Appendix A
The elements of matrix \(\mathbf{K}\)s1 in Eq. (1) are expressed as:
$$ \mathbf{K}_{\mathrm{s}1}\!=\! \left [\!\! \textstyle\begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} K_{\mathrm{s}1}^{\left ( 1,1 \right )} & 0 & K_{\mathrm{s}1}^{\left ( 1,3 \right )} & K_{\mathrm{s}1}^{\left ( 1,4 \right )} & 0 & 0 & K_{\mathrm{s}1}^{\left ( 1,7 \right )} & 0 & K_{\mathrm{s}1}^{\left ( 1,9 \right )} & 0 & 0 & 0 \\ 0 & K_{\mathrm{s}1}^{\left ( 2,2 \right )} & K_{\mathrm{s}1}^{\left ( 2,3 \right )} & 0 & K_{\mathrm{s}1}^{\left ( 2,5 \right )} & K_{\mathrm{s}1}^{\left ( 2,6 \right )} & 0 & 0 & 0 & 0 & 0 & 0 \\ K_{\mathrm{s}1}^{\left ( 3,1 \right )} & K_{\mathrm{s}1}^{\left ( 3,2 \right )} & K_{\mathrm{s}1}^{\left ( 3,3 \right )} & 0 & K_{\mathrm{s}1}^{\left ( 3,5 \right )} & K_{\mathrm{s}1}^{\left ( 3,6 \right )} & K_{\mathrm{s}1}^{\left ( 3,7 \right )} & 0 & K_{\mathrm{s}1}^{\left ( 3,9 \right )} & 0 & 0 & 0 \\ K_{\mathrm{s}1}^{\left ( 4,1 \right )} & 0 & 0 & K_{\mathrm{s}1}^{\left ( 4,4 \right )} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & K_{\mathrm{s}1}^{\left ( 5,2 \right )} & K_{\mathrm{s}1}^{\left ( 5,3 \right )} & 0 & K_{\mathrm{s}1}^{\left ( 5,5 \right )} & K_{\mathrm{s}1}^{\left ( 5,6 \right )} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & K_{\mathrm{s}1}^{\left ( 6,2 \right )} & K_{\mathrm{s}1}^{\left ( 6,3 \right )} & 0 & K_{\mathrm{s}1}^{\left ( 6,5 \right )} & K_{\mathrm{s}1}^{\left ( 6,6 \right )} & 0 & 0 & 0 & 0 & 0 & 0 \\ K_{\mathrm{s}1}^{\left ( 7,1 \right )} & 0 & K_{\mathrm{s}1}^{\left ( 7,3 \right )} & 0 & 0 & 0 & K_{\mathrm{s}1}^{\left ( 7,7 \right )} & 0 & K_{\mathrm{s}1}^{\left ( 7,9 \right )} & K_{\mathrm{s}1}^{\left ( 7,10 \right )} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & K_{\mathrm{s}1}^{\left ( 8,8 \right )} & K_{\mathrm{s}1}^{\left ( 8,9 \right )} & 0 & K_{\mathrm{s}1}^{\left ( 8,11 \right )} & K_{\mathrm{s}1}^{\left ( 8,12 \right )} \\ K_{\mathrm{s}1}^{\left ( 9,1 \right )} & 0 & K_{\mathrm{s}1}^{\left ( 9,3 \right )} & 0 & 0 & 0 & K_{\mathrm{s}1}^{\left ( 9,7 \right )} & K_{\mathrm{s}1}^{\left ( 9,8 \right )} & K_{\mathrm{s}1}^{\left ( 9,9 \right )} & 0 & K_{\mathrm{s}1}^{\left ( 9,11 \right )} & K_{\mathrm{s}1}^{\left ( 9,12 \right )} \\ 0 & 0 & 0 & 0 & 0 & 0 & K_{\mathrm{s}1}^{\left ( 10,7 \right )} & 0 & 0 & K_{\mathrm{s}1}^{\left ( 10,10 \right )} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & K_{\mathrm{s}1}^{\left ( 11,8 \right )} & K_{\mathrm{s}1}^{\left ( 11,9 \right )} & 0 & K_{\mathrm{s}1}^{\left ( 11,11 \right )} & K_{\mathrm{s}1}^{\left ( 11,12 \right )} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & K_{\mathrm{s}1}^{\left ( 12,8 \right )} & K_{\mathrm{s}1}^{\left ( 12,9 \right )} & 0 & K_{\mathrm{s}1}^{\left ( 12,11 \right )} & K_{\mathrm{s}1}^{\left ( 12,12 \right )} \end{array}\displaystyle \!\! \right ]\!, $$
(A.1)
where
$$ K_{\mathrm{s}1}^{\left ( 1,1 \right )} = K_{\mathrm{s}1}^{\left ( 7,7 \right )} = 2k_{\mathrm{n}1} + k_{\mathrm{n}2} + k_{\mathrm{s}3}, $$
(A.2)
$$\begin{aligned} K_{\mathrm{s}1}^{\left ( 3,9 \right )} =& K_{\mathrm{s}1}^{\left ( 9,3 \right )} = \frac{a}{2}K_{\mathrm{s}1}^{\left ( 3,7 \right )} = \frac{a}{2}K_{\mathrm{s}1}^{\left ( 7,3 \right )} = \frac{a}{2}K_{\mathrm{s}1}^{\left ( 7,9 \right )} = \frac{a}{2}K_{\mathrm{s}1}^{\left ( 9,7 \right )} = - \frac{a}{2}K_{\mathrm{s}1}^{\left ( 1,3 \right )} = - \frac{a}{2}K_{\mathrm{s}1}^{\left ( 3,1 \right )} \\ =& - \frac{a}{2}K_{\mathrm{s}1}^{\left ( 1,9 \right )} = - \frac{a}{2}K_{\mathrm{s}1}^{\left ( 9,1 \right )} = - \frac{a^{2}}{4}K_{\mathrm{s}1}^{\left ( 1,7 \right )} = - \frac{a^{2}}{4}K_{\mathrm{s}1}^{\left ( 7,1 \right )} = \frac{k_{\mathrm{s}3}a^{2}}{4}, \end{aligned}$$
(A.3)
$$ K_{\mathrm{s}1}^{\left ( 4,4 \right )} = K_{\mathrm{s}1}^{\left ( 10,10 \right )} = - K_{\mathrm{s}1}^{\left ( 1,4 \right )} = - K_{\mathrm{s}1}^{\left ( 4,1 \right )} = - K_{\mathrm{s}1}^{\left ( 7,10 \right )} = - K_{\mathrm{s}1}^{\left ( 10,7 \right )} = k_{\mathrm{n}2}, $$
(A.4)
$$ K_{\mathrm{s}1}^{\left ( 2,2 \right )} = K_{\mathrm{s}1}^{\left ( 8,8 \right )} = 2k_{\mathrm{s}1} + k_{\mathrm{s}2}, $$
(A.5)
$$ K_{\mathrm{s}1}^{\left ( 6,6 \right )} = K_{\mathrm{s}1}^{\left ( 12,12 \right )} = - K_{\mathrm{s}1}^{\left ( 3,6 \right )} = - K_{\mathrm{s}1}^{\left ( 6,3 \right )} = - K_{\mathrm{s}1}^{\left ( 12,9 \right )} = - K_{\mathrm{s}1}^{\left ( 9,12 \right )} = k_{\mathrm{r}2} + \frac{k_{\mathrm{s}2}r^{2}}{4}, $$
(A.6)
$$\begin{aligned} K_{\mathrm{s}1}^{\left ( 2,3 \right )} =& K_{\mathrm{s}1}^{\left ( 3,2 \right )} = K_{\mathrm{s}1}^{\left ( 5,6 \right )} = K_{\mathrm{s}1}^{\left ( 6,5 \right )} = K_{\mathrm{s}1}^{\left ( 8,9 \right )} = K_{\mathrm{s}1}^{\left ( 9,8 \right )} = K_{\mathrm{s}1}^{\left ( 12,11 \right )} = K_{\mathrm{s}1}^{\left ( 11,12 \right )} = - K_{\mathrm{s}1}^{\left ( 2,6 \right )} \\ =& - K_{\mathrm{s}1}^{\left ( 6,2 \right )} = - K_{\mathrm{s}1}^{\left ( 3,5 \right )} = - K_{\mathrm{s}1}^{\left ( 5,3 \right )} = - K_{\mathrm{s}1}^{\left ( 8,12 \right )} = - K_{\mathrm{s}1}^{\left ( 12,8 \right )} = - K_{\mathrm{s}1}^{\left ( 9,11 \right )} = - K_{\mathrm{s}1}^{\left ( 11,9 \right )} \\ =& \frac{r}{2}K_{\mathrm{s}1}^{\left ( 5,5 \right )} = \frac{r}{2}K_{\mathrm{s}1}^{\left ( 11,11 \right )} = - \frac{r}{2}K_{\mathrm{s}1}^{\left ( 2,5 \right )} = - \frac{r}{2}K_{\mathrm{s}1}^{\left ( 5,2 \right )} \\ =& - \frac{r}{2}K_{\mathrm{s}1}^{\left ( 8,11 \right )} = - \frac{r}{2}K_{\mathrm{s}1}^{\left ( 11,8 \right )} = \frac{k_{\mathrm{s}2}r}{2}, \end{aligned}$$
(A.7)
$$ K_{\mathrm{s}1}^{\left ( 3,3 \right )} = K_{\mathrm{s}1}^{\left ( 9,9 \right )} = 2k_{\mathrm{r}1} + k_{\mathrm{r}2} + \frac{k_{\mathrm{s}1}a^{2}}{2} + \frac{k_{\mathrm{s}2}r^{2}}{4} + \frac{k_{\mathrm{s}3}a^{2}}{4}. $$
(A.8)
Appendix B
The elements of matrix \(\mathbf{K}\)s2 in Eq. (1) are expressed as:
$$ \mathbf{K}_{\mathrm{s}2} = \left [ \textstyle\begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} K_{\mathrm{s}2}^{\left ( 1,1 \right )} & K_{\mathrm{s}2}^{\left ( 1,1 \right )} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & K_{\mathrm{s}2}^{\left ( 2,3 \right )} & K_{\mathrm{s}2}^{\left ( 2,4 \right )} & K_{\mathrm{s}2}^{\left ( 2,5 \right )} & K_{\mathrm{s}2}^{\left ( 2,6 \right )} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & K_{\mathrm{s}2}^{\left ( 3,3 \right )} & K_{\mathrm{s}2}^{\left ( 3,4 \right )} & K_{\mathrm{s}2}^{\left ( 3,5 \right )} & K_{\mathrm{s}2}^{\left ( 3,6 \right )} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & K_{\mathrm{s}2}^{\left ( 7,7 \right )} & K_{\mathrm{s}2}^{\left ( 7,8 \right )} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & K_{\mathrm{s}2}^{\left ( 8,9 \right )} & K_{\mathrm{s}2}^{\left ( 8,10 \right )} & K_{\mathrm{s}2}^{\left ( 8,11 \right )} & K_{\mathrm{s}2}^{\left ( 8,12 \right )} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & K_{\mathrm{s}2}^{\left ( 9,9 \right )} & K_{\mathrm{s}2}^{\left ( 9,10 \right )} & K_{\mathrm{s}2}^{\left ( 9,11 \right )} & K_{\mathrm{s}2}^{\left ( 9,12 \right )} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array}\displaystyle \right ], $$
(B.1)
where
$$ K_{\mathrm{s}2}^{\left ( 1,1 \right )} = K_{\mathrm{s}2}^{\left ( 1,2 \right )} = K_{\mathrm{s}2}^{\left ( 7,7 \right )} = K_{\mathrm{s}2}^{\left ( 7,8 \right )} = - k_{\mathrm{n}1}, $$
(B.2)
$$\begin{aligned} K_{\mathrm{s}2}^{\left ( 2,6 \right )} =& K_{\mathrm{s}2}^{\left ( 3,3 \right )} = K_{\mathrm{s}2}^{\left ( 8,12 \right )} = K_{\mathrm{s}2}^{\left ( 9,9 \right )} = - K_{\mathrm{s}2}^{\left ( 8,11 \right )} = - K_{\mathrm{s}2}^{\left ( 3,4 \right )} = - K_{\mathrm{s}2}^{\left ( 2,5 \right )} = - K_{\mathrm{s}2}^{\left ( 9,10 \right )} = - \frac{a}{2}K_{\mathrm{s}2}^{\left ( 2,3 \right )} \\ =& - \frac{a}{2}K_{\mathrm{s}2}^{\left ( 2,4 \right )} = - \frac{a}{2}K_{\mathrm{s}2}^{\left ( 8,9 \right )} = - \frac{a}{2}K_{\mathrm{s}2}^{\left ( 8,10 \right )} = \frac{k_{\mathrm{s}1}a}{2}, \end{aligned}$$
(B.3)
$$ K_{\mathrm{s}2}^{\left ( 3,5 \right )} = K_{\mathrm{s}2}^{\left ( 3,6 \right )} = K_{\mathrm{s}2}^{\left ( 9,11 \right )} = K_{\mathrm{s}2}^{\left ( 9,12 \right )} = - k_{\mathrm{r}1} + \frac{k_{\mathrm{s}1}a^{2}}{4}. $$
(B.4)
Appendix C
The elements of matrix \(\mathbf{D}\)s1 in Eq. (3) are expressed as:
$$ \mathbf{D}_{\mathrm{s}1}\! =\! \left [\!\! \textstyle\begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} D_{\mathrm{s}1}^{\left ( 1,1 \right )} & 0 & D_{\mathrm{s}1}^{\left ( 1,3 \right )} & D_{\mathrm{s}1}^{\left ( 1,4 \right )} & 0 & 0 & D_{\mathrm{s}1}^{\left ( 1,7 \right )} & 0 & D_{\mathrm{s}1}^{\left ( 1,9 \right )} & 0 & 0 & 0 \\ 0 & D_{\mathrm{s}1}^{\left ( 2,2 \right )} & D_{\mathrm{s}1}^{\left ( 2,3 \right )} & 0 & D_{\mathrm{s}1}^{\left ( 2,5 \right )} & D_{\mathrm{s}1}^{\left ( 2,6 \right )} & 0 & 0 & 0 & 0 & 0 & 0 \\ D_{\mathrm{s}1}^{\left ( 3,1 \right )} & D_{\mathrm{s}1}^{\left ( 3,2 \right )} & D_{\mathrm{s}1}^{\left ( 3,3 \right )} & 0 & D_{\mathrm{s}1}^{\left ( 3,5 \right )} & D_{\mathrm{s}1}^{\left ( 3,6 \right )} & D_{\mathrm{s}1}^{\left ( 3,7 \right )} & 0 & D_{\mathrm{s}1}^{\left ( 3,9 \right )} & 0 & 0 & 0 \\ D_{\mathrm{s}1}^{\left ( 4,1 \right )} & 0 & 0 & D_{\mathrm{s}1}^{\left ( 4,2 \right )} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & D_{\mathrm{s}1}^{\left ( 5,2 \right )} & D_{\mathrm{s}1}^{\left ( 5,3 \right )} & 0 & D_{\mathrm{s}1}^{\left ( 5,5 \right )} & D_{\mathrm{s}1}^{\left ( 5,6 \right )} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & D_{\mathrm{s}1}^{\left ( 6,2 \right )} & D_{\mathrm{s}1}^{\left ( 6,3 \right )} & 0 & D_{\mathrm{s}1}^{\left ( 6,5 \right )} & D_{\mathrm{s}1}^{\left ( 6,6 \right )} & 0 & 0 & 0 & 0 & 0 & 0 \\ D_{\mathrm{s}1}^{\left ( 7,1 \right )} & 0 & D_{\mathrm{s}1}^{\left ( 7,2 \right )} & 0 & 0 & 0 & D_{\mathrm{s}1}^{\left ( 7,7 \right )} & 0 & D_{\mathrm{s}1}^{\left ( 7,9 \right )} & D_{\mathrm{s}1}^{\left ( 7,10 \right )} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & D_{\mathrm{s}1}^{\left ( 8,8 \right )} & D_{\mathrm{s}1}^{\left ( 8,9 \right )} & 0 & D_{\mathrm{s}1}^{\left ( 8,11 \right )} & D_{\mathrm{s}1}^{\left ( 8,12 \right )} \\ D_{\mathrm{s}1}^{\left ( 9,1 \right )} & 0 & D_{\mathrm{s}1}^{\left ( 9,3 \right )} & 0 & 0 & 0 & D_{\mathrm{s}1}^{\left ( 9,7 \right )} & D_{\mathrm{s}1}^{\left ( 9,8 \right )} & D_{\mathrm{s}1}^{\left ( 9,9 \right )} & 0 & D_{\mathrm{s}1}^{\left ( 9,11 \right )} & D_{\mathrm{s}1}^{\left ( 9,12 \right )} \\ 0 & 0 & 0 & 0 & 0 & 0 & D_{\mathrm{s}1}^{\left ( 10,7 \right )} & 0 & 0 & D_{\mathrm{s}1}^{\left ( 10,10 \right )} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & D_{\mathrm{s}1}^{\left ( 11,8 \right )} & D_{\mathrm{s}1}^{\left ( 11,9 \right )} & 0 & D_{\mathrm{s}1}^{\left ( 11,11 \right )} & D_{\mathrm{s}1}^{\left ( 11,12 \right )} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & D_{\mathrm{s}1}^{\left ( 12,8 \right )} & D_{\mathrm{s}1}^{\left ( 12,9 \right )} & 0 & D_{\mathrm{s}1}^{\left ( 12,11 \right )} & D_{\mathrm{s}1}^{\left ( 12,12 \right )} \end{array}\displaystyle \!\! \right ]\!, $$
(C.1)
where
$$ D_{\mathrm{s}1}^{\left ( 1,1 \right )} = D_{\mathrm{s}1}^{\left ( 7,7 \right )} = - m_{1}w^{2} + k_{\mathrm{n}2} + k_{\mathrm{s}3} + 2k_{\mathrm{n}1}[1 - \cos (ka)], $$
(C.2)
$$\begin{aligned} D_{\mathrm{s}1}^{\left ( 3,9 \right )} =& D_{\mathrm{s}1}^{\left ( 9,3 \right )} = \frac{a}{2}D_{\mathrm{s}1}^{\left ( 9,7 \right )} = \frac{a}{2}D_{\mathrm{s}1}^{\left ( 7,9 \right )} = \frac{a}{2}D_{\mathrm{s}1}^{\left ( 7,3 \right )} = \frac{a}{2}D_{\mathrm{s}1}^{\left ( 3,7 \right )} = - \frac{a}{2}D_{\mathrm{s}1}^{\left ( 1,3 \right )} = - \frac{a}{2}D_{\mathrm{s}1}^{\left ( 3,1 \right )} \\ =& - \frac{a}{2}D_{\mathrm{s}1}^{\left ( 9,1 \right )} = - \frac{a}{2}D_{\mathrm{s}1}^{\left ( 1,9 \right )} = - \frac{a^{2}}{4}D_{\mathrm{s}1}^{\left ( 7,1 \right )} = - \frac{a^{2}}{4}D_{\mathrm{s}1}^{\left ( 1,7 \right )} = \frac{k_{\mathrm{s}3}a^{2}}{4}, \end{aligned}$$
(C.3)
$$ D_{\mathrm{s}1}^{\left ( 2,2 \right )} = D_{\mathrm{s}1}^{\left ( 8,8 \right )} = - m_{1}w^{2} + k_{\mathrm{s}2} + 2k_{\mathrm{s}1}[1 - \cos (ka)], $$
(C.4)
$$ D_{\mathrm{s}1}^{\left ( 1,4 \right )} = D_{\mathrm{s}1}^{\left ( 4,1 \right )} = D_{\mathrm{s}1}^{\left ( 7,10 \right )} = D_{\mathrm{s}1}^{\left ( 10,7 \right )} = - k_{\mathrm{n}2}, $$
(C.5)
$$\begin{aligned} D_{\mathrm{s}1}^{\left ( 5,6 \right )} =& D_{\mathrm{s}1}^{\left ( 6,5 \right )} = D_{\mathrm{s}1}^{\left ( 12,11 \right )} = D_{\mathrm{s}1}^{\left ( 11,12 \right )} = - D_{\mathrm{s}1}^{\left ( 6,2 \right )} = - D_{\mathrm{s}1}^{\left ( 2,6 \right )} = - D_{\mathrm{s}1}^{\left ( 5,3 \right )} = - D_{\mathrm{s}1}^{\left ( 3,5 \right )} = - D_{\mathrm{s}1}^{\left ( 9,11 \right )} \\ =& - D_{\mathrm{s}1}^{\left ( 11,9 \right )} = - D_{\mathrm{s}1}^{\left ( 12,8 \right )} = - D_{\mathrm{s}1}^{\left ( 8,12 \right )} = - \frac{r}{2}D_{\mathrm{s}1}^{\left ( 5,2 \right )} = - \frac{r}{2}D_{\mathrm{s}1}^{\left ( 2,5 \right )} = - \frac{r}{2}D_{\mathrm{s}1}^{\left ( 8,11 \right )} \\ =& - \frac{r}{2}D_{\mathrm{s}1}^{\left ( 11,8 \right )} = \frac{k_{\mathrm{s}2}r}{2}, \end{aligned}$$
(C.6)
$$ D_{\mathrm{s}1}^{\left ( 3,3 \right )} = D_{\mathrm{s}1}^{\left ( 9,9 \right )} = - I_{1}w^{2} + k_{\mathrm{r}2} + \frac{k_{\mathrm{s}2}r^{2}}{4} + \frac{k_{\mathrm{s}3}a^{2}}{4} + 2k_{\mathrm{r}1}[1 - \cos (ka)] + \frac{k_{\mathrm{s}1}a^{2}}{2}[1 + \cos (ka)], $$
(C.7)
$$ D_{\mathrm{s}1}^{\left ( 6,6 \right )} = D_{\mathrm{s}1}^{\left ( 12,12 \right )} = - I_{2}w^{2} + k_{\mathrm{r}2} + \frac{k_{\mathrm{s}2}r^{2}}{4}, $$
(C.8)
$$ D_{\mathrm{s}1}^{\left ( 2,3 \right )} = D_{\mathrm{s}1}^{\left ( 8,9 \right )} = \frac{k_{\mathrm{s}2}r}{2} - \mathrm{i}a\sin (ka)k_{\mathrm{s}1}, $$
(C.9)
$$ D_{\mathrm{s}1}^{\left ( 3,2 \right )} = D_{\mathrm{s}1}^{\left ( 9,8 \right )} = \frac{k_{\mathrm{s}2}r}{2} + \mathrm{i}a\sin (ka)k_{\mathrm{s}1}, $$
(C.10)
$$ D_{\mathrm{s}1}^{\left ( 3,6 \right )} = D_{\mathrm{s}1}^{\left ( 6,3 \right )} = D_{\mathrm{s}1}^{\left ( 9,12 \right )} = D_{\mathrm{s}1}^{\left ( 12,9 \right )} = - k_{\mathrm{r}2} - \frac{k_{\mathrm{s}2}r^{2}}{4}, $$
(C.11)
$$ D_{\mathrm{s}1}^{\left ( 5,5 \right )} = D_{\mathrm{s}1}^{\left ( 11,11 \right )} = k_{\mathrm{s}2} - m_{2}w^{2}, $$
(C.12)
$$ D_{\mathrm{s}1}^{\left ( 4,4 \right )} = D_{\mathrm{s}1}^{\left ( 10,10 \right )} = k_{\mathrm{n}2} - m_{2}w^{2}. $$
(C.13)
Appendix D
The elements of matrix \(\mathbf{G}_{\mathrm{s}}\) in Eq. (13) are expressed as:
$$ \mathbf{G}_{\mathrm{s}} = \left [ \textstyle\begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} G_{\mathrm{s}}^{\left ( 1,1 \right )} & 0 & 0 & G_{\mathrm{s}}^{\left ( 1,4 \right )} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & G_{\mathrm{s}}^{\left ( 2,2 \right )} & G_{\mathrm{s}}^{\left ( 2,3 \right )} & 0 & G_{\mathrm{s}}^{\left ( 2,5 \right )} & G_{\mathrm{s}}^{\left ( 2,6 \right )} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & G_{\mathrm{s}}^{\left ( 3,2 \right )} & G_{\mathrm{s}}^{\left ( 3,3 \right )} & 0 & G_{\mathrm{s}}^{\left ( 3,5 \right )} & G_{\mathrm{s}}^{\left ( 3,6 \right )} & 0 & 0 & 0 & 0 & 0 & 0 \\ G_{\mathrm{s}}^{\left ( 4,1 \right )} & 0 & 0 & G_{\mathrm{s}}^{\left ( 4,4 \right )} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & G_{\mathrm{s}}^{\left ( 5,2 \right )} & G_{\mathrm{s}}^{\left ( 5,3 \right )} & 0 & G_{\mathrm{s}}^{\left ( 5,5 \right )} & G_{\mathrm{s}}^{\left ( 5,6 \right )} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & G_{\mathrm{s}}^{\left ( 6,2 \right )} & G_{\mathrm{s}}^{\left ( 6,3 \right )} & 0 & G_{\mathrm{s}}^{\left ( 6,5 \right )} & G_{\mathrm{s}}^{\left ( 6,6 \right )} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & G_{\mathrm{s}}^{\left ( 7,7 \right )} & 0 & 0 & G_{\mathrm{s}}^{\left ( 10,7 \right )} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & G_{\mathrm{s}}^{\left ( 8,8 \right )} & G_{\mathrm{s}}^{\left ( 8,9 \right )} & 0 & G_{\mathrm{s}}^{\left ( 8,11 \right )} & G_{\mathrm{s}}^{\left ( 8,12 \right )} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & G_{\mathrm{s}}^{\left ( 9,8 \right )} & G_{\mathrm{s}}^{\left ( 9,9 \right )} & 0 & G_{\mathrm{s}}^{\left ( 9,11 \right )} & G_{\mathrm{s}}^{\left ( 9,12 \right )} \\ 0 & 0 & 0 & 0 & 0 & 0 & G_{\mathrm{s}}^{\left ( 7,10 \right )} & 0 & 0 & G_{\mathrm{s}}^{\left ( 10,10 \right )} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & G_{\mathrm{s}}^{\left ( 11,8 \right )} & G_{\mathrm{s}}^{\left ( 11,9 \right )} & 0 & G_{\mathrm{s}}^{\left ( 11,11 \right )} & G_{\mathrm{s}}^{\left ( 11,12 \right )} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & G_{\mathrm{s}}^{\left ( 12,8 \right )} & G_{\mathrm{s}}^{\left ( 12,9 \right )} & 0 & G_{\mathrm{s}}^{\left ( 12,11 \right )} & G_{\mathrm{s}}^{\left ( 12,12 \right )} \end{array}\displaystyle \right ]\!, $$
(D.1)
where
$$ G_{\mathrm{s}}^{\left ( 1,1 \right )} = G_{\mathrm{s}}^{\left ( 7,7 \right )} = 2k_{\mathrm{n}1} + k_{\mathrm{n}2}, $$
(D.2)
$$ G_{\mathrm{s}}^{\left ( 4,4 \right )} = G_{\mathrm{s}}^{\left ( 10,10 \right )} = - G_{\mathrm{s}}^{\left ( 1,4 \right )} = - G_{\mathrm{s}}^{\left ( 4,1 \right )} = - G_{\mathrm{s}}^{\left ( 7,10 \right )} = - G_{\mathrm{s}}^{\left ( 10,7 \right )} = k_{\mathrm{n}2}, $$
(D.3)
$$ G_{\mathrm{s}}^{\left ( 2,2 \right )} = G_{\mathrm{s}}^{\left ( 8,8 \right )} = 2k_{\mathrm{s}1} + k_{\mathrm{s}2}, $$
(D.4)
$$ G_{\mathrm{s}}^{\left ( 6,6 \right )} = G_{\mathrm{s}}^{\left ( 12,12 \right )} = - G_{\mathrm{s}}^{\left ( 3,6 \right )} = - G_{\mathrm{s}}^{\left ( 6,3 \right )} = - G_{\mathrm{s}}^{\left ( 12,9 \right )} = - G_{\mathrm{s}}^{\left ( 9,12 \right )} = k_{\mathrm{r}2} + \frac{k_{\mathrm{s}2}r^{2}}{4}, $$
(D.5)
$$\begin{aligned} G_{\mathrm{s}}^{\left ( 2,3 \right )} =& G_{\mathrm{s}}^{\left ( 3,2 \right )} = G_{\mathrm{s}}^{\left ( 5,6 \right )} = G_{\mathrm{s}}^{\left ( 6,5 \right )} = G_{\mathrm{s}}^{\left ( 8,9 \right )} = G_{\mathrm{s}}^{\left ( 9,8 \right )} = G_{\mathrm{s}}^{\left ( 12,11 \right )} = G_{\mathrm{s}}^{\left ( 11,12 \right )} = - G_{\mathrm{s}}^{\left ( 2,6 \right )} \\ =& - G_{\mathrm{s}}^{\left ( 6,2 \right )} = - G_{\mathrm{s}}^{\left ( 3,5 \right )} = - G_{\mathrm{s}}^{\left ( 5,3 \right )} = - G_{\mathrm{s}}^{\left ( 8,12 \right )} = - G_{\mathrm{s}}^{\left ( 12,8 \right )} = - G_{\mathrm{s}}^{\left ( 9,11 \right )} = - G_{\mathrm{s}}^{\left ( 11,9 \right )} \\ =& \frac{r}{2}G_{\mathrm{s}}^{\left ( 5,5 \right )} = \frac{r}{2}G_{\mathrm{s}}^{\left ( 11,11 \right )} = - \frac{r}{2}G_{\mathrm{s}}^{\left ( 2,5 \right )} = - \frac{r}{2}G_{\mathrm{s}}^{\left ( 5,2 \right )} = - \frac{r}{2}G_{\mathrm{s}}^{\left ( 8,11 \right )} = - \frac{r}{2}G_{\mathrm{s}}^{\left ( 11,8 \right )} = \frac{k_{\mathrm{s}2}r}{2}, \end{aligned}$$
(D.6)
$$ G_{\mathrm{s}}^{\left ( 3,3 \right )} = G_{\mathrm{s}}^{\left ( 9,9 \right )} = 2k_{\mathrm{r}1} + k_{\mathrm{r}2} + \frac{k_{\mathrm{s}1}a^{2}}{2} + \frac{k_{\mathrm{s}2}r^{2}}{4}. $$
(D.7)
Appendix E
The elements of matrix \(\mathbf{G}\)s1 in Eq. (13) are expressed as:
$$ \mathbf{G}_{\mathrm{s}1} = \left [ \textstyle\begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} G_{\mathrm{s}1}^{\left ( 1,1 \right )} & G_{\mathrm{s}1}^{\left ( 1,2 \right )} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & G_{\mathrm{s}1}^{\left ( 2,3 \right )} & G_{\mathrm{s}1}^{\left ( 2,4 \right )} & G_{\mathrm{s}1}^{\left ( 2,5 \right )} & G_{\mathrm{s}1}^{\left ( 2,6 \right )} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & G_{\mathrm{s}1}^{\left ( 3,3 \right )} & G_{\mathrm{s}1}^{\left ( 3,4 \right )} & G_{\mathrm{s}1}^{\left ( 3,5 \right )} & G_{\mathrm{s}1}^{\left ( 3,6 \right )} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & G_{\mathrm{s}1}^{\left ( 7,7 \right )} & G_{\mathrm{s}1}^{\left ( 7,8 \right )} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & G_{\mathrm{s}1}^{\left ( 8,9 \right )} & G_{\mathrm{s}1}^{\left ( 8,10 \right )} & G_{\mathrm{s}1}^{\left ( 8,11 \right )} & G_{\mathrm{s}1}^{\left ( 8,12 \right )} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & G_{\mathrm{s}1}^{\left ( 9,9 \right )} & G_{\mathrm{s}1}^{\left ( 9,10 \right )} & G_{\mathrm{s}1}^{\left ( 9,11 \right )} & G_{\mathrm{s}1}^{\left ( 9,12 \right )} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array}\displaystyle \right ], $$
(E.1)
where
$$ G_{\mathrm{s}1}^{\left ( 1,1 \right )} = G_{\mathrm{s}1}^{\left ( 1,2 \right )} = G_{\mathrm{s}1}^{\left ( 7,7 \right )} = G_{\mathrm{s}1}^{\left ( 7,8 \right )} = - k_{\mathrm{n}1}, $$
(E.2)
$$\begin{aligned} G_{\mathrm{s}1}^{\left ( 2,6 \right )} =& G_{\mathrm{s}1}^{\left ( 3,3 \right )} = G_{\mathrm{s}1}^{\left ( 8,12 \right )} = G_{\mathrm{s}1}^{\left ( 9,9 \right )} = - G_{\mathrm{s}1}^{\left ( 8,11 \right )} = - G_{\mathrm{s}1}^{\left ( 3,4 \right )} = - G_{\mathrm{s}1}^{\left ( 2,5 \right )} = - G_{\mathrm{s}1}^{\left ( 9,10 \right )} = - \frac{a}{2}G_{\mathrm{s}1}^{\left ( 2,3 \right )} \\ =& - \frac{a}{2}G_{\mathrm{s}1}^{\left ( 2,4 \right )} = - \frac{a}{2}G_{\mathrm{s}1}^{\left ( 8,9 \right )} = - \frac{a}{2}G_{\mathrm{s}1}^{\left ( 8,10 \right )} = \frac{k_{\mathrm{s}1}a}{2}, \end{aligned}$$
(E.3)
$$ G_{\mathrm{s}1}^{\left ( 3,5 \right )} = G_{\mathrm{s}1}^{\left ( 3,6 \right )} = G_{\mathrm{s}1}^{\left ( 9,11 \right )} = G_{\mathrm{s}1}^{\left ( 9,12 \right )} = - k_{\mathrm{r}1} + \frac{k_{\mathrm{s}1}a^{2}}{4}. $$
(E.4)
Appendix F
The elements of matrix \(\mathbf{G}\)s3 in Eq. (19) are expressed as:
$$ \mathbf{G}_{\mathrm{s}3}\! = \!\left [\!\! \textstyle\begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} G_{\mathrm{s}3}^{\left ( 1,1 \right )} & 0 & 0 & G_{\mathrm{s}3}^{\left ( 1,4 \right )} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & G_{\mathrm{s}3}^{\left ( 2,2 \right )} & G_{\mathrm{s}3}^{\left ( 2,3 \right )} & 0 & G_{\mathrm{s}3}^{\left ( 2,5 \right )} & G_{\mathrm{s}3}^{\left ( 2,6 \right )} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & G_{\mathrm{s}3}^{\left ( 3,2 \right )} & G_{\mathrm{s}3}^{\left ( 3,3 \right )} & 0 & G_{\mathrm{s}3}^{\left ( 3,5 \right )} & G_{\mathrm{s}3}^{\left ( 3,6 \right )} & 0 & 0 & 0 & 0 & 0 & 0 \\ G_{\mathrm{s}3}^{\left ( 4,1 \right )} & 0 & 0 & G_{\mathrm{s}3}^{\left ( 4,4 \right )} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & G_{\mathrm{s}3}^{\left ( 5,2 \right )} & G_{\mathrm{s}3}^{\left ( 5,3 \right )} & 0 & G_{\mathrm{s}3}^{\left ( 5,5 \right )} & G_{\mathrm{s}3}^{\left ( 5,6 \right )} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & G_{\mathrm{s}3}^{\left ( 6,2 \right )} & G_{\mathrm{s}3}^{\left ( 6,3 \right )} & 0 & G_{\mathrm{s}3}^{\left ( 6,5 \right )} & G_{\mathrm{s}3}^{\left ( 6,6 \right )} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & G_{\mathrm{s}3}^{\left ( 7,7 \right )} & 0 & 0 & G_{\mathrm{s}3}^{\left ( 7,10 \right )} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & G_{\mathrm{s}3}^{\left ( 8,8 \right )} & G_{\mathrm{s}3}^{\left ( 8,9 \right )} & 0 & G_{\mathrm{s}3}^{\left ( 8,11 \right )} & G_{\mathrm{s}3}^{\left ( 8,12 \right )} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & G_{\mathrm{s}3}^{\left ( 9,8 \right )} & G_{\mathrm{s}3}^{\left ( 9,9 \right )} & 0 & G_{\mathrm{s}3}^{\left ( 9,11 \right )} & G_{\mathrm{s}3}^{\left ( 9,12 \right )} \\ 0 & 0 & 0 & 0 & 0 & 0 & G_{\mathrm{s}3}^{\left ( 7,10 \right )} & 0 & 0 & G_{\mathrm{s}3}^{\left ( 10,10 \right )} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & G_{\mathrm{s}3}^{\left ( 11,8 \right )} & G_{\mathrm{s}3}^{\left ( 11,9 \right )} & 0 & G_{\mathrm{s}3}^{\left ( 11,11 \right )} & G_{\mathrm{s}3}^{\left ( 11,12 \right )} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & G_{\mathrm{s}3}^{\left ( 12,8 \right )} & G_{\mathrm{s}3}^{\left ( 12,9 \right )} & 0 & G_{\mathrm{s}3}^{\left ( 12,11 \right )} & G_{\mathrm{s}3}^{\left ( 12,12 \right )} \end{array}\displaystyle \!\! \right ]\!, $$
(F.1)
where
$$ G_{\mathrm{s}3}^{\left ( 1,1 \right )} = G_{\mathrm{s}3}^{\left ( 7,7 \right )} = m_{1}(0 + \mathrm{i}k v)^{2} + 2k_{\mathrm{n}1}[1 - \cos (ka)] + k_{\mathrm{n}2}, $$
(F.2)
$$ G_{\mathrm{s}3}^{\left ( 4,4 \right )} = G_{\mathrm{s}3}^{\left ( 10,10 \right )} = m_{2}(0 + \mathrm{i}k v)^{2} + k_{\mathrm{n}2}, $$
(F.3)
$$ G_{\mathrm{s}3}^{\left ( 2,2 \right )} = G_{\mathrm{s}3}^{\left ( 8,8 \right )} = m_{1}(0 + \mathrm{i}k v)^{2} + 2k_{\mathrm{s}1}[1 - \cos (ka)] + k_{\mathrm{s}2}, $$
(F.4)
$$ G_{\mathrm{s}3}^{\left ( 6,6 \right )} = G_{\mathrm{s}3}^{\left ( 12,12 \right )} = I_{2}(0 + \mathrm{i}k v)^{2} + k_{\mathrm{r}2} + \frac{k_{\mathrm{s}2}r^{2}}{4}, $$
(F.5)
$$ G_{\mathrm{s}3}^{\left ( 5,5 \right )} = G_{\mathrm{s}3}^{\left ( 11,11 \right )} = m_{2}(0 + \mathrm{i}k v)^{2} + k_{\mathrm{s}2}, $$
(F.6)
$$ G_{\mathrm{s}3}^{\left ( 3,3 \right )} = G_{\mathrm{s}3}^{\left ( 9,9 \right )} = I_{1}(0 + \mathrm{i}k v)^{2} + 2k_{\mathrm{r}1}[1 - \cos (ka)] + \frac{k_{\mathrm{s}1}a^{2}}{2}[1 + \cos (ka)] + \frac{k_{\mathrm{s}2}r^{2}}{4} + k_{\mathrm{r}2}, $$
(F.7)
$$ G_{\mathrm{s}3}^{\left ( 1,4 \right )} = G_{\mathrm{s}3}^{\left ( 4,1 \right )} = G_{\mathrm{s}3}^{\left ( 7,10 \right )} = G_{\mathrm{s}3}^{\left ( 10,7 \right )} = - k_{\mathrm{n} 2}, $$
(F.8)
$$ G_{\mathrm{s}3}^{\left ( 2,3 \right )} = G_{\mathrm{s}3}^{\left ( 8,9 \right )} = \frac{k_{\mathrm{s}2}r}{2} - \mathrm{i}a\sin (ka)k_{\mathrm{s}1}, $$
(F.9)
$$ G_{\mathrm{s}3}^{\left ( 3,2 \right )} = G_{\mathrm{s}3}^{\left ( 9,8 \right )} = \frac{k_{\mathrm{s}2}r}{2} + \mathrm{i}a\sin (ka)k_{\mathrm{s}1}, $$
(F.10)
$$ G_{\mathrm{s}3}^{\left ( 3,6 \right )} = G_{\mathrm{s}3}^{\left ( 6,3 \right )} = G_{\mathrm{s}3}^{\left ( 9,12 \right )} = G_{\mathrm{s}3}^{\left ( 12,9 \right )} = - k_{\mathrm{r}2} - \frac{k_{\mathrm{s}2}r^{2}}{4}, $$
(F.11)
$$\begin{aligned} G_{\mathrm{s}3}^{\left ( 5,6 \right )} =& G_{\mathrm{s}3}^{\left ( 6,5 \right )} = G_{\mathrm{s}3}^{\left ( 12,11 \right )} = G_{\mathrm{s}3}^{\left ( 11,12 \right )} = - G_{\mathrm{s}3}^{\left ( 6,2 \right )} = - G_{\mathrm{s}3}^{\left ( 2,6 \right )} = - G_{\mathrm{s}3}^{\left ( 5,3 \right )} = - G_{\mathrm{s}3}^{\left ( 3,5 \right )} = - G_{\mathrm{s}3}^{\left ( 9,11 \right )} \\ =& - G_{\mathrm{s}3}^{\left ( 11,9 \right )} = - G_{\mathrm{s}3}^{\left ( 12,8 \right )} = - G_{\mathrm{s}3}^{\left ( 8,12 \right )} = - \frac{r}{2}G_{\mathrm{s}3}^{\left ( 5,2 \right )} = - \frac{r}{2}G_{\mathrm{s}3}^{\left ( 2,5 \right )} = - \frac{r}{2}G_{\mathrm{s}3}^{\left ( 8,11 \right )} \\ =& - \frac{r}{2}G_{\mathrm{s}3}^{\left ( 11,8 \right )} = \frac{k_{\mathrm{s}2}r}{2}. \end{aligned}$$
(F.12)
