Analysis of Vibration Isolation Effect of Double-Layer WIB Based on Wave Impedance Ratio

Abstract

Based on single-phase elastic medium and unsaturated porous medium theory, the vibration isolation effect of double-layer wave impeding block (WIB) in unsaturated ground under an underground dynamic load is studied. The results show that the optimal vibration isolation effect can be obtained by designing the wave impedance ratio at the intersection between the layers of the double-layer WIB. In the case of the same thickness, the vibration isolation effect of the designed double-layer WIB is not only much better than that of the homogeneous WIB, but also improves the characteristic that the homogeneous WIB is only good at low frequency vibration isolation. Soil saturation has a significant effect on the vibration isolation effect of double-layer WIB in unsaturated soil foundation, and a better vibration isolation effect can be achieved in the case of lower saturation.

Appendices

Appendix 1

$$\begin{aligned} \beta _{1} = & - B_{1} a_{{12}} a_{{23}} + B_{1} a_{{13}} a_{{22}} + B_{2} a_{{11}} a_{{23}} - B_{2} a_{{13}} a_{{21}} \\ & + \,B_{3} a_{{11}} a_{{22}} - B_{3} a_{{12}} a_{{21}} - \mu _{p} a_{{12}} a_{{23}} + \mu _{p} a_{{12}} a_{{22}} \\ \end{aligned}$$

$$\begin{aligned} \beta_{2} = & - B_{1} a_{12} b_{33} + B_{1} a_{13} b_{32} + B_{1} a_{22} b_{23} - B_{1} a_{23} b_{22} + B_{2} a_{11} b_{33} - B_{2} a_{13} b_{31} - B_{2} a_{21} b_{23} + B_{2} a_{23} b_{21} - B_{3} a_{11} b_{32} + B_{3} a_{12} b_{31} + B_{3} a_{21} b_{22} - B_{3} a_{22} b_{21} \\ & - a_{11} a_{22} b_{13} + a_{11} a_{23} b_{12} + a_{12} a_{21} b_{13} - a_{12} a_{23} b_{11} - a_{13} a_{21} b_{12} + a_{13} a_{22} b_{11} - \mu_{p} a_{12} b_{33} + \mu_{p} a_{13} b_{32} + \mu_{p} a_{22} b_{23} - \mu_{p} a_{23} b_{22} \\ \end{aligned}$$

$$\begin{aligned} \beta_{3} = & - \,B_{1} b_{22} b_{33} + B_{1} b_{23} b_{32} + B_{2} b_{21} b_{33} - B_{2} b_{23} b_{31} - B_{3} b_{21} b_{32} + B_{3} b_{22} b_{31} + a_{11} b_{12} b_{33} - a_{11} b_{13} b_{32} - a_{12} b_{11} b_{33} + a_{12} b_{13} b_{31} + a_{13} b_{11} b_{32} - a_{13} b_{12} b_{31} \\ & - \,a_{21} b_{12} b_{23} + a_{21} b_{13} b_{22} + a_{22} b_{11} b_{23} - a_{22} b_{13} b_{21} - a_{23} b_{11} b_{22} + a_{23} b_{12} b_{21} - \mu_{p} b_{22} b_{33} + \mu_{p} b_{23} b_{32} \\ \end{aligned}$$

$$\beta_{4} = - b_{11} b_{22} b_{33} + b_{11} b_{23} b_{32} + b_{12} b_{21} b_{33} - b_{12} b_{23} b_{31} - b_{13} b_{21} b_{32} + b_{13} b_{22} b_{31}$$

$$\beta_{5} = - \mu_{p} d_{22} d_{33}$$

$$\beta_{6} = - \,d_{11} d_{22} d_{33} + d_{21} d_{12} d_{33} + d_{13} d_{22} d_{31}$$

Appendix 2

The expression of the system of Eqs. (37) are as follows:

$$\begin{gathered} T_{28 \times 28} [A_{tp1}^{{\text{I}}} \quad A_{tp2}^{{\text{I}}} \quad A_{tp3}^{{\text{I}}} \quad A_{rp1}^{{\text{I}}} \quad A_{rp2}^{{\text{I}}} \quad A_{rp3}^{{\text{I}}} \quad B_{ts}^{{\text{I}}} \quad B_{rs}^{{\text{I}}} \quad A_{1te} \quad A_{1re} \quad B_{1te} \quad B_{1re} \quad A_{2te} \quad A_{2re} \hfill \\ B_{2te} \quad B_{2re} \quad A_{tp1}^{{{\text{II}}}} \quad A_{tp2}^{{{\text{II}}}} \quad A_{tp3}^{{{\text{II}}}} \quad A_{rp1}^{{{\text{II}}}} \quad A_{rp2}^{{{\text{II}}}} \quad A_{rp3}^{{{\text{II}}}} \quad B_{ts}^{{{\text{II}}}} \quad B_{rs}^{{{\text{II}}}} \quad A_{tp1}^{{{\text{III}}}} \quad A_{tp2}^{{{\text{III}}}} \quad A_{tp3}^{{{\text{III}}}} \quad B_{ts}^{{{\text{III}}}} ]^{{\text{T}}} \hfill \\ = [0\quad 0\quad 0\quad 0\quad 0\quad 0\quad 0\quad 0\quad 0\quad 0\quad 0\quad 0\quad 0\quad 0\quad 0\quad 0\quad 0\quad 0\quad 0\quad 0\quad 0\quad 0\quad 0\quad 0 \hfill \\ q_{0} \frac{\sin (\xi l)}{{\xi l}}\quad 0\quad 0\quad 0]^{{\text{T}}} \hfill \\ \end{gathered}$$