A Details of Derivation
1.1 A.1 Generalised Coordinate \(\beta _0(t)\)
The generalised coordinate \(\beta _0(t)\) follows from the volume conservation condition appearing in the sloshing problem as the geometric constraint
$$\begin{aligned} \displaystyle \int _0^{2\pi }\!\int _{0}^{x_{20}} x_2\left( x_{10}^2 f + x_{10} f^2 +\frac{1}{3}f^3\right) dx_2 dx_3=0. \end{aligned}$$
(61)
Resolving this constraint makes this generalised coordinate \(\beta _0(t)\) an explicitly-given function of other generalised coordinates, \(p_{Mi}(t)\) and \(r_{mi}(t)\). The function can be found in an asymptotic sense keeping up to the \(O(\epsilon ^3)\)-order terms (here, all generalised coordinates have the first order of smallness)
$$\begin{aligned} \beta _0=\sum _{Mi} \beta _{Mi,Mi}^{pp} p_{Mi}^2 +\sum _{mi} \beta _{mi,mi}^{rr} r_{mi}^2 +\sum _{MNLijk} \beta _{Mi,Nj,Lk}^{ppp} p_{Mi} p_{Nj} p_{Lk} \nonumber \\ +\sum _{Mnlijk} \beta _{Mi,nj,lk}^{prr} p_{Mi} r_{nj} r_{lk}, \end{aligned}$$
(62)
The \(\beta \)-coefficients in (62) are as follows
$$\begin{aligned}&\beta _{Mi,Mi}^{pp}=-\frac{\Lambda _{MM}^{cc} \lambda _{Mi,Mi}}{\pi x_{10} x_{20}^2},\ \ \ \beta _{mi,mi}^{rr}=-\frac{\Lambda _{mm}^{ss} \lambda _{mi,mi}}{\pi x_{10} x_{20}^2},\nonumber \\&\quad \qquad \beta _{Mi,Nj,Lk}^{ppp}=-\frac{\Lambda _{MNL}^{ccc} \lambda _{Mi,Nj,Lk}}{3 \pi x_{10}^2 x_{20}^2},\ \ \ \beta _{Mi,nj,lk}^{prr}=-\frac{\Lambda _{Mnl}^{css} \lambda _{Mi,nj,lk}}{\pi x_{10}^2 x_{20}^2}, \end{aligned}$$
(63)
where we introduced the tensor-type coefficient
$$\begin{aligned} \Lambda _{\underbrace{i..j}_{N_1}\underbrace{k...l}_{N_2}}^{\overbrace{c...c}^{N_1}\overbrace{s...s}^{N_2}}= \!\int _{-\pi }^{\pi }\! \underbrace{\cos \left( ix_3\right) \!\cdot \!\ldots \!\cdot \!\cos \left( jx_3\right) }_{N_1}\!\cdot \! \underbrace{\sin \left( kx_3\right) \!\cdot \!\ldots \!\cdot \!\sin \left( lx_3\right) }_{N_2}dx_3 \end{aligned}$$
(64)
for the angular coordinate and the tensor-type coefficients are responsible for the radial direction
$$\begin{aligned} \lambda _{\underbrace{Mi,\ldots ,Nj}_{N_3}}=\int _{0}^{x_{20}} x_2 \underbrace{f_{Mi}\left( x_2\right) \cdot \ldots \cdot f_{Nj}\left( x_2\right) }_{N_3} dx_2. \end{aligned}$$
(65)
1.2 A.2 Integrals \(A_{Mi}^{p}\) and \(A_{mi}^{r}\) Defined by (28)
Expanding \(A_{Mi}^{p}\) and \(A_{mi}^{r}\) up to the third polynomial order in \(p_{Mi}\) and \(r_{mi}\) gives
$$\begin{aligned} A_{Ab}^p&=\mathbb {A}_{Ab}^p +\mathbb {A}_{Ab,Ab}^{p.p}\,p_{Ab} +\!\sum _{MNij}\!\!\mathbb {A}_{Ab,Mi,Nj}^{p.pp}p_{Mi}p_{Nj} +\!\sum _{mnij}\!\!\mathbb {A}_{Ab,mi,nj}^{p.rr}r_{mi}r_{nj} \nonumber \\&+\sum _{MNLijk}\!\!\mathbb {A}_{Ab,Mi,Nj,Lk}^{p.ppp}p_{Mi}p_{Nj}p_{Lk} +\sum _{Mnlijk}\!\!\mathbb {A}_{Ab,Mi,nj,lk}^{p.prr}p_{Mi}r_{nj}r_{lk}, \end{aligned}$$
(66a)
$$\begin{aligned} A_{ab}^r= \mathbb {A}_{ab,ab}^{r.r}\,r_{ab} +\!\sum _{Mnij}\mathbb {A}_{ab,Mi,nj}^{r.pr} p_{Mi} r_{nj}&+\!\sum _{MNlijk}\!\!\mathbb {A}_{ab,Mi,Nj,lk}^{r.ppr}p_{Mi}p_{Nj}r_{lk}\nonumber \\&+\!\sum _{mnlijk}\!\!\mathbb {A}_{ab,mi,nj,lk}^{r.rrr}r_{mi}r_{nj}r_{lk}. \end{aligned}$$
(66b)
All generalised coordinates have the first order of smallness (\(p_{Mi}\sim {r}_{mi}\sim \epsilon \)). The \(\mathbb {A}\)-coefficients take the following form
$$\begin{aligned} \begin{aligned}&\mathbb {A}_{Ab}^p=\Lambda _A^c \hat{\mathcal {E}}^{Ab,0}, \ \ \mathbb {A}_{Ab,Mi,Nj}^{p.pp}= \Lambda _{AMN}^{ccc} \hat{\mathcal {E}}^{Ab,2}_{Mi,Nj} +\delta _{MN} \delta _{ij} \Lambda _A^c \hat{\mathcal {E}}^{Ab,1} \beta _{Mi,Nj}^{pp}, \\&\mathbb {A}_{Ab,Ab}^{p.p}=\Lambda _{AA}^{cc} \hat{\mathcal {E}}^{Ab,1}_{Ab}, \ \ \mathbb {A}_{Ab,mi,nj}^{p.rr}= \Lambda _{Amn}^{css} \hat{\mathcal {E}}^{Ab,2}_{mi,nj} +\delta _{mn} \delta _{ij} \Lambda _A^c \hat{\mathcal {E}}^{Ab,1} \beta _{mi,nj}^{rr}, \\ \end{aligned} \end{aligned}$$
$$\begin{aligned} \begin{aligned} \mathbb {A}_{Ab,Mi,Nj,Lk}^{p.ppp}&= \Lambda _A^c \hat{\mathcal {E}}^{Ab,1} \beta _{Mi,Nj,Lk}^{ppp} +\Lambda _{AMNL}^{cccc} \hat{\mathcal {E}}^{Ab,3}_{Mi,Nj,Lk}\\&+2 \delta _{MA} \delta _{ib}\Lambda _{AM}^{cc} \hat{\mathcal {E}}^{Ab,2}_{Mi} \delta _{NL} \delta _{jk} \beta _{Nj,Lk}^{pp},\\ \mathbb {A}_{Ab,Mi,nj,lk}^{p.prr}&= \Lambda _A^c \hat{\mathcal {E}}^{Ab,1} \beta _{Mi,nj,lk}^{prr} +3 \Lambda _{AMnl}^{ccss} \hat{\mathcal {E}}^{Ab,3}_{Mi,nj,lk}\\&+2 \delta _{MA} \delta _{ib} \Lambda _{AM}^{cc} \hat{\mathcal {E}}^{Ab,2}_{Mi} \delta _{nl} \delta _{jk} \beta _{nj,lk}^{rr}, \end{aligned} \end{aligned}$$
(67a)
$$\begin{aligned} \begin{aligned} \mathbb {A}_{ab,ab}^{r.r}&=\Lambda _{aa}^{ss} \hat{\mathcal {E}}^{ab,1}_{ab}, \ \ \mathbb {A}_{ab,Mi,nj}^{r.pr}= 2 \Lambda _{Mna}^{css} \hat{\mathcal {E}}^{ab,2}_{Mi,nj}, \\ \mathbb {A}_{ab,Mi,Nj,lk}^{r.ppr}&= 3 \Lambda _{MNla}^{ccss} \hat{\mathcal {E}}^{ab,3}_{Mi,Nj,lk} +2 \delta _{al} \delta _{bk} \Lambda _{al}^{ss} \hat{\mathcal {E}}^{ab,2}_{lk} \delta _{MN} \delta _{ij} \beta _{Mi,Nj}^{pp}, \\ \mathbb {A}_{ab,mi,nj,lk}^{r.rrr}&= \Lambda _{mnla}^{ssss} \hat{\mathcal {E}}^{ab,3}_{mi,nj,lk} +2 \delta _{al} \delta _{bk} \Lambda _{al}^{ss} \hat{\mathcal {E}}^{ab,2}_{lk} \delta _{mn} \delta _{ij} \beta _{mi,nj}^{rr}, \end{aligned} \end{aligned}$$
(67b)
where
$$\begin{aligned} \hat{\mathcal {E}}^{Ab,e}_{\underbrace{Mi,\ldots ,Nj}_{N_3}}= \int _{0}^{x_{20}}x_2 B_e^{Ab}\left( x_2\right) \underbrace{f_{Mi}\left( x_2\right) \cdot \ldots \cdot f_{Nj}\left( x_2\right) }_{N_3}dx_2. \end{aligned}$$
(68)
Partial derivatives of \(A_{Ab}^p\) and \(A_{ab}^r\) and \(A_{N}\) (\(A_{N}=\left\{ \{A_{Ab}^{p}\}, \{A_{ab}^{r}\}\right\} \)) by the generalised coordinates \(p_{Mi}\) and \(r_{mi}\) take the following form
$$\begin{aligned} \begin{aligned} \frac{\partial A_{Ab}^p}{\partial p_{Eh}}&=\mathbb {V}_{Ab,Eh}^p +\!\sum _{Mi} \mathbb {V}_{Ab,Eh,Mi}^{p.p} p_{Mi}\\&+\!\sum _{MNij} \mathbb {V}_{Ab,Eh,Mi,Nj}^{p.pp} p_{Mi} p_{Nj} +\sum _{mnij} \mathbb {V}_{Ab,Eh,mi,nj}^{p.rr} r_{mi} r_{nj},\\ \frac{\partial A_{Ab}^p}{\partial r_{eh}}&= \sum _{mi} \mathbb {V}_{Ab,mi,eh}^{p.r} r_{mi} +\sum _{Mnij} \mathbb {V}_{Ab,Mi,nj,eh}^{p.pr} p_{Mi} r_{nj},\\ \frac{\partial A_{ab}^r}{\partial p_{Eh}}&= \sum _{mi} \mathbb {V}_{ab,Eh,mi}^{r.r} r_{mi} +\sum _{Mnij} \mathbb {V}_{ab,Eh,Mi,nj}^{r.pr} p_{Mi} r_{nj},\\ \frac{\partial A_{ab}^r}{\partial r_{eh}}&= \mathbb {V}_{ab,eh}^r +\sum _{Mi} \mathbb {V}_{ab,Mi,eh}^{r.p} p_{Mi} \\&+\sum _{MNij} \mathbb {V}_{ab,Mi,Nj,eh}^{r.pp} p_{Mi} p_{Nj} +\sum _{mnij} \mathbb {V}_{ab,mi,nj,eh}^{r.rr} r_{mi} r_{nj}, \end{aligned} \end{aligned}$$
(69)
where \(\mathbb {V}\)-coefficients are expressed in terms of (67) as follows
$$\begin{aligned}&\mathbb {V}_{Ab,Eh}^p=\mathbb {A}_{Ab,Eh}^{p.p},\ \ \mathbb {V}_{Ab,Eh,Mi}^{p.p}= 2 \mathbb {A}_{Ab,Eh,Mi}^{p.pp}p_{Mi}, \nonumber \\&\mathbb {V}_{ab,eh}^r=\mathbb {A}_{ab,eh}^{r.r}, \ \ \mathbb {V}_{Ab,Eh,Mi,Nj}^{p.pp}= \mathbb {A}_{Ab,Eh,Mi,Nj}^{p.ppp}+2 \mathbb {A}_{Ab,Mi,Eh,Nj}^{p.ppp},\nonumber \\&\mathbb {V}_{Ab,Eh,mi,nj}^{p.rr}=\mathbb {A}_{Ab,Eh,mi,nj}^{p.prr},\ \ \mathbb {V}_{Ab,Mi,nj,eh}^{p.pr}=2 \mathbb {A}_{Ab,Mi,nj,eh}^{p.prr}p_{Mi} r_{nj},\nonumber \\&\mathbb {V}_{Ab,mi,eh}^{p.r}=2 \mathbb {A}_{Ab,mi,eh}^{p.rr},\ \ \mathbb {V}_{ab,Eh,Mi,Nj}^{r.pr}=2 \mathbb {A}_{ab,Eh,Mi,nj}^{r.ppr}p_{Mi} r_{nj},\\&\mathbb {V}_{ab,Mi,eh}^{r.p}=\mathbb {A}_{ab,Mi,eh}^{r.pr}, \ \ \mathbb {V}_{ab,mi,nj,eh}^{r.rr}=2 \mathbb {A}_{ab,eh,mi,nj}^{r.rrr}+\mathbb {A}_{ab,mi,nj,eh}^{r.rrr},\nonumber \\&\mathbb {V}_{ab,Eh,mi}^{r.r}=\mathbb {A}_{ab,Eh,mi}^{r.pr},\ \ \mathbb {V}_{ab,Mi,Nj,eh}^{r.pp}=\mathbb {A}_{ab,Mi,Nj,eh}^{r.ppr}.\nonumber \end{aligned}$$
(70)
1.3 A.3 Integrals \(A_{NK}\) Defined by (29)
By expanding elements of (29) (\(A_{NK}=\left\{ \{A_{NK}^{pp}, A_{NK}^{pr}\},\right. \) \(\left. \{A_{NK}^{pr}, A_{NK}^{rr}\}\right\} \)) to the second polynomial order by the generalised coordinates \(p_{Mi} \) and \(r_{mi}\), we get the following expressions
$$\begin{aligned} \begin{aligned} A_{Ab,Cd}^{pp}&=\mathbb {B}_{Ab,Cd}^{pp.0} +\sum _{Mi}\mathbb {B}_{Ab,Cd,Mi}^{pp.p} p_{Mi} \\&+\sum _{MNij}\!\!\mathbb {B}_{Ab,Cd,Mi,Nj}^{pp.pp} p_{Mi} p_{Nj} +\sum _{mnij}\!\!\mathbb {B}_{Ab,Cd,mi,nj}^{pp.rr} r_{mi} r_{nj},\\ A_{ab,cd}^{rr}&=\mathbb {B}_{ab,cd}^{rr.0} +\sum _{Mi}\mathbb {B}_{ab,cd,Mi}^{rr.p} p_{Mi} \\&+\sum _{MNij}\!\!\mathbb {B}_{ab,cd,Mi,Nj}^{rr.pp} p_{Mi} p_{Nj} +\sum _{mnij}\!\!\mathbb {B}_{ab,cd,mi,nj}^{rr.rr} r_{mi} r_{nj},\\ A_{Ab,cd}^{pr}&= \sum _{mi}\mathbb {B}_{Ab,cd,mi}^{pr.r} r_{mi} +\sum _{Mnij}\!\!\mathbb {B}_{Ab,cd,Mi,nj}^{pr.pr}p_{Mi} r_{nj}. \end{aligned} \end{aligned}$$
(71)
The \(\mathbb {B}\)-coefficients are as follows
$$\begin{aligned} \begin{aligned}&\mathbb {B}_{Ab,Cd}^{pp.0}= \Lambda _{AC}^{cc} \tilde{\mathcal {E}}^{Ab,Cd,0} +\Lambda _{AC}^{ss} \bar{\mathcal {E}}^{Ab,Cd,0}, \\&\mathbb {B}_{Ab,Cd,Mi}^{pp.p}= \Lambda _{ACM}^{ccc} \tilde{\mathcal {E}}^{Ab,Cd,1}_{Mi} +\Lambda _{MAC}^{css} \bar{\mathcal {E}}^{Ab,Cd,1}_{Mi},\\&\mathbb {B}_{Ab,Cd,Mi,Nj}^{pp.pp}= \Lambda _{ACMN}^{cccc} \tilde{\mathcal {E}}^{Ab,Cd,2}_{Mi,Nj} +\Lambda _{MNAC}^{ccss} \bar{\mathcal {E}}^{Ab,Cd,2}_{Mi,Nj} \\&\qquad +\left( \Lambda _{AC}^{cc} \tilde{\mathcal {E}}^{Ab,Cd,1} +\Lambda _{AC}^{ss} \bar{\mathcal {E}}^{Ab,Cd,1}\right) \delta _{MN} \delta _{ij} \beta _{Mi,Nj}^{pp}, \\&\mathbb {B}_{Ab,Cd,mi,nj}^{pp.rr}= \Lambda _{A,C,m,n}^{ccss} \tilde{\mathcal {E}}^{Ab,Cd,2}_{mi,nj} +\Lambda _{A,C,m,n}^{ssss} \bar{\mathcal {E}}^{Ab,Cd,2}_{mi,nj} \\&\qquad +\left( \Lambda _{AC}^{cc} \tilde{\mathcal {E}}^{Ab,Cd,1} +\Lambda _{AC}^{ss} \bar{\mathcal {E}}^{Ab,Cd,1}\right) \delta _{mn} \delta _{ij} \beta _{mi,nj}^{rr}, \end{aligned} \end{aligned}$$
(72a)
$$\begin{aligned} \begin{aligned}&\mathbb {B}_{ab,cd}^{rr.0}= \delta _{ac} \Lambda _{ac}^{ss} \tilde{\mathcal {E}}^{ab,cd,0} +\delta _{ac}\Lambda _{ac}^{cc} \bar{\mathcal {E}}^{ab,cd,0},\\&\mathbb {B}_{ab,cd,Mi}^{rr.p}= \Lambda _{Mac}^{css} \tilde{\mathcal {E}}^ {ab,cd,1}_{Mi} +\Lambda _{acM}^{ccc}\bar{\mathcal {E}}^{ab,cd,1}_{Mi},\\&\mathbb {B}_{ab,cd,Mi,Nj}^{rr.pp}= \Lambda _{MNac}^{ccss} \tilde{\mathcal {E}}^{ab,cd,2}_{Mi,Nj} +\Lambda _{acMN}^{cccc} \bar{\mathcal {E}}^{ab,cd,2}_{Mi,Nj} \\&\qquad +\left( \Lambda _{ac}^{ss} \tilde{\mathcal {E}}^{ab,cd,1} +\Lambda _{ac}^{cc} \bar{\mathcal {E}}^{ab,cd,1}\right) \delta _{m1} \delta _{i1} \delta _{MN} \delta _{ij} \beta _{Mi,Nj}^{pp},\\&\mathbb {B}_{ab,cd,mi,nj}^{rr.rr}= \Lambda _{mnac}^{ssss} \tilde{\mathcal {E}}^{ab,cd,2}_{mi,nj} +\Lambda _{acmn}^{ccss} \bar{\mathcal {E}}^{ab,cd,2}_{mi,nj} \\&\qquad +\left( \Lambda _{ac}^{ss} \tilde{\mathcal {E}}^{ab,cd,1} +\Lambda _{ac}^{cc} \bar{\mathcal {E}}^{ab,cd,1}\right) \delta _{m1} \delta _{i1} \delta _{mn} \delta _{ij} \beta _{mi,nj}^{rr}, \end{aligned} \end{aligned}$$
(72b)
$$\begin{aligned} \begin{aligned}&\mathbb {B}_{Ab,cd,mi}^{pr.r}= \Lambda _{Acm}^{css} \tilde{\mathcal {E}}^{Ab,cd,1}_{mi} -\Lambda _{cAm}^{css}\bar{\mathcal {E}}^{Ab,cd,1}_{mi}, \\&\mathbb {B}_{Ab,cd,Mi,nj}^{pr.pr}= 2 \left( \Lambda _{AMcn}^{ccss} \tilde{\mathcal {E}}^{Ab,cd,2}_{Mi,nj} -\Lambda _{cMAn}^{ccss} \bar{\mathcal {E}}^{Ab,cd,2}_{Mi,nj}\right) , \end{aligned} \end{aligned}$$
(72c)
where
$$\begin{aligned} \tilde{\mathcal {E}}^{Ab,Cd,e}_{\underbrace{Mi,\ldots ,Nj}_{N_3}}= \int _{0}^{x_{20}} F_e^{AbCd}\left( x_2\right) \underbrace{f_{Mi}\left( x_2\right) \cdot \ldots \cdot f_{Nj}\left( x_2\right) }_{N_3} dx_2, \end{aligned}$$
(73a)
$$\begin{aligned} \bar{\mathcal {E}}^{Ab,Cd,e}_{\underbrace{Mi,\ldots ,Nj}_{N_3}}= A C \int _{0}^{x_{20}} \frac{1}{x_2} {B_e^{AbCd}\left( x_2\right) \underbrace{f_{Mi}\left( x_2\right) \cdot \ldots \cdot f_{Nj}\left( x_2\right) }}_{{N_3}} dx_2. \end{aligned}$$
(73b)
The partial derivatives of \(A_{MiNj}^{pp}\), \(A_{minj}^{rr}\) and \(A_{Minj}^{pr}\) by \(p_{Mi}\), \(r_{mi}\) are
$$\begin{aligned} \begin{aligned}&\frac{\partial A_{Ab,Cd}^{pp}}{\partial p_{Eh}}= \mathbb {W}_{Ab,Cd,Eh}^{pp.p} +\sum _{Mi} \mathbb {W}_{Ab,Cd,Eh,Mi}^{pp.pp}p_{Mi},\\&\frac{\partial A_{Ab,Cd}^{pp}}{\partial r_{eh}}= \sum _{mi} \mathbb {W}_{Ab,Cd,mi,eh}^{pp.rr} r_{mi},\\&\frac{\partial A_{ab,cd}^{rr}}{\partial p_{Eh}}= \mathbb {W}_{ab,cd,Eh}^{rr.p} +\sum _{Mi} \mathbb {W}_{ab,cd,Eh,Mi}^{rr.pp} p_{Mi},\\&\frac{\partial A_{ab,cd}^{rr}}{\partial r_{eh}}=\sum _{m,i} \mathbb {W}_{ab,cd,mi,eh}^{rr.rr} r_{mi},\\&\frac{\partial A_{Ab,cd}^{pr}}{\partial p_{Eh}}=\sum _{mi} \mathbb {W}_{Ab,cd,Eh,mi}^{pr.pr} r_{mi},\\&\frac{\partial A_{Ab,cd}^{pr}}{\partial r_{eh}} =\mathbb {W}_{eh}^{pr.r} +\sum _{Mi} \mathbb {W}_{Ab,cd,Mi,eh}^{pr.pr}p_{Mi}, \end{aligned} \end{aligned}$$
(74)
where the \(\mathbb {W}\)-coefficients are expressed in terms of the matrix \(A_{NK}\) (72)
$$\begin{aligned} \begin{aligned}&\mathbb {W}_{Ab,Cd,Eh}^{pp.p}=\mathbb {B}_{Ab,Cd,Eh}^{pp.p}, \\&\mathbb {W}_{Ab,Cd,Eh,Mi}^{pp.pp}= 2\mathbb {B}_{Ab,Cd,Eh,Mi}^{pp.pp} =2 \mathbb {B}_{Ab,Cd,Mi,Eh}^{pp.pp},\\&\mathbb {W}_{Ab,Cd,mi,eh}^{pp.rr} =2\mathbb {B}_{Ab,Cd,eh,mi}^{pp.rr} =2 \mathbb {B}_{Ab,Cd,mi,eh}^{pp.rr},\\&\mathbb {W}_{ab,cd,Eh,Mi}^{rr.pp}= 2\mathbb {B}_{ab,cd,Eh,Mi}^{rr.pp}= 2 \mathbb {B}_{ab,cd,Mi,Eh}^{rr.pp},\\&\mathbb {W}_{ab,cd,mi,eh}^{rr.rr}= 2\mathbb {B}_{ab,cd,eh,mi}^{rr.rr}= 2 \mathbb {B}_{ab,cd,mi,eh}^{rr.rr},\\&\mathbb {W}_{ab,cd,Eh}^{rr.p}=\mathbb {B}_{ab,cd,Eh}^{rr.p}, \ \ \mathbb {W}_{Ab,cd,Eh,mi}^{pr.pr}=\mathbb {B}_{Ab,cd,Eh,mi}^{pr.pr},\\&\mathbb {W}_{Ab,cd,eh}^{pr.r}=\mathbb {B}_{Ab,cd,eh}^{pr.r}, \ \ \mathbb {W}_{Ab,cd,Mi,eh}^{pr.pr}=\mathbb {B}_{Ab,cd,Mi,eh}^{pr.pr}. \end{aligned} \end{aligned}$$
(75)
1.4 A.4 Generalised Velocities \(P_{Cd}\) and \(R_{cd}\)
After substituting expressions for the generalised velocities (33) into the kinematic equation (25), accounting for the derivatives (69) and (74) and collecting similar terms, we derive the \(\mathbb {Z}\)-coefficients as follows
$$\begin{aligned}&\mathbb {Z}_{Ab}^p=\frac{\mathbb {V}_{Ab,Ab}^p}{\mathbb {B}_{Ab,Ab}^{pp.0}},\ \ \mathbb {Z}_{Mi,Nj}^{pp,Ab}=\frac{\mathbb {V}_{Ab,Nj,Mi}^{p.p}-\mathbb {B}_{Ab,Nj,Mi}^{pp.p} \mathbb {Z}_{Nj}^p}{\mathbb {B}_{Ab,Ab}^{pp.0}},\nonumber \\&\mathbb {Z}_{Mi,Nj,Lk}^{ppp,Ab}=\frac{\mathbb {V}_{Ab,Lk,Mi,Nj}^{p.pp} -\mathbb {B}_{Ab,Lk,Mi,Nj}^{pp.pp} \mathbb {Z}_{Lk}^p -\sum _{Cd} \mathbb {B}_{Ab,Cd,Mi}^{pp.p} \mathbb {Z}_{Nj,Lk}^{pp,Cd}}{\mathbb {B}_{Ab,Ab}^{pp.0}},\nonumber \\&\qquad \qquad \qquad \mathbb {Z}_{mi,nj}^{rr,Ab}= \frac{\mathbb {V}_{Ab,mi,nj}^{p.r}-\mathbb {B}_{Ab,nj,mi}^{pr.r}\mathbb {Z}_{nj}^r}{\mathbb {B}_{Ab,Ab}^{pp.0}},\ \ \ \mathbb {Z}_{Mi,nj,lk}^{prr,Ab}= \nonumber \\&\frac{\mathbb {V}_{Ab,Mi,nj,lk}^{p.pr} -\mathbb {B}_{Ab,lk,Mi,nj}^{pr.pr} \mathbb {Z}_{lk}^r -\mathbb {B}_{Ab,cd,nj}^{pr.r} \mathbb {Z}_{Mi,lk}^{pr,cd} -\sum _{Cd} \mathbb {B}_{Ab,Cd,Mi}^{pp.p} \mathbb {Z}_{nj,lk}^{rr,Cd}}{\mathbb {B}_{Ab,Ab}^{pp.0}},\nonumber \\&\qquad \quad \mathbb {Z}_{mi,nj,Lk}^{rrp,Ab}=\frac{\mathbb {V}_{Ab,Lk,mi,nj}^{p.rr}- \mathbb {B}_{Ab,Lk,mi,nj}^{pp.rr}\mathbb {Z}_{Lk}^p -\mathbb {B}_{Ab,cd,nj}^{pr.r} \mathbb {Z}_{mi,Lk}^{rp,cd}}{\mathbb {B}_{Ab,Ab}^{pp.0}}, \end{aligned}$$
(76a)
$$\begin{aligned} \mathbb {Z}_{ab}^r=\frac{\mathbb {V}_{ab,ab}^r}{\mathbb {B}_{ab,ab}^{rr.0}},\, \, \,&\mathbb {Z}_{Mi,nj}^{pr,ab} =\frac{\mathbb {V}_{ab,Mi,nj}^{r.p} -\mathbb {B}_{ab,nj,Mi}^{rr.p} \mathbb {Z}_{nj}^r}{\mathbb {B}_{ab,ab}^{rr.0}},\nonumber \\&\quad \mathbb {Z}_{mi,Nj}^{rp,ab}=\frac{\mathbb {V}_{ab,Nj,mi}^{r.r} -\mathbb {B}_{Nj,ab,mi}^{pr.r} \mathbb {Z}_{Nj}^p}{\mathbb {B}_{ab,ab}^{rr.0}},\nonumber \\ \mathbb {Z}_{Mi,Nj,lk}^{ppr,ab}=&\; \frac{\mathbb {V}_{ab,Mi,Nj,lk}^{r.pp}-\mathbb {B}_{ab,lk,Mi,Nj}^{rr.pp}\mathbb {Z}_{lk}^r -\sum _{cd} \mathbb {B}_{ab,cd,Mi}^{rr.p} \mathbb {Z}_{Nj,lk}^{pr,cd}}{\mathbb {B}_{ab,ab}^{rr.0}},\nonumber \\ \mathbb {Z}_{mi,nj,lk}^{rrr,ab}=&\; \frac{\mathbb {V}_{ab,mi,nj,lk}^{r.rr}-\mathbb {B}_{ab,lk,mi,nj}^{rr.rr} \mathbb {Z}_{lk}^r -\sum _{Cd} \mathbb {B}_{Cd,ab,nj}^{pr.r} \mathbb {Z}_{mi,lk}^{rr,Cd}}{\mathbb {B}_{ab,ab}^{rr.0}},\nonumber \\ \mathbb {Z}_{Mi,nj,Lk}^{prp,ab}=&\; \bigg (\mathbb {V}_{ab,Lk,Mi,nj}^{r.pr} -\mathbb {B}_{Lk,ab,Mi,nj}^{pr.pr} \mathbb {Z}_{Lk}^p -\sum _{Cd} \mathbb {B}_{Cd,ab,nj}^{pr.r} \mathbb {Z}_{Mi,Lk}^{pp,Cd} \nonumber \\&\qquad \qquad \qquad \qquad \qquad -\sum _{cd} \mathbb {B}_{ab,cd,Mi}^{rr.p} \mathbb {Z}_{nj,Lk}^{rp,cd}\bigg )\bigg / {\mathbb {B}_{ab,ab}^{rr.0}}. \end{aligned}$$
(76b)
1.5 A.5 Integrals \(l_{i}\)
Expressions for \(\varvec{l}\) (see, (13)) appearing in the dynamic equations (26) take the form
$$\begin{aligned} \begin{aligned}&l_1=\rho \int _0^{2 \pi }\int _{0}^{x_{20}}\int _{0}^{f^*\left( x_2,x_3,t\right) +x_{10}}x_1^3 x_2dx_1dx_2dx_3,\\&l_2=\rho \int _0^{2 \pi }\int _{0}^{x_{20}}\int _{0}^{f^*\left( x_2,x_3,t\right) +x_{10}}x_1^3 x_2^2 \cos \left( x_3\right) dx_1dx_2dx_3,\\&l_3=\rho \int _0^{2 \pi }\int _{0}^{x_{20}}\int _{0}^{f^*\left( x_2,x_3,t\right) +x_{10}}x_1^3 x_2^2 \sin \left( x_3\right) dx_1dx_2dx_3. \end{aligned} \end{aligned}$$
(77)
Coefficients \(\hat{\mathbf {l}}_{Mi}^{\mathbf {r}\beta }\), \(\hat{\mathbf {l}}_{Mi,Nj}^{\mathbf {r}\beta \beta }\), \(\hat{\mathbf {l}}_{Mi,Nj,Lk}^{\mathbf {r}\beta \beta \beta }\) in (34) are determined by the following expressions (\(h_t\) and \(h_b\) are distances from the cone vertex to the unperturbed free surface and the bottom, respectively; the \(\beta _{Mi,Nj}^{pp}\) coefficients appear in expression for \(\beta _0\) (20), and \(\delta _{ij}\) is the Kronecker delta):
$$\begin{aligned} \begin{aligned}&\mathbf {l}^x=\frac{\pi }{4}\left( h_t^4-h_b^4\right) x_{20}^2,\ \ \mathbf {l}_{Mi,Nj}^{xpp}=\frac{h_t^2}{2} \delta _{MN} \delta _{ij} \Lambda _{MN}^{cc} \lambda _{Mi,Nj},\\&\mathbf {l}_{mi,nj}^{xrr}=\frac{h_t^2}{2} \delta _{mn} \delta _{ij} \Lambda _{mn}^{ss} \lambda _{mi,nj},\ \ \mathbf {l}_{Mi,Nj,Lk}^{xppp}=\frac{2}{3} h_t \Lambda _{MNL}^{ccc} \lambda _{Mi,Nj,Lk},\\&\mathbf {l}_{Mi,nj,lk}^{xprr}=2 h_t \Lambda _{Mnl}^{css} \lambda _{Mi,nj,lk},\quad \hat{\mathbf {l}}_{Mi}^{yp}=h_t^3 \delta _{1,M} \Lambda _{1M}^{cc} \hat{\lambda }_{Mi},\\&\hat{\mathbf {l}}_{Mi,Nj,Lk}^{yppp}=h_t \Lambda _{1MNL}^{cccc} \hat{\lambda }_{Mi,Nj,Lk} +3 h_t^2 \delta _{1M} \Lambda _{1M}^{cc} \hat{\lambda }_{Mi} \delta _{NL} \delta _{jk} \beta _{Nj,Lk}^{pp},\\&\hat{\mathbf {l}}_{Mi,Nj}^{ypp}=\frac{3}{2} h_t^2 \Lambda _{1MN}^{ccc} \hat{\lambda }_{Mi,Nj},\ \ \hat{\mathbf {l}}_{mi,nj}^{yrr}=\frac{3}{2} h_t^2 \Lambda _{1mn}^{css} \hat{\lambda }_{mi,nj},\\&\hat{\mathbf {l}}_{Mi,nj,lk}^{yprr}=3 h_t \Lambda _{1Mnl}^{ccss} \hat{\lambda }_{Mi,nj,lk} +3 h_t^2 \delta _{1M} \Lambda _{1M}^{cc} \hat{\lambda }_{Mi} \delta _{nl} \delta _{jk} \beta _{nj,lk}^{rr},\\&\hat{\mathbf {l}}_{mi}^{zp}=h_t^3 \delta _{1m} \Lambda _{m1}^{ss} \hat{\lambda }_{mi},\ \ \hat{\mathbf {l}}_{Mi,nj}^{zpr}=3 h_t^2 \Lambda _{Mn1}^{css} \hat{\lambda }_{Mi,nj},\\&\hat{\mathbf {l}}_{Mi,Nj,lk}^{zppr}=3 h_t \Lambda _{MNl1}^{ccss} \hat{\lambda }_{Mi,Nj,lk} +3 h_t^2 \delta _{1l} \Lambda _{l1}^{ss} \hat{\lambda }_{lk} \delta _{MN} \delta _{ij} \beta _{Mi,Nj}^{pp},\\&\hat{\mathbf {l}}_{mi,nj,lk}^{zrrr}=h_t \Lambda _{mnl1}^{ssss} \hat{\lambda }_{mi,nj,lk} +3 h_t^2 \delta _{1l} \Lambda _{l1}^{ss} \hat{\lambda }_{lk} \delta _{mn} \delta _{ij} \beta _{mi,nj}^{rr}. \end{aligned} \end{aligned}$$
(78)
The following notation is adopted
$$\begin{aligned} \displaystyle \hat{\lambda }_{\underbrace{Mi,\ldots ,Nj}_{N_3}}\!=\!\!\int _{0}^{x_{20}}\!\!\! x_2^2\underbrace{f_{Mi}\left( x_2\right) \cdot \ldots \cdot {f}_{Nj}\left( x_2\right) }_{N_3}dx_2, \end{aligned}$$
(79)
in addition to (64) and (65).
When using the Moiseev-Narimanov asymptotics (37) in (34), we deduce that only the following components should be kept
$$\begin{aligned} \begin{aligned} \displaystyle l_1&= \mathbf {l}^x+\mathbf {l}_{11,11}^{xpp} p_{11}^2 +\mathbf {l}_{11,11}^{xrr} r_{11}^2 +\mathbf {l}_{11,11,11}^{xprr} p_{11} r_{11}^2 +\mathbf {l}_{11,11,11}^{xppp} p_{11}^3,\\ \displaystyle l_2&= \hat{\mathbf {l}}_{11,11}^{ypp} p_{11}^2 +\hat{\mathbf {l}}_{11,11}^{yrr} r_{11}^2 +\hat{\mathbf {l}}_{11,11,11}^{yprr} p_{11} r_{11}^2 +\hat{\mathbf {l}}_{11,11,11}^{yppp} p_{11}^3\\ \displaystyle&+\sum _{i}\hat{\mathbf {l}}_{1i}^{yp} p_{1i} +\sum _{i}\!\left( \hat{\mathbf {l}}_{0i,11}^{ypp} +\hat{\mathbf {l}}_{11,0i}^{ypp}\right) p_{11} p_{0i}\\ \displaystyle&+\sum _{i}\!\left( \hat{\mathbf {l}}_{2i,11}^{ypp} +\hat{\mathbf {l}}_{11,2i}^{ypp}\right) p_{11} p_{2i} +\sum _{i}\!\left( \hat{\mathbf {l}}_{2i,11}^{yrr}\! +\hat{\mathbf {l}}_{11,2i}^{yrr}\right) r_{11} r_{2i},\\ \displaystyle l_3&=\hat{\mathbf {l}}_{11,11}^{zpr} p_{11} r_{11} +\hat{\mathbf {l}}_{11,11,11}^{zrrr} r_{11}^3 +\hat{\mathbf {l}}_{11,11,11}^{zppr} p_{11}^2 r_{11}\\ \displaystyle&+\sum _{i}\hat{\mathbf {l}}_{1i}^{zp} r_{1i} +\sum _{i}\hat{\mathbf {l}}_{0i,11}^{zpr} r_{11} p_{0i} +\sum _{i}\hat{\mathbf {l}}_{11,2i}^{zpr} p_{11} r_{2i} +\sum _{i}\hat{\mathbf {l}}_{2i,11}^{zpr} r_{11} p_{2i}. \end{aligned} \end{aligned}$$
(80)
The derivatives \({\partial l_1}/{\partial \beta _{N}}\) by \(p_{Mi}\) and \(r_{mi}\) take the following form
$$\begin{aligned} \begin{aligned} \displaystyle \frac{\partial l_1}{\partial p_{Eh}}&= \bar{\mathbf {l}}_{Eh,Eh}^{xpp} p_{Eh} +\sum _{MNij}\!\bar{\mathbf {l}}_{Eh,Mi,Nj}^{xppp} p_{Mi} p_{Nj} +\sum _{mnij}\!\bar{\mathbf {l}}_{Eh,mi,nj}^{xprr} r_{mi} r_{nj}\\ \displaystyle&+\sum _{MNLijk}\!\!\bar{\mathbf {l}}_{Eh,Mi,Nj,Lk}^{xpppp} p_{Mi} p_{Nj} p_{Lk} +\sum _{Mnlijk}\!\!\bar{\mathbf {l}}_{Eh,Mi,nj,lk}^{xpprr} p_{Mi} r_{nj} r_{lk}, \end{aligned} \end{aligned}$$
(81a)
$$\begin{aligned} \begin{aligned} \displaystyle \frac{\partial l_1}{\partial r_{eh}}&= \bar{\mathbf {l}}_{eh,eh}^{xrr} r_{eh} +\sum _{Mnij}\!\bar{\mathbf {l}}_{Mi,nj,eh}^{xprr} p_{Mi} r_{nj}\\ \displaystyle&+\sum _{MNlijk}\!\!\bar{\mathbf {l}}_{Mi,Nj,lk,eh}^{xpprr} p_{Mi} p_{Nj} r_{lk} +\sum _{mnlijk}\!\!\bar{\mathbf {l}}_{mi,nj,lk,eh}^{xrrrr} r_{mi} r_{nj} r_{lk}, \end{aligned} \end{aligned}$$
(81b)
where the derived \(\bar{\mathbf {l}}\)-coefficients are expressed in terms of \(l_1\) as follows
$$\begin{aligned} \begin{aligned} \displaystyle&\bar{\mathbf {l}}_{Eh,Eh}^{xpp}=2 \mathbf {l}_{Eh,Eh}^{xpp},\ \ \bar{\mathbf {l}}_{Eh,Mi,Nj}^{xppp}=3 \mathbf {l}_{Eh,Mi,Nj}^{xppp},\ \ \bar{\mathbf {l}}_{Eh,mi,nj}^{xprr}=\mathbf {l}_{Eh,mi,nj}^{xprr},\\ \displaystyle&\bar{\mathbf {l}}_{Eh,Mi,Nj,Lk}^{xpppp}=4 \mathbf {l}_{Eh,Mi,Nj,Lk}^{xpppp},\ \ \bar{\mathbf {l}}_{Eh,Mi,nj,lk}^{xpprr}=2 \mathbf {l}_{Eh,Mi,nj,lk}^{xpprr},\\ \displaystyle&\bar{\mathbf {l}}_{Mi,nj,eh}^{xprr}=2 \mathbf {l}_{Mi,nj,eh}^{xprr},\ \ \bar{\mathbf {l}}_{Mi,Nj,lk,eh}^{xpprr}=2 \mathbf {l}_{Mi,Nj,lk,eh}^{xpprr},\\ \displaystyle&\bar{\mathbf {l}}_{eh,eh}^{xrr}=2 \mathbf {l}_{eh,eh}^{xrr},\ \ \bar{\mathbf {l}}_{mi,nj,lk,eh}^{xrrrr}=4 \mathbf {l}_{mi,nj,lk,eh}^{xrrrr}. \end{aligned} \end{aligned}$$
(82)
For the steady-state sloshing regimes (53), (55), using the Moiseev-Narimanov asymptotics derives the second time derivative for horizontal components of the vector \(\varvec{l}\) as
$$\begin{aligned} \begin{aligned}&\ddot{l}_2 = B_s \left( \lambda _{y1}^{s} +A_c^2 \lambda _{y1}^{ccs}+B_s^2 \lambda _{y1}^{sss}\right) \sigma ^2 \sin \sigma t + B_s \left( A_c^2-B_s^2\right) \lambda _{y3}^{sss} \sigma ^2 \sin 3 \sigma t,\\&\ddot{l}_3 = A_c \left( \lambda _{z1}^c +A_c^2 \lambda _{z1}^{ccc}+B_s^2 \lambda _{z1}^{css}\right) \sigma ^2 \cos \sigma t + A_c \left( A_c^2-B_s^2\right) \lambda _{z3}^{ccc} \sigma ^2 \cos 3 \sigma t, \end{aligned} \end{aligned}$$
(83)
where coefficients \(\lambda _{ijk}\) are
$$\begin{aligned}{}\begin{array}[c]{l} \displaystyle \lambda _{y1}^s = \lambda _{z1}^c = -\pi h_t^3 \hat{\lambda }_{11},\quad \hat{\lambda }_{111} = \frac{{x_{20}^2 \hat{\lambda }_{11,11,11} -4 \hat{\lambda }_{11} \lambda _{11,11}}}{{4 h_t x_{20}^2}},\\ \lambda _{y1}^{sss}=\lambda _{yo1}^{sss}+\lambda _{yn1}^{sss},\ \lambda _{y1}^{ccs}=\lambda _{yo1}^{ccs}+\lambda _{yn1}^{ccs},\ \lambda _{y3}^{sss}=\lambda _{yo3}^{sss}+\lambda _{yn3}^{sss},\\ \lambda _{z1}^{ccc}=\lambda _{zo1}^{ccc}+\lambda _{zn1}^{ccc},\ \lambda _{z1}^{css}=\lambda _{zo1}^{css}+\lambda _{zn1}^{css},\ \lambda _{z3}^{ccc}=\lambda _{zo3}^{ccc}+\lambda _{zn3}^{ccc},\\ \lambda _{yo1}^{sss} = \lambda _{zo1}^{ccc} = -\frac{3}{4} \pi h_t^2 \left( 3 \hat{\lambda }_{111} + 2\left( 2\mathbf {o}_{010} + \mathbf {o}_{012}\right) \hat{\lambda }_{01,11} \right. \\ \left. \qquad + 2\left( \mathbf {o}_{210} + \mathbf {o}_{212} \right) \hat{\lambda }_{21,11}\right) ,\\ \lambda _{yo1}^{ccs} = \lambda _{zo1}^{css} = -\frac{3}{4} \pi h_t^2 \left( \hat{\lambda }_{111} + 2\left( 2\mathbf {o}_{010} - \mathbf {o}_{012}\right) \hat{\lambda }_{01,11} \right. \\ \left. \qquad - \left( 2 \mathbf {o}_{210} - 3 \mathbf {o}_{212}\right) \hat{\lambda }_{21,11}\right) ,\\ \lambda _{yo3}^{sss} = \lambda _{z3}^{ccc} = -\frac{27}{4} \pi h_t^2 \left( \hat{\lambda }_{111} + 2\mathbf {o}_{012} \hat{\lambda }_{01,11} + \mathbf {o}_{212} \hat{\lambda }_{21,11} \right) ,\\ \lambda _{yn1}^{ccc} = \frac{1}{2}\pi h_t^2 \left( 2 h_t G_{11}^{\hat{\lambda }_1} -3\left( 2 C_0^{\hat{\lambda }_{01}} + C_2^{\hat{\lambda }_{01}} + S_0^{\hat{\lambda }_{21}} + \frac{1}{2}{S_2^{\hat{\lambda }_{21}}}\right) \right) ,\\ \lambda _{yn1}^{css}= \frac{1}{2}\pi h_t^2 \left( 2 h_t G_{12}^{\hat{\lambda }_1} -3\left( 2 C_0^{\hat{\lambda }_{01}} + C_2^{\hat{\lambda }_{01}} + S_0^{\hat{\lambda }_{21}} - \frac{3}{2}{S_2^{\hat{\lambda }_{21}}}\right) \right) ,\\ \lambda _{yn3}^{ccc} = \frac{9}{2} \pi h_t^2 \left( 2 h_t G_3^{\hat{\lambda }_1} -3 C_2^{\hat{\lambda }_{01}} -\frac{3}{2}{S_2^{\hat{\lambda }_{21}}}\right) , \end{array} \end{aligned}$$
(84)
and
$$\begin{aligned}{}\begin{array}[c]{l} \displaystyle C_j^{\hat{\lambda }_{k1}} = \sum _{i=2}^{\infty } \hat{\lambda }_{ki11} \mathbf {o}_{0ij},\quad S_j^{\hat{\lambda }_{k1}} = \sum _{i=2}^{\infty } \hat{\lambda }_{ki11} \mathbf {o}_{2ij},\\ [1.2mm] \displaystyle G_3^{\hat{\lambda }_1} = \sum _{i=2}^{\infty } \hat{\lambda }_{1i} \mathbf {o}_{1i3},\quad G_{jk}^{\hat{\lambda }_1} = \sum _{i=2}^{\infty } \hat{\lambda }_{1i} \mathbf {o}_{1ijk}. \end{array} \end{aligned}$$
(85)
1.6 A.6 The \(\mathbf {d}\)-, \(\mathbf {g}\)-, \(\mathbf {t}\)-Coefficients in (35)
The \(\mathbf {d}\)-, \(\mathbf {g}\)-, \(\mathbf {t}\)-coefficients of the infinite-dimensional nonlinear modal equation (35) are computed by the formulas
$$\begin{aligned}&\mathbf {d}_{Mi}^{p,Eh}=\delta _{M,E} \delta _{i,h} \mathbb {V}_{Mi,Eh}^p \mathbb {Z}_{Mi}^p,\ \ \ \mathbf {g}_{Mi}^{p,Eh}=\delta _{M,E} \delta _{i,h} {\bar{\mathbf {l}}}_{Eh,Mi}^{opp},\\&\mathbf {g}_{Mi,Nj}^{pp,Eh}={\bar{\mathbf {l}}}_{Eh,Mi,Nj}^{oppp},\ \ \ \mathbf {g}_{Mi,nj,lk}^{prr,Eh}={\bar{\mathbf {l}}}_{Eh,Mi,nj,lk}^{opprr},\\&\mathbf {d}_{Mi,Nj}^{pp,Eh}= \mathbb {V}_{Nj,Eh,Mi}^{p.p} \mathbb {Z}_{Nj}^p +\sum _{Ab} \delta _{A,E} \delta _{b,h} \mathbb {V}_{Ab,Eh}^p \mathbb {Z}_{Mi,Nj}^{pp,Ab},\\&\mathbf {d}_{mi,nj}^{rr,Eh}= \mathbb {V}_{nj,Eh,mi}^{r.r} \mathbb {Z}_{nj}^r+\sum _{Ab} \delta _{A,E} \delta _{b,h} \mathbb {V}_{Ab,Eh}^p \mathbb {Z}_{mi,nj}^{rr,Ab}, \\&\mathbf {t}_{Mi,Nj}^{pp,Eh}=\frac{1}{2} \mathbb {W}_{Mi,Nj,Eh}^{pp.p} \mathbb {Z}_{Mi}^p \mathbb {Z}_{Nj}^p +\sum _{Ab} \delta _{A,E} \delta _{b,h} \mathbb {V}_{Ab,Eh}^p \mathbb {Z}_{Mi,Nj}^{pp,Ab},\\&\mathbf {t}_{mi,nj}^{rr,Eh}= \frac{1}{2} \mathbb {W}_{mi,nj,Eh}^{rr.p} \mathbb {Z}_{mi}^r \mathbb {Z}_{nj}^r +\sum _{Ab} \delta _{AE} \delta _{bh} \mathbb {V}_{Ab,Eh}^p \mathbb {Z}_{mi,nj}^{rr,Ab},\\&\mathbf {d}_{Mi,Nj,Lk}^{ppp,Eh}= \mathbb {V}_{Lk,Eh,Mi,Nj}^{p.pp} \mathbb {Z}_{Lk}^p +\sum _{Ab} {\mathbb {V}}_{Ab,Eh,Mi}^{p.p} \mathbb {Z}_{Nj,Lk}^{pp,Ab}\\&\qquad +\sum _{Ab} \delta _{AE} \delta _{bh} \mathbb {V}_{Ab,Eh}^p \mathbb {Z}_{Mi,Nj,Lk}^{ppp,Ab},\\&\mathbf {d}_{Mi,nj,lk}^{prr,Eh}= \mathbb {V}_{lk,Eh,Mi,nj}^{r.pr} \mathbb {Z}_{lk}^r +\sum _{ab} \mathbb {V}_{ab,Eh,nj}^{r.r} \mathbb {Z}_{Mi,lk}^{pr,ab}\\&\qquad +\sum _{Ab} \mathbb {V}_{Ab,Eh,Mi}^{p.p} \mathbb {Z}_{nj,lk}^{rr,Ab} +\sum _{Ab} \delta _{AE} \delta _{bh} \mathbb {V}_{Ab,Eh}^p \mathbb {Z}_{Mi,nj,lk}^{prr,Ab}, \\&\mathbf {g}_{mi,nj}^{rr,Eh}={\bar{\mathbf {l}}}_{Eh,mi,nj}^{oprr},\ \ \mathbf {g}_{Mi,Nj,Lk}^{ppp,Eh}=\bar{\mathbf {l}}_{Eh,Mi,Nj,Lk}^{opppp}, \end{aligned}$$
$$\begin{aligned}&\mathbf {d}_{mi,nj,Lk}^{rrp,Eh}\!=\mathbb {V}_{Lk,Eh,mi,nj}^{p.rr}\mathbb {Z}_{Lk}^p +\!\sum _{ab} \mathbb {V}_{ab,Eh,mi}^{r.r} \mathbb {Z}_{nj,Lk}^{rp,ab}\\&\qquad +\sum _{Ab} \delta _{AE} \delta _{bh} \mathbb {V}_{Ab,Eh}^p \mathbb {Z}_{mi,nj,Lk}^{rrp,Ab},\\&\mathbf {t}_{Mi,Nj,Lk}^{ppp,Eh}= \frac{1}{2} \mathbb {W}_{Nj,Lk,Eh,Mi}^{pp.pp} \mathbb {Z}_{Nj}^p \mathbb {Z}_{Lk}^p +\sum _{Ab} \mathbb {V}_{Ab,Eh,Mi}^{p.p} \mathbb {Z}_{Nj,Lk}^{pp,Ab}\\&\qquad +\sum _{Cd} \frac{1}{2} \left( \mathbb {W}_{Cd,Nj,Eh}^{pp.p} +\mathbb {W}_{Nj,Cd,Eh}^{pp.p}\right) \mathbb {Z}_{Nj}^p \mathbb {Z}_{Mi,Lk}^{pp,Cd} \\&\qquad +\sum _{Ab} \delta _{AE} \delta _{bh} \mathbb {V}_{Ab,Eh}^p \left( \mathbb {Z}_{Mi,Nj,Lk}^{ppp,Ab} +\mathbb {Z}_{Nj,Mi,Lk}^{ppp,Ab}\right) , \end{aligned}$$
$$\begin{aligned}&\mathbf {t}_{Mi,nj,lk}^{prr,Eh}= \sum _{Ab} \mathbb {V}_{Ab,Eh,Mi}^{p.p} \mathbb {Z}_{nj,lk}^{rr,Ab} +\sum _{Ab} \delta _{AE} \delta _{bh} \mathbb {V}_{Ab,Eh}^p \mathbb {Z}_{Mi,nj,lk}^{prr,Ab}\\&+\frac{1}{2} \mathbb {W}_{nj,lk,Eh,Mi}^{rr.pp} \mathbb {Z}_{nj}^r \mathbb {Z}_{lk}^r +\sum _{cd} \frac{1}{2} \left( \mathbb {W}_{cd,lk,Eh}^{rr.p} +\mathbb {W}_{lk,cd,Eh}^{rr.p}\right) \mathbb {Z}_{lk}^r \mathbb {Z}_{Mi,nj}^{pr,cd},\\&\mathbf {t}_{mi,Nj,lk}^{rpr,Eh}= \mathbb {W}_{Nj,lk,Eh,mi}^{pr.pr} \mathbb {Z}_{Nj}^p \mathbb {Z}_{lk}^r +\sum _{Cd} \frac{1}{2} \left( \mathbb {W}_{Cd,Nj,Eh}^{pp.p} +\mathbb {W}_{Nj,Cd,Eh}^{pp.p}\right) \\&\qquad \times \mathbb {Z}_{Nj}^p \mathbb {Z}_{mi,lk}^{rr,Cd} +\sum _{cd} \frac{1}{2} \left( \mathbb {W}_{cd,lk,Eh}^{rr.p} +\mathbb {W}_{lk,cd,Eh}^{rr.p}\right) \mathbb {Z}_{lk}^r \mathbb {Z}_{mi,Nj}^{rp,cd} \\&\qquad +\sum _{ab} \mathbb {V}_{ab,Eh,mi}^{r.r} \left( \mathbb {Z}_{Nj,lk}^{pr,ab}+\mathbb {Z}_{lk,Nj}^{rp,ab}\right) \\&\qquad +\sum _{Ab} \delta _{AE} \delta _{bh} \mathbb {V}_{Ab,Eh}^p \left( \mathbb {Z}_{Nj,mi,lk}^{prr,Ab} +\mathbb {Z}_{mi,lk,Nj}^{rrp,Ab}+\mathbb {Z}_{lk,mi,Nj}^{rrp,Ab}\right) , \end{aligned}$$
$$\begin{aligned}&\mathbf {d}_{mi}^{r,eh}=\delta _{m,e} \delta _{i,h} \mathbb {V}_{mi,eh}^r \mathbb {Z}_{mi}^r,\ \ \mathbf {g}_{mi}^{r,eh}=\delta _{m,e} \delta _{i,h} {\bar{\mathbf {l}}}_{mi,eh}^{orr},\\&\mathbf {g}_{Mi,nj}^{pr,eh}={\bar{\mathbf {l}}}_{Mi,nj,eh}^{oprr},\ \ \mathbf {g}_{Mi,Nj,lk}^{ppr,eh}={\bar{\mathbf {l}}}_{Mi,Nj,lk,eh}^{opprr},\ \ \ \mathbf {g}_{mi,nj,lk}^{rrr,eh}={\bar{\mathbf {l}}}_{mi,nj,lk,eh}^{orrrr},\\&\mathbf {t}_{Mi,nj}^{pr,eh}= \mathbb {W}_{eh}^{pr.r} \mathbb {Z}_{Mi}^p \mathbb {Z}_{nj}^r +\sum _{ab} \delta _{ae} \delta _{bh} \mathbb {V}_{ab,eh}^r \left( \mathbb {Z}_{Mi,nj}^{pr,ab} +\mathbb {Z}_{nj,Mi}^{rp,ab}\right) ,\\&\mathbf {d}_{Mi,nj}^{pr,eh}= \mathbb {V}_{nj,Mi,eh}^{r.p} \mathbb {Z}_{nj}^r +\sum _{ab} \delta _{ae} \delta _{bh} \mathbb {V}_{ab,eh}^r \mathbb {Z}_{Mi,nj}^{pr,ab},\\&\mathbf {d}_{mi,Nj}^{rp,eh}= \mathbb {V}_{Nj,mi,eh}^{p.r} \mathbb {Z}_{Nj}^p +\sum _{ab} \delta _{ae} \delta _{bh} \mathbb {V}_{ab,eh}^r \mathbb {Z}_{mi,Nj}^{rp,ab}, \end{aligned}$$
$$\begin{aligned}&\mathbf {d}_{Mi,nj,Lk}^{prp,eh}\!= \mathbb {V}_{Lk,Mi,nj,eh}^{p.pr}\mathbb {Z}_{Lk}^p +\sum _{Ab}\!\mathbb {V}_{Ab,nj,eh}^{p.r} \mathbb {Z}_{Mi,Lk}^{pp,Ab}\\&\qquad +\sum _{ab}\!\mathbb {V}_{ab,Mi,eh}^{r.p} \mathbb {Z}_{nj,Lk}^{rp,ab} +\sum _{ab} \delta _{ae} \delta _{bh} \mathbb {V}_{ab,eh}^r \mathbb {Z}_{Mi,nj,Lk}^{prp,ab},\\&\mathbf {d}_{Mi,Nj,lk}^{ppr,eh}= \mathbb {V}_{lk,Mi,Nj,eh}^{r.pp} \mathbb {Z}_{lk}^r +\sum _{ab} \mathbb {V}_{ab,Mi,eh}^{r.p}\mathbb {Z}_{Nj,lk}^{pr,ab}\\&\qquad +\sum _{ab} \delta _{ae} \delta _{bh} \mathbb {V}_{ab,eh}^r \mathbb {Z}_{Mi,Nj,lk}^{ppr,ab},\\&\mathbf {d}_{mi,nj,lk}^{rrr,eh} =\mathbb {V}_{lk,mi,nj,eh}^{r.rr} \mathbb {Z}_{lk}^r +\sum _{Ab} \mathbb {V}_{Ab,mi,eh}^{p.r} \mathbb {Z}_{nj,lk}^{rr,Ab}\\&\qquad +\sum _{ab} \delta _{ae} \delta _{bh} \mathbb {V}_{ab,eh}^r \mathbb {Z}_{mi,nj,lk}^{rrr,ab}, \end{aligned}$$
$$\begin{aligned}&\mathbf {t}_{mi,Nj,Lk}^{rpp,eh}= \frac{1}{2} \mathbb {W}_{Nj,Lk,mi,eh}^{pp.rr} \mathbb {Z}_{Nj}^p \mathbb {Z}_{Lk}^p +\sum _{Ab} \mathbb {V}_{Ab,mi,eh}^{p.r} \mathbb {Z}_{Nj,Lk}^{pp,Ab}\\&\qquad +\sum _{cd} \mathbb {W}_{Nj,cd,eh}^{pr.r} \mathbb {Z}_{mi,Lk}^{rp,cd} \mathbb {Z}_{Nj}^p +\sum _{ab} \delta _{ae} \delta _{bh} \mathbb {V}_{ab,eh}^r \mathbb {Z}_{Nj,mi,Lk}^{prp,ab},\\&\mathbf {t}_{Mi,Nj,lk}^{ppr,eh}= \mathbb {W}_{Nj,lk,Mi,eh}^{pr.pr} \mathbb {Z}_{Nj}^p \mathbb {Z}_{lk}^r +\sum _{cd}\mathbb {W}_{Nj,cd,eh}^{pr.r} \mathbb {Z}_{Mi,lk}^{pr,cd} \mathbb {Z}_{Nj}^p\\&\qquad +\sum _{Ab}\mathbb {W}_{Ab,lk,eh}^{pr.r} \mathbb {Z}_{Mi,Nj}^{pp,Ab} \mathbb {Z}_{lk}^r +\sum _{ab}\mathbb {V}_{ab,Mi,eh}^{r.p} \left( \mathbb {Z}_{Nj,lk}^{pr,ab}+\mathbb {Z}_{lk,Nj}^{rp,ab}\right) \\&\qquad +\sum _{ab}\delta _{ae}\delta _{bh}\mathbb {V}_{ab,eh}^r \left( \mathbb {Z}_{Mi,Nj,lk}^{ppr,ab}+\mathbb {Z}_{Nj,Mi,lk}^{ppr,ab}+\mathbb {Z}_{Mi,lk,Nj}^{prp,ab}\right) ,\\&\mathbf {t}_{mi,nj,lk}^{rrr,eh}= \frac{1}{2} \mathbb {W}_{nj,lk,mi,eh}^{rr.rr} \mathbb {Z}_{nj}^r \mathbb {Z}_{lk}^r +\sum _{Ab} \mathbb {W}_{Ab,lk,eh}^{pr.r} \mathbb {Z}_{mi,nj}^{rr,Ab} \mathbb {Z}_{lk}^r\\&\qquad +\sum _{Ab} \mathbb {V}_{Ab,mi,eh}^{p.r} \mathbb {Z}_{nj,lk}^{rr,Ab} +\sum _{ab} \delta _{ae} \delta _{bh} \mathbb {V}_{ab,eh}^r \left( \mathbb {Z}_{mi,nj,lk}^{rrr,ab} +\mathbb {Z}_{nj,mi,lk}^{rrr,ab}\right) . \end{aligned}$$
1.7 A.7 Coefficients of the Modal System (38)
The nonzero hydrodynamic coefficients in (38) take the form
$$\begin{aligned}&\mu _{0h}^p=\mathbf {d}_{1i}^{p,1i}= \mu _{0h}^r=\mathbf {d}_{1i}^{r,li}, \ \sigma _{0h}^2=\mathbf {g}_{1i}^{p,1i}/\mathbf {d}_{1i}^{p,1i},\ \mathcal {G}_{0h}=\mathbf {g}_{11,11}^{pp,1i}=\mathbf {g}_{11,11}^{rr,1i},\\&d _{8,h}=\mathbf {t}_{11,11}^{pp.1i}=\mathbf {t}_{11,11}^{rr,1i},\ \ d _{10,h}=\mathbf {d}_{11,11}^{pp,1i}=\mathbf {d}_{11,11}^{rr,1i},\\&\mu _{2h}^p=\mathbf {d}_{2h}^{p,2h}= \mu _{1k}^r=\mathbf {d}_{2h}^{r,2h},\ \ \sigma _{2h}^2= \mathbf {g}_{2h}^{p,2h}/\mathbf {d}_{2h}^{p,2h} =\mathbf {g}_{2h}^{r,2h}/\mathbf {d}_{2h}^{r,2h},\\&\mathcal {G}_{4,h}=\mathbf {g}_{11,11}^{pp,2h}= -\mathbf {g}_{11,11}^{rr,2h}= \tfrac{1}{2}\mathbf {g}_{11,11}^{pr,2h},\ \ d _{7,h}=\mathbf {t}_{11,11}^{pp,2h}=-\mathbf {t}_{11,11}^{rr,2h}= \tfrac{1}{2}\mathbf {t}_{11,11}^{pr,2h}, \end{aligned}$$
$$\begin{aligned}&d _{9,h}=\mathbf {d}_{11,11}^{pp,2h}=-\mathbf {d}_{11,11}^{rr,2h}= \mathbf {d}_{11,11}^{pr,2h}=\mathbf {d}_{11,11}^{rp,2h},\\&\mu _{11}^p=\mathbf {d}_{11}^{p,11} =\mu _{1k}^r=\mathbf {d}_{11}^{r,11},\ \sigma _{11}^2=\mathbf {g}_{11}^{p,11}/\mathbf {d}_{11}^{p,11} =\mathbf {g}_{11}^{r,11}/\mathbf {d}_{11}^{r,11},\\&\mathcal {G}_1=\mathbf {g}_{11,11,11}^{ppp,11}=\mathbf {g}_{11,11,11}^{prr,11}= \mathbf {g}_{11,11,11}^{ppr,11}=\mathbf {g}_{11,11,11}^{rrr,11},\ \ \mathcal {G}_2^j=\mathbf {g}_{0j,11}^{pp,11}+\mathbf {g}_{11,0j}^{pp,11}= \mathbf {g}_{0j,11}^{pr,11},\\&\mathcal {G}_3^j=\mathbf {g}_{11,2j}^{pp,11}+\mathbf {g}_{2j,11}^{pp,11} = \mathbf {g}_{11,2j}^{rr,11}+\mathbf {g}_{2j,11}^{rr,11}=\mathbf {g}_{11,2j}^{pr,11}= -\mathbf {g}_{2j,11}^{pr,11}, \end{aligned}$$
$$\begin{aligned}&d _1=\mathbf {d}_{11,11,11}^{ppp,11}=\mathbf {d}_{11,11,11}^{prr,11} =\mathbf {t}_{11,11,11}^{ppp,11}=\mathbf {t}_{11,11,11}^{prr,11} =\mathbf {d}_{11,11,11}^{prp,11}=\mathbf {d}_{11,11,11}^{rrr,11}\\& = \mathbf {t}_{11,11,11}^{rpp,11}=\mathbf {t}_{11,11,11}^{rrr,11},\\&d _2=\mathbf {d}_{11,11,11}^{rrp,11}=-\mathbf {d}_{11,11,11}^{prr,11}= \tfrac{1}{2} \mathbf {t}_{11,11,11}^{rpr,11}= -\tfrac{1}{2} \mathbf {t}_{11,11,11}^{prr,11} =\mathbf {d}_{11,11,11}^{ppr,11}\\&\qquad =-\mathbf {d}_{11,11,11}^{prp,11}= \tfrac{1}{2} \mathbf {t}_{11,11,11}^{ppr,11}= -\tfrac{1}{2} \mathbf {t}_{11,11,11}^{rpp,11},\\&d _3^j=\mathbf {d}_{2j,11}^{pp,11}=\mathbf {d}_{2j,11}^{rr,11} =\mathbf {t}_{2j,11}^{pp,11}+\mathbf {t}_{11,2j}^{pp,11}= \mathbf {t}_{2j,11}^{rr,11}+\mathbf {t}_{11,2j}^{rr,11} =\mathbf {d}_{2j,11}^{rp,11}\\&=-\mathbf {d}_{2j,11}^{pr,11}= \mathbf {t}_{11,2j}^{pr,11} =-\mathbf {t}_{2j,11}^{pr,11},\\&d _4^j=\mathbf {d}_{11,2j}^{pp,11}=\mathbf {d}_{11,2j}^{rr,11}= \mathbf {d}_{11,2j}^{pr,11}=-\mathbf {d}_{11,2j}^{rp,11}, \end{aligned}$$
$$\begin{aligned}&d _5^j=\mathbf {d}_{0j,11}^{pp,11} =\mathbf {t}_{0j,11}^{pp,11}+\mathbf {t}_{11,0j}^{pp,11}= \mathbf {d}_{0j,11,11}^{pr,11}=\mathbf {t}_{0j,11,11}^{pr},\\&d _6^j=\mathbf {d}_{11,0j}^{pp,11}=\mathbf {d}_{11,0j}^{rp,11},\\&\mu _{3h}^p=\mathbf {d}_{3h}^{p,3h} =\mu _{3h}^r=\mathbf {d}_{3h}^{r,3h},\ \sigma _{3h}^2=\mathbf {g}_{3h}^{p,3h}/\mathbf {d}_{3h}^{p,3h} =\mathbf {g}_{3h}^{r,3h}/\mathbf {d}_{3h}^{r,3h},\\&\mathcal {G}_{6,h}=\mathbf {g}_{11,11,11}^{ppp,3h}= -\tfrac{1}{3} \mathbf {g}_{11,11,11}^{prr,3h}=\tfrac{1}{3}\mathbf {g}_{11,11,11}^{ppr,3h}= -\mathbf {g}_{11,11,11}^{rrr,3h},\\&\mathcal {G}_{5,h}^j= \mathbf {g}_{11,2j}^{pp,3h}+\mathbf {g}_{2j,11}^{pp,3h}= -\mathbf {g}_{11,2j}^{rr,3h}-\mathbf {g}_{2j,11}^{rr,3h}= \mathbf {g}_{11,2j}^{pr,3h}=\mathbf {g}_{2j,11}^{pr,3h}, \end{aligned}$$
$$\begin{aligned}&d _{11,h}= \mathbf {d}_{11,11,11}^{ppp,3h}= -\mathbf {d}_{11,11,11}^{rrp,3h}= -\tfrac{1}{2} \mathbf {d}_{11,11,11}^{prr,3h}=\mathbf {d}_{11,11,11}^{ppr,3h}= -\mathbf {d}_{11,11,11}^{rrr,3h} =\tfrac{1}{2} \mathbf {d}_{11,11,11}^{prp,3h},\\&d _{12,h}= \mathbf {t}_{11,11,11}^{ppp}= -\mathbf {t}_{11,11,11}^{prr,3h}= -\tfrac{1}{2} \mathbf {t}_{11,11,11}^{rpr,3h}= \mathbf {t}_{11,11,11}^{rpp,3h} = -\mathbf {t}_{11,11,11}^{rrr,3h}=\tfrac{1}{2} \mathbf {t}_{11,11,11}^{ppr,3h}, \end{aligned}$$
$$\begin{aligned}&d _{13,h}^j=\mathbf {d}_{2j,11}^{pp,3h}=-\mathbf {d}_{2j,11}^{rr,3h}= \mathbf {d}_{2j,11}^{rp,3h}=\mathbf {d}_{2j,11}^{pr,3h},\\&d _{14,h}^j=\mathbf {d}_{11,2j}^{pp,3h}=-\mathbf {d}_{11,2j}^{rr,3h}= \mathbf {d}_{11,2j}^{pr,3h}=\mathbf {d}_{11,2j}^{rp,3h},\\&d _{15,h}^j=\mathbf {t}_{2j,11}^{pp,3h}+\mathbf {t}_{11,2j}^{pp,3h} = -\mathbf {t}_{2j,11}^{rr,3h}-\mathbf {t}_{11,2j}^{rr,3h}=\mathbf {t}_{11,2j}^{pr,3h}= \mathbf {t}_{2j,11}^{pr,3h}, \end{aligned}$$
$$\begin{aligned}&\mu _{1k}^p=\mathbf {d}_{1k}^{p,1k} =\mu _{1k}^r=\mathbf {d}_{1k}^{r,1k},\ \ \sigma _{1k}^2=\mathbf {g}_{1k}^{p,1k}/\mathbf {d}_{1k}^{p,1k} =\mathbf {g}_{1k}^{r,1k}/\mathbf {d}_{1k}^{r,1k},\\&\mathcal {G}_{1k}= \mathbf {g}_{11,11,11}^{ppp,1k}=\mathbf {g}_{11,11,11}^{prr,1k}= \mathbf {g}_{11,11,11}^{ppr,1k}=\mathbf {g}_{11,11,11}^{rrr,1k},\\&\mathcal {G}_{2,k}^j= \mathbf {g}_{11,2j}^{pp,1k}+\mathbf {g}_{2j,11}^{pp,1k}= \mathbf {g}_{11,2j}^{rr,1k}+\mathbf {g}_{2j,11}^{rr,1k}= \mathbf {g}_{1k,11,2j}^{pr,1k}=-\mathbf {g}_{2j,11}^{pr,1k},\\&\mathcal {G}_{3,k}^j= \mathbf {g}_{0j,11}^{pp,1k}+\mathbf {g}_{11,0j}^{pp,1k}= \mathbf {g}_{1k,0j,11}^{pr}, \end{aligned}$$
$$\begin{aligned}&d _{16,k}^j= \mathbf {d}_{11,11,11}^{ppp,1k}=\mathbf {d}_{11,11,11}^{prr,1k} = \mathbf {d}_{11,11,11}^{prp,1k}=\mathbf {d}_{11,11,11}^{rrr,1k},\\&d _{17,k}^j= \mathbf {d}_{11,11,11}^{rrp,1k}=-\mathbf {d}_{11,11,11}^{prr,1k} =\mathbf {d}_{11,11,11}^{ppr,1k} =-\mathbf {d}_{11,11,11}^{prp,1k},\\&d _{18,k}^j= \mathbf {t}_{11,11,11}^{ppp,1k}=\mathbf {t}_{11,11,11}^{prr,1k}= \mathbf {t}_{11,11,11}^{rpp,1k}=\mathbf {t}_{11,11,11}^{rrr,1k},\\&d _{19,k}^j= \mathbf {t}_{11,11,11}^{rpr,1k}=-\mathbf {t}_{11,11,11}^{prr,1k}= \mathbf {t}_{11,11,11}^{ppr,1k}=-\mathbf {t}_{11,11,11}^{rpp,1k}, \end{aligned}$$
$$\begin{aligned}&d _{20,k}^j= \mathbf {d}_{2j,11}^{pp,1k} =\mathbf {d}_{2j,11}^{rr,1k}= \mathbf {d}_{2j,11}^{rp,1k}=-\mathbf {d}_{2j,11}^{pr,1k},\\&d _{21k}^j= \mathbf {d}_{11,2j}^{pp,1k}=\mathbf {d}_{11,2j}^{rr,1k}= -\mathbf {d}_{11,2j}^{rp,1k} =\mathbf {d}_{11,2j}^{pr,1k},\\&d _{22,k}^j= \mathbf {t}_{2j,11}^{pp,1k}+\mathbf {t}_{11,2j}^{pp,1k}= \mathbf {t}_{2j,11}^{rr,1k}+\mathbf {t}_{11,2j}^{rr,1k}= \mathbf {t}_{11,2j}^{pr,1k}=-\mathbf {t}_{2j,11}^{pr,1k},\\&d _{23,k}^j=\mathbf {d}_{0j,11}^{pp,1k}=\mathbf {d}_{0j,11}^{pr,1k},\ \ d _{24,k}^j=\mathbf {d}_{11,0j}^{pp,1k}=\mathbf {d}_{11,0j}^{rp,1k},\\&d _{25,k}^j=\mathbf {t}_{0j,11}^{pp,1k}+\mathbf {t}_{11,0j}^{pp,1k}= \mathbf {t}_{0j,11}^{pr,1k}. \end{aligned}$$
