Appendix 1
The mass matrix M is as follows:
$$ {\varvec{M}} = \left[ {\begin{array}{*{20}c} {M_{11} } & 0 & 0 & 0 & 0 & {M_{{1{6}}} } & {{\varvec{M}}_{17} } \\ 0 & {M_{22} } & 0 & 0 & 0 & {M_{{{26}}} } & {{\varvec{M}}_{27} } \\ 0 & 0 & {M_{33} } & {M_{34} } & {M_{{3{5}}} } & 0 & {{\varvec{M}}_{37} } \\ 0 & 0 & {M_{34}^{{}} } & {M_{44} } & {M_{{{45}}} } & 0 & {{\varvec{M}}_{47} } \\ 0 & 0 & {M_{{3{5}}}^{{}} } & {M_{{4{5}}}^{{}} } & {M_{55} } & {0} & {{\varvec{M}}_{57} } \\ {M_{{{16}}}^{{}} } & {M_{{{26}}}^{{}} } & 0 & 0 & {0} & {M_{66} } & {{\varvec{M}}_{67} } \\ {{\varvec{M}}_{17}^{{\text{T}}} } & {{\varvec{M}}_{27}^{{\text{T}}} } & {{\varvec{M}}_{37}^{{\text{T}}} } & {{\varvec{M}}_{47}^{{\text{T}}} } & {{\varvec{M}}_{57}^{{\text{T}}} } & {{\varvec{M}}_{67}^{{\text{T}}} } & {{\varvec{M}}_{77} } \\ \end{array} } \right] $$
The elements of the mass matrix M are as follows:
$$ M_{11} = 2N \cdot 4\rho hab + m_{R} $$
$$ M_{16} = - 2\rho h\sum\limits_{i = 1}^{N} {\int_{0}^{a} {\int_{ - b}^{b} {y_{{R_{i} }} } } } {\text{d}}y_{{R_{i} }} {\text{d}}x_{{R_{i} }} - 2\rho h\sum\limits_{i = 1}^{N} {\int_{ - a}^{0} {\int_{ - b}^{b} {y_{{L_{i} }} } } } {\text{d}}y_{{L_{i} }} {\text{d}}x_{{L_{i} }} $$
$$ M_{22} = 2N \cdot 4\rho hab + m_{R} $$
$$ M_{26} = \rho \sum\limits_{i = 1}^{N} {\int_{0}^{a} {\int_{ - b}^{b} {2h} [x_{{R_{i} }} + r_{0} { + }a{(}i - 1{\text{)] d}}y_{{R_{i} }} {\text{d}}x_{{R_{i} }} } } + \rho \sum\limits_{i = 1}^{N} {\int_{ - a}^{0} {\int_{ - b}^{b} {2h} [x_{{L_{i} }} - r_{0} - a(i - 1){\text{] d}}y_{{L_{i} }} {\text{d}}x_{{L_{i} }} } } $$
$$ M_{33} = 2N \cdot 4\rho hab + m_{R} $$
$$ M_{34} = \sum\limits_{i = 1}^{N} {\int_{0}^{a} {\int_{ - b}^{b} {{2}hy_{{R_{i} }} } {\text{ d}}y_{{R_{i} }} {\text{d}}x_{{R_{i} }} } } + \rho \sum\limits_{i = 1}^{N} {\int_{ - a}^{0} {\int_{ - b}^{b} {{2}hy_{{L_{i} }} } {\text{ d}}y_{{L_{i} }} {\text{d}}x_{{L_{i} }} } } $$
$$ M_{{{35}}} = \rho \sum\limits_{i = 1}^{N} {\int_{0}^{a} {\int_{ - b}^{b} { - 2h} [x_{{R_{i} }} + r_{0} + a(i - 1)]{\text{ d}}y_{{R_{i} }} {\text{d}}x_{{R_{i} }} } } + \rho \sum\limits_{i = 1}^{N} {\int_{ - a}^{0} {\int_{ - b}^{b} { - 2h} [x_{{L_{i} }} - r_{0} - a(i - 1)]{\text{ d}}y_{{L_{i} }} {\text{d}}x_{{L_{i} }} } } $$
$$ M_{44} = \rho \sum\limits_{i = 1}^{N} {\int_{0}^{a} {\int_{ - b}^{b} {\frac{2}{3}h^{3} } { + }2hy_{{R_{i} }}^{2} {\text{d}}y_{{R_{i} }} {\text{d}}x_{{R_{i} }} } } + \rho \sum\limits_{i = 1}^{N} {\int_{ - a}^{0} {\int_{ - b}^{b} {\frac{2}{3}h^{3} } { + }2hy_{{L_{i} }}^{2} {\text{d}}y_{{L_{i} }} {\text{d}}x_{{L_{i} }} } } + J_{x} $$
$$ M_{45} = \rho \sum\limits_{i = 1}^{N} {\int_{0}^{a} {\int_{ - b}^{b} { - 2h} y_{{R_{i} }} [x_{{R_{i} }} + r_{0} + a(i - 1)]{\text{ d}}y_{{R_{i} }} {\text{d}}x_{{R_{i} }} } } + \rho \sum\limits_{i = 1}^{N} {\int_{ - a}^{0} {\int_{ - b}^{b} { - 2hy_{{L_{i} }} [x_{{L_{i} }} - r_{0} - a(i - 1)]} {\text{ d}}y_{{L_{i} }} {\text{d}}x_{{L_{i} }} } } $$
$$ \begin{aligned} M_{55} & = \rho \sum\limits_{i = 1}^{N} {\int_{0}^{a} {\int_{ - b}^{b} {\frac{2}{3}h^{3} + 2h[x_{{R_{i} }} + r_{0} + a(i - 1)]^{2} } {\text{ d}}y_{{R_{i} }} {\text{d}}x_{{R_{i} }} } } \\ & \quad + \rho \sum\limits_{i = 1}^{N} {\int_{ - a}^{0} {\int_{ - b}^{b} {\frac{2}{3}h^{3} + 2h[x_{{L_{i} }} - r_{0} - a(i - 1)]^{2} } {\text{ d}}y_{{L_{i} }} {\text{d}}x_{{L_{i} }} } } { + }J_{y} \, \\ \end{aligned} $$
$$ \begin{aligned} M_{66} & = \rho \sum\limits_{i = 1}^{N} {\int_{0}^{a} {\int_{ - b}^{b} {2h} \left\{ {[x_{{R_{i} }} + r_{0} + a(i - 1)]^{2} + y_{{R_{i} }}^{2} } \right\}{\text{d}}y_{{R_{i} }} {\text{d}}x_{{R_{i} }} } } \\ & \quad + \rho \sum\limits_{i = 1}^{N} {\int_{ - a}^{0} {\int_{ - b}^{b} {2h\left\{ {[x_{{L_{i} }} + r_{0} + a(i - 1)]^{2} + y_{{L_{i} }}^{2} } \right\}} {\text{ d}}y_{{L_{i} }} {\text{d}}x_{{L_{i} }} } } { + }J_{z} \, \\ \end{aligned} $$
$$ \begin{aligned} {\varvec{M}}_{17} & = \left( {2N \cdot 4\rho hab + m_{R} } \right){\text{X}}_{0} \\ & \quad - \left( {2\rho h\sum\limits_{i = 1}^{N} {\int_{0}^{a} {\int_{ - b}^{b} {y_{{R_{i} }} } } } {\text{d}} y_{{R_{i} }} {\text{d}} x_{{R_{i} }} + 2\rho h\sum\limits_{i = 1}^{N} {\int_{ - a}^{0} {\int_{ - b}^{b} {y_{{L_{i} }} } } } {\text{d}} y_{{L_{i} }} {\text{d}} x_{{L_{i} }} } \right){\uptheta }_{0}^{(z)} \\ \end{aligned} $$
$$ \begin{aligned} {\varvec{M}}_{27} & = \left( {2N \cdot 4\rho hab + m_{R} } \right){\text{Y}}_{0} + \left\{ {\rho \sum\limits_{i = 1}^{N} {\int_{0}^{a} {\int_{ - b}^{b} {2h} [x_{{R_{i} }} + r_{0} { + }a{(}i - 1{)] }{\text{d}} y_{{R_{i} }} {\text{d}} x_{{R_{i} }} } } } \right. \\ & \quad \left. { + \rho \sum\limits_{i = 1}^{N} {\int_{ - a}^{0} {\int_{ - b}^{b} {2h} [x_{{L_{i} }} - r_{0} - a(i - 1)] \, {\text{d}} y_{{L_{i} }} {\text{d}} x_{{L_{i} }} } } } \right\}{\uptheta }_{0}^{(z)} \\ \end{aligned} $$
$$ \begin{aligned} {\varvec{M}}_{37} & = \left( {2N \cdot 4\rho hab + m_{R} } \right){\text{Z}}_{0} \\ & \quad + \left( {\rho \sum\limits_{i = 1}^{N} {\int_{0}^{a} {\int_{ - b}^{b} {2h} y_{{R_{i} }} {\text{d}} y_{{R_{i} }} {\text{d}} x_{{R_{i} }} } } } \right.\left. { + \rho \sum\limits_{i = 1}^{N} {\int_{ - a}^{0} {\int_{ - b}^{b} {2h} y_{{L_{i} }} {\text{d}} y_{{L_{i} }} {\text{d}} x_{{L_{i} }} } } } \right){\uptheta }_{0}^{(x)} \\ & \quad - \left\{ {\rho \sum\limits_{i = 1}^{N} {\int_{0}^{a} {\int_{ - b}^{b} {2h} [x_{{R_{i} }} + r_{0} + a(i - 1)] \, {\text{d}} y_{{R_{i} }} {\text{d}} x_{{R_{i} }} } } } \right. \\ & \quad \left. { + \rho \sum\limits_{i = 1}^{N} {\int_{ - a}^{0} {\int_{ - b}^{b} {2h} [x_{{L_{i} }} - r_{0} - a(i - 1)]{\text{d}} y_{{L_{i} }} {\text{d}} x_{{L_{i} }} } } } \right\}{\uptheta }_{0}^{(y)} \\ & \quad + \rho \sum\limits_{i = 1}^{N} {\int_{0}^{a} {\int_{ - b}^{b} {2h} {\text{W}}_{{R_{i} }} {\text{d}} y_{{R_{i} }} {\text{d}} x_{{R_{i} }} } } + \rho \sum\limits_{i = 1}^{N} {\int_{ - a}^{0} {\int_{ - b}^{b} {2h} {\text{W}}_{{L_{i} }} {\text{d}} y_{{L_{i} }} {\text{d}} x_{{L_{i} }} } } \\ \end{aligned} $$
$$ \begin{aligned} {\varvec{M}}_{47} & = \left( {2N \cdot \frac{4}{3}\rho hab^{3} + 2N \cdot \frac{4}{3}\rho h^{3} ab + J_{x} } \right){\uptheta }_{0}^{(x)} \\ & \quad + \left( {\rho \sum\limits_{i = 1}^{N} {\int_{0}^{a} {\int_{ - b}^{b} {2h} y_{{R_{i} }} {\text{d}} y_{{R_{i} }} {\text{d}} x_{{R_{i} }} } } } \right.\left. { + \rho \sum\limits_{i = 1}^{N} {\int_{ - a}^{0} {\int_{ - b}^{b} {2h} y_{{L_{i} }} {\text{d}} y_{{L_{i} }} {\text{d}} x_{{L_{i} }} } } } \right){\text{Z}}_{0} \\ & \quad - \left\{ {\rho \sum\limits_{i = 1}^{N} {\int_{0}^{a} {\int_{ - b}^{b} {2h} [x_{{R_{i} }} + r_{0} + a(i - 1)] \, y_{{R_{i} }} {\text{d}} y_{{R_{i} }} {\text{d}} x_{{R_{i} }} } } } \right. \\ & \quad \left. { + \rho \sum\limits_{i = 1}^{N} {\int_{ - a}^{0} {\int_{ - b}^{b} {2h} [x_{{L_{i} }} - r_{0} - a(i - 1)]y_{{L_{i} }} {\text{d}} y_{{L_{i} }} {\text{d}} x_{{L_{i} }} } } } \right\}{\uptheta }_{0}^{(y)} \\ & \quad + \rho \sum\limits_{i = 1}^{N} {\int_{0}^{a} {\int_{ - b}^{b} {2h} y_{{R_{i} }} {\text{W}}_{{R_{i} }} {\text{d}} y_{{R_{i} }} {\text{d}} x_{{R_{i} }} } } + \rho \sum\limits_{i = 1}^{N} {\int_{ - a}^{0} {\int_{ - b}^{b} {2h} y_{{R_{i} }} {\text{W}}_{{L_{i} }} {\text{d}} y_{{L_{i} }} {\text{d}} x_{{L_{i} }} } } \\ & \quad + \rho \sum\limits_{i = 1}^{N} {\int_{0}^{a} {\int_{ - b}^{b} {\frac{2}{3}h^{3} } \frac{{\partial {\text{W}}_{{R_{i} }} }}{{\partial y_{{R_{i} }} }}{\text{d}} y_{{R_{i} }} {\text{d}} x_{{R_{i} }} } } + \rho \sum\limits_{i = 1}^{N} {\int_{ - a}^{0} {\int_{ - b}^{b} {\frac{2}{3}h^{3} } \frac{{\partial {\text{W}}_{{L_{i} }} }}{{\partial y_{{L_{i} }} }}{\text{d}} y_{{L_{i} }} {\text{d}} x_{{L_{i} }} } } \\ \end{aligned} $$
$$ \begin{aligned} {\varvec{M}}_{57} & = - \left\{ {\rho \sum\limits_{i = 1}^{N} {\int_{0}^{a} {\int_{ - b}^{b} {2h} [x_{{R_{i} }} + r_{0} + a(i - 1)]{\text{d}} y_{{R_{i} }} {\text{d}} x_{{R_{i} }} } } } \right. \\ & \quad \left. { + \rho \sum\limits_{i = 1}^{N} {\int_{ - a}^{0} {\int_{ - b}^{b} {2h} [x_{{L_{i} }} - r_{0} - a(i - 1)]{\text{d}} y_{{L_{i} }} {\text{d}} x_{{L_{i} }} } } } \right\}{\text{Z}}_{0} \\ & \quad - \left\{ {\rho \sum\limits_{i = 1}^{N} {\int_{0}^{a} {\int_{ - b}^{b} {2h} [x_{{R_{i} }} + r_{0} + a(i - 1)] \, y_{{R_{i} }} {\text{d}} y_{{R_{i} }} {\text{d}} x_{{R_{i} }} } } } \right. \\ & \quad \left. { + \rho \sum\limits_{i = 1}^{N} {\int_{ - a}^{0} {\int_{ - b}^{b} {2h} [x_{{L_{i} }} - r_{0} - a(i - 1)]y_{{L_{i} }} {\text{d}} y_{{L_{i} }} {\text{d}} x_{{L_{i} }} } } } \right\}{\uptheta }_{0}^{(x)} \\ & \quad { + }\left\{ {\rho \sum\limits_{i = 1}^{N} {\int_{0}^{a} {\int_{ - b}^{b} {2h} [x_{{R_{i} }} + r_{0} + a(i - 1)]^{2} {\text{d}} y_{{R_{i} }} {\text{d}} x_{{R_{i} }} } } } \right. \\ & \quad \left. { + \rho \sum\limits_{i = 1}^{N} {\int_{ - a}^{0} {\int_{ - b}^{b} {2h} [x_{{L_{i} }} - r_{0} - a(i - 1)]^{2} {\text{d}} y_{{L_{i} }} {\text{d}} x_{{L_{i} }} } } + 2N \cdot \frac{4}{3}\rho h^{3} ab + J_{y} } \right\}{\uptheta }_{0}^{(y)} \\ & \quad - \left\{ {\rho \sum\limits_{i = 1}^{N} {\int_{0}^{a} {\int_{ - b}^{b} {2h} [x_{{R_{i} }} + r_{0} + a(i - 1)]{\text{ W}}_{{R_{i} }} {\text{d}} y_{{R_{i} }} {\text{d}} x_{{R_{i} }} } } } \right. \\ & \quad \left. { + \rho \sum\limits_{i = 1}^{N} {\int_{ - a}^{0} {\int_{ - b}^{b} {2h} [x_{{L_{i} }} - r_{0} - a(i - 1)]{\text{W}}_{{L_{i} }} {\text{d}} y_{{L_{i} }} {\text{d}} x_{{L_{i} }} } } } \right\} \\ & \quad + \rho \sum\limits_{i = 1}^{N} {\int_{0}^{a} {\int_{ - b}^{b} {\frac{2}{3}h^{3} } \frac{{\partial {\text{W}}_{{R_{i} }} }}{{\partial x_{{R_{i} }} }}{\text{d}} y_{{R_{i} }} {\text{d}} x_{{R_{i} }} } } + \rho \sum\limits_{i = 1}^{N} {\int_{ - a}^{0} {\int_{ - b}^{b} {\frac{2}{3}h^{3} } \frac{{\partial {\text{W}}_{{L_{i} }} }}{{\partial x_{{L_{i} }} }}{\text{d}} y_{{L_{i} }} {\text{d}}
x_{{L_{i} }} } } \\ \end{aligned} $$
$$ \begin{gathered} {\varvec{M}}_{67} = - \left( {\rho \sum\limits_{i = 1}^{N} {\int_{0}^{a} {\int_{ - b}^{b} {2h} \, y_{{R_{i} }} {\text{d}} y_{{R_{i} }} {\text{d}} x_{{R_{i} }} + } } \rho \sum\limits_{i = 1}^{N} {\int_{ - a}^{0} {\int_{ - b}^{b} {2h} y_{{L_{i} }} {\text{d}} y_{{L_{i} }} {\text{d}} x_{{L_{i} }} } } } \right){\text{X}}_{0} \hfill \\ \, - \left\{ {\rho \sum\limits_{i = 1}^{N} {\int_{0}^{a} {\int_{ - b}^{b} {2h} [x_{{R_{i} }} + r_{0} + a(i - 1)]{\text{d}} y_{{R_{i} }} {\text{d}} x_{{R_{i} }} } } } \right. \hfill \\ \, \left. { \, + \rho \sum\limits_{i = 1}^{N} {\int_{ - a}^{0} {\int_{ - b}^{b} {2h} [x_{{L_{i} }} - r_{0} - a(i - 1)]{\text{d}} y_{{L_{i} }} {\text{d}} x_{{L_{i} }} } } } \right\}{\text{Y}}_{0} \hfill \\ { + }\left\{ {\rho \sum\limits_{i = 1}^{N} {\int_{0}^{a} {\int_{ - b}^{b} {2h} [x_{{R_{i} }} + r_{0} + a(i - 1)]^{2} {\text{d}} y_{{R_{i} }} {\text{d}} x_{{R_{i} }} } } } \right. \hfill \\ \, \left. { \, + \rho \sum\limits_{i = 1}^{N} {\int_{ - a}^{0} {\int_{ - b}^{b} {2h} [x_{{L_{i} }} - r_{0} - a(i - 1)]^{2} {\text{d}} y_{{L_{i} }} {\text{d}} x_{{L_{i} }} } } + 2N \cdot \frac{4}{3}\rho hab^{3} + J_{z} } \right\}{\uptheta }_{0}^{(z)} \hfill \\ \end{gathered} $$
$$ \begin{aligned} {M}_{{77}} & = M_{{11}} {\text{X}}_{0}^{T} {\text{X}}_{0} + M_{{22}} {\text{Y}}_{0}^{T} {\text{Y}}_{0} + M_{{33}} {\text{Z}}_{0}^{T} {\text{Z}}_{0} + M_{{44}} {{\theta }}_{0}^{{(x)^{T} }} {{\theta }}_{0}^{{(x)}} + M_{{55}} {{\theta }}_{0}^{{(y)^{T} }} {{\theta }}_{0}^{{(y)}} + M_{{66}} {{\theta }}_{0}^{{(z)^{T} }} {{\theta }}_{0}^{{(z)}} \\ & \quad + \rho \sum\limits_{{i = 1}}^{N} {\int_{0}^{a} {\int_{{ - b}}^{b} {\frac{2}{3}h^{3} } \frac{{\partial {\text{W}}_{{R_{i} }}^{T} }}{{\partial x_{{R_{i} }} }}\frac{{\partial {\text{W}}_{{R_{i} }} }}{{\partial x_{{R_{i} }} }}\text{d} y_{{R_{i} }} \text{d} x_{{R_{i} }} } } + \rho \sum\limits_{{i = 1}}^{N} {\int_{{ - a}}^{0} {\int_{{ - b}}^{b} {\frac{2}{3}h^{3} } \frac{{\partial {\text{W}}_{{L_{i} }}^{T} }}{{\partial x_{{L_{i} }} }}\frac{{\partial {\text{W}}_{{L_{i} }} }}{{\partial x_{{L_{i} }} }}\text{d} y_{{L_{i} }} \text{d} x_{{L_{i} }} } } \\ & \quad + \rho \sum\limits_{{i = 1}}^{N} {\int_{0}^{a} {\int_{{ - b}}^{b} {\frac{2}{3}h^{3} } \frac{{\partial {\text{W}}_{{R_{i} }}^{T} }}{{\partial y_{{R_{i} }} }}\frac{{\partial {\text{W}}_{{R_{i} }} }}{{\partial y_{{R_{i} }} }}\text{d} y_{{R_{i} }} \text{d} x_{{R_{i} }} } } + \rho \sum\limits_{{i = 1}}^{N} {\int_{{ - a}}^{0} {\int_{{ - b}}^{b} {\frac{2}{3}h^{3} } \frac{{\partial {\text{W}}_{{L_{i} }}^{T} }}{{\partial y_{{L_{i} }} }}\frac{{\partial {\text{W}}_{{L_{i} }} }}{{\partial y_{{L_{i} }} }}\text{d} y_{{L_{i} }} \text{d} x_{{L_{i} }} } } \\ & \quad + \rho \sum\limits_{{i = 1}}^{N} {\int_{0}^{a} {\int_{{ - b}}^{b} {2h} {\text{W}}_{{R_{i} }}^{T} {\text{W}}_{{R_{i} }} \text{d} y_{{R_{i} }} \text{d} x_{{R_{i} }} } } + \rho \sum\limits_{{i = 1}}^{N} {\int_{{ - a}}^{0} {\int_{{ - b}}^{b} {2h} {\text{W}}_{{L_{i} }}^{T} {\text{W}}_{{L_{i} }} \text{d} y_{{L_{i} }} \text{d} x_{{L_{i} }} } } \\ & \quad + \rho \sum\limits_{{i = 1}}^{N} {\int_{0}^{a} {\int_{{ - b}}^{b} {2h} [x_{{R_{i} }} + r_{0} + a(i - 1)]\text{d} y_{{R_{i} }} \text{d} x_{{R_{i} }} } } \left( {{{\theta }}_{0}^{{(z)^{T} }} {\text{Y}}_{0} + {\text{Y}}_{0}^{T} {{\theta }}_{0}^{{(z)}} } \right) \\ & \quad + \rho \sum\limits_{{i = 1}}^{N} {\int_{{ - a}}^{0} {\int_{{ - b}}^{b} {2h} [x_{{L_{i} }} - r_{0} - a(i - 1)]\text{d} y_{{L_{i} }} \text{d} x_{{L_{i} }} } } \left( {{{\theta }}_{0}^{{(z)^{T} }} {\text{Y}}_{0} + {\text{Y}}_{0}^{T} {{\theta }}_{0}^{{(z)}} } \right) \\ & \quad + \rho \sum\limits_{{i = 1}}^{N} {\int_{0}^{a} {\int_{{ - b}}^{b} {2h{\kern 1pt} } \left( {{\text{Z}}_{0}^{T} {\text{W}}_{{R_{i} }} {\text{ + W}}_{{R_{i} }}^{T} {\text{Z}}_{0} } \right)\text{d} y_{{R_{i} }} \text{d} x_{{R_{i} }} } } + \rho \sum\limits_{{i = 1}}^{N} {\int_{{ - a}}^{0} {\int_{{ - b}}^{b} {2h{\kern 1pt} } \left( {{\text{Z}}_{0}^{T} {\text{W}}_{{L_{i} }} {\text{ + W}}_{{L_{i} }}^{T} {\text{Z}}_{0} } \right)\text{d} y_{{L_{i} }} \text{d} x_{{L_{i} }} } } \\ & \quad - \rho \sum\limits_{{i = 1}}^{N} {\int_{0}^{a} {\int_{{ - b}}^{b} {2h} [x_{{R_{i} }} + r_{0} + a(i - 1)]\text{d} y_{{R_{i} }} \text{d} x_{{R_{i} }} } } \left( {{\text{Z}}_{0}^{T} {{\theta }}_{0}^{{(y)}} + {{\theta }}_{0}^{{(y)^{T} }} {\text{Z}}_{0} } \right) \\ & \quad - \rho \sum\limits_{{i = 1}}^{N} {\int_{{ - a}}^{0} {\int_{{ - b}}^{b} {2h} [x_{{L_{i} }} - r_{0} - a(i - 1)]\text{d} y_{{L_{i} }} \text{d} x_{{L_{i} }} } } \left( {{\text{Z}}_{0}^{T} {{\theta }}_{0}^{{(y)}} + {{\theta }}_{0}^{{(y)^{T} }} {\text{Z}}_{0} } \right) \\ & \quad + \rho \sum\limits_{{i = 1}}^{N} {\int_{0}^{a} {\int_{{ - b}}^{b} {2h} y_{{R_{i} }} \text{d} y_{{R_{i} }} \text{d} x_{{R_{i} }} } } \left( {{\text{Z}}_{0}^{T} {{\theta }}_{0}^{{(x)}} + {{\theta }}_{0}^{{(x)^{T} }} {\text{Z}}_{0} } \right) + \rho \sum\limits_{{i = 1}}^{N} {\int_{{ - a}}^{0} {\int_{{ - b}}^{b} {2h} y_{{L_{i} }} \text{d} y_{{L_{i} }} \text{d} x_{{L_{i} }} } } \left( {{\text{Z}}_{0}^{T} {{\theta }}_{0}^{{(x)}} + {{\theta }}_{0}^{{(x)^{T} }} {\text{Z}}_{0} } \right) \\ & \quad - \rho \sum\limits_{{i = 1}}^{N} {\int_{0}^{a} {\int_{{ - b}}^{b} {2h} [x_{{R_{i} }} + r_{0} + a(i - 1)]y_{{R_{i} }} \text{d} y_{{R_{i} }} \text{d} x_{{R_{i} }} } } \left( {{{\theta }}_{0}^{{(x)^{T} }} {{\theta }}_{0}^{{(y)}} + {{\theta }}_{0}^{{(y)^{T} }} {{\theta }}_{0}^{{(x)}} } \right) \\ & \quad - \rho \sum\limits_{{i = 1}}^{N} {\int_{{ - a}}^{0} {\int_{{ - b}}^{b} {2h} [x_{{L_{i} }} - r_{0} - a(i - 1)]y_{{L_{i} }} \text{d} y_{{L_{i} }} \text{d} x_{{L_{i} }} } } \left( {{{\theta }}_{0}^{{(x)^{T} }} {{\theta }}_{0}^{{(y)}} + {{\theta }}_{0}^{{(y)^{T} }} {{\theta }}_{0}^{{(x)}} } \right) \\ & \quad - \rho \sum\limits_{{i = 1}}^{N} {\int_{0}^{a} {\int_{{ - b}}^{b} {2h} y_{{R_{i} }} \text{d} y_{{R_{i} }} \text{d} x_{{R_{i} }} } } \left( {{{\theta }}_{0}^{{(z)^{T} }} {\text{X}}_{0} + {\text{X}}_{0}^{T} {{\theta }}_{0}^{{(z)}} } \right) - \rho \sum\limits_{{i = 1}}^{N} {\int_{{ - a}}^{0} {\int_{{ - b}}^{b} {2h} y_{{L_{i} }} \text{d} y_{{L_{i} }} \text{d} x_{{L_{i} }} } } \left( {{{\theta }}_{0}^{{(z)^{T} }} {\text{X}}_{0} + {\text{X}}_{0}^{T} {{\theta }}_{0}^{{(z)}} } \right) \\ & \quad - \rho \sum\limits_{{i = 1}}^{N} {\int_{0}^{a} {\int_{{ - b}}^{b} {2h} [x_{{R_{i} }} + r_{0} + a(i - 1)]\left( {{\text{W}}_{{_{{R_{i} }} }}^{T} {{\theta }}_{0}^{{(y)}} + {{\theta }}_{0}^{{(y)^{T} }} {\text{W}}_{{R_{i} }} } \right)\text{d} y_{{R_{i} }} \text{d} x_{{R_{i} }} } } \\ & \quad - \rho \sum\limits_{{i = 1}}^{N} {\int_{{ - a}}^{0} {\int_{{ - b}}^{b} {2h} [x_{{L_{i} }} - r_{0} - a(i - 1)]\left( {{\text{W}}_{{L_{i} }}^{T} {{\theta }}_{0}^{{(y)}} + {{\theta }}_{0}^{{(y)^{T} }} {\text{W}}_{{L_{i} }} } \right)\text{d} y_{{L_{i} }} \text{d} x_{{L_{i} }} } } \\ & \quad + \rho \sum\limits_{{i = 1}}^{N} {\int_{0}^{a} {\int_{{ - b}}^{b} {2h} y_{{R_{i} }} \left( {{\text{W}}_{{R_{i} }}^{T} {{\theta }}_{0}^{{(x)}} + {{\theta }}_{0}^{{(x)^{T} }} {\text{W}}_{{R_{i} }} } \right)\text{d} y_{{R_{i} }} \text{d} x_{{R_{i} }} } } \\ & \quad + \rho \sum\limits_{{i = 1}}^{N} {\int_{{ - a}}^{0} {\int_{{ - b}}^{b} {2hy_{{L_{i} }} } \left( {{\text{W}}_{{L_{i} }}^{T} {{\theta }}_{0}^{{(x)}} + {{\theta }}_{0}^{{(x)^{T} }} {\text{W}}_{{L_{i} }} } \right)\text{d} y_{{L_{i} }} \text{d} x_{{L_{i} }} } } \\ & \quad - \rho \sum\limits_{{i = 1}}^{N} {\int_{0}^{a} {\int_{{ - b}}^{b} {\frac{2}{3}h^{3} } \left( {\frac{{\partial {\text{W}}_{{R_{i} }}^{T} }}{{\partial x_{{R_{i} }} }}{{\theta }}_{0}^{{(y)}} + {{\theta }}_{0}^{{(y)^{T} }} \frac{{\partial {\text{W}}_{{R_{i} }} }}{{\partial x_{{R_{i} }} }}} \right)\text{d} y_{{R_{i} }} \text{d} x_{{R_{i} }} } } \\ & \quad - \rho \sum\limits_{{i = 1}}^{N} {\int_{{ - a}}^{0} {\int_{{ - b}}^{b} {\frac{2}{3}h^{3} } \left( {\frac{{\partial {\text{W}}_{{L_{i} }}^{T} }}{{\partial x_{{L_{i} }} }}{{\theta }}_{0}^{{(y)}} + {{\theta }}_{0}^{{(y)^{T} }} \frac{{\partial {\text{W}}_{{L_{i} }} }}{{\partial x_{{L_{i} }} }}} \right)\text{d} y_{{L_{i} }} \text{d} x_{{L_{i} }} } } \\ & \quad + \rho \sum\limits_{{i = 1}}^{N} {\int_{0}^{a} {\int_{{ - b}}^{b} {\frac{2}{3}h^{3} } \left( {\frac{{\partial {\text{W}}_{{R_{i} }}^{T} }}{{\partial y_{{R_{i} }} }}{{\theta }}_{0}^{{(x)}} + {{\theta }}_{0}^{{(x)^{T} }} \frac{{\partial {\text{W}}_{{R_{i} }} }}{{\partial y_{{R_{i} }} }}} \right)\text{d} y_{{R_{i} }} \text{d} x_{{R_{i} }} } } \\ & \quad + \rho \sum\limits_{{i = 1}}^{N} {\int_{{ - a}}^{0} {\int_{{ - b}}^{b} {\frac{2}{3}h^{3} } \left( {\frac{{\partial {\text{W}}_{{L_{i} }}^{T} }}{{\partial y_{{L_{i} }} }}{{\theta }}_{0}^{{(x)}} + {{\theta }}_{0}^{{(x)^{T} }} \frac{{\partial {\text{W}}_{{L_{i} }} }}{{\partial y_{{L_{i} }} }}} \right)\text{d} y_{{L_{i} }} \text{d} x_{{L_{i} }} } } \\ \end{aligned} $$
$$ {\mathbf{W}}_{{R_{i} }} \left( {x_{{R_{i} }} ,y_{{R_{i} }} } \right) = \left[ {W_{{R_{i} ,1}} \left( {x_{{R_{i} }} ,y_{{R_{i} }} } \right), \cdots ,W_{{R_{i} ,n}} \left( {x_{{R_{i} }} ,y_{{R_{i} }} } \right)} \right] $$
where
$$ {\mathbf{W}}_{{L_{i} }} \left( {x_{{L_{i} }} ,y_{{L_{i} }} } \right) = \left[ {W_{{L_{i} ,1}} \left( {x_{{L_{i} }} ,y_{{L_{i} }} } \right), \cdots ,W_{{L_{i} ,n}} \left( {x_{{L_{i} }} ,y_{{L_{i} }} } \right)} \right] $$
$$ {\text{X}}_{0}^{{}} = \left[ {X_{0,1}^{{}} , \ldots ,X_{0,n}^{{}} } \right],\,{\text{Y}}_{0}^{{}} = \left[ {Y_{0,1}^{{}} , \ldots ,Y_{0,n}^{{}} } \right],\,{\text{Z}}_{0}^{{}} = \left[ {Z_{0,1}^{{}} , \ldots ,Z_{0,n}^{{}} } \right] $$
$$ {\uptheta }_{0}^{(x)} = \left[ {\theta_{0,1}^{(x)} , \ldots
,\theta_{0,n}^{(x)} } \right],\,{\uptheta }_{0}^{(y)} = \left[ {\theta_{0,1}^{(y)} , \ldots ,\theta_{0,n}^{(y)} } \right],\,{\uptheta }_{1}^{(z)} = \left[ {\theta_{0,1}^{(z)} , \ldots ,\theta_{0,n}^{(z)} } \right] $$
Appendix 2
The stiffness and damping matrices K and C are as follows:
$$ {\varvec{K}} = \left[ {\begin{array}{*{20}c} {{\mathbf{0}}_{{{6} \times {6}}} } & {{\mathbf{0}}_{{{6} \times n}} } \\ {{\mathbf{0}}_{{n \times {6}}} } & {{\varvec{K}}_{77} } \\ \end{array} } \right], \, {\varvec{C}} = \left[ {\begin{array}{*{20}c} {{\mathbf{0}}_{{{6} \times {6}}} } & {{\mathbf{0}}_{{{6} \times n}} } \\ {{\mathbf{0}}_{{n \times {6}}} } & {{\varvec{C}}_{77} } \\ \end{array} } \right] $$
where
$$ \begin{gathered} {\varvec{K}}_{77} = \Re + \ell \hfill \\ \Re = D\sum\limits_{i = 1}^{N} {\int_{0}^{a} {\int_{ - b}^{b} {\frac{{\partial^{2} {\text{W}}_{{R_{i} }}^{T} }}{{\partial x_{{R_{i} }}^{2} }}\frac{{\partial^{2} {\text{W}}_{{R_{i} }} }}{{\partial x_{{R_{i} }}^{2} }}} {\text{d}} y_{{R_{i} }} {\text{d}} x_{{R_{i} }} } } \hfill \\ \, + vD\sum\limits_{i = 1}^{N} {\int_{0}^{a} {\int_{ - b}^{b} {\frac{{\partial^{2} {\text{W}}_{{R_{i} }}^{T} }}{{\partial x_{{R_{i} }}^{2} }}\frac{{\partial^{2} {\text{W}}_{{R_{i} }} }}{{\partial y_{{R_{i} }}^{2} }}} + \frac{{\partial^{2} {\text{W}}_{{R_{i} }}^{T} }}{{\partial y_{{R_{i} }}^{2} }}\frac{{\partial^{2} {\text{W}}_{{R_{i} }} }}{{\partial x_{{R_{i} }}^{2} }}{\text{d}} y_{{R_{i} }} {\text{d}} x_{{R_{i} }} } } \hfill \\ \, + 2\left( {1 - v} \right)D\sum\limits_{i = 1}^{N} {\int_{0}^{a} {\int_{ - b}^{b} {\frac{{\partial^{2} {\text{W}}_{{R_{i} }}^{T} }}{{\partial x_{{R_{i} }} \partial y_{{R_{i} }} }}\frac{{\partial^{2} {\text{W}}_{{R_{i} }} }}{{\partial x_{{R_{i} }} \partial y_{{R_{i} }} }}} {\text{d}} y_{{R_{i} }} {\text{d}} x_{{R_{i} }} } } \hfill \\ \, + D\sum\limits_{i = 1}^{N} {\int_{0}^{a} {\int_{ - b}^{b} {\frac{{\partial^{2} {\text{W}}_{{R_{i} }}^{T} }}{{\partial y_{{R_{i} }}^{2} }}\frac{{\partial^{2} {\text{W}}_{{R_{i} }} }}{{\partial y_{{R_{i} }}^{2} }}} {\text{d}} y_{{R_{i} }} {\text{d}} x_{{R_{i} }} } } \hfill \\ \, + k\sum\limits_{i = 1}^{N} {\left( {\Delta \Theta_{{R_{{A_{i} }} }} } \right)^{T} \Delta \Theta_{{R_{{A_{i} }} }} } + k\sum\limits_{i = 1}^{N} {\left( {\Delta \Theta_{{R_{{B_{i} }} }} } \right)^{T} \Delta \Theta_{{R_{{B_{i} }} }} } \hfill \\ \end{gathered} $$
where \(\Delta \Theta_{{R_{{A_{1} }} }} = \left. {\frac{{\partial {\text{W}}_{{R_{1} }}^{T} }}{{\partial x_{{R_{1} }} }}} \right|_{\begin{subarray}{l} x_{{R_{1} }} = 0 \\ y_{{R_{1} }} = y{}_{a} \end{subarray} } \left. {\frac{{\partial {\text{W}}_{{R_{1} }} }}{{\partial x_{{R_{1} }} }}} \right|_{\begin{subarray}{l} x_{{R_{1} }} = 0 \\ y_{{R_{1} }} = y{}_{a} \end{subarray} }\), \(\Delta \Theta_{{R_{{B_{1} }} }} = \left. {\frac{{\partial {\text{W}}_{{R_{1} }}^{T} }}{{\partial x_{{R_{1} }} }}} \right|_{\begin{subarray}{l} x_{{R_{1} }} = 0 \\ y_{{R_{1} }} = y{}_{b} \end{subarray} } \left. {\frac{{\partial {\text{W}}_{{R_{1} }} }}{{\partial x_{{R_{1} }} }}} \right|_{\begin{subarray}{l} x_{{R_{1} }} = 0 \\ y_{{R_{1} }} = y{}_{b} \end{subarray} }\),
\(\Delta \Theta_{{L_{{A_{1} }} }} = \left. {\frac{{\partial {\text{W}}_{{L_{1} }}^{T} }}{{\partial x_{{L_{1} }} }}} \right|_{\begin{subarray}{l} x_{{L_{1} }} = 0 \\ y_{{L_{1} }} = y{}_{a} \end{subarray} } \left. {\frac{{\partial {\text{W}}_{{L_{1} }} }}{{\partial x_{{L_{1} }} }}} \right|_{\begin{subarray}{l} x_{{L_{1} }} = 0 \\ y_{{L_{1} }} = y{}_{a} \end{subarray} }\), \(\Delta \Theta_{{L_{{B_{1} }} }} = \left. {\frac{{\partial {\text{W}}_{{L_{1} }}^{T} }}{{\partial x_{{L_{1} }} }}} \right|_{\begin{subarray}{l} x_{{L_{1} }} = 0 \\ y_{{L_{1} }} = y{}_{b} \end{subarray} } \left. {\frac{{\partial {\text{W}}_{{L_{1} }} }}{{\partial x_{{L_{1} }} }}} \right|_{\begin{subarray}{l} x_{{L_{1} }} = 0 \\ y_{{L_{1} }} = y{}_{b} \end{subarray} }\), and when \(i = 2,3, \ldots ,N\),
\(\Delta \Theta_{{R_{{A_{i} }} }} = \left. {\frac{{\partial {\mathbf{W}}_{{R_{i} }} }}{{\partial x_{{R_{i} }} }}} \right|_{\begin{subarray}{l} x_{{R_{i} }} = 0 \\ y_{{R_{i} }} = y_{a} \end{subarray} } - \left. {\frac{{\partial {\mathbf{W}}_{{R_{i - 1} }} }}{{\partial x_{{R_{i - 1} }} }}} \right|_{\begin{subarray}{l} x_{{R_{i - 1} }} = a \\ y_{{R_{i - 1} }} = y_{a} \end{subarray} }\), \(\Delta \Theta_{{R_{{B_{i} }} }} = \left. {\frac{{\partial {\mathbf{W}}_{{R_{i} }} }}{{\partial x_{{R_{i} }} }}} \right|_{\begin{subarray}{l} x_{{R_{i} }} = 0 \\ y_{{R_{i} }} = y_{b} \end{subarray} } - \left. {\frac{{\partial {\mathbf{W}}_{{R_{i - 1} }} }}{{\partial x_{{R_{i - 1} }} }}} \right|_{\begin{subarray}{l} x_{{R_{i - 1} }} = a \\ y_{{R_{i - 1} }} = y_{b} \end{subarray} }\),
\(\Delta \Theta_{{L_{{A_{i} }} }} = \left. {\frac{{\partial {\mathbf{W}}_{{L_{i} }} }}{{\partial x_{{L_{i} }} }}} \right|_{\begin{subarray}{l} x_{{L_{i} }} = 0 \\ y_{{L_{i} }} = y_{a} \end{subarray} } - \left. {\frac{{\partial {\mathbf{W}}_{{L_{i - 1} }} }}{{\partial x_{{L_{i - 1} }} }}} \right|_{\begin{subarray}{l} x_{{L_{i - 1} }} = a \\ y_{{L_{i - 1} }} = y_{a} \end{subarray} }\), \(\Delta \Theta_{{L_{{B_{i} }} }} = \left. {\frac{{\partial {\mathbf{W}}_{{L_{i} }} }}{{\partial x_{{L_{i} }} }}} \right|_{\begin{subarray}{l} x_{{L_{i} }} = 0 \\ y_{{L_{i} }} = y_{b} \end{subarray} } - \left. {\frac{{\partial {\mathbf{W}}_{{L_{i - 1} }} }}{{\partial x_{{L_{i - 1} }} }}} \right|_{\begin{subarray}{l} x_{{L_{i - 1} }} = a \\ y_{{L_{i - 1} }} = y_{b} \end{subarray} }\).
The expression for \(\ell\) can be obtained from \(\Re\) by replacing all instances of \(R_{i}\) with \(L_{i}\).
The damping matrix C77 is as follows
$$ {\varvec{C}}_{77} = \kappa_{M} {\varvec{M}} + \kappa_{K} {\varvec{K}}_{p} + c{\varvec{C}}_{j} $$
where the coefficients \(\kappa_{M}\) and \(\kappa_{K}\) are proportionality constants and
$$ \begin{aligned} {\varvec{K}}_{p} & = \Re_{p} + \ell_{p} \\ \Re_{p} & = D\sum\limits_{i = 1}^{N} {\int_{0}^{a} {\int_{ - b}^{b} {\frac{{\partial^{2} {\text{W}}_{{R_{i} }}^{T} }}{{\partial x_{{R_{i} }}^{2} }}\frac{{\partial^{2} {\text{W}}_{{R_{i} }} }}{{\partial x_{{R_{i} }}^{2} }}} {\text{d}} y_{{R_{i} }} {\text{d}} x_{{R_{i} }} } } \\ & \quad + vD\sum\limits_{i = 1}^{N} {\int_{0}^{a} {\int_{ - b}^{b} {\frac{{\partial^{2} {\text{W}}_{{R_{i} }}^{T} }}{{\partial x_{{R_{i} }}^{2} }}\frac{{\partial^{2} {\text{W}}_{{R_{i} }} }}{{\partial y_{{R_{i} }}^{2} }}} + \frac{{\partial^{2} {\text{W}}_{{R_{i} }}^{T} }}{{\partial y_{{R_{i} }}^{2} }}\frac{{\partial^{2} {\text{W}}_{{R_{i} }} }}{{\partial x_{{R_{i} }}^{2} }}{\text{d}} y_{{R_{i} }} {\text{d}} x_{{R_{i} }} } } \\ & \quad + 2\left( {1 - v} \right)D\sum\limits_{i = 1}^{N} {\int_{0}^{a} {\int_{ - b}^{b} {\frac{{\partial^{2} {\text{W}}_{{R_{i} }}^{T} }}{{\partial x_{{R_{i} }} \partial y_{{R_{i} }} }}\frac{{\partial^{2} {\text{W}}_{{R_{i} }} }}{{\partial x_{{R_{i} }} \partial y_{{R_{i} }} }}} {\text{d}} y_{{R_{i} }} {\text{d}} x_{{R_{i} }} } } \\ & \quad + D\sum\limits_{i = 1}^{N} {\int_{0}^{a} {\int_{ - b}^{b} {\frac{{\partial^{2} {\text{W}}_{{R_{i} }}^{T} }}{{\partial y_{{R_{i} }}^{2} }}\frac{{\partial^{2} {\text{W}}_{{R_{i} }} }}{{\partial y_{{R_{i} }}^{2} }}} {\text{d}} y_{{R_{i} }} {\text{d}} x_{{R_{i} }} } } \\ \end{aligned} $$
The expression for \(\ell_{p}\) can be obtained from \(\Re_{p}\) by replacing \(R_{i}\) with \(L_{i}\).
$$ \begin{aligned} C_{j} & = \sum\limits_{i = 1}^{N} {\left( {\Delta \Theta_{{R_{{A_{i} }} }} } \right)^{T} \Delta \Theta_{{R_{{A_{i} }} }} } + \sum\limits_{i = 1}^{N} {\left( {\Delta \Theta_{{R_{{B_{i} }} }} } \right)^{T} \Delta \Theta_{{R_{{B_{i} }} }} } \\ & \quad + \sum\limits_{i = 1}^{N} {\left( {\Delta \Theta_{{L_{{A_{i} }} }} } \right)^{T} \Delta \Theta_{{L_{{A_{i} }} }} } + \sum\limits_{i = 1}^{N} {\left( {\Delta \Theta_{{L_{{B_{i} }} }} } \right)^{T} \Delta \Theta_{{L_{{B_{i} }} }} } \\ \end{aligned} $$
Appendix 3
The nonlinear stiffness \({\varvec{K}}_{n} \left( {\varvec{q}} \right)\) is as follows:
$$ {\varvec{K}}_{n} ({\varvec{q}}) = \Re_{n} + \ell_{n} $$
$$ \begin{aligned} \Re_{n} & = k_{n} \left. {\frac{{\partial {\text{W}}_{{R_{1} }}^{T} }}{{\partial x_{{R_{1} }} }}} \right|_{\begin{subarray}{l} x_{{R_{1} }} = 0 \\ y_{{R_{1} }} = y{}_{a} \end{subarray} } \left( {\left. {\frac{{\partial w_{{R_{1} }} }}{{\partial x_{{R_{1} }} }}} \right|_{\begin{subarray}{l} x_{{R_{1} }} = 0 \\ y_{{R_{1} }} = y{}_{a} \end{subarray} } } \right)^{3} + k_{n} \left. {\frac{{\partial {\text{W}}_{{R_{1} }}^{T} }}{{\partial x_{{R_{1} }} }}} \right|_{\begin{subarray}{l} x_{{R_{1} }} = 0 \\ y_{{R_{1} }} = y{}_{b} \end{subarray} } \left( {\left. {\frac{{\partial w_{{R_{1} }} }}{{\partial x_{{R_{1} }} }}} \right|_{\begin{subarray}{l} x_{{R_{1} }} = 0 \\ y_{{R_{1} }} = y{}_{b} \end{subarray} } } \right)^{3} \\ & \quad + k_{n} \sum\limits_{i = 2}^{N} {\left( {\left. {\frac{{\partial {\text{W}}_{{R_{i} }} }}{{\partial x_{{R_{i} }} }}} \right|_{\begin{subarray}{l} x_{{R_{i} }} = 0 \\ y_{{R_{i} }} = y_{a} \end{subarray} } - \left. {\frac{{\partial {\text{W}}_{{R_{i - 1} }} }}{{\partial x_{{R_{i - 1} }} }}} \right|_{\begin{subarray}{l} x_{{R_{i - 1} }} = a \\ y_{{R_{i - 1} }} = y_{a} \end{subarray} } } \right)^{T} \left( {\left. {\frac{{\partial w_{{R_{i} }} }}{{\partial x_{{R_{i} }} }}} \right|_{\begin{subarray}{l} x_{{R_{i} }} = 0 \\ y_{{R_{i} }} = y_{a} \end{subarray} } - \left. {\frac{{\partial w_{{R_{i - 1} }} }}{{\partial x_{{R_{i - 1} }} }}} \right|_{\begin{subarray}{l} x_{{R_{i - 1} }} = a \\ y_{{R_{i - 1} }} = y_{a} \end{subarray} } } \right)^{3} } \\ & \quad + k_{n} \sum\limits_{i = 2}^{N} {\left( {\left. {\frac{{\partial {\text{W}}_{{R_{i} }} }}{{\partial x_{{R_{i} }} }}} \right|_{\begin{subarray}{l} x_{{R_{i} }} = 0 \\ y_{{R_{i} }} = y_{b} \end{subarray} } - \left. {\frac{{\partial {\text{W}}_{{R_{i - 1} }} }}{{\partial x_{{R_{i - 1} }} }}} \right|_{\begin{subarray}{l} x_{{R_{i - 1} }} = a \\ y_{{R_{i - 1} }} = y_{b} \end{subarray} } } \right)^{T} \left( {\left. {\frac{{\partial w_{{R_{i} }} }}{{\partial x_{{R_{i} }} }}} \right|_{\begin{subarray}{l} x_{{R_{i} }} = 0 \\ y_{{R_{i} }} = y_{a} \end{subarray} } - \left. {\frac{{\partial w_{{R_{i - 1} }} }}{{\partial x_{{R_{i - 1} }} }}} \right|_{\begin{subarray}{l} x_{{R_{i - 1} }} = a \\ y_{{R_{i - 1} }} = y_{b} \end{subarray} } } \right)^{3} } \\ \end{aligned} $$
The expression for \(\ell_{n}\) can be obtained from \(\Re_{n}\) by replacing \(R_{i}\) with \(L_{i}\).
Appendix 4
The Coulomb friction matrix is as follows:
$$ {{\varvec{\upmu}}}(\dot{\user2{q}}) = \left[ {\begin{array}{*{20}c} {{0}_{6 \times 1} } \\ {{{\varvec{\upmu}}}(\dot{\user2{p}})} \\ \end{array} } \right] $$
where
$$ \begin{aligned} {{\varvec{\upmu}}}(\dot{\user2{p}}) & = \mu \sum\limits_{i = 1}^{N} {\left\{ {{\text{sign}} \left( {\Delta \Theta_{{R_{{A_{i} }} }} \dot{\user2{p}}} \right)\left( {\Delta \Theta_{{R_{{A_{i} }} }} } \right)^{\text{T}} } \right.} + {\text{sign}} \left( {\Delta \Theta_{{R_{{B_{i} }} }} \dot{\user2{p}}} \right)\left( {\Delta \Theta_{{R_{{B_{i} }} }} } \right)^{\text{T}} \\ & \quad + {\text{sign}} \left( {\Delta \Theta_{{L_{{A_{i} }} }} \dot{\user2{p}}} \right)\left( {\Delta \Theta_{{L_{{A_{i} }} }} } \right)^{\text{T}} \left. { + {\text{sign}} \left( {\Delta \Theta_{{L_{{B_{i} }} }} \dot{\user2{p}}} \right)\left( {\Delta \Theta_{{L_{{B_{i} }} }} } \right)^{\text{T}} } \right\} \\ \end{aligned} $$
