Sound Propagation in Rib-Stiffened Sandwich Structures with Cavity Absorption

Abstract

This chapter is organized as two parts: in the first part, a comprehensive theoretical model is developed for the radiation of sound from an infinite orthogonally rib-stiffened sandwich structure filled with fibrous sound absorptive material in the partitioned cavity, when excited by a time-harmonic point force. The vibrations of the rib-stiffeners are accounted for by considering all possible motions. Built upon the concepts of dynamic density and bulk modulus, both frequency dependent, an equivalent fluid model is employed to characterize the absorption of sound in the fibrous material. Given the periodicity of the sandwich structure, Fourier transform technique is employed to solve the series of panel vibration equations and acoustic equations. In the absence of fibrous sound absorptive material, the model can be favorably degraded to the case of an infinite rib-stiffened structure with air or vacuum cavity. Validation of the model is performed by comparing the present model predictions with previously published data, with excellent agreements achieved. The influences of air-structure coupling effect and cavity-filling fibrous material on the sound radiation are systematically examined. The physical features associated with sound penetration across these sandwich structures are interpreted by considering the combined effects of fiberglass stiffness and damping, the balance of which is significantly affected by stiffener separation. The proposed model provides a convenient and efficient tool for the factual engineering design of this kind of sandwich structures.

In the second part, the transmission loss of sound through infinite orthogonally rib-stiffened double-panel structures having cavity-filling fibrous sound absorptive materials is theoretically investigated. The propagation of sound across the fibrous material is characterized using an equivalent fluid model, and the motions of the rib-stiffeners are described by including all possible vibrations, i.e., tensional displacements, bending, and torsional rotations. The effects of fluid-structure coupling are accounted for by enforcing velocity continuity conditions at fluid-panel interfaces. By fully taking advantage of the periodic nature of the double panel, the space-harmonic approach and virtual work principle are applied to solve the sets of resultant governing equations, which are eventually truncated as a finite system of simultaneous algebraic equations and numerically solved insofar as the solution converges. To validate the proposed model, a comparison between the present model predictions and existing numerical and experimental results for a simplified version of the double-panel structure is carried out, with overall agreement achieved. The model is subsequently employed to explore the influence of the fluid-structure coupling between fluid in the cavity and the two panels on sound transmission across the orthogonally rib-stiffened double-panel structure. Obtained results demonstrate that this fluid-structure coupling affects significantly sound transmission loss (STL) at low frequencies and cannot be ignored when the rib-stiffeners are sparsely distributed. As a highlight of this research, an integrated optimal algorithm toward lightweight, high-stiffness, and superior sound insulation capability is proposed, based on which a preliminary optimal design of the double-panel structure is performed.

Appendices

Appendices

6.1.1 Appendix A

Taking into account the inertial effects (due to stiffener mass) and applying both Hooke’s law and Newton’s second law, one can express the tensional forces arising from the rib-stiffeners as [18]

$$ {Q}_x^{+}=-{R}_{Q1}{w}_1+{R}_{Q2}{w}_2,\kern1em {Q}_x^{-}=-{R}_{Q2}{w}_1+{R}_{Q1}{w}_2 $$

(6.A.1)

$$ {Q}_y^{+}=-{R}_{Q3}{w}_1+{R}_{Q4}{w}_2,\kern1em {Q}_y^{-}=-{R}_{Q4}{w}_1+{R}_{Q3}{w}_2 $$

(6.A.2)

$$ {R}_{Q1}=\frac{K_x\left({K}_x-{m}_x{\omega}^2\right)}{\left(2{K}_x-{m}_x{\omega}^2\right)},\kern1em {R}_{Q2}=\frac{K_x^2}{\left(2{K}_x-{m}_x{\omega}^2\right)} $$

(6.A.3)

$$ {R}_{Q3}=\frac{K_y\left({K}_y-{m}_y{\omega}^2\right)}{\left(2{K}_y-{m}_y{\omega}^2\right)},\kern1em {R}_{Q4}=\frac{K_y^2}{\left(2{K}_y-{m}_y{\omega}^2\right)} $$

(6.A.4)

where ω is the circle frequency and (K _x , K _y ) are the tensional stiffness of half the rib-stiffeners per unit length.

Likewise, the bending moments of the rib-stiffeners can be expressed as [18]

$$ {M}_x^{+}={R}_{M1}\frac{\partial^2{w}_1}{\partial {x}^2}-{R}_{M2}\frac{\partial^2{w}_2}{\partial {x}^2},\kern1em {M}_x^{-}={R}_{M2}\frac{\partial^2{w}_1}{\partial {x}^2}-{R}_{M1}\frac{\partial^2{w}_2}{\partial {x}^2} $$

(6.A.5)

$$ {M}_y^{+}={R}_{M3}\frac{\partial^2{w}_1}{\partial {y}^2}-{R}_{M4}\frac{\partial^2{w}_2}{\partial {y}^2},\kern1em {M}_y^{-}={R}_{M4}\frac{\partial^2{w}_1}{\partial {y}^2}-{R}_{M3}\frac{\partial^2{w}_2}{\partial {y}^2} $$

(6.A.6)

$$ {R}_{M1}=\frac{E_x{I}_x^{*}\left({E}_x{I}_x^{*}-{\rho}_x{I}_x{\omega}^2\right)}{\left(2{E}_x{I}_x^{*}-{\rho}_x{I}_x{\omega}^2\right)},\kern1em {R}_{M2}=\frac{E_x^2{I_x^{*}}^2}{\left(2{E}_x{I}_x^{*}-{\rho}_x{I}_x{\omega}^2\right)} $$

(6.A.7)

$$ {R}_{M3}=\frac{E_y{I}_y^{*}\left({E}_y{I}_y^{*}-{\rho}_y{I}_y{\omega}^2\right)}{\left(2{E}_y{I}_y^{*}-{\rho}_y{I}_y{\omega}^2\right)},\kern1em {R}_{M4}=\frac{E_y^2{I_y^{*}}^2}{\left(2{E}_y{I}_y^{*}-{\rho}_y{I}_y{\omega}^2\right)} $$

(6.A.8)

where (E _x I ^* _x , E _y I ^* _y ) are the bending stiffness of half the rib-stiffeners and (ρ _x , ρ _y ) and (I _x , I _y ) are mass density and polar moment of inertia for the rib-stiffeners, with subscripts x and y indicating the direction of the stiffeners.

In a similar manner, the torsional moments of the rib-stiffeners are obtained as [18]

$$ {M}_{T x}^{+}={R}_{T1}\frac{\partial^2{w}_1}{\partial x\partial y}-{R}_{T2}\frac{\partial^2{w}_2}{\partial x\partial y},\kern1em {M}_{T x}^{-}={R}_{T2}\frac{\partial^2{w}_1}{\partial x\partial y}-{R}_{T1}\frac{\partial^2{w}_2}{\partial x\partial y} $$

(6.A.9)

$$ {M}_{T y}^{+}={R}_{T3}\frac{\partial^2{w}_1}{\partial y\partial x}-{R}_{T4}\frac{\partial^2{w}_2}{\partial y\partial x},\kern1em {M}_{T y}^{-}={R}_{T4}\frac{\partial^2{w}_1}{\partial y\partial x}-{R}_{T3}\frac{\partial^2{w}_2}{\partial y\partial x} $$

(6.A.10)

$$ {R}_{T1}=\frac{G_x{J}_x^{*}\left({G}_x{J}_x^{*}-{\rho}_x{J}_x{\omega}^2\right)}{\left(2{G}_x{J}_x^{*}-{\rho}_x{J}_x{\omega}^2\right)},\kern1em {R}_{T2}=\frac{G_x^2{J_x^{*}}^2}{\left(2{G}_x{J}_x^{*}-{\rho}_x{J}_x{\omega}^2\right)} $$

(6.A.11)

$$ {R}_{T3}=\frac{G_y{J}_y^{*}\left({G}_y{J}_y^{*}-{\rho}_y{J}_y{\omega}^2\right)}{\left(2{G}_y{J}_y^{*}-{\rho}_y{J}_y{\omega}^2\right)},\kern1em {R}_{T4}=\frac{G_y^2{J_y^{*}}^2}{\left(2{G}_y{J}_y^{*}-{\rho}_y{J}_y{\omega}^2\right)} $$

(6.A.12)

where (G _x J ^* _x , G _y J ^* _y ) are the torsional stiffness of half the rib-stiffeners and (J _x , J _y ) are the torsional moment of inertia for the rib-stiffeners.

The Fourier transforms of the tensional forces, bending moments, and torsional moments are listed below:

• 1. Fourier transforms of tensional forces

  $$ {\tilde{Q}}_x^{+}={R}_{Q2}{\tilde{w}}_2\left( a,{\beta}_n\right)-{R}_{Q1}{\tilde{w}}_1\left( a,{\beta}_n\right),\kern0.6em {\tilde{Q}}_x^{-}={R}_{Q1}{\tilde{w}}_2\left( a,{\beta}_n\right)-{R}_{Q2}{\tilde{w}}_1\left( a,{\beta}_n\!\right) $$

  (6.A.13)
  $$ {\tilde{Q}}_y^{+}{=}{R}_{Q4}{\tilde{w}}_2\left({a}_m,\beta \right){-}{R}_{Q3}{\tilde{w}}_1\left({a}_m,\beta \right),\kern.5em {\tilde{Q}}_y^{-}={R}_{Q3}{\tilde{w}}_2\left({a}_m,\beta \right){-}{R}_{Q4}{\tilde{w}}_1\left({a}_m,\beta \right) $$

  (6.A.14)

• 2. Fourier transforms of bending moments

  $$ \begin{array} {llll} {\tilde{M}}_x^{+}&={\alpha}^2\left[{R}_{M2}{\tilde{w}}_2\left( a,{\beta}_n\right)-{R}_{M1}{\tilde{w}}_1\left( a,{\beta}_n\right)\right],\\ {\tilde{M}}_x^{-}&={\alpha}^2\left[{R}_{M1}{\tilde{w}}_2\left( a,{\beta}_n\right)-{R}_{M2}{\tilde{w}}_1\left( a,{\beta}_n\right)\right] \end{array} $$

  (6.A.15)
  $$ \begin{array} {llll} {\tilde{M}}_y^{+}&={\beta}^2\left[{R}_{M4}{\tilde{w}}_2\left({a}_m,\beta \right)-{R}_{M3}{\tilde{w}}_1\left({a}_m,\beta \right)\right],\\ {\tilde{M}}_y^{-}&={\beta}^2\left[{R}_{M3}{\tilde{w}}_2\left({a}_m,\beta \right)-{R}_{M4}{\tilde{w}}_1\left({a}_m,\beta \right)\right] \end{array} $$

  (6.A.16)

• 3. Fourier transforms of torsional moments

  $$ \begin{array} {llll} {\tilde{M}}_{T x}^{+}&=\alpha {\beta}_n\left[{R}_{T2}{\tilde{w}}_2\left( a,{\beta}_n\right)-{R}_{T1}{\tilde{w}}_1\left( a,{\beta}_n\right)\right],\\ {\tilde{M}}_{T x}^{-}&=\alpha {\beta}_n\left[{R}_{T1}{\tilde{w}}_2\left( a,{\beta}_n\right)-{R}_{T2}{\tilde{w}}_1\left( a,{\beta}_n\right)\right] \end{array} $$

  (6.A.17)
  $$ \begin{array} {llll} {\tilde{M}}_{T y}^{+}&={\alpha}_m\beta \left[{R}_{T4}{\tilde{w}}_2\left({a}_m,\beta \right)-{R}_{T3}{\tilde{w}}_1\left({a}_m,\beta \right)\right],\\ {\tilde{M}}_{T y}^{-}&={\alpha}_m\beta \left[{R}_{T3}{\tilde{w}}_2\left({a}_m,\beta \right)-{R}_{T4}{\tilde{w}}_1\left({a}_m,\beta \right)\right] \end{array} $$

  (6.A.18)

6.1.2 Appendix B

Adopting the similar procedure of Takahashi’s beam model [11] for rib-stiffeners, taking the inertial effects of the rib-stiffeners into consideration, and applying Hooke’s law and the Newton’s second law, one can obtain the tensional forces of the rib-stiffeners as [18]

$$ {Q}_x^{+}=-\frac{K_x\left({K}_x-{m}_x{\omega}^2\right)}{2{K}_x-{m}_x{\omega}^2}{w}_1+\frac{K_x^2}{2{K}_x-{m}_x{\omega}^2}{w}_2 $$

(6.B.1)

$$ {Q}_x^{-}=-\frac{K_x^2}{2{K}_x-{m}_x{\omega}^2}{w}_1+\frac{K_x\left({K}_x-{m}_x{\omega}^2\right)}{2{K}_x-{m}_x{\omega}^2}{w}_2 $$

(6.B.2)

$$ {Q}_y^{+}=-\frac{K_y\left({K}_y-{m}_y{\omega}^2\right)}{2{K}_y-{m}_y{\omega}^2}{w}_1+\frac{K_y^2}{2{K}_y-{m}_y{\omega}^2}{w}_2 $$

(6.B.3)

$$ {Q}_y^{-}=-\frac{K_y^2}{2{K}_y-{m}_y{\omega}^2}{w}_1+\frac{K_y\left({K}_y-{m}_y{\omega}^2\right)}{2{K}_y-{m}_y{\omega}^2}{w}_2 $$

(6.B.4)

where ω is the circular frequency, (K _x , K _y ) are the tensional stiffness of half rib-stiffeners per unit length, and (m _x , m _y ) are the line mass density of the x- and y-wise stiffeners, respectively.

Likewise, the bending moments of the rib-stiffeners can be expressed as [18]

$$ {M}_x^{+}=\frac{E_x{I}_x^{*}\left({E}_x{I}_x^{*}-{\rho}_x{I}_x{\omega}^2\right)}{2{E}_x{I}_x^{*}-{\rho}_x{I}_x{\omega}^2}\frac{\partial^2{w}_1}{\partial {x}^2}-\frac{E_x^2{I_x^{*}}^2}{2{E}_x{I}_x^{*}-{\rho}_x{I}_x{\omega}^2}\frac{\partial^2{w}_2}{\partial {x}^2} $$

(6.B.5)

$$ {M}_x^{-}=\frac{E_x^2{I_x^{*}}^2}{2{E}_x{I}_x^{*}-{\rho}_x{I}_x{\omega}^2}\frac{\partial^2{w}_1}{\partial {x}^2}-\frac{E_x{I}_x^{*}\left({E}_x{I}_x^{*}-{\rho}_x{I}_x{\omega}^2\right)}{2{E}_x{I}_x^{*}-{\rho}_x{I}_x{\omega}^2}\frac{\partial^2{w}_2}{\partial {x}^2} $$

(6.B.6)

$$ {M}_y^{+}=\frac{E_y{I}_y^{*}\left({E}_y{I}_y^{*}-{\rho}_y{I}_y{\omega}^2\right)}{2{E}_y{I}_y^{*}-{\rho}_y{I}_y{\omega}^2}\frac{\partial^2{w}_1}{\partial {y}^2}-\frac{E_y^2{I_y^{*}}^2}{2{E}_y{I}_y^{*}-{\rho}_y{I}_y{\omega}^2}\frac{\partial^2{w}_2}{\partial {y}^2} $$

(6.B.7)

$$ {M}_y^{-}=\frac{E_y^2{I_y^{*}}^2}{2{E}_y{I}_y^{*}-{\rho}_y{I}_y{\omega}^2}\frac{\partial^2{w}_1}{\partial {y}^2}-\frac{E_y{I}_y^{*}\left({E}_y{I}_y^{*}-{\rho}_y{I}_y{\omega}^2\right)}{2{E}_y{I}_y^{*}-{\rho}_y{I}_y{\omega}^2}\frac{\partial^2{w}_2}{\partial {y}^2} $$

(6.B.8)

where (E _x I ^* _x , E _y I ^* _y ) are the bending stiffness of half rib-stiffeners per unit length and (ρ _x , ρ _y ), (I _x , I _y ) are the mass density and polar moment of inertia of the stiffeners, respectively, with subscripts x and y indicating the corresponding orientations of the stiffeners.

In a similar scheme, the torsional moments of the rib-stiffeners are given by [18]

$$ {M}_{\mathrm{T} x}^{+}=\frac{G_x{J}_x^{*}\left({G}_x{J}_x^{*}-{\rho}_x{J}_x{\omega}^2\right)}{2{G}_x{J}_x^{*}-{\rho}_x{J}_x{\omega}^2}\frac{\partial^2{w}_1}{\partial x\partial y}-\frac{G_x^2{J_x^{*}}^2}{2{G}_x{J}_x^{*}-{\rho}_x{J}_x{\omega}^2}\frac{\partial^2{w}_2}{\partial x\partial y} $$

(6.B.9)

$$ {M}_{\mathrm{T} x}^{-}=\frac{G_x^2{J_x^{*}}^2}{2{G}_x{J}_x^{*}-{\rho}_x{J}_x{\omega}^2}\frac{\partial^2{w}_1}{\partial x\partial y}-\frac{G_x{J}_x^{*}\left({G}_x{J}_x^{*}-{\rho}_x{J}_x{\omega}^2\right)}{2{G}_x{J}_x^{*}-{\rho}_x{J}_x{\omega}^2}\frac{\partial^2{w}_2}{\partial x\partial y} $$

(6.B.10)

$$ {M}_{\mathrm{T} y}^{+}=\frac{G_y{J}_y^{*}\left({G}_y{J}_y^{*}-{\rho}_y{J}_y{\omega}^2\right)}{2{G}_y{J}_y^{*}-{\rho}_y{J}_y{\omega}^2}\frac{\partial^2{w}_1}{\partial y\partial x}-\frac{G_y^2{J_y^{*}}^2}{2{G}_y{J}_y^{*}-{\rho}_y{J}_y{\omega}^2}\frac{\partial^2{w}_2}{\partial y\partial x} $$

(6.B.11)

$$ {M}_{\mathrm{T} y}^{-}=\frac{G_y^2{J_y^{*}}^2}{2{G}_y{J}_y^{*}-{\rho}_y{J}_y{\omega}^2}\frac{\partial^2{w}_1}{\partial y\partial x}-\frac{G_y{J}_y^{*}\left({G}_y{J}_y^{*}-{\rho}_y{J}_y{\omega}^2\right)}{2{G}_y{J}_y^{*}-{\rho}_y{J}_y{\omega}^2}\frac{\partial^2{w}_2}{\partial y\partial x} $$

(6.B.12)

where (G _x J ^* _x , G _y J ^* _y ) are the torsional stiffness of half rib-stiffeners per unit length and (J _x , J _y ) are the torsional moments of inertia of the stiffeners.

To simplify Eqs. (6.B.1), (6.B.2), (6.B.3), (6.B.4), (6.B.5), (6.B.6), (6.B.7), (6.B.8), (6.B.9), (6.B.10), (6.B.11), and (6.B.12), the following sets of specified characteristics are utilized to replace the coefficients of the general displacements:

• 1. Replacement of tensional force coefficients

  $$ {R}_{Q1}=\frac{K_x\left({K}_x-{m}_x{\omega}^2\right)}{2{K}_x-{m}_x{\omega}^2},\kern1em {R}_{Q2}=\frac{K_x^2}{2{K}_x-{m}_x{\omega}^2} $$

  (6.B.13)
  $$ {R}_{Q3}=\frac{K_y\left({K}_y-{m}_y{\omega}^2\right)}{2{K}_y-{m}_y{\omega}^2},\kern1em {R}_{Q4}=\frac{K_y^2}{2{K}_y-{m}_y{\omega}^2} $$

  (6.B.14)

• 2. Replacement of bending moment coefficients

  $$ {R}_{M1}=\frac{E_x{I}_x^{*}\left({E}_x{I}_x^{*}-{\rho}_x{I}_x{\omega}^2\right)}{2{E}_x{I}_x^{*}-{\rho}_x{I}_x{\omega}^2},\kern1em {R}_{M2}=\frac{E_x^2{I_x^{*}}^2}{2{E}_x{I}_x^{*}-{\rho}_x{I}_x{\omega}^2} $$

  (6.B.15)
  $$ {R}_{M3}=\frac{E_y{I}_y^{*}\left({E}_y{I}_y^{*}-{\rho}_y{I}_y{\omega}^2\right)}{2{E}_y{I}_y^{*}-{\rho}_y{I}_y{\omega}^2},\kern1em {R}_{M4}=\frac{E_y^2{I_y^{*}}^2}{2{E}_y{I}_y^{*}-{\rho}_y{I}_y{\omega}^2} $$

  (6.B.16)

• 3. Replacement of torsional moment coefficients

  $$ {R}_{T1}=\frac{G_x{J}_x^{*}\left({G}_x{J}_x^{*}-{\rho}_x{J}_x{\omega}^2\right)}{2{G}_x{J}_x^{*}-{\rho}_x{J}_x{\omega}^2},\kern1em {R}_{T2}=\frac{G_x^2{J_x^{*}}^2}{2{G}_x{J}_x^{*}-{\rho}_x{J}_x{\omega}^2} $$

  (6.B.17)
  $$ {R}_{T3}=\frac{G_y{J}_y^{*}\left({G}_y{J}_y^{*}-{\rho}_y{J}_y{\omega}^2\right)}{2{G}_y{J}_y^{*}-{\rho}_y{J}_y{\omega}^2},\kern1em {R}_{T4}=\frac{G_y^2{J_y^{*}}^2}{2{G}_y{J}_y^{*}-{\rho}_y{J}_y{\omega}^2} $$

  (6.B.18)

In terms of space-harmonic series, the expressions of the tensional forces, bending moments, and torsional moments can be simplified as follows:

• 1. Tensional forces

  $$ {Q}_x^{+}={\displaystyle \sum_{m=-\infty}^{+\infty }{\displaystyle \sum_{n=-\infty}^{+\infty}\left(-{R}_{Q1}{\alpha}_{1, mn}+{R}_{Q2}{\alpha}_{2, mn}\right){\mathrm{e}}^{- i\left({\alpha}_m x+{\beta}_n y\right)}}} $$

  (6.B.19)
  $$ {Q}_x^{-}={\displaystyle \sum_{m=-\infty}^{+\infty }{\displaystyle \sum_{n=-\infty}^{+\infty}\left(-{R}_{Q2}{\alpha}_{1, mn}+{R}_{Q1}{\alpha}_{2, mn}\right){\mathrm{e}}^{- i\left({\alpha}_m x+{\beta}_n y\right)}}} $$

  (6.B.20)
  $$ {Q}_y^{+}={\displaystyle \sum_{m=-\infty}^{+\infty }{\displaystyle \sum_{n=-\infty}^{+\infty}\left(-{R}_{Q3}{\alpha}_{1, mn}+{R}_{Q4}{\alpha}_{2, mn}\right){\mathrm{e}}^{- i\left({\alpha}_m x+{\beta}_n y\right)}}} $$

  (6.B.21)
  $$ {Q}_y^{-}={\displaystyle \sum_{m=-\infty}^{+\infty }{\displaystyle \sum_{n=-\infty}^{+\infty}\left(-{R}_{Q4}{\alpha}_{1, mn}+{R}_{Q3}{\alpha}_{2, mn}\right){\mathrm{e}}^{- i\left({\alpha}_m x+{\beta}_n y\right)}}} $$

  (6.B.22)

• 2. Bending moments

  $$ {M}_x^{+}={\displaystyle \sum_{m=-\infty}^{+\infty }{\displaystyle \sum_{n=-\infty}^{+\infty}\left(-{R}_{M1}{\alpha}_{1, mn}+{R}_{M2}{\alpha}_{2, mn}\right){\alpha}_m^2{\mathrm{e}}^{- i\left({\alpha}_m x+{\beta}_n y\right)}}} $$

  (6.B.23)
  $$ {M}_x^{-}={\displaystyle \sum_{m=-\infty}^{+\infty }{\displaystyle \sum_{n=-\infty}^{+\infty}\left(-{R}_{M2}{\alpha}_{1, mn}+{R}_{M1}{\alpha}_{2, mn}\right){\alpha}_m^2{\mathrm{e}}^{- i\left({\alpha}_m x+{\beta}_n y\right)}}} $$

  (6.B.24)
  $$ {M}_y^{+}={\displaystyle \sum_{m=-\infty}^{+\infty }{\displaystyle \sum_{n=-\infty}^{+\infty}\left(-{R}_{M3}{\alpha}_{1, mn}+{R}_{M4}{\alpha}_{2, mn}\right){\beta}_n^2{\mathrm{e}}^{- i\left({\alpha}_m x+{\beta}_n y\right)}}} $$

  (6.B.25)
  $$ {M}_y^{-}={\displaystyle \sum_{m=-\infty}^{+\infty }{\displaystyle \sum_{n=-\infty}^{+\infty}\left(-{R}_{M4}{\alpha}_{1, mn}+{R}_{M3}{\alpha}_{2, mn}\right){\beta}_n^2{\mathrm{e}}^{- i\left({\alpha}_m x+{\beta}_n y\right)}}} $$

  (6.B.26)

• 3. Torsional moments

  $$ {M}_{\mathrm{T} x}^{+}={\displaystyle \sum_{m=-\infty}^{+\infty }{\displaystyle \sum_{n=-\infty}^{+\infty}\left(-{R}_{T1}{\alpha}_{1, mn}+{R}_{T2}{\alpha}_{2, mn}\right){\alpha}_m{\beta}_n{\mathrm{e}}^{- i\left({\alpha}_m x+{\beta}_n y\right)}}} $$

  (6.B.27)
  $$ {M}_{\mathrm{T} x}^{-}={\displaystyle \sum_{m=-\infty}^{+\infty }{\displaystyle \sum_{n=-\infty}^{+\infty}\left(-{R}_{T2}{\alpha}_{1, mn}+{R}_{T1}{\alpha}_{2, mn}\right){\alpha}_m{\beta}_n{\mathrm{e}}^{- i\left({\alpha}_m x+{\beta}_n y\right)}}} $$

  (6.B.28)
  $$ {M}_{\mathrm{T} y}^{+}={\displaystyle \sum_{m=-\infty}^{+\infty }{\displaystyle \sum_{n=-\infty}^{+\infty}\left(-{R}_{T3}{\alpha}_{1, mn}+{R}_{T4}{\alpha}_{2, mn}\right){\alpha}_m{\beta}_n{\mathrm{e}}^{- i\left({\alpha}_m x+{\beta}_n y\right)}}} $$

  (6.B.29)
  $$ {M}_{\mathrm{T} y}^{-}={\displaystyle \sum_{m=-\infty}^{+\infty }{\displaystyle \sum_{n=-\infty}^{+\infty}\left(-{R}_{T4}{\alpha}_{1, mn}+{R}_{T3}{\alpha}_{2, mn}\right){\alpha}_m{\beta}_n{\mathrm{e}}^{- i\left({\alpha}_m x+{\beta}_n y\right)}}} $$

  (6.B.30)

6.1.3 Appendix C

The deflection coefficients of the two face panels are

$$ \left\{{\alpha}_{1, kl}\right\}={\left[\begin{array}{cccccccccc}\hfill {\alpha}_{1,11}\hfill & \hfill {\alpha}_{1,21}\hfill & \hfill \cdot \cdot \cdot \hfill & \hfill {\alpha}_{1, K1}\hfill & \hfill {\alpha}_{1,12}\hfill & \hfill {\alpha}_{1,22}\hfill & \hfill \cdot \cdot \cdot \hfill & \hfill {\alpha}_{1, K2}\hfill & \hfill \cdot \cdot \cdot \hfill & \hfill {\alpha}_{1, K L}\hfill \end{array}\right]}_{KL\times 1}^{\boldsymbol{T}} $$

(6.C.1)

$$ \left\{{\alpha}_{2, kl}\right\}={\left[\begin{array}{cccccccccc}\hfill {\alpha}_{2,11}\hfill & \hfill {\alpha}_{2,21}\hfill & \hfill \cdot \cdot \cdot \hfill & \hfill {\alpha}_{2, K1}\hfill & \hfill {\alpha}_{2,12}\hfill & \hfill {\alpha}_{2,22}\hfill & \hfill \cdot \cdot \cdot \hfill & \hfill {\alpha}_{2, K2}\hfill & \hfill \cdot \cdot \cdot \hfill & \hfill {\alpha}_{2, K L}\hfill \end{array}\right]}_{KL\times 1}^{\boldsymbol{T}} $$

(6.C.2)

The right-hand side of Eq. (6.70) represents the generalized force, that is,

$$ \left\{{F}_{kl}\right\}={\left[\begin{array}{cccccccccc}\hfill {F}_{11}\hfill & \hfill {F}_{21}\hfill & \hfill \cdot \cdot \cdot \hfill & \hfill {F}_{K1}\hfill & \hfill {F}_{12}\hfill & \hfill {F}_{22}\hfill & \hfill \cdot \cdot \cdot \hfill & \hfill {F}_{K2}\hfill & \hfill \cdot \cdot \cdot \hfill & \hfill {F}_{K L}\hfill \end{array}\right]}_{K L\times 1}^{\boldsymbol{T}} $$

(6.C.3)

where

$$ {F}_{kl}=\left\{\begin{array}{l@{\quad}l@{\quad}l}2 I{l}_x{l}_y & \mathrm{at} & k=\frac{K+1}{2}\ \&\ l=\frac{L+1}{2} \\[6pt] 0 & \mathrm{at} & k\ne \frac{K+1}{2}\ \parallel\ l\ne \frac{L+1}{2} \end{array}\right. $$

(6.C.4)

$$ {\lambda}_{k l}^{11,1}=\left[{D}_1{\left({\alpha}_k^2+{\beta}_l^2\right)}^2-{m}_1{\omega}^2-\frac{\omega^2{\rho}_0}{i{ k}_{z, kl}}+\frac{\omega^2{\rho}_{\mathrm{cav}} \cos \left({k}_{z,\mathrm{cav}, kl} d\right)}{k_{z,\mathrm{cav}, kl} \sin \left({k}_{z,\mathrm{cav}, kl} d\right)}\right]\cdot {l}_x{l}_y $$

(6.C.5)

$$ {T}_{11,1}=\mathrm{diag}{\left[\begin{array}{cccccccccc}\hfill {\lambda}_{11}^{11,1}\hfill & \hfill {\lambda}_{21}^{11,1}\hfill & \hfill \cdots \hfill & \hfill {\lambda}_{K1}^{11,1}\hfill & \hfill {\lambda}_{12}^{11,1}\hfill & \hfill {\lambda}_{22}^{11,1}\hfill & \hfill \cdots \hfill & \hfill {\lambda}_{K2}^{11,1}\hfill & \hfill \cdots \hfill & \hfill {\lambda}_{K L}^{11,1}\hfill \end{array}\right]}_{K L\times KL} $$

(6.C.6)

$$ {\lambda}_{K L}^{11,2}={l}_x\times \mathrm{diag}{\left[\begin{array}{cccc}\hfill {R}_{Q1}- i{\alpha}_1^3{R}_{M1}\hfill & \hfill {R}_{Q1}- i{\alpha}_2^3{R}_{M1}\hfill & \hfill \cdots \hfill & \hfill {R}_{Q1}- i{\alpha}_K^3{R}_{M1}\hfill \end{array}\right]}_{K\times L} $$

(6.C.7)

$$ {T}_{11,2}={\left[\begin{array}{cccc} {\lambda}_{KL}^{11,2} & {\lambda}_{KL}^{11,2} & \cdots & {\lambda}_{KL}^{11,2} \\[6pt] {\lambda}_{KL}^{11,2} & {\lambda}_{KL}^{11,2} & \cdots & {\lambda}_{KL}^{11,2} \\[3pt] \vdots & \vdots & \ddots & \vdots \\[3pt] {\lambda}_{KL}^{11,2} & {\lambda}_{KL}^{11,2} & \cdots & {\lambda}_{KL}^{11,2} \end{array}\right]}_{KL\times KL} $$

(6.C.8)

$$ {\lambda}_{K l, n}^{11,3}={l}_x\times \mathrm{diag}{\left[\begin{array}{cccc} - i{\alpha}_1{\beta}_l{\beta}_n{R}_{T1} & - i{\alpha}_2{\beta}_l{\beta}_n{R}_{T1} & \cdots & - i{\alpha}_K{\beta}_l{\beta}_n{R}_{T1} \end{array}\right]}_{K\times L} $$

(6.C.9)

$$ {T}_{11,3}={\left[\begin{array}{cccc} {\lambda}_{K1,1}^{11,3} & {\lambda}_{K1,2}^{11,3} & \cdots & {\lambda}_{K1, L}^{11,3} \\[6pt] {\lambda}_{K2,1}^{11,3} & {\lambda}_{K2,2}^{11,3} & \cdots & {\lambda}_{K2, L}^{11,3} \\[3pt] \vdots & \vdots & \ddots & \vdots \\[3pt] {\lambda}_{K L,1}^{11,3} & {\lambda}_{K L,2}^{11,3} & \cdots & {\lambda}_{K L, L}^{11,3} \end{array}\right]}_{K L\times KL} $$

(6.C.10)

$$ {\lambda}_{K l}^{11,4}={l}_y\times {\left[\begin{array}{cccc} {R}_{Q3}- i{\beta}_l^3{R}_{M3} & {R}_{Q3}- i{\beta}_l^3{R}_{M3} & \cdots & {R}_{Q3}- i{\beta}_l^3{R}_{M3} \\[6pt] {R}_{Q3}- i{\beta}_l^3{R}_{M3} & {R}_{Q3}- i{\beta}_l^3{R}_{M3} & \cdots & {R}_{Q3}- i{\beta}_l^3{R}_{M3} \\[3pt] \vdots & \vdots & \ddots & \vdots \\[3pt] {R}_{Q3}- i{\beta}_l^3{R}_{M3} & {R}_{Q3}- i{\beta}_l^3{R}_{M3} & \cdots & {R}_{Q3}- i{\beta}_l^3{R}_{M3} \end{array}\right]}_{K\times L} $$

(6.C.11)

$$ {T}_{11,4}=\mathrm{diag}{\left[\begin{array}{cccc} {\lambda}_{K1}^{11,4} & {\lambda}_{K2}^{11,4} & \cdots & {\lambda}_{K L}^{11,4} \end{array}\right]}_{K L\times KL} $$

(6.C.12)

$$ {\lambda}_{K l}^{11,5}={l}_y\times {\left[\begin{array}{cccc} - i{\alpha}_1{\beta}_l{\alpha}_1{R}_{T3} & - i{\alpha}_1{\beta}_l{\alpha}_2{R}_{T3} & \cdots & - i{\alpha}_1{\beta}_l{\alpha}_K{R}_{T3} \\[6pt] - i{\alpha}_2{\beta}_l{\alpha}_1{R}_{T3} & - i{\alpha}_2{\beta}_l{\alpha}_2{R}_{T3} & \cdots & - i{\alpha}_2{\beta}_l{\alpha}_K{R}_{T3} \\[3pt] \vdots & \vdots & \ddots & \vdots \\[3pt] - i{\alpha}_K{\beta}_l{\alpha}_1{R}_{T3} & - i{\alpha}_K{\beta}_l{\alpha}_2{R}_{T3} & \cdots & - i{\alpha}_K{\beta}_l{\alpha}_K{R}_{T3} \end{array}\right]}_{K\times L} $$

(6.C.13)

$$ {T}_{11,5}=\mathrm{diag}{\left[\begin{array}{cccc} {\lambda}_{K1}^{11,5} & {\lambda}_{K2}^{11,5} & \cdots & {\lambda}_{K L}^{11,5} \end{array}\right]}_{K L\times KL} $$

(6.C.14)

$$ {\lambda}_{k l}^{12,1}=-\frac{\omega^2{\rho}_{cav}}{k_{z, cav, kl} \sin \left({k}_{z, cav, kl} d\right)}\cdot {l}_x{l}_y $$

(6.C.15)

$$ {T}_{12,1}=\mathrm{diag}{\left[\begin{array}{cccccccccc} {\lambda}_{11}^{12,1} & {\lambda}_{21}^{12,1} & \cdots & {\lambda}_{K1}^{12,1} & {\lambda}_{12}^{12,1} & {\lambda}_{22}^{12,1} & \cdots & {\lambda}_{K2}^{12,1} & \cdots & {\lambda}_{K L}^{12,1} \end{array}\right]}_{K L\times KL} $$

(6.C.16)

$$ {\lambda}_{K L}^{12,2}\!=\!{l}_x\!\times\! \mathrm{diag}{\left[\begin{array}{cccc} -{R}_{Q2}\!+\! i{\alpha}_1^3{R}_{M2} & -{R}_{Q2}\!+\! i{\alpha}_2^3{R}_{M2} & \cdots & -{R}_{Q2}\!+\! i{\alpha}_K^3{R}_{M2} \end{array}\right]}_{K\times L} $$

(6.C.17)

$$ {T}_{12,2}={\left[\begin{array}{cccc} {\lambda}_{KL}^{12,2} & {\lambda}_{KL}^{12,2} & \cdots & {\lambda}_{KL}^{12,2} \\[6pt] {\lambda}_{KL}^{12,2} & {\lambda}_{KL}^{12,2} & \cdots & {\lambda}_{KL}^{12,2} \\[3pt] \vdots & \vdots & \ddots & \vdots \\[3pt] {\lambda}_{KL}^{12,2} & {\lambda}_{KL}^{12,2} & \cdots & {\lambda}_{KL}^{12,2} \end{array}\right]}_{KL\times KL} $$

(6.C.18)

$$ {\lambda}_{K l, n}^{12,3}={l}_x\times \mathrm{diag}{\left[\begin{array}{cccc} i{\alpha}_1{\beta}_l{\beta}_n{R}_{T2} & i{\alpha}_2{\beta}_l{\beta}_n{R}_{T2} & \cdots & i{\alpha}_K{\beta}_l{\beta}_n{R}_{T2} \end{array}\right]}_{K\times L} $$

(6.C.19)

$$ {T}_{12,3}={\left[\begin{array}{cccc} {\lambda}_{K1,1}^{12,3} & {\lambda}_{K1,2}^{12,3} & \cdots & {\lambda}_{K1, L}^{12,3} \\[6pt] {\lambda}_{K2,1}^{12,3} & {\lambda}_{K2,2}^{12,3} & \cdots & {\lambda}_{K2, L}^{12,3} \\[3pt] \vdots & \vdots & \ddots & \vdots \\[3pt] {\lambda}_{K L,1}^{12,3} & {\lambda}_{K L,2}^{12,3} & \cdots & {\lambda}_{K L, L}^{12,3} \end{array}\right]}_{K L\times KL} $$

(6.C.20)

$$ {\lambda}_{K l}^{12,4}={l}_y\times {\left[\begin{array}{cccc} -{R}_{Q4}+ i{\beta}_l^3{R}_{M4} & -{R}_{Q4}+ i{\beta}_l^3{R}_{M4} & \cdots & -{R}_{Q4}+ i{\beta}_l^3{R}_{M4} \\[6pt] -{R}_{Q4}+ i{\beta}_l^3{R}_{M4} & -{R}_{Q4}+ i{\beta}_l^3{R}_{M4} & \cdots & -{R}_{Q4}+ i{\beta}_l^3{R}_{M4} \\[3pt] \vdots & \vdots & \ddots & \vdots \\[3pt] -{R}_{Q4}+ i{\beta}_l^3{R}_{M4} & -{R}_{Q4}+ i{\beta}_l^3{R}_{M4} & \cdots & -{R}_{Q4}+ i{\beta}_l^3{R}_{M4} \end{array}\right]}_{K\times L} $$

(6.C.21)

$$ {T}_{12,4}=\mathrm{diag}{\left[\begin{array}{cccc} {\lambda}_{K1}^{12,4} & {\lambda}_{K2}^{12,4} & \cdots & {\lambda}_{K L}^{12,4} \end{array}\right]}_{K L\times KL} $$

(6.C.22)

$$ {\lambda}_{K l}^{12,5}={l}_y\times {\left[\begin{array}{cccc} i{\alpha}_1{\beta}_l{\alpha}_1{R}_{T4} & i{\alpha}_1{\beta}_l{\alpha}_2{R}_{T4} & \cdots & i{\alpha}_1{\beta}_l{\alpha}_K{R}_{T4} \\[6pt] i{\alpha}_2{\beta}_l{\alpha}_1{R}_{T4} & i{\alpha}_2{\beta}_l{\alpha}_2{R}_{T4} & \cdots & i{\alpha}_2{\beta}_l{\alpha}_K{R}_{T4} \\[3pt] \vdots & \vdots & \ddots & \vdots \\[3pt] i{\alpha}_K{\beta}_l{\alpha}_1{R}_{T4} & i{\alpha}_K{\beta}_l{\alpha}_2{R}_{T4} & \cdots & i{\alpha}_K{\beta}_l{\alpha}_K{R}_{T4} \end{array}\right]}_{K\times L} $$

(6.C.23)

$$ {T}_{12,5}=\mathrm{diag}{\left[\begin{array}{cccc} {\lambda}_{K1}^{12,5} & {\lambda}_{K2}^{12,5} & \cdots & {\lambda}_{K L}^{12,5} \end{array}\right]}_{K L\times KL} $$

(6.C.24)

$$ {\lambda}_{k l}^{21,1}=-\frac{\omega^2{\rho}_{\mathrm{cav}}}{k_{z,\mathrm{cav}, kl} \sin \left({k}_{z,\mathrm{cav}, kl} d\right)}\cdot {l}_x{l}_y $$

(6.C.25)

$$ {T}_{21,1}=\mathrm{diag}{\left[\begin{array}{cccccccccc} {\lambda}_{11}^{21,1} & {\lambda}_{21}^{21,1} & \cdots & {\lambda}_{K1}^{21,1} & {\lambda}_{12}^{21,1} & {\lambda}_{22}^{21,1} & \cdots & {\lambda}_{K2}^{21,1} & \cdots & {\lambda}_{K L}^{21,1} \end{array}\right]}_{K L\times KL} $$

(6.C.26)

$$ {\lambda}_{K L}^{21,2}\!=\!{l}_x\!\times\! \mathrm{diag}{\left[\begin{array}{cccc} -{R}_{Q2}\!+\! i{\alpha}_1^3{R}_{M2} & -{R}_{Q2}\!+\! i{\alpha}_2^3{R}_{M2} & \cdots & -{R}_{Q2}\!+\! i{\alpha}_K^3{R}_{M2} \end{array}\right]}_{K\times L} $$

(6.C.27)

$$ {T}_{21,2}={\left[\begin{array}{cccc} {\lambda}_{KL}^{21,2} & {\lambda}_{KL}^{21,2} & \cdots & {\lambda}_{KL}^{21,2} \\[6pt] {\lambda}_{KL}^{21,2} & {\lambda}_{KL}^{21,2} & \cdots & {\lambda}_{KL}^{21,2} \\[3pt] \vdots & \vdots & \ddots & \vdots \\[3pt] {\lambda}_{KL}^{21,2} & {\lambda}_{KL}^{21,2} & \cdots & {\lambda}_{KL}^{21,2} \end{array}\right]}_{KL\times KL} $$

(6.C.28)

$$ {\lambda}_{K l, n}^{21,3}={l}_x\times \mathrm{diag}{\left[\begin{array}{cccc} i{\alpha}_1{\beta}_l{\beta}_n{R}_{T2} & i{\alpha}_2{\beta}_l{\beta}_n{R}_{T2} & \cdots & i{\alpha}_K{\beta}_l{\beta}_n{R}_{T2} \end{array}\right]}_{K\times L} $$

(6.C.29)

$$ {T}_{21,3}={\left[\begin{array}{cccc} {\lambda}_{K1,1}^{21,3} & {\lambda}_{K1,2}^{21,3} & \cdots & {\lambda}_{K1, L}^{21,3} \\[6pt] {\lambda}_{K2,1}^{21,3} & {\lambda}_{K2,2}^{21,3} & \cdots & {\lambda}_{K2, L}^{21,3} \\[3pt] \vdots & \vdots & \ddots & \vdots \\[3pt] {\lambda}_{K L,1}^{21,3} & {\lambda}_{K L,2}^{21,3} & \cdots & {\lambda}_{K L, L}^{21,3} \end{array}\right]}_{K L\times KL} $$

(6.C.30)

$$ {\lambda}_{K l}^{21,4}={l}_y\times {\left[\begin{array}{cccc} -{R}_{Q4}+ i{\beta}_l^3{R}_{M4} & -{R}_{Q4}+ i{\beta}_l^3{R}_{M4} & \cdots & -{R}_{Q4}+ i{\beta}_l^3{R}_{M4} \\[6pt] -{R}_{Q4}+ i{\beta}_l^3{R}_{M4} & -{R}_{Q4}+ i{\beta}_l^3{R}_{M4} & \cdots & -{R}_{Q4}+ i{\beta}_l^3{R}_{M4} \\[3pt] \vdots & \vdots & \ddots & \vdots \\[3pt] -{R}_{Q4}+ i{\beta}_l^3{R}_{M4} & -{R}_{Q4}+ i{\beta}_l^3{R}_{M4} & \cdots & -{R}_{Q4}+ i{\beta}_l^3{R}_{M4} \end{array}\right]}_{K\times L} $$

(6.C.31)

$$ {T}_{21,4}=\mathrm{diag}{\left[\begin{array}{cccc} {\lambda}_{K1}^{21,4} & {\lambda}_{K2}^{21,4} & \cdots & {\lambda}_{K L}^{21,4} \end{array}\right]}_{K L\times KL} $$

(6.C.32)

$$ {\lambda}_{K l}^{21,5}={l}_y\times {\left[\begin{array}{cccc} i{\alpha}_1{\beta}_l{\alpha}_1{R}_{T4} & i{\alpha}_1{\beta}_l{\alpha}_2{R}_{T4} & \cdots & i{\alpha}_1{\beta}_l{\alpha}_K{R}_{T4} \\ i{\alpha}_2{\beta}_l{\alpha}_1{R}_{T4} & i{\alpha}_2{\beta}_l{\alpha}_2{R}_{T4} & \cdots & i{\alpha}_2{\beta}_l{\alpha}_K{R}_{T4} \\ \vdots & \vdots & \ddots & \vdots \\ i{\alpha}_K{\beta}_l{\alpha}_1{R}_{T4} & i{\alpha}_K{\beta}_l{\alpha}_2{R}_{T4} & \cdots & i{\alpha}_K{\beta}_l{\alpha}_K{R}_{T4} \end{array}\right]}_{K\times L} $$

(6.C.33)

$$ {T}_{21,5}=\mathrm{diag}{\left[\begin{array}{cccc} {\lambda}_{K1}^{21,5} & {\lambda}_{K2}^{21,5} & \cdots & {\lambda}_{K L}^{21,5} \end{array}\right]}_{K L\times KL} $$

(6.C.34)

$$ {\lambda}_{k l}^{22,1}=\left[{D}_2{\left({\alpha}_k^2+{\beta}_l^2\right)}^2-{m}_2{\omega}^2-\frac{\omega^2{\rho}_0}{i{ k}_{z, kl}}+\frac{\omega^2{\rho}_{\mathrm{cav}} \cos \left({k}_{z,\mathrm{cav}, kl} d\right)}{k_{z,\mathrm{cav}, kl} \sin \left({k}_{z,\mathrm{cav}, kl} d\right)}\right]\cdot {l}_x{l}_y $$

(6.C.35)

$$ {T}_{22,1}=\mathrm{diag}{\left[\begin{array}{cccccccccc} {\lambda}_{11}^{22,1} & {\lambda}_{21}^{22,1} & \cdots & {\lambda}_{K1}^{22,1} & {\lambda}_{12}^{22,1} & {\lambda}_{22}^{22,1} & \cdots & {\lambda}_{K2}^{22,1} & \cdots & {\lambda}_{K L}^{22,1} \end{array}\right]}_{K L\times KL} $$

(6.C.36)

$$ {\lambda}_{K L}^{22,2}={l}_x\times \mathrm{diag}{\left[\begin{array}{cccc} {R}_{Q1}- i{\alpha}_1^3{R}_{M1} & {R}_{Q1}- i{\alpha}_2^3{R}_{M1} & \cdots & {R}_{Q1}- i{\alpha}_K^3{R}_{M1} \end{array}\right]}_{K\times L} $$

(6.C.37)

$$ {T}_{22,2}={\left[\begin{array}{cccc} {\lambda}_{KL}^{22,2} & {\lambda}_{KL}^{22,2} & \cdots & {\lambda}_{KL}^{22,2} \\[6pt] {\lambda}_{KL}^{22,2} & {\lambda}_{KL}^{22,2} & \cdots & {\lambda}_{KL}^{22,2} \\[3pt] \vdots & \vdots & \ddots & \vdots \\[3pt] {\lambda}_{KL}^{22,2} & {\lambda}_{KL}^{22,2} & \cdots & {\lambda}_{KL}^{22,2} \end{array}\right]}_{KL\times KL} $$

(6.C.38)

$$ {\lambda}_{K l, n}^{22,3}={l}_x\times \mathrm{diag}{\left[\begin{array}{cccc} - i{\alpha}_1{\beta}_l{\beta}_n{R}_{T1} & - i{\alpha}_2{\beta}_l{\beta}_n{R}_{T1} & \cdots & - i{\alpha}_K{\beta}_l{\beta}_n{R}_{T1} \end{array}\right]}_{K\times L} $$

(6.C.39)

$$ {T}_{22,3}={\left[\begin{array}{cccc} {\lambda}_{K1,1}^{22,3} & {\lambda}_{K1,2}^{22,3} & \cdots & {\lambda}_{K1, L}^{22,3} \\ {\lambda}_{K2,1}^{22,3} & {\lambda}_{K2,2}^{22,3} & \cdots & {\lambda}_{K2, L}^{22,3} \\[3pt] \vdots & \vdots & \ddots & \vdots \\[3pt] {\lambda}_{K L,1}^{22,3} & {\lambda}_{K L,2}^{22,3} & \cdots & {\lambda}_{K L, L}^{22,3} \end{array}\right]}_{K L\times KL} $$

(6.C.40)

$$ {\lambda}_{K l}^{22,4}={l}_y\times {\left[\begin{array}{cccc} {R}_{Q3}- i{\beta}_l^3{R}_{M3} & {R}_{Q3}- i{\beta}_l^3{R}_{M3} & \cdots & {R}_{Q3}- i{\beta}_l^3{R}_{M3} \\[6pt] {R}_{Q3}- i{\beta}_l^3{R}_{M3} & {R}_{Q3}- i{\beta}_l^3{R}_{M3} & \cdots & {R}_{Q3}- i{\beta}_l^3{R}_{M3} \\[3pt] \vdots & \vdots & \ddots & \vdots \\[3pt] {R}_{Q3}- i{\beta}_l^3{R}_{M3} & {R}_{Q3}- i{\beta}_l^3{R}_{M3} & \cdots & {R}_{Q3}- i{\beta}_l^3{R}_{M3} \end{array}\right]}_{K\times L} $$

(6.C.41)

$$ {T}_{22,4}=\mathrm{diag}{\left[\begin{array}{cccc} {\lambda}_{K1}^{22,4} & {\lambda}_{K2}^{22,4} & \cdots & {\lambda}_{K L}^{22,4} \end{array}\right]}_{K L\times KL} $$

(6.C.42)

$$ {\lambda}_{K l}^{22,5}={l}_y\times {\left[\begin{array}{cccc} - i{\alpha}_1{\beta}_l{\alpha}_1{R}_{T3} & - i{\alpha}_1{\beta}_l{\alpha}_2{R}_{T3} & \cdots & - i{\alpha}_1{\beta}_l{\alpha}_K{R}_{T3} \\[6pt] - i{\alpha}_2{\beta}_l{\alpha}_1{R}_{T3} & - i{\alpha}_2{\beta}_l{\alpha}_2{R}_{T3} & \cdots & - i{\alpha}_2{\beta}_l{\alpha}_K{R}_{T3} \\[3pt] \vdots & \vdots & \ddots & \vdots \\[3pt] - i{\alpha}_K{\beta}_l{\alpha}_1{R}_{T3} & - i{\alpha}_K{\beta}_l{\alpha}_2{R}_{T3} & \cdots & - i{\alpha}_K{\beta}_l{\alpha}_K{R}_{T3} \end{array}\right]}_{K\times L} $$

(6.C.43)

$$ {T}_{22,5}=\mathrm{diag}{\left[\begin{array}{cccc} {\lambda}_{K1}^{22,5} & {\lambda}_{K2}^{22,5} & \cdots & {\lambda}_{K L}^{22,5} \end{array}\right]}_{K L\times KL} $$

(6.C.44)

Using the definition of the sub-matrices presented above, one obtains

$$ {T}_{11}={T}_{11,1}{+}{T}_{11,2}{+}{T}_{11,3}{+}{T}_{11,4}{+}{T}_{11,5},\kern1em {T}_{22}={T}_{22,1}{+}{T}_{22,2}{+}{T}_{22,3}{+}{T}_{22,4}{+}{T}_{22,5} $$

(6.C.45)

$$ {T}_{12}={T}_{12,1}{+}{T}_{12,2}{+}{T}_{12,3}{+}{T}_{12,4}{+}{T}_{12,5},\kern1em {T}_{21}={T}_{21,1}{+}{T}_{21,2}{+}{T}_{21,3}{+}{T}_{21,4}{+}{T}_{21,5} $$

(6.C.46)