Appendix
The element stiffness matrix of the TELW is given by
$$\begin{aligned} K_{11\zeta \eta kemn}^{\mathrm {uu}}&= \psi _{m,x}A_{11\zeta \eta ke}^{1}\psi _{n,x}+\psi _{m,x}A_{16\zeta \eta ke}^{1}\psi _{n,y}+\psi _{m}A_{55\zeta \eta ke}^{4}\psi _{n}\nonumber \\&\quad +\psi _{m,y}A_{16\zeta \eta ke}^{1}\psi _{n,x}+\psi _{m,y}A_{66\zeta \eta ke}^{1}\psi _{n,y}\nonumber \\ K_{12\zeta \eta kemn}^{\mathrm {uu}}&= \psi _{m,x}A_{16\zeta \eta ke}^{1}\psi _{n,x}+\psi _{m,y}A_{66\zeta \eta ke}^{1}\psi _{n,x}+\psi _{m,y}A_{26\zeta \eta ke}^{1}\psi _{n,y}\nonumber \\&\quad + \psi _{m,x}A_{12\zeta \eta ke}^{1}\psi _{n,y}+\psi _{m}A_{45\zeta \eta ke}^{4}\psi _{n}\nonumber \\ K_{13\zeta \eta kemn}^{\mathrm {uu}}&= \psi _{m,x}A_{13\zeta \eta ke}^{2}\psi _{n}+\psi _{m,y}A_{36\zeta \eta ke}^{2}\psi _{n}+\psi _{m}A_{55\zeta \eta ke}^{3}\psi _{n,x}+\psi _{m}A_{45\zeta \eta ke}^{3}\psi _{n,y}\nonumber \\ K_{1\zeta \eta kemn}^{\mathrm {\theta \theta }}&= -\psi _{m,x}B_{11\zeta \eta ke}^{1}\psi _{n}-\psi _{m,y}B_{12\zeta \eta ke}^{1}\psi _{n} \end{aligned}$$
(34a)
$$\begin{aligned} K_{21\zeta \eta kemn}^{\mathrm {uu}}&= \psi _{m,y}A_{12\zeta \eta ke}^{1}\psi _{n,x}+\psi _{m,y}A_{26\zeta \eta ke}^{1}\psi _{n,y}+\psi _{m,x}A_{16\zeta \eta ke}^{1}\psi _{n,x}\nonumber \\&\quad + \psi _{m,x}A_{66\zeta \eta ke}^{1}\psi _{n,y}+\psi _{m}A_{45\zeta \eta ke}^{4}\psi _{n}\nonumber \\ K_{22\zeta \eta kemn}^{\mathrm {uu}}&= \psi _{m,y}A_{26\zeta \eta ke}^{1}\psi _{n,x}+\psi _{m,y}A_{22\zeta \eta ke}^{1}\psi _{n,y}+\psi _{m,x}A_{66\zeta \eta ke}^{1}\psi _{n,x}\nonumber \\&\quad +\psi _{m,x}A_{26\zeta \eta ke}^{1}\psi _{n,y}+\psi _{m}A_{44\zeta \eta ke}^{4}\psi _{n}\nonumber \\ K_{23\zeta \eta kemn}^{\mathrm {uu}}&= \psi _{m,y}A_{23\zeta \eta ke}^{2}\psi _{n}+\psi _{m,x}A_{36\zeta \eta ke}^{2}\psi _{n}+\psi _{m}A_{45\zeta \eta ke}^{3}\psi _{n,x}+\psi _{m}A_{44\zeta \eta ke}^{3}\psi _{n,y}\nonumber \\ K_{2\zeta \eta kemn}^{\mathrm {u \theta }}&= -\psi _{m,y}B_{22\zeta \eta ke}^{1}\psi _{n}-\psi _{m,x}B_{12\zeta \eta ke}^{1}\psi _{n} \end{aligned}$$
(34b)
$$\begin{aligned} K_{31\zeta \eta kemn}^{\mathrm {uu}}&= \psi _{m,y}A_{45\zeta \eta ke}^{2}\psi _{n}+\psi _{m}A_{13\zeta \eta ke}^{3}\psi _{n,x}+\psi _{m}A_{36\zeta \eta ke}^{3}\psi _{n,y}+\psi _{m,x}A_{55\zeta \eta ke}^{2}\psi _{n}\nonumber \\ K_{32\zeta \eta kemn}^{\mathrm {uu}}&= \psi _{m,y}A_{44\zeta \eta ke}^{2}\psi _{n}+\psi _{m,x}A_{45\zeta \eta ke}^{2}\psi _{n}+\psi _{m}A_{36\zeta \eta ke}^{3}\psi _{n,x}+\psi _{m}A_{23\zeta \eta ke}^{3}\psi _{n,y}\nonumber \\ K_{33\zeta \eta kemn}^{\mathrm {uu}}&= \psi _{m,y}A_{45\zeta \eta ke}^{1}\psi _{n,x}+\psi _{m,y}A_{44\zeta \eta ke}^{1}\psi _{n,y}+\psi _{m,x}A_{55\zeta \eta ke}^{1}\psi _{n,x}\nonumber \\&\quad +\psi _{m,x}A_{45\zeta \eta ke}^{1}\psi _{n,y}+\psi _{m}A_{33\zeta \eta ke}^{4}\psi _{n}\nonumber \\ K_{3\zeta \eta kemn}^{\mathrm {u \theta }}&= -\psi _{m}B_{33\zeta \eta ke}^{1}\psi _{n} \end{aligned}$$
(34c)
$$\begin{aligned} K_{\zeta \eta kemn}^{\mathrm {\theta \theta }}&= -\psi _{m,x}D_{11\zeta \eta ke}^{1}\psi _{n,x}-\psi _{m,x}D_{12\zeta \eta ke}^{1}\psi _{n,y}-\psi _{m,x}D_{13\zeta \eta ke}^{2}\psi _{n}\nonumber \\&\quad -\psi _{m,y}D_{21\zeta \eta ke}^{1}\psi _{n,x}-\psi _{m,y}D_{22\zeta \eta ke}^{1}\psi _{n,y}-\psi _{m,y}D_{23\zeta \eta ke}^{2}\psi _{n}\nonumber \\&\quad -\psi _{m}D_{31\zeta \eta ke}^{3}\psi _{n,x}-\psi _{m}D_{32\zeta \eta ke}^{3}\psi _{n,y}-\psi _{m}D_{33\zeta \eta ke}^{4}\psi _{n} \end{aligned}$$
(34d)
The element damping matrix of the TELW is given by
$$\begin{aligned} C_{1\zeta \eta kemn}^{\mathrm {\theta u}}&= -\psi _{m}E_{11\zeta \eta ke}^{1}\psi _{n,x}-\psi _{m}E_{12\zeta \eta ke}^{1}\psi _{n,y}\nonumber \\ C_{2\zeta \eta kemn}^{\mathrm {\theta u}}&= -\psi _{m}E_{22\zeta \eta ke}^{1}\psi _{n,y}-\psi _{m}E_{12\zeta \eta ke}^{1}\psi _{n,x}\nonumber \\ C_{3\zeta \eta kemn}^{\mathrm {\theta u}}&= -\psi _{m}E_{33\zeta \eta ke}^{3}\psi _{n}\nonumber \\ C_{\zeta \eta kemn}^{\mathrm {\theta \theta }}&= -\psi _{m}G_{\zeta \eta ke}^{1}\psi _{n} \end{aligned}$$
(35)
where the laminate stiffness coefficients \(A_{pq\zeta \eta ke}^{1},\,A_{pq\zeta \eta ke}^{2},\,A_{pq\zeta \eta ke}^{3},\,A_{pq\zeta \eta ke}^{4}\) are given in terms of modified elastic constants and the through-thickness interpolation polynomials as
$$\begin{aligned} A_{pq\zeta \eta ke}^{1}&=\displaystyle \int _{-H/2}^{H/2}\varPhi _{\zeta k}\bar{C}_{pq}\varPhi _{\eta e}dz,\nonumber \\ A_{pq\zeta \eta ke}^{2}&=\displaystyle \int _{-H/2}^{H/2}\varPhi _{\zeta k,z}\bar{C}_{pq}\varPhi _{\eta e}dz,\nonumber \\ A_{pq\zeta \eta ke}^{3}&=\displaystyle \int _{-H/2}^{H/2}\varPhi _{\zeta k}\bar{C}_{pq}\varPhi _{\eta e,z}dz,\nonumber \\ A_{pq\zeta \eta ke}^{4}&=\displaystyle \int _{-H/2}^{H/2}\varPhi _{\zeta k,z}\bar{C}_{pq}\varPhi _{\eta e,z}dz, \end{aligned}$$
(36)
the laminate stress–temperature coefficients \(B_{pq\zeta \eta ke}^{1}\) and \(B_{pq\zeta \eta ke}^{3}\) are given by
$$\begin{aligned} B_{pq\zeta \eta ke}^{1}&=\displaystyle \int _{-H/2}^{H/2}\varPhi _{\zeta k}\bar{\lambda }_{pq}\varPhi _{\eta e}dz,\nonumber \\ B_{pq\zeta \eta ke}^{3}&=\displaystyle \int _{-H/2}^{H/2}\varPhi _{\zeta k}\bar{\lambda }_{pq}\varPhi _{\eta e,z}dz, \end{aligned}$$
(37)
$$\begin{aligned} E_{pq\zeta \eta ke}^{1}&=\displaystyle \int _{-H/2}^{H/2}\varPhi _{\zeta k}\varTheta ^{0}\bar{\lambda }_{pq}\varPhi _{\eta e}dz,\nonumber \\ E_{pq\zeta \eta ke}^{3}&=\displaystyle \int _{-H/2}^{H/2}\varPhi _{\zeta k}\varTheta ^{0}\bar{\lambda }_{pq}\varPhi _{\eta e,z}dz,\nonumber \\ F_{\zeta \eta ke}&=\displaystyle \int _{-H/2}^{H/2}\varPhi _{\zeta k}\bar{s}dz,\nonumber \\ G_{\zeta \eta ke}^{1}&=\displaystyle \int _{-H/2}^{H/2}\varPhi _{\zeta k}\varTheta ^{0}\bar{\varrho }\varPhi _{\eta e}dz, \end{aligned}$$
(38)
the laminate heat conduction coefficients \(D_{pq\zeta \eta ke}^{1}\) are
$$\begin{aligned} D_{pq\zeta \eta ke}^{1}&=\displaystyle \int _{-H/2}^{H/2}\varPhi _{\zeta k}\bar{\kappa }_{pq}\varPhi _{\eta e}dz,\nonumber \\ D_{pq\zeta \eta ke}^{2}&=\displaystyle \int _{-H/2}^{H/2}\varPhi _{\zeta k,z}\bar{\kappa }_{pq}\varPhi _{\eta e}dz,\nonumber \\ D_{pq\zeta \eta ke}^{3}&=\displaystyle \int _{-H/2}^{H/2}\varPhi _{\zeta k}\bar{\kappa }_{pq}\varPhi _{\eta e,z}dz,\nonumber \\ D_{pq\zeta \eta ke}^{4}&=\displaystyle \int _{-H/2}^{H/2}\varPhi _{\zeta k,z}\bar{\kappa }_{pq}\varPhi _{\eta e,z}dz, \end{aligned}$$
(39)
The element stiffness matrices of the composite sandwich structures with n layer cores and \(n+1\) facesheets are given by
$$\begin{aligned} \varvec{K}_{uu}^{\mathrm {ee}}&= \left[ \begin{array}{cccc} ^{1}\varvec{K}_{uu,11}^{c}+^{1}\varvec{K}_{uu,11}^{f} &{} ^{1}\varvec{K}_{uu,12}^{c} &{} \cdots &{} 0\\ ^{1}\varvec{K}_{uu,21}^{c} &{} ^{1}\varvec{K}_{uu,22}^{c}+^{2}\varvec{K}_{uu,11}^{f} &{} \cdots &{} 0\\ \vdots &{} \vdots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} ^{j}\varvec{K}_{uu,11}^{c}+^{j}\varvec{K}_{uu,11}^{f}\\ 0 &{} 0 &{} \cdots &{} ^{j}\varvec{K}_{uu,21}^{c}\\ \vdots &{} \vdots &{} \cdots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} 0\\ 0 &{} 0 &{} \cdots &{} 0 \end{array}\right. \nonumber \\&\qquad \left. \begin{array}{cccc} 0 &{} \cdots &{} 0 &{} 0\\ 0 &{} \cdots &{} 0 &{} 0\\ \vdots &{} \cdots &{} \vdots &{} \vdots \\ \varvec{K}_{uu,12}^{c} &{} \cdots &{} 0 &{} 0\\ ^{j}\varvec{K}_{uu,22}^{c}+^{j+1}\varvec{K}_{uu,11}^{f} &{} \cdots &{} 0 &{} 0\\ \vdots &{} \vdots &{} \vdots &{} \vdots \\ 0 &{} \cdots &{} ^{n}\varvec{K}_{uu,11}^{c}+^{n}\varvec{K}_{uu,11}^{f} &{} ^{n}\varvec{K}_{uu,12}^{c}\\ 0 &{} \cdots &{} ^{n}\varvec{K}_{uu,21}^{c} &{} ^{n}\varvec{K}_{uu,22}^{c}+^{n+1}\varvec{K}_{uu,11}^{f} \end{array}\right] _{2n\times 2n} \end{aligned}$$
(40a)
$$\begin{aligned} \varvec{K}_{uu}^{\mathrm {ei}}=\left[ \varvec{K}_{uu}^{\mathrm {ie}}\right] ^{T}&= \left[ \begin{array}{ccccccc} ^{1}\varvec{K}_{uu,12}^{f} &{} ^{1}\varvec{K}_{uu,13}^{c} &{} 0 &{} \cdots &{} 0 &{} 0 &{} 0\\ 0 &{} ^{1}\varvec{K}_{uu,23}^{c} &{} ^{2}\varvec{K}_{uu,13}^{f} &{} \cdots &{} 0 &{} 0 &{} 0\\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} 0 &{} \cdots &{} ^{j}\varvec{K}_{uu,23}^{f} &{} ^{j}\varvec{K}_{uu,13}^{c} &{} 0\\ 0 &{} 0 &{} 0 &{} \cdots &{} 0 &{} ^{j}\varvec{K}_{uu,23}^{c} &{} ^{j+1}\varvec{K}_{uu,13}^{f}\\ \vdots &{} \vdots &{} \vdots &{} \cdots &{} \vdots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} 0 &{} \cdots &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} \cdots &{} 0 &{} 0 &{} 0 \end{array}\right. \nonumber \\&\qquad \left. \begin{array}{cccc} \cdots &{} 0 &{} 0 &{} 0\\ \cdots &{} 0 &{} 0 &{} 0\\ \cdots &{} \vdots &{} \vdots &{} \vdots \\ \cdots &{} 0 &{} 0 &{} 0\\ \cdots &{} 0 &{} 0 &{} 0\\ \vdots &{} \vdots &{} \vdots &{} \vdots \\ \cdots &{} ^{n-1}\varvec{K}_{uu,23}^{f} &{} ^{n}\varvec{K}_{uu,13}^{c} &{} 0\\ \cdots &{} 0 &{} ^{n}\varvec{K}_{uu,13}^{c} &{} ^{n+1}\varvec{K}_{uu,12}^{f} \end{array}\right] _{2n\times \left( 2n+1\right) } \end{aligned}$$
(40b)
$$\begin{aligned} \varvec{K}_{uu}^{\mathrm {ii}}&=\begin{bmatrix}^{1}\varvec{K}_{uu,22}^{f} &{} 0 &{} \cdots &{} 0 &{} 0 &{} \cdots &{} 0 &{} 0\\ 0 &{} ^{1}\varvec{K}_{uu,33}^{c} &{} \cdots &{} 0 &{} 0 &{} \cdots &{} 0 &{} 0\\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \cdots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} ^{j}\varvec{K}_{uu,33}^{f} &{} 0 &{} \cdots &{} 0 &{} 0\\ 0 &{} 0 &{} \cdots &{} 0 &{} ^{j}\varvec{K}_{uu,33}^{c} &{} \cdots &{} 0 &{} 0\\ \vdots &{} \vdots &{} \cdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} 0 &{} 0 &{} \cdots &{} ^{n}\varvec{K}_{uu,33}^{c} &{} 0\\ 0 &{} 0 &{} \cdots &{} 0 &{} 0 &{} \cdots &{} 0 &{} ^{n+1}\varvec{K}_{uu,22}^{f} \end{bmatrix}_{\left( 2n+1\right) \times \left( 2n+1\right) } \end{aligned}$$
(40c)
$$\begin{aligned} \varvec{K}_{u\theta }^{\mathrm {ee}}&= \left[ \begin{array}{ccccc} ^{1}\varvec{K}_{u\theta ,11}^{c}+^{1}\varvec{K}_{u\theta ,11}^{f} &{} ^{1}\varvec{K}_{u\theta ,12}^{c} &{} \cdots &{} 0 &{} 0\\ ^{1}\varvec{K}_{u\theta ,21}^{c} &{} ^{1}\varvec{K}_{u\theta ,22}^{c}+^{2}\varvec{K}_{u\theta ,11}^{f} &{} \cdots &{} 0 &{} 0\\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} ^{j}\varvec{K}_{u\theta ,11}^{c}+^{j}\varvec{K}_{u\theta ,11}^{f} &{} \varvec{K}_{u\theta ,12}^{c}\\ 0 &{} 0 &{} \cdots &{} ^{j}\varvec{K}_{u\theta ,21}^{c} &{} ^{j}\varvec{K}_{u\theta ,22}^{c}+^{j+1}\varvec{K}_{u\theta ,11}^{f}\\ \vdots &{} \vdots &{} \cdots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} 0 &{} 0\\ 0 &{} 0 &{} \cdots &{} 0 &{} 0 \end{array}\right. \nonumber \\&\qquad \left. \begin{array}{ccc} \cdots &{} 0 &{} 0\\ \cdots &{} 0 &{} 0\\ \cdots &{} \vdots &{} \vdots \\ \cdots &{} 0 &{} 0\\ \cdots &{} 0 &{} 0\\ \vdots &{} \vdots &{} \vdots \\ \cdots &{} ^{n}\varvec{K}_{u\theta ,11}^{c}+^{n}\varvec{K}_{u\theta ,11}^{f} &{} ^{n}\varvec{K}_{u\theta ,12}^{c}\\ \cdots &{} ^{n}\varvec{K}_{u\theta ,21}^{c} &{} ^{n}\varvec{K}_{u\theta ,22}^{c}+^{n+1}\varvec{K}_{u\theta ,11}^{f} \end{array}\right] _{2n\times 2n} \end{aligned}$$
(40d)
$$\begin{aligned} \varvec{K}_{u\theta }^{\mathrm {ei}}=\left[ \varvec{K}_{u\theta }^{\mathrm {ie}}\right] ^{T}&= \left[ \begin{array}{ccccccc} ^{1}\varvec{K}_{u\theta ,12}^{f} &{} ^{1}\varvec{K}_{u\theta ,13}^{c} &{} 0 &{} \cdots &{} 0 &{} 0 &{} 0\\ 0 &{} ^{1}\varvec{K}_{u\theta ,23}^{c} &{} ^{2}\varvec{K}_{u\theta ,13}^{f} &{} \cdots &{} 0 &{} 0 &{} 0\\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} 0 &{} \cdots &{} ^{j}\varvec{K}_{u\theta ,23}^{f} &{} ^{j}\varvec{K}_{u\theta ,13}^{c} &{} 0\\ 0 &{} 0 &{} 0 &{} \cdots &{} 0 &{} ^{j}\varvec{K}_{u\theta ,23}^{c} &{} ^{j+1}\varvec{K}_{u\theta ,13}^{f}\\ \vdots &{} \vdots &{} \vdots &{} \cdots &{} \vdots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} 0 &{} \cdots &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} \cdots &{} 0 &{} 0 &{} 0 \end{array}\right. \nonumber \\&\qquad \left. \begin{array}{cccc} \cdots &{} 0 &{} 0 &{} 0\\ \cdots &{} 0 &{} 0 &{} 0\\ \cdots &{} \vdots &{} \vdots &{} \vdots \\ \cdots &{} 0 &{} 0 &{} 0\\ \cdots &{} 0 &{} 0 &{} 0\\ \vdots &{} \vdots &{} \vdots &{} \vdots \\ \cdots &{} ^{n-1}\varvec{K}_{u\theta ,23}^{f} &{} ^{n}\varvec{K}_{u\theta ,13}^{c} &{} 0\\ \cdots &{} 0 &{} ^{n}\varvec{K}_{u\theta ,23}^{c} &{} ^{n+1}\varvec{K}_{u\theta ,12}^{f} \end{array}\right] _{2n\times \left( 2n+1\right) } \end{aligned}$$
(40e)
$$\begin{aligned} \varvec{K}_{u\theta }^{\mathrm {ii}}&=\begin{bmatrix}^{1}\varvec{K}_{u\theta ,22}^{f} &{} 0 &{} \cdots &{} 0 &{} 0 &{} \cdots &{} 0 &{} 0\\ 0 &{} ^{1}\varvec{K}_{u\theta ,33}^{c} &{} \cdots &{} 0 &{} 0 &{} \cdots &{} 0 &{} 0\\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \cdots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} ^{j}\varvec{K}_{u\theta ,33}^{f} &{} 0 &{} \cdots &{} 0 &{} 0\\ 0 &{} 0 &{} \cdots &{} 0 &{} ^{j}\varvec{K}_{u\theta ,33}^{c} &{} \cdots &{} 0 &{} 0\\ \vdots &{} \vdots &{} \cdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} 0 &{} 0 &{} \cdots &{} ^{n}\varvec{K}_{u\theta ,33}^{c} &{} 0\\ 0 &{} 0 &{} \cdots &{} 0 &{} 0 &{} \cdots &{} 0 &{} ^{n+1}\varvec{K}_{u\theta ,22}^{f} \end{bmatrix}_{\left( 2n+1\right) \times \left( 2n+1\right) } \end{aligned}$$
(40f)
$$\begin{aligned} \varvec{K}_{\theta \theta }^{\mathrm {ee}}&= \left[ \begin{array}{cccc} ^{1}\varvec{K}_{\theta \theta ,11}^{c}+^{1}\varvec{K}_{\theta \theta ,11}^{f} &{} ^{1}\varvec{K}_{\theta \theta ,12}^{c} &{} \cdots &{} 0\\ ^{1}\varvec{K}_{\theta \theta ,21}^{c} &{} ^{1}\varvec{K}_{\theta \theta ,22}^{c}+^{2}\varvec{K}_{\theta \theta ,11}^{f} &{} \cdots &{} 0\\ \vdots &{} \vdots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} ^{j}\varvec{K}_{\theta \theta ,11}^{c}+^{j}\varvec{K}_{\theta \theta ,11}^{f}\\ 0 &{} 0 &{} \cdots &{} ^{j}\varvec{K}_{\theta \theta ,21}^{c}\\ \vdots &{} \vdots &{} \cdots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} 0\\ 0 &{} 0 &{} \cdots &{} 0 \end{array}\right. \nonumber \\&\qquad \left. \begin{array}{cccc} 0 &{} \cdots &{} 0 &{} 0\\ 0 &{} \cdots &{} 0 &{} 0\\ \vdots &{} \cdots &{} \vdots &{} \vdots \\ \varvec{K}_{\theta \theta ,12}^{c} &{} \cdots &{} 0 &{} 0\\ ^{j}\varvec{K}_{\theta \theta ,22}^{c}+^{j+1}\varvec{K}_{\theta \theta ,11}^{f} &{} \cdots &{} 0 &{} 0\\ \vdots &{} \vdots &{} \vdots &{} \vdots \\ 0 &{} \cdots &{} ^{n}\varvec{K}_{\theta \theta ,11}^{c}+^{n}\varvec{K}_{\theta \theta ,11}^{f} &{} ^{n}\varvec{K}_{\theta \theta ,12}^{c}\\ 0 &{} \cdots &{} ^{n}\varvec{K}_{\theta \theta ,21}^{c} &{} ^{n}\varvec{K}_{\theta \theta ,22}^{c}+^{n+1}\varvec{K}_{\theta \theta ,11}^{f} \end{array}\right] _{2n\times 2n} \end{aligned}$$
(40g)
$$\begin{aligned} \varvec{K}_{\theta \theta }^{\mathrm {ei}} =\left[ \varvec{K}_{\theta \theta }^{\mathrm {ie}}\right] ^{T}&= \left[ \begin{array}{ccccccc} ^{1}\varvec{K}_{\theta \theta ,12}^{f} &{} ^{1}\varvec{K}_{\theta \theta ,13}^{c} &{} 0 &{} \cdots &{} 0 &{} 0 &{} 0\\ 0 &{} ^{1}\varvec{K}_{\theta \theta ,23}^{c} &{} ^{2}\varvec{K}_{\theta \theta ,13}^{f} &{} \cdots &{} 0 &{} 0 &{} 0\\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} 0 &{} \cdots &{} ^{j}\varvec{K}_{\theta \theta ,23}^{f} &{} ^{j}\varvec{K}_{\theta \theta ,13}^{c} &{} 0\\ 0 &{} 0 &{} 0 &{} \cdots &{} 0 &{} ^{j}\varvec{K}_{\theta \theta ,23}^{c} &{} ^{j+1}\varvec{K}_{\theta \theta ,13}^{f}\\ \vdots &{} \vdots &{} \vdots &{} \cdots &{} \vdots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} 0 &{} \cdots &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} \cdots &{} 0 &{} 0 &{} 0 \end{array}\right. \nonumber \\&\qquad \left. \begin{array}{cccc} \cdots &{} 0 &{} 0 &{} 0\\ \cdots &{} 0 &{} 0 &{} 0\\ \cdots &{} \vdots &{} \vdots &{} \vdots \\ \cdots &{} 0 &{} 0 &{} 0\\ \cdots &{} 0 &{} 0 &{} 0\\ \vdots &{} \vdots &{} \vdots &{} \vdots \\ \cdots &{} ^{n-1}\varvec{K}_{\theta \theta ,23}^{f} &{} ^{n}\varvec{K}_{\theta \theta ,13}^{c} &{} 0\\ \cdots &{} 0 &{} ^{n}\varvec{K}_{\theta \theta ,23}^{c} &{} ^{n+1}\varvec{K}_{\theta \theta ,12}^{f} \end{array}\right] _{2n\times \left( 2n+1\right) } \end{aligned}$$
(40h)
$$\begin{aligned} \varvec{K}_{\theta \theta }^{\mathrm {ii}}&=\begin{bmatrix}^{1}\varvec{K}_{\theta \theta ,22}^{f} &{} 0 &{} \cdots &{} 0 &{} 0 &{} \cdots &{} 0 &{} 0\\ 0 &{} ^{1}\varvec{K}_{\theta \theta ,33}^{c} &{} \cdots &{} 0 &{} 0 &{} \cdots &{} 0 &{} 0\\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \cdots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} ^{j}\varvec{K}_{\theta \theta ,33}^{f} &{} 0 &{} \cdots &{} 0 &{} 0\\ 0 &{} 0 &{} \cdots &{} 0 &{} ^{j}\varvec{K}_{\theta \theta ,33}^{c} &{} \cdots &{} 0 &{} 0\\ \vdots &{} \vdots &{} \cdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} 0 &{} 0 &{} \cdots &{} ^{n}\varvec{K}_{\theta \theta ,33}^{c} &{} 0\\ 0 &{} 0 &{} \cdots &{} 0 &{} 0 &{} \cdots &{} 0 &{} ^{n+1}\varvec{K}_{\theta \theta ,22}^{f} \end{bmatrix}_{\left( 2n+1\right) \times \left( 2n+1\right) } \end{aligned}$$
(40i)
The element mass matrices of the composite sandwich structures with n layer cores and \(n+1\) facesheets are given by
$$\begin{aligned} \varvec{M}^{\mathrm {ee}}&= \left[ \begin{array}{ccccc} ^{1}\varvec{M}_{11}^{c}+^{1}\varvec{M}_{11}^{f} &{} ^{1}\varvec{M}_{12}^{c} &{} \cdots &{} 0 &{} 0\\ ^{1}\varvec{M}_{21}^{c} &{} ^{1}\varvec{M}_{22}^{c}+^{2}\varvec{M}_{11}^{f} &{} \cdots &{} 0 &{} 0\\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} ^{j}\varvec{M}_{11}^{c}+^{j}\varvec{M}_{11}^{f} &{} \varvec{M}_{12}^{c}\\ 0 &{} 0 &{} \cdots &{} ^{j}\varvec{M}_{21}^{c} &{} ^{j}\varvec{M}_{22}^{c}+^{j+1}\varvec{M}_{11}^{f}\\ \vdots &{} \vdots &{} \cdots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} 0 &{} 0\\ 0 &{} 0 &{} \cdots &{} 0 &{} 0 \end{array}\right. \nonumber \\&\qquad \left. \begin{array}{ccc} \cdots &{} 0 &{} 0\\ \cdots &{} 0 &{} 0\\ \cdots &{} \vdots &{} \vdots \\ \cdots &{} 0 &{} 0\\ \cdots &{} 0 &{} 0\\ \vdots &{} \vdots &{} \vdots \\ \cdots &{} ^{n}\varvec{M}_{11}^{c}+^{n}\varvec{M}_{11}^{f} &{} ^{n}\varvec{M}_{12}^{c}\\ \cdots &{} ^{n}\varvec{M}_{21}^{c} &{} ^{n}\varvec{M}_{22}^{c}+^{n+1}\varvec{M}_{11}^{f} \end{array}\right] _{2n\times 2n} \end{aligned}$$
(40j)
$$\begin{aligned} \varvec{M}^{\mathrm {ei}}=\left[ \varvec{M}^{\mathrm {ie}}\right] ^{T}&= \left[ \begin{array}{ccccccc} ^{1}\varvec{M}_{12}^{f} &{} ^{1}\varvec{M}_{13}^{c} &{} 0 &{} \cdots &{} 0 &{} 0 &{} 0\\ 0 &{} ^{1}\varvec{M}_{23}^{c} &{} ^{2}\varvec{M}_{13}^{f} &{} \cdots &{} 0 &{} 0 &{} 0\\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} 0 &{} \cdots &{} ^{j}\varvec{M}_{23}^{f} &{} ^{j}\varvec{M}_{13}^{c} &{} 0\\ 0 &{} 0 &{} 0 &{} \cdots &{} 0 &{} ^{j}\varvec{M}_{23}^{c} &{} ^{j+1}\varvec{M}_{13}^{f}\\ \vdots &{} \vdots &{} \vdots &{} \cdots &{} \vdots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} 0 &{} \cdots &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} \cdots &{} 0 &{} 0 &{} 0 \end{array}\right. \nonumber \\&\qquad \left. \begin{array}{cccc} \cdots &{} 0 &{} 0 &{} 0\\ \cdots &{} 0 &{} 0 &{} 0\\ \cdots &{} \vdots &{} \vdots &{} \vdots \\ \cdots &{} 0 &{} 0 &{} 0\\ \cdots &{} 0 &{} 0 &{} 0\\ \vdots &{} \vdots &{} \vdots &{} \vdots \\ \cdots &{} ^{n-1}\varvec{M}_{23}^{f} &{} ^{n}\varvec{M}_{13}^{c} &{} 0\\ \cdots &{} 0 &{} ^{n}\varvec{M}_{23}^{c} &{} ^{n+1}\varvec{M}_{12}^{f} \end{array}\right] _{2n\times \left( 2n+1\right) } \end{aligned}$$
(40k)
$$\begin{aligned} \varvec{M}^{\mathrm {ii}}&=\begin{bmatrix}^{1}\varvec{M}_{22}^{f} &{} 0 &{} \cdots &{} 0 &{} 0 &{} \cdots &{} 0 &{} 0\\ 0 &{} ^{1}\varvec{M}_{33}^{c} &{} \cdots &{} 0 &{} 0 &{} \cdots &{} 0 &{} 0\\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \cdots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} ^{j}\varvec{M}_{33}^{f} &{} 0 &{} \cdots &{} 0 &{} 0\\ 0 &{} 0 &{} \cdots &{} 0 &{} ^{j}\varvec{M}_{33}^{c} &{} \cdots &{} 0 &{} 0\\ \vdots &{} \vdots &{} \cdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} 0 &{} 0 &{} \cdots &{} ^{n}\varvec{M}_{33}^{c} &{} 0\\ 0 &{} 0 &{} \cdots &{} 0 &{} 0 &{} \cdots &{} 0 &{} ^{n+1}\varvec{M}_{22}^{f} \end{bmatrix}_{\left( 2n+1\right) \times \left( 2n+1\right) } \end{aligned}$$
(40l)
The element damping matrices of the composite sandwich structures with n layer cores and \(n+1\) facesheets are given by
$$\begin{aligned} \varvec{C}_{\theta u}^{\mathrm {ee}}&= \left[ \begin{array}{ccccc} ^{1}\varvec{C}_{\theta u,11}^{c}+^{1}\varvec{C}_{\theta u,11}^{f} &{} ^{1}\varvec{C}_{\theta u,12}^{c} &{} \cdots &{} 0 &{} 0\\ ^{1}\varvec{C}_{\theta u,21}^{c} &{} ^{1}\varvec{C}_{\theta u,22}^{c}+^{2}\varvec{C}_{\theta u,11}^{f} &{} \cdots &{} 0 &{} 0\\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} ^{j}\varvec{C}_{\theta u,11}^{c}+^{j}\varvec{C}_{\theta u,11}^{f} &{} \varvec{C}_{\theta u,12}^{c}\\ 0 &{} 0 &{} \cdots &{} ^{j}\varvec{C}_{\theta u,21}^{c} &{} ^{j}\varvec{C}_{\theta u,22}^{c}+^{j+1}\varvec{C}_{\theta u,11}^{f}\\ \vdots &{} \vdots &{} \cdots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} 0 &{} 0\\ 0 &{} 0 &{} \cdots &{} 0 &{} 0 \end{array}\right. \nonumber \\&\qquad \left. \begin{array}{ccc} \cdots &{} 0 &{} 0\\ \cdots &{} 0 &{} 0\\ \cdots &{} \vdots &{} \vdots \\ \cdots &{} 0 &{} 0\\ \cdots &{} 0 &{} 0\\ \vdots &{} \vdots &{} \vdots \\ \cdots &{} ^{n}\varvec{C}_{\theta u,11}^{c}+^{n}\varvec{C}_{\theta u,11}^{f} &{} ^{n}\varvec{C}_{\theta u,12}^{c}\\ \cdots &{} ^{n}\varvec{C}_{\theta u,21}^{c} &{} ^{n}\varvec{C}_{\theta u,22}^{c}+^{n+1}\varvec{C}_{\theta u,11}^{f} \end{array}\right] _{2n\times 2n} \end{aligned}$$
(40m)
$$\begin{aligned} \varvec{C}_{\theta u}^{\mathrm {ei}}=\left[ \varvec{C}_{\theta u}^{\mathrm {ie}}\right] ^{T}&= \left[ \begin{array}{ccccccc} ^{1}\varvec{C}_{\theta u,12}^{f} &{} ^{1}\varvec{C}_{\theta u,13}^{c} &{} 0 &{} \cdots &{} 0 &{} 0 &{} 0\\ 0 &{} ^{1}\varvec{C}_{\theta u,23}^{c} &{} ^{2}\varvec{C}_{\theta u,13}^{f} &{} \cdots &{} 0 &{} 0 &{} 0\\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} 0 &{} \cdots &{} ^{j}\varvec{C}_{\theta u,23}^{f} &{} ^{j}\varvec{C}_{\theta u,13}^{c} &{} 0\\ 0 &{} 0 &{} 0 &{} \cdots &{} 0 &{} ^{j}\varvec{C}_{\theta u,23}^{c} &{} ^{j+1}\varvec{C}_{\theta u,13}^{f}\\ \vdots &{} \vdots &{} \vdots &{} \cdots &{} \vdots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} 0 &{} \cdots &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} \cdots &{} 0 &{} 0 &{} 0 \end{array}\right. \nonumber \\&\qquad \left. \begin{array}{cccc} \cdots &{} 0 &{} 0 &{} 0\\ \cdots &{} 0 &{} 0 &{} 0\\ \cdots &{} \vdots &{} \vdots &{} \vdots \\ \cdots &{} 0 &{} 0 &{} 0\\ \cdots &{} 0 &{} 0 &{} 0\\ \vdots &{} \vdots &{} \vdots &{} \vdots \\ \cdots &{} ^{n-1}\varvec{C}_{\theta u,23}^{f} &{} ^{n}\varvec{C}_{\theta u,13}^{c} &{} 0\\ \cdots &{} 0 &{} ^{n}\varvec{C}_{\theta u,23}^{c} &{} ^{n+1}\varvec{C}_{\theta u,12}^{f} \end{array}\right] _{2n\times \left( 2n+1\right) } \end{aligned}$$
(40n)
$$\begin{aligned} \varvec{C}_{\theta u}^{\mathrm {ii}}&=\begin{bmatrix}^{1}\varvec{C}_{\theta u,22}^{f} &{} 0 &{} \cdots &{} 0 &{} 0 &{} \cdots &{} 0 &{} 0\\ 0 &{} ^{1}\varvec{C}_{\theta u,33}^{c} &{} \cdots &{} 0 &{} 0 &{} \cdots &{} 0 &{} 0\\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \cdots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} ^{j}\varvec{C}_{\theta u,33}^{f} &{} 0 &{} \cdots &{} 0 &{} 0\\ 0 &{} 0 &{} \cdots &{} 0 &{} ^{j}\varvec{C}_{\theta u,33}^{c} &{} \cdots &{} 0 &{} 0\\ \vdots &{} \vdots &{} \cdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} 0 &{} 0 &{} \cdots &{} ^{n}\varvec{C}_{\theta u,33}^{c} &{} 0\\ 0 &{} 0 &{} \cdots &{} 0 &{} 0 &{} \cdots &{} 0 &{} ^{n+1}\varvec{C}_{\theta u,22}^{f} \end{bmatrix}_{\left( 2n+1\right) \times \left( 2n+1\right) } \end{aligned}$$
(40o)
$$\begin{aligned} \varvec{C}_{\theta \theta }^{\mathrm {ee}}&= \left[ \begin{array}{ccccc} ^{1}\varvec{C}_{\theta \theta ,11}^{c}+^{1}\varvec{C}_{\theta \theta ,11}^{f} &{} ^{1}\varvec{C}_{\theta \theta ,12}^{c} &{} \cdots &{} 0 &{} 0\\ ^{1}\varvec{C}_{\theta \theta ,21}^{c} &{} ^{1}\varvec{C}_{\theta \theta ,22}^{c}+^{2}\varvec{C}_{\theta \theta ,11}^{f} &{} \cdots &{} 0 &{} 0\\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} ^{j}\varvec{C}_{\theta \theta ,11}^{c}+^{j}\varvec{C}_{\theta \theta ,11}^{f} &{} \varvec{C}_{\theta \theta ,12}^{c}\\ 0 &{} 0 &{} \cdots &{} ^{j}\varvec{C}_{\theta \theta ,21}^{c} &{} ^{j}\varvec{C}_{\theta \theta ,22}^{c}+^{j+1}\varvec{C}_{\theta \theta ,11}^{f}\\ \vdots &{} \vdots &{} \cdots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} 0 &{} 0\\ 0 &{} 0 &{} \cdots &{} 0 &{} 0 \end{array}\right. \nonumber \\&\qquad \left. \begin{array}{ccc} \cdots &{} 0 &{} 0\\ \cdots &{} 0 &{} 0\\ \cdots &{} \vdots &{} \vdots \\ \cdots &{} 0 &{} 0\\ \cdots &{} 0 &{} 0\\ \vdots &{} \vdots &{} \vdots \\ \cdots &{} ^{n}\varvec{C}_{\theta \theta ,11}^{c}+^{n}\varvec{C}_{\theta \theta ,11}^{f} &{} ^{n}\varvec{C}_{\theta \theta ,12}^{c}\\ \cdots &{} ^{n}\varvec{C}_{\theta \theta ,21}^{c} &{} ^{n}\varvec{C}_{\theta \theta ,22}^{c}+^{n+1}\varvec{C}_{\theta \theta ,11}^{f} \end{array}\right] _{2n\times 2n} \end{aligned}$$
(40p)
$$\begin{aligned} \varvec{C}_{\theta \theta }^{\mathrm {ei}}=\left[ \varvec{C}_{\theta \theta }^{\mathrm {ie}}\right] ^{T}&= \left[ \begin{array}{ccccccc} ^{1}\varvec{C}_{\theta \theta ,12}^{f} &{} ^{1}\varvec{C}_{\theta \theta ,13}^{c} &{} 0 &{} \cdots &{} 0 &{} 0 &{} 0\\ 0 &{} ^{1}\varvec{C}_{\theta \theta ,23}^{c} &{} ^{2}\varvec{C}_{\theta \theta ,13}^{f} &{} \cdots &{} 0 &{} 0 &{} 0\\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} 0 &{} \cdots &{} ^{j}\varvec{C}_{\theta \theta ,23}^{f} &{} ^{j}\varvec{C}_{\theta \theta ,13}^{c} &{} 0\\ 0 &{} 0 &{} 0 &{} \cdots &{} 0 &{} ^{j}\varvec{C}_{\theta \theta ,23}^{c} &{} ^{j+1}\varvec{C}_{\theta \theta ,13}^{f}\\ \vdots &{} \vdots &{} \vdots &{} \cdots &{} \vdots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} 0 &{} \cdots &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} \cdots &{} 0 &{} 0 &{} 0 \end{array}\right. \nonumber \\&\qquad \left. \begin{array}{cccc} \cdots &{} 0 &{} 0 &{} 0\\ \cdots &{} 0 &{} 0 &{} 0\\ \cdots &{} \vdots &{} \vdots &{} \vdots \\ \cdots &{} 0 &{} 0 &{} 0\\ \cdots &{} 0 &{} 0 &{} 0\\ \vdots &{} \vdots &{} \vdots &{} \vdots \\ \cdots &{} ^{n-1}\varvec{C}_{\theta \theta ,23}^{f} &{} ^{n}\varvec{C}_{\theta \theta ,13}^{c} &{} 0\\ \cdots &{} 0 &{} ^{n}\varvec{C}_{\theta \theta ,23}^{c} &{} ^{n+1}\varvec{C}_{\theta \theta ,12}^{f} \end{array}\right] _{2n\times \left( 2n+1\right) } \end{aligned}$$
(40q)
$$\begin{aligned} \varvec{C}_{\theta \theta }^{\mathrm {ii}}&=\begin{bmatrix}^{1}\varvec{C}_{22}^{f} &{} 0 &{} \cdots &{} 0 &{} 0 &{} \cdots &{} 0 &{} 0\\ 0 &{} ^{1}\varvec{C}_{\theta \theta ,33}^{c} &{} \cdots &{} 0 &{} 0 &{} \cdots &{} 0 &{} 0\\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \cdots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} ^{j}\varvec{C}_{\theta \theta ,33}^{f} &{} 0 &{} \cdots &{} 0 &{} 0\\ 0 &{} 0 &{} \cdots &{} 0 &{} ^{j}\varvec{C}_{\theta \theta ,33}^{c} &{} \cdots &{} 0 &{} 0\\ \vdots &{} \vdots &{} \cdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} 0 &{} 0 &{} \cdots &{} ^{n}\varvec{C}_{\theta \theta ,33}^{c} &{} 0\\ 0 &{} 0 &{} \cdots &{} 0 &{} 0 &{} \cdots &{} 0 &{} ^{n+1}\varvec{C}_{\theta \theta ,22}^{f} \end{bmatrix}_{\left( 2n+1\right) \times \left( 2n+1\right) } \end{aligned}$$
(40r)
The displacement, temperature and load vectors of the composite sandwich structures with n layer cores and \(n+1\) facesheets are given by
$$\begin{aligned} \ddot{\varvec{u}}^{\mathrm {e}}&=\begin{Bmatrix}^{1}\ddot{\varvec{u}}_{1}^{f}\\ ^{2}\ddot{\varvec{u}}_{1}^{f,1}\\ ^{2}\ddot{\varvec{u}}_{1}^{f,2}\\ \vdots \\ ^{j}\ddot{\varvec{u}}_{1}^{f,j}\\ ^{j}\ddot{\varvec{u}}_{1}^{f,j+1}\\ \vdots \\ ^{n}\ddot{\varvec{u}}_{1}^{f,n+1} \end{Bmatrix},\,\ddot{\varvec{u}}^{i}=\begin{Bmatrix}^{1}\ddot{\varvec{u}}_{2}^{f}\\ ^{1}\ddot{\varvec{u}}_{2}^{c}\\ \vdots \\ ^{j}\ddot{\varvec{u}}_{2}^{f}\\ ^{j}\ddot{\varvec{u}}_{2}^{c}\\ \vdots \\ ^{n}\ddot{\varvec{u}}_{2}^{c}\\ ^{n+1}\ddot{\varvec{u}}_{2}^{f} \end{Bmatrix},\,\ddot{\varvec{\theta }}^{\mathrm {e}}=\begin{Bmatrix}^{1}\ddot{\varvec{\theta }}_{1}^{f}\\ ^{2}\ddot{\varvec{\theta }}_{1}^{f,1}\\ ^{2}\ddot{\varvec{\theta }}_{1}^{f,2}\\ \vdots \\ ^{j}\ddot{\varvec{\theta }}_{1}^{f,j}\\ ^{j}\ddot{\varvec{\theta }}_{1}^{f,j+1}\\ \vdots \\ ^{n}\ddot{\varvec{\theta }}_{1}^{F,n+1} \end{Bmatrix},\,\ddot{\varvec{\theta }}^{i}=\begin{Bmatrix}^{1}\ddot{\varvec{\theta }}_{2}^{f}\\ ^{1}\ddot{\varvec{\theta }}_{2}^{c}\\ \vdots \\ ^{j}\ddot{\varvec{\theta }}_{2}^{f}\\ ^{j}\ddot{\varvec{\theta }}_{2}^{c}\\ \vdots \\ ^{n}\ddot{\varvec{\theta }}_{2}^{f}\\ ^{n+1}\ddot{\varvec{\theta }}_{2}^{c} \end{Bmatrix} \end{aligned}$$
(40s)
$$\begin{aligned} \dot{\varvec{u}}^{\mathrm {e}}&=\begin{Bmatrix}^{1}\dot{\varvec{u}}_{1}^{f}\\ ^{2}\dot{\varvec{u}}_{1}^{f,1}\\ ^{2}\dot{\varvec{u}}_{1}^{f,2}\\ \vdots \\ ^{j}\dot{\varvec{u}}_{1}^{f,j}\\ \dot{^{j}\varvec{u}}_{1}^{f,j+1}\\ \vdots \\ ^{n}\dot{\varvec{u}}_{1}^{f,n+1} \end{Bmatrix},\,\dot{\varvec{u}}^{i}=\begin{Bmatrix}^{1}\dot{\varvec{u}}_{2}^{f}\\ ^{1}\dot{\varvec{u}}_{2}^{c}\\ \vdots \\ ^{j}\dot{\varvec{u}}_{2}^{f}\\ ^{j}\dot{\varvec{u}}_{2}^{c}\\ \vdots \\ ^{n}\dot{\varvec{u}}_{2}^{c}\\ ^{n+1}\dot{\varvec{u}}_{2}^{f} \end{Bmatrix},\,\dot{\varvec{\theta }}^{\mathrm {e}}=\begin{Bmatrix}^{1}\dot{\varvec{\theta }}_{1}^{f}\\ ^{2}\dot{\varvec{\theta }}_{1}^{f,1}\\ ^{2}\dot{\varvec{\theta }}_{1}^{f,2}\\ \vdots \\ ^{j}\dot{\varvec{\theta }}_{1}^{f,j}\\ ^{j}\dot{\varvec{\theta }}_{1}^{f,j+1}\\ \vdots \\ ^{n}\dot{\varvec{\theta }}_{1}^{f,n+1} \end{Bmatrix},\,\dot{\varvec{\theta }}^{i}=\begin{Bmatrix}^{1}\dot{\varvec{\theta }}_{2}^{f}\\ ^{1}\dot{\varvec{\theta }}_{2}^{c}\\ \vdots \\ ^{j}\dot{\varvec{\theta }}_{2}^{f}\\ ^{j}\dot{\varvec{\theta }}_{2}^{c},\\ \vdots \\ ^{n}\dot{\varvec{\theta }}_{2}^{c}\\ ^{n+1}\dot{\varvec{\theta }}_{2}^{f} \end{Bmatrix} \end{aligned}$$
(40t)
$$\begin{aligned} \varvec{u}^{\mathrm {e}}&=\begin{Bmatrix}^{1}\varvec{u}_{1}^{f}\\ ^{2}\varvec{u}_{1}^{f,1}\\ ^{2}\varvec{u}_{1}^{f,2}\\ \vdots \\ ^{j}\varvec{u}_{1}^{f,j}\\ ^{j}\varvec{u}_{1}^{f,j+1}\\ \vdots \\ ^{n}\varvec{u}_{1}^{f,n+1} \end{Bmatrix},\,\varvec{u}^{i}=\begin{Bmatrix}^{1}\varvec{u}_{2}^{f}\\ ^{1}\varvec{u}_{2}^{c}\\ \vdots \\ ^{j}\varvec{u}_{2}^{f}\\ ^{j}\varvec{u}_{2}^{c}\\ \vdots \\ ^{n}\varvec{u}_{2}^{c}\\ ^{n+1}\varvec{u}_{2}^{f} \end{Bmatrix},\,\varvec{\theta }^{\mathrm {e}}=\begin{Bmatrix}^{1}\varvec{\theta }_{1}^{f}\\ ^{2}\varvec{\theta }_{1}^{f,1}\\ ^{2}\varvec{\theta }_{1}^{f,2}\\ \vdots \\ ^{j}\varvec{\theta }_{1}^{f,j}\\ ^{j}\varvec{\theta }_{1}^{f,j+1}\\ \vdots \\ ^{n}\varvec{\theta }_{1}^{f,n+1} \end{Bmatrix} \end{aligned}$$
(40u)
$$\begin{aligned} \varvec{\theta }^{i}&=\begin{Bmatrix}^{1}\varvec{\theta }_{2}^{f}\\ ^{1}\varvec{\theta }_{2}^{c}\\ \vdots \\ ^{j}\varvec{\theta }_{2}^{f}\\ ^{j}\varvec{\theta }_{2}^{c},\\ \vdots \\ ^{n}\varvec{\theta }_{2}^{c},\\ ^{n+1}\varvec{\theta }_{2}^{f} \end{Bmatrix}\,\varvec{F}=\begin{Bmatrix}^{1}\varvec{F}_{2}^{f}\\ ^{1}\varvec{F}_{2}^{c}\\ \vdots \\ ^{j}\varvec{F}_{2}^{f}\\ ^{j}\varvec{F}_{2}^{c}\\ \vdots \\ ^{n}\varvec{F}_{2}^{c}\\ ^{n+1}\varvec{F}_{2}^{f} \end{Bmatrix},\,\,\varvec{Q}=\begin{Bmatrix}^{1}\varvec{Q}_{2}^{f}\\ ^{1}\varvec{Q}_{2}^{c}\\ \vdots \\ ^{j}\varvec{Q}_{2}^{f}\\ ^{j}\varvec{Q}_{2}^{c}\\ \vdots \\ ^{n}\varvec{Q}_{2}^{c}\\ ^{n+1}\varvec{Q}_{2}^{f} \end{Bmatrix} \end{aligned}$$
(40v)
