_{20 \times 20} \left[ \begin{gathered} \begin{array}{*{20}c} {A_{tp1}^{I} } & {A_{tp2}^{I} } & {A_{tp3}^{I} } & {A_{rp1}^{I} } & {A_{rp2}^{I} } & {A_{rp3}^{I} } & {B_{ts}^{I} } & {B_{rs}^{I} } & {A_{1te} } & {A_{1re} } \\ \end{array} \hfill \\ \begin{array}{*{20}c} {B_{1te} } & {B_{1re} } & {A_{2te} } & {A_{2re} } & {B_{2te} } & {B_{2re} } & {A_{tp1}^{II} } & {A_{tp2}^{II} } & {A_{tp3}^{II} } & {B_{ts}^{{II}{}} } \\ \end{array} \hfill \\ \end{gathered} \right]^{T} = \hfill \\ \left[ \begin{gathered} \begin{array}{*{20}c} {q_{0} \frac{\sin (\xi l)}{{\xi l}}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} \hfill \\ \begin{array}{*{20}c} 0 & 0 & 0 & 0 \\ \end{array} \hfill \\ \end{gathered} \right]^{T} \hfill \\ \end{gathered} $$

The elements not 0 in matrix T are:

$T_{0101} = T_{0104} = \chi_{1}$, $T_{0102} = T_{0105} = \chi_{2}$, $T_{0103} = T_{0106} = \chi_{3}$, $T_{0107} = { - }2\mu_{p} {\text{i}}\xi r$, $T_{0108} = 2\mu_{p} {\text{i}}\xi r$, $T_{0201} = { - }2\mu_{p} {\text{i}}\xi \lambda_{1}$, $T_{0202} = { - }2\mu_{p} {\text{i}}\xi \lambda_{2}$, $T_{0203} = { - }2\mu_{p} {\text{i}}\xi \lambda_{3}$, $T_{0204} = 2\mu_{p} {\text{i}}\xi \lambda_{1}$, $T_{0205} = 2\mu_{p} {\text{i}}\xi \lambda_{2}$, $T_{0206} = 2\mu_{p} {\text{i}}\xi \lambda_{3}$, $T_{0207} = T_{0208} { = - }\mu_{p} \left( {r^{2} + \xi^{2} } \right)$, $T_{0301} = T_{0304} { = }\left( {a_{11} + a_{12} \delta_{p1}^{l} + a_{13} \delta_{p1}^{g} } \right)(\xi^{2} - \lambda_{1}^{2} )$, $T_{0302} = T_{0305} { = }\left( {a_{11} + a_{12} \delta_{p2}^{l} + a_{13} \delta_{p2}^{g} } \right)(\xi^{2} - \lambda_{2}^{2} )$, $T_{0303} = T_{0306} { = }\left( {a_{11} + a_{12} \delta_{p3}^{l} + a_{13} \delta_{p3}^{g} } \right)(\xi^{2} - \lambda_{3}^{2} )$, $T_{0401} = T_{0304} { = }\left( {a_{21} + a_{22} \delta_{p1}^{l} + a_{23} \delta_{p1}^{g} } \right)(\xi^{2} - \lambda_{1}^{2} )$, $T_{0402} = T_{0405} { = }\left( {a_{21} + a_{22} \delta_{p2}^{l} + a_{23} \delta_{p2}^{g} } \right)(\xi^{2} - \lambda_{2}^{2} )$, $T_{0403} = T_{0406} { = }\left( {a_{21} + a_{22} \delta_{p3}^{l} + a_{23} \delta_{p3}^{g} } \right)(\xi^{2} - \lambda_{3}^{2} )$, $T_{0501} { = }\chi_{1} {\text{e}}^{{{ - }\lambda_{1} H}}$, $T_{0502} { = }\chi_{2} {\text{e}}^{{{ - }\lambda_{2} H}}$, $T_{0503} { = }\chi_{3} {\text{e}}^{{{ - }\lambda_{3} H}}$, $T_{0504} { = }\chi_{1} {\text{e}}^{{\lambda_{1} H}}$, $T_{0505} { = }\chi_{2} {\text{e}}^{{\lambda_{2} H}}$, $T_{0506} { = }\chi_{3} {\text{e}}^{{\lambda_{3} H}}$, $T_{0507} = { - }2\mu_{p} {\text{i}}\xi r{\text{e}}^{ - rH}$, $T_{0508} = 2\mu_{p} {\text{i}}\xi r{\text{e}}^{rH}$, $T_{0509} = \left[ {\lambda_{1e} \xi^{2} + \alpha_{1e}^{2} (\lambda_{1e} + 2\mu_{1e} )} \right]{\text{e}}^{{ - {\text{i}}\alpha_{1e} H}}$, $T_{0510} = \left[ {\lambda_{1e} \xi^{2} + \alpha_{1e}^{2} (\lambda_{1e} + 2\mu_{1e} )} \right]{\text{e}}^{{{\text{i}}\alpha_{1e} H}}$, $T_{0511} = { - 2}\mu_{1e} \xi \beta_{1e} {\text{e}}^{{ - {\text{i}}\beta_{1e} H}}$, $T_{0512} = {2}\mu_{1e} \xi \beta_{1e} {\text{e}}^{{{\text{i}}\beta_{1e} H}}$, $T_{0601} = { - }2\mu_{p} {\text{i}}\xi \lambda_{1} {\text{e}}^{{ - \lambda_{1} H}}$, $T_{0602} = { - }2\mu_{p} {\text{i}}\xi \lambda_{2} {\text{e}}^{{ - \lambda_{2} H}}$, $T_{0603} = { - }2\mu_{p} {\text{i}}\xi \lambda_{3} {\text{e}}^{{ - \lambda_{3} H}}$, $T_{0604} = 2\mu_{p} {\text{i}}\xi \lambda_{1} {\text{e}}^{{\lambda_{1} H}}$, $T_{0605} = 2\mu_{p} {\text{i}}\xi \lambda_{2} {\text{e}}^{{\lambda_{2} H}}$, $T_{0606} = 2\mu_{p} {\text{i}}\xi \lambda_{3} {\text{e}}^{{\lambda_{3} H}}$, $T_{0607} = { - }\mu_{p} \left( {r^{2} + \xi^{2} } \right){\text{e}}^{{{ - }rH}}$, $T_{0608} = { - }\mu_{p} \left( {r^{2} + \xi^{2} } \right){\text{e}}^{rH}$, $T_{0609} = { - 2}\mu_{1e} \xi \alpha_{1e} {\text{e}}^{{{\text{ - i}}\alpha_{1e} H}}$, $T_{0610} = {2}\mu_{1e} \xi \alpha_{1e} {\text{e}}^{{{\text{i}}\alpha_{1e} H}}$, $T_{0611} = \mu_{1e} \left( {\xi^{2} { - }\beta_{1e}^{2} } \right){\text{e}}^{{{\text{ - i}}\beta_{1e} H}}$, $T_{0612} = \mu_{1e} \left( {\xi^{2} { - }\beta_{1e}^{2} } \right){\text{e}}^{{{\text{i}}\beta_{1e} H}}$, $T_{0701} = { - }\lambda_{1} {\text{e}}^{{ - \lambda_{1} H}}$, $T_{0701} = { - }\lambda_{2} {\text{e}}^{{ - \lambda_{2} H}}$, $T_{0703} = { - }\lambda_{3} {\text{e}}^{{ - \lambda_{3} H}}$, $T_{0704} = \lambda_{1} {\text{e}}^{{\lambda_{1} H}}$, $T_{0705} = \lambda_{2} {\text{e}}^{{\lambda_{2} H}}$, $T_{0706} = \lambda_{3} {\text{e}}^{{\lambda_{3} H}}$, $T_{0707} = {\text{i}}\xi {\text{e}}^{ - rH}$, $T_{0708} = {\text{i}}\xi {\text{e}}^{rH}$, $T_{0709} = {\text{i}}\alpha_{1e} {\text{e}}^{{ - {\text{i}}\alpha_{1e} H}}$, $T_{0710} = {\text{ - i}}\alpha_{1e} {\text{e}}^{{{\text{i}}\alpha_{1e} H}}$, $T_{0711} = {\text{ - i}}\xi {\text{e}}^{{{\text{ - i}}\beta_{1e} H}}$, $T_{0712} = {\text{ - i}}\xi {\text{e}}^{{{\text{i}}\beta_{1e} H}}$, $T_{0801} = {\text{i}}\xi {\text{e}}^{{ - \lambda_{1} H}}$, $T_{0802} = {\text{i}}\xi {\text{e}}^{{ - \lambda_{2} H}}$, $T_{0803} = {\text{i}}\xi {\text{e}}^{{ - \lambda_{3} H}}$, $T_{0804} = {\text{i}}\xi {\text{e}}^{{\lambda_{1} H}}$, $T_{0805} = {\text{i}}\xi {\text{e}}^{{\lambda_{2} H}}$, $T_{0806} = {\text{i}}\xi {\text{e}}^{{\lambda_{3} H}}$, $T_{0807} = r{\text{e}}^{ - rH}$, $T_{0808} = { - }r{\text{e}}^{rH}$, $T_{0809} = {\text{ - i}}\xi {\text{e}}^{{{\text{ - i}}\alpha_{1e} H}}$, $T_{0810} = {\text{ - i}}\xi {\text{e}}^{{{\text{i}}\alpha_{1e} H}}$, $T_{0811} = {\text{ - i}}\beta_{1e} {\text{e}}^{{ - {\text{i}}\beta_{1e} H}}$, $T_{0812} = {\text{i}}\beta_{1e} {\text{e}}^{{{\text{i}}\beta_{1e} H}}$, $T_{0901} = \lambda_{1} (1 - \delta_{p1}^{l} ){\text{e}}^{{ - \lambda_{1} H}}$, $T_{0902} = \lambda_{2} (1 - \delta_{p2}^{l} ){\text{e}}^{{ - \lambda_{2} H}}$, $T_{0903} = \lambda_{3} (1 - \delta_{p3}^{l} ){\text{e}}^{{ - \lambda_{3} H}}$, $T_{0904} = \lambda_{1} (\delta_{p1}^{l} - 1){\text{e}}^{{\lambda_{1} H}}$, $T_{0905} = \lambda_{2} (\delta_{p2}^{l} - 1){\text{e}}^{{\lambda_{2} H}}$, $T_{0906} = \lambda_{3} (\delta_{p3}^{l} - 1){\text{e}}^{{\lambda_{3} H}}$, $T_{0907} = {\text{i}}\xi \left( {\delta_{s}^{l} - 1} \right){\text{e}}^{ - rH}$, $T_{0908} = {\text{i}}\xi \left( {\delta_{s}^{l} - 1} \right){\text{e}}^{rH}$, $T_{1001} = \lambda_{1} (1 - \delta_{p1}^{g} ){\text{e}}^{{ - \lambda_{1} H}}$, $T_{1002} = \lambda_{2} (1 - \delta_{p2}^{g} ){\text{e}}^{{ - \lambda_{2} H}}$, $T_{1003} = \lambda_{3} (1 - \delta_{p3}^{g} ){\text{e}}^{{ - \lambda_{3} H}}$, $T_{1004} = \lambda_{1} (\delta_{p1}^{g} - 1){\text{e}}^{{\lambda_{1} H}}$, $T_{1005} = \lambda_{2} (\delta_{p2}^{g} - 1){\text{e}}^{{\lambda_{2} H}}$, $T_{1006} = \lambda_{3} (\delta_{p3}^{g} - 1){\text{e}}^{{\lambda_{3} H}}$, $T_{1007} = {\text{i}}\xi \left( {\delta_{s}^{g} - 1} \right){\text{e}}^{ - rH}$, $T_{1008} = {\text{i}}\xi \left( {\delta_{s}^{g} - 1} \right){\text{e}}^{rH}$, $T_{1109} = - \left[ {\lambda_{1e} \xi^{2} + \alpha_{1e}^{2} (\lambda_{1e} + 2\mu_{1e} )} \right]{\text{e}}^{{ - {\text{i}}\alpha_{1e} \left( {H + h_{w1} } \right)}}$, $T_{1110} = - \left[ {\lambda_{1e} \xi^{2} + \alpha_{1e}^{2} (\lambda_{1e} + 2\mu_{1e} )} \right]{\text{e}}^{{{\text{i}}\alpha_{1e} \left( {H + h_{w1} } \right)}}$, $T_{1111} = 2\mu_{1e} \xi \beta_{1e} {\text{e}}^{{ - {\text{i}}\beta_{1e} \left( {H + h_{w1} } \right)}}$, $T_{1112} = - 2\mu_{1e} \xi \beta_{1e} {\text{e}}^{{ - {\text{i}}\beta_{1e} \left( {H + h_{w1} } \right)}}$, $T_{1113} = \left[ {\lambda_{2e} \xi^{2} + \alpha_{2e}^{2} (\lambda_{2e} + 2\mu_{2e} )} \right]{\text{e}}^{{ - {\text{i}}\alpha_{2e} \left( {H + h_{w1} } \right)}}$, $T_{1114} = \left[ {\lambda_{2e} \xi^{2} + \alpha_{2e}^{2} (\lambda_{2e} + 2\mu_{2e} )} \right]{\text{e}}^{{{\text{i}}\alpha_{2e} \left( {H + h_{w1} } \right)}}$, $T_{1115} = - 2\mu_{2e} \xi \beta_{2e} {\text{e}}^{{ - {\text{i}}\beta_{2e} \left( {H + h_{w1} } \right)}}$, $T_{1115} = - 2\mu_{2e} \xi \beta_{2e} {\text{e}}^{{ - {\text{i}}\beta_{2e} \left( {H + h_{w1} } \right)}}$, $T_{1116} = 2\mu_{2e} \xi \beta_{2e} {\text{e}}^{{{\text{i}}\beta_{2e} \left( {H + h_{w1} } \right)}}$, $T_{1209} = 2\mu_{1e} \xi \alpha_{1e} {\text{e}}^{{ - {\text{i}}\alpha_{1e} \left( {H + h_{w1} } \right)}}$, $T_{1210} = - 2\mu_{1e} \xi \alpha_{1e} {\text{e}}^{{{\text{i}}\alpha_{1e} \left( {H + h_{w1} } \right)}}$, $T_{1211} = \mu_{1e} \left( {\beta_{1e}^{2} - \xi^{2} } \right){\text{e}}^{{{\text{ - i}}\beta_{1e} \left( {H + h_{w1} } \right)}}$, $T_{1212} = \mu_{1e} \left( {\beta_{1e}^{2} - \xi^{2} } \right){\text{e}}^{{{\text{i}}\beta_{1e} \left( {H + h_{w1} } \right)}}$, $T_{1213} = - 2\mu_{2e} \xi \alpha_{2e} {\text{e}}^{{ - {\text{i}}\alpha_{2e} \left( {H + h_{w1} } \right)}}$, $T_{1214} = 2\mu_{2e} \xi \alpha_{2e} {\text{e}}^{{{\text{i}}\alpha_{2e} \left( {H + h_{w1} } \right)}}$, $T_{1215} = \mu_{2e} \left( {\xi^{2} - \beta_{2e}^{2} } \right){\text{e}}^{{{\text{ - i}}\beta_{2e} \left( {H + h_{w1} } \right)}}$, $T_{1216} = \mu_{2e} \left( {\xi^{2} - \beta_{2e}^{2} } \right){\text{e}}^{{{\text{i}}\beta_{2e} \left( {H + h_{w1} } \right)}}$, $T_{1309} = - {\text{i}}\alpha_{1e} {\text{e}}^{{ - {\text{i}}\alpha_{1e} \left( {H + h_{w1} } \right)}}$, $T_{1310} = {\text{i}}\alpha_{1e} {\text{e}}^{{{\text{i}}\alpha_{1e} \left( {H + h_{w1} } \right)}}$, $T_{1311} = {\text{i}}\xi {\text{e}}^{{ - {\text{i}}\beta_{1e} \left( {H + h_{w1} } \right)}}$, $T_{1312} = {\text{i}}\xi {\text{e}}^{{{\text{i}}\beta_{1e} \left( {H + h_{w1} } \right)}}$, $T_{1313} = {\text{i}}\alpha_{2e} {\text{e}}^{{ - {\text{i}}\alpha_{2e} \left( {H + h_{w1} } \right)}}$, $T_{1314} = - {\text{i}}\alpha_{2e} {\text{e}}^{{{\text{i}}\alpha_{2e} \left( {H + h_{w1} } \right)}}$, $T_{1315} = - {\text{i}}\xi {\text{e}}^{{ - {\text{i}}\beta_{2e} \left( {H + h_{w1} } \right)}}$, $T_{1316} = - {\text{i}}\xi {\text{e}}^{{{\text{i}}\beta_{2e} \left( {H + h_{w1} } \right)}}$, $T_{1409} = {\text{i}}\xi {\text{e}}^{{ - {\text{i}}\alpha_{1e} \left( {H + h_{w1} } \right)}}$, $T_{1410} = {\text{i}}\xi {\text{e}}^{{{\text{i}}\alpha_{1e} \left( {H + h_{w1} } \right)}}$, $T_{1411} = {\text{i}}\beta_{1e} {\text{e}}^{{ - {\text{i}}\beta_{1e} \left( {H + h_{w1} } \right)}}$, $T_{1412} = - {\text{i}}\beta_{1e} {\text{e}}^{{{\text{i}}\beta_{1e} \left( {H + h_{w1} } \right)}}$, $T_{1413} = - {\text{i}}\xi {\text{e}}^{{{\text{ - i}}\alpha_{2e} \left( {H + h_{w1} } \right)}}$, $T_{1414} = - {\text{i}}\xi {\text{e}}^{{{\text{i}}\alpha_{2e} \left( {H + h_{w1} } \right)}}$, $T_{1415} = - {\text{i}}\beta_{2e} {\text{e}}^{{ - {\text{i}}\beta_{2e} \left( {H + h_{w1} } \right)}}$, $T_{1416} = {\text{i}}\beta_{2e} {\text{e}}^{{{\text{i}}\beta_{2e} \left( {H + h_{w1} } \right)}}$, $T_{1513} = { - }\left[ {\lambda_{2e} \xi^{2} { + }\alpha_{2e}^{2} (\lambda_{2e} + 2\mu_{2e} )} \right]{\text{e}}^{{ - {\text{i}}\alpha_{2e} \left( {H{ + }h_{w1} + h_{w2} } \right)}}$, $T_{1514} = { - }\left[ {\lambda_{2e} \xi^{2} { + }\alpha_{2e}^{2} (\lambda_{2e} + 2\mu_{2e} )} \right]{\text{e}}^{{{\text{i}}\alpha_{2e} \left( {H{ + }h_{w1} + h_{w2} } \right)}}$, $T_{1515} = {2}\mu_{2e} \xi \beta_{2e} {\text{e}}^{{ - {\text{i}}\beta_{2e} \left( {H{ + }h_{w1} + h_{w2} } \right)}}$, $T_{1516} = - {2}\mu_{2e} \xi \beta_{2e} {\text{e}}^{{{\text{i}}\beta_{2e} \left( {H{ + }h_{w1} + h_{w2} } \right)}}$, $T_{1517} = - \chi_{1} {\text{e}}^{{{ - }\lambda_{1} \left( {H{ + }h_{w1} + h_{w2} } \right)}}$, $T_{1518} = - \chi_{2} {\text{e}}^{{{ - }\lambda_{2} \left( {H{ + }h_{w1} + h_{w2} } \right)}}$, $T_{1519} = - \chi_{3} {\text{e}}^{{{ - }\lambda_{3} \left( {H{ + }h_{w1} + h_{w2} } \right)}}$, $T_{1520} = 2\mu_{p} {\text{i}}\xi r{\text{e}}^{{ - r\left( {H{ + }h_{w1} + h_{w2} } \right)}}$, $T_{1613} = {2}\mu_{2e} \xi \alpha_{2e} {\text{e}}^{{{\text{ - i}}\alpha_{2e} \left( {H{ + }h_{w1} + h_{w2} } \right)}}$, $T_{1614} = - {2}\mu_{2e} \xi \alpha_{2e} {\text{e}}^{{{\text{i}}\alpha_{2e} \left( {H{ + }h_{w1} + h_{w2} } \right)}}$, $T_{1615} = \mu_{2e} \left( {\beta_{2e}^{2} { - }\xi^{2} } \right){\text{e}}^{{{\text{ - i}}\beta_{2e} \left( {H{ + }h_{w1} + h_{w2} } \right)}}$, $T_{1616} = \mu_{2e} \left( {\beta_{2e}^{2} { - }\xi^{2} } \right){\text{e}}^{{{\text{i}}\beta_{2e} \left( {H{ + }h_{w1} + h_{w2} } \right)}}$, $T_{1617} = 2\mu_{p} {\text{i}}\xi \lambda_{1} {\text{e}}^{{ - \lambda_{1} \left( {H{ + }h_{w1} + h_{w2} } \right)}}$, $T_{1618} = 2\mu_{p} {\text{i}}\xi \lambda_{2} {\text{e}}^{{ - \lambda_{2} \left( {H{ + }h_{w1} + h_{w2} } \right)}}$, $T_{1619} = 2\mu_{p} {\text{i}}\xi \lambda_{3} {\text{e}}^{{ - \lambda_{3} \left( {H{ + }h_{w1} + h_{w2} } \right)}}$, $T_{1620} = \mu_{p} \left( {r^{2} + \xi^{2} } \right){\text{e}}^{{{ - }r\left( {H{ + }h_{w1} + h_{w2} } \right)}}$, $T_{1713} = {\text{ - i}}\alpha_{2e} {\text{e}}^{{ - {\text{i}}\alpha_{2e} \left( {H{ + }h_{w1} + h_{w2} } \right)}}$, $T_{1714} = {\text{i}}\alpha_{2e} {\text{e}}^{{{\text{i}}\alpha_{2e} \left( {H{ + }h_{w1} + h_{w2} } \right)}}$, $T_{1715} = {\text{i}}\xi {\text{e}}^{{{\text{ - i}}\beta_{2e} \left( {H{ + }h_{w1} + h_{w2} } \right)}}$, $T_{1716} = {\text{i}}\xi {\text{e}}^{{{\text{i}}\beta_{2e} \left( {H{ + }h_{w1} + h_{w2} } \right)}}$, $T_{1717} = \lambda_{1} {\text{e}}^{{ - \lambda_{1} \left( {H{ + }h_{w1} + h_{w2} } \right)}}$, $T_{1718} = \lambda_{2} {\text{e}}^{{ - \lambda_{2} \left( {H{ + }h_{w1} + h_{w2} } \right)}}$, $T_{1719} = \lambda_{3} {\text{e}}^{{ - \lambda_{3} \left( {H{ + }h_{w1} + h_{w2} } \right)}}$, $T_{1720} = - {\text{i}}\xi {\text{e}}^{{ - r\left( {H{ + }h_{w1} + h_{w2} } \right)}}$, $T_{1813} = {\text{i}}\xi {\text{e}}^{{{\text{ - i}}\alpha_{2e} \left( {H{ + }h_{w1} + h_{w2} )} \right)}}$, $T_{1814} = {\text{i}}\xi {\text{e}}^{{{\text{i}}\alpha_{2e} \left( {H{ + }h_{w1} + h_{w2} } \right)}}$, $T_{1815} = {\text{i}}\beta_{2e} {\text{e}}^{{ - {\text{i}}\beta_{2e} \left( {H{ + }h_{w1} + h_{w2} } \right)}}$, $T_{1816} = - {\text{i}}\beta_{2e} {\text{e}}^{{{\text{i}}\beta_{2e} \left( {H{ + }h_{w1} + h_{w2} } \right)}}$, $T_{1817} = {\text{ - i}}\xi {\text{e}}^{{ - \lambda_{1} \left( {H{ + }h_{w1} + h_{w2} } \right)}}$, $T_{1818} = {\text{ - i}}\xi {\text{e}}^{{ - \lambda_{2} \left( {H{ + }h_{w1} + h_{w2} } \right)}}$, $T_{1819} = {\text{ - i}}\xi {\text{e}}^{{ - \lambda_{3} \left( {H{ + }h_{w1} + h_{w2} } \right)}}$, $T_{1820} = { - }r{\text{e}}^{{ - r\left( {H{ + }h_{w1} + h_{w2} } \right)}}$, $T_{1917} = \lambda_{1} {\text{e}}^{{ - \lambda_{1} \left( {H{ + }h_{w1} + h_{w2} } \right)}} (\delta_{p1}^{l} { - }1)$, $T_{1918} = \lambda_{2} {\text{e}}^{{ - \lambda_{2} \left( {H{ + }h_{w1} + h_{w2} } \right)}} (\delta_{p2}^{l} { - }1)$, $T_{1919} = \lambda_{3} {\text{e}}^{{ - \lambda_{3} \left( {H{ + }h_{w1} + h_{w2} } \right)}} (\delta_{p3}^{l} { - }1)$, $T_{1920} = {\text{i}}\xi {\text{e}}^{{ - r\left( {H{ + }h_{w1} + h_{w2} } \right)}} \left( {1{ - }\delta_{s}^{l} } \right)$, $T_{2017} = \lambda_{1} {\text{e}}^{{ - \lambda_{1} \left( {H{ + }h_{w1} + h_{w2} } \right)}} (\delta_{p1}^{g} { - }1)$, $T_{2018} = \lambda_{2} {\text{e}}^{{ - \lambda_{2} \left( {H{ + }h_{w1} + h_{w2} } \right)}} (\delta_{p2}^{g} { - }1)$, $T_{2019} = \lambda_{3} {\text{e}}^{{ - \lambda_{3} \left( {H{ + }h_{w1} + h_{w2} } \right)}} (\delta_{p3}^{g} { - }1)$, $T_{2020} = {\text{i}}\xi {\text{e}}^{{ - r\left( {H{ + }h_{w1} + h_{w2} } \right)}} \left( {1{ - }\delta_{s}^{g} } \right)$.