Conflict of interest
The authors declare no potential conflict of interests.
Appendix 1 General solutions of displacements and stresses for poroelastic soils
$$ \begin{aligned} u_{sk} = & T_{2k} b_{1k} \left[ { - A_{1k} K_{1} (b_{1k} r) + A_{2k} I_{1} (b_{1k} r)} \right]\left[ {B_{1k} \sin (b_{2k} z) + B_{2k} \cos (b_{2k} z)} \right] \\ & + T_{3k} b_{3k} \left[ { - A_{3k} K_{1} (b_{3k} r) + A_{4k} I_{1} (b_{3k} r)} \right]\left[ {B_{3k} \sin (b_{3k} z) + B_{4k} \cos (b_{3k} z)} \right], \\ & + \left[ {A_{5k} K_{1} (b_{4k} r) + A_{6k} I_{1} (b_{4k} r)} \right]\left[ {B_{5k} \sin (b_{5k} z) + B_{6k} \cos (b_{5k} z)} \right] \\ \end{aligned} $$
(45)
$$ \begin{aligned} w_{sk} = & T_{4k} b_{2k} \left[ {A_{1k} K_{0} (b_{1k} r) + A_{2k} I_{0} (b_{1k} r)} \right]\left[ {B_{1k} \cos (b_{2k} z) - B_{2k} \sin (b_{2k} z)} \right] \\ {\kern 1pt} & + T_{5k} b_{3k} \left[ {A_{3k} K_{0} (b_{3k} r) + A_{4k} I_{0} (b_{3k} r)} \right]\left[ {B_{3k} \cos (b_{3k} z) - B_{4k} \sin (b_{3k} z)} \right], \\ & + \left[ {A_{7k} K_{0} (b_{6k} r) + A_{8k} I_{0} (b_{6k} r)} \right]\left[ {B_{7k} \sin (b_{7k} z) + B_{8k} \cos (b_{7k} z)} \right] \\ \end{aligned} $$
(46)
$$ \begin{aligned} u_{fk} = & \left\{ {\frac{{i\omega s_{vk} \alpha_{k}^{2} T_{2k} b_{1k} + n^{fk} a_{3k} b_{1k} }}{{(i\omega s_{vk} - \rho^{fk} \omega^{2} )\alpha_{k}^{2} }}} \right\}\left[ { - A_{1k} K_{1} (b_{1k} r) + A_{2k} I_{1} (b_{1k} r)} \right]\left[ {B_{1k} \sin (b_{2k} z) + B_{2k} \cos (b_{2k} z)} \right] \\ & + \left\{ {\frac{{i\omega s_{vk} T_{3k} b_{3k} - n^{fk} b_{3k} }}{{(i\omega s_{vk} - \rho^{fk} \omega^{2} )}}} \right\}\left[ { - A_{3k} K_{1} (b_{3k} r) + A_{4k} I_{1} (b_{3k} r)} \right]\left[ {B_{3k} \sin (b_{3k} z) + B_{4k} \cos (b_{3k} z)} \right], \\ {\kern 1pt} & + \frac{{i\omega s_{vk} }}{{(i\omega s_{vk} - \rho^{fk} \omega^{2} )}}\left[ {A_{5k} K_{1} (b_{4k} r) + A_{6k} I_{1} (b_{4k} r)} \right]\left[ {B_{5k} \sin (b_{5k} z) + B_{6k} \cos (b_{5k} z)} \right] \\ \end{aligned} $$
(47)
$$ \begin{aligned} w_{fk} = & \left\{ {\frac{{i\omega s_{vk} \alpha_{k}^{2} T_{4k} b_{2k} + n^{fk} a_{3k} b_{2k} }}{{(i\omega s_{vk} - \rho^{fk} \omega^{2} )\alpha_{k}^{2} }}} \right\}\left[ {A_{1k} K_{0} (b_{1k} r) + A_{2k} I_{0} (b_{1k} r)} \right]\left[ {B_{1k} \cos (b_{2k} z) - B_{2k} \sin (b_{2k} z)} \right] \\ & + \left\{ {\frac{{i\omega s_{vk} T_{5k} b_{3k} - n^{fk} b_{3k} }}{{(i\omega s_{vk} - \rho^{fk} \omega^{2} )}}} \right\}\left[ {A_{3k} K_{0} (b_{3k} r) + A_{4k} I_{0} (b_{3k} r)} \right]\left[ {B_{3k} \cos (b_{3k} z) - B_{4k} \sin (b_{3k} z)} \right], \\ & + \frac{{i\omega s_{vk} }}{{(i\omega s_{vk} - \rho^{fk} \omega^{2} )}}\left[ {A_{7k} K_{0} (b_{6k} r) + A_{8k} I_{0} (b_{6k} r)} \right]\left[ {B_{7k} \sin (b_{7k} z) + B_{8k} \cos (b_{7k} z)} \right] \\ \end{aligned} $$
(48)
$$ \begin{aligned} \sigma_{rrk}^{s} = & \left\{ \begin{gathered} (\lambda^{sk} + 2\mu^{sk} T_{2k} b_{1k}^{2} )\left[ {A_{1k} K_{0} (b_{1k} r) + A_{2k} I_{0} (b_{1k} r)} \right] \hfill \\ + 2\mu^{sk} T_{2k} b_{1k} \frac{1}{r}\left[ {A_{1k} K_{1} (b_{1k} r) - A_{2k} I_{1} (b_{1k} r)} \right] \hfill \\ \end{gathered} \right\}\left[ {B_{1k} \sin (b_{2k} z) + B_{2k} \cos (b_{2k} z)} \right] \\ {\kern 1pt} & + 2\mu^{sk} T_{3k} b_{3k} \left\{ \begin{gathered} - A_{3k} \left[ { - b_{3k} K_{0} (b_{3k} r) - \frac{1}{r}K_{1} (b_{3k} r)} \right] \hfill \\ + A_{4k} \left[ {b_{3k} I_{0} (b_{3k} r) - \frac{1}{r}I_{1} (b_{3k} r)} \right] \hfill \\ \end{gathered} \right\}\left[ {B_{3k} \sin (b_{3k} z) + B_{4k} \cos (b_{3k} z)} \right], \\ & + 2\mu^{sk} \left\{ \begin{gathered} A_{5k} \left[ { - b_{4k} K_{0} (b_{4k} r) - \frac{1}{r}K_{1} (b_{4k} r)} \right] \hfill \\ + A_{6k} \left[ {b_{4k} I_{0} (b_{4k} r) - \frac{1}{r}I_{1} (b_{4k} r)} \right] \hfill \\ \end{gathered} \right\}\left[ {B_{5k} \sin (b_{5k} z) + B_{6k} \cos (b_{5k} z)} \right] \\ \end{aligned} $$
(49)
$$ \begin{aligned} \sigma_{zzk}^{s} = & (\lambda^{sk} - 2\mu^{sk} T_{4k} b_{2k}^{2} )\left[ {A_{1k} K_{0} (b_{1k} r) + A_{2k} I_{0} (b_{1k} r)} \right]\left[ {B_{1k} \sin (b_{2k} z) + B_{2k} \cos (b_{2k} z)} \right] \\ {\kern 1pt} & - 2\mu^{sk} T_{5k} b_{3k}^{2} \left[ {A_{3k} K_{0} (b_{3k} r) + A_{4k} I_{0} (b_{3k} r)} \right]\left[ {B_{3k} \sin (b_{3k} z) + B_{4k} \cos (b_{3k} z)} \right], \\ {\kern 1pt} & + 2\mu^{sk} b_{7k} \left[ {A_{7k} K_{0} (b_{6k} r) + A_{8k} I_{0} (b_{6k} r)} \right]\left[ {B_{7k} \cos (b_{7k} z) - B_{8k} \sin (b_{7k} z)} \right] \\ \end{aligned} $$
(50)
$$ \begin{aligned} \sigma_{rzk}^{s} = \sigma_{zrk}^{s} = & \mu^{sk} (T_{2k} + T_{4k} )b_{1k} b_{2k} \left[ { - A_{1k} K_{1} (b_{1k} r) + A_{2k} I_{1} (b_{1k} r)} \right]\left[ {B_{1k} \cos (b_{2k} z) - B_{2k} \sin (b_{2k} z)} \right] \\ {\kern 1pt} & + \mu^{sk} (T_{3k} + T_{5k} )b_{3k}^{2} \left[ { - A_{3k} K_{1} (b_{3k} r) + A_{4k} I_{1} (b_{3k} r)} \right]\left[ {B_{3k} \cos (b_{3k} z) - B_{4k} \sin (b_{3k} z)} \right], \\ {\kern 1pt} & + \mu^{sk} b_{5k} \left[ {A_{5k} K_{1} (b_{4k} r) + A_{6k} I_{1} (b_{4k} r)} \right]\left[ {B_{5k} \cos (b_{5k} z) - B_{6k} \sin (b_{5k} z)} \right] \\ {\kern 1pt} & + \mu^{sk} b_{6k} \left[ { - A_{7k} K_{1} (b_{6k} r) + A_{8k} I_{1} (b_{6k} r)} \right]\left[ {B_{7k} \sin (b_{7k} z) + B_{8k} \cos (b_{7k} z)} \right] \\ \end{aligned} $$
(51)
where \(b_{1k} \sim b_{7k}\), \(A_{1k} \sim A_{8k}\), and \(B_{1k} \sim B_{8k}\) are undetermined constants; \(I_{1} ( \cdot )\) and \(K_{1} ( \cdot )\) are the first-order modified Bessel functions of the first and second types, respectively; and
$$ b_{5k}^{2} - b_{4k}^{2} = \frac{1}{{\mu^{sk} }}\left[ {\rho^{sk} \omega^{2} + \frac{{\rho^{fk} \omega^{2} i\omega s_{vk} }}{{(i\omega s_{vk} - \rho^{fk} \omega^{2} )}}} \right];\;b_{7k}^{2} - b_{6k}^{2} = \frac{1}{{\mu^{sk} }}\left[ {\rho^{sk} \omega^{2} + \frac{{\rho^{fk} \omega^{2} i\omega s_{vk} }}{{(i\omega s_{vk} - \rho^{fk} \omega^{2} )}}} \right]; $$
$$ T_{2k} = \frac{{\left\{ {(\lambda^{sk} + \mu^{sk} ) + \frac{{a_{3k} }}{{\alpha_{k}^{2} }}\left[ {1 + \frac{{\rho^{fk} \omega^{2} n^{fk} }}{{(i\omega s_{vk} - \rho^{fk} \omega^{2} )}}} \right]} \right\}}}{{\left\{ {\mu^{sk} \alpha_{k}^{2} - \left[ {\rho^{sk} \omega^{2} + \frac{{\rho^{fk} \omega^{2} i\omega s_{vk} }}{{(i\omega s_{vk} - \rho^{fk} \omega^{2} )}}} \right]} \right\}}};\;T_{3k} = \frac{{\left[ {1 + \frac{{\rho^{fk} \omega^{2} n^{fk} }}{{(i\omega s_{vk} - \rho^{fk} \omega^{2} )}}} \right]}}{{\left[ {\rho^{sk} \omega^{2} + \frac{{\rho^{fk} \omega^{2} i\omega s_{vk} }}{{(i\omega s_{vk} - \rho^{fk} \omega^{2} )}}} \right]}}; $$
$$ T_{4k} = \frac{{\frac{{a_{3k} }}{{\alpha_{k}^{2} }}\left[ {1 + \frac{{\rho^{fk} \omega^{2} n^{fk} }}{{(i\omega s_{vk} - \rho^{fk} \omega^{2} )}}} \right] + (\lambda^{sk} + \mu^{sk} )}}{{\mu^{sk} \alpha_{k}^{2} - \left[ {\rho^{sk} \omega^{2} + \frac{{\rho^{fk} \omega^{2} i\omega s_{vk} }}{{(i\omega s_{vk} - \rho^{fk} \omega^{2} )}}} \right]}};\;T_{5k} = \frac{{\left[ {1 + \frac{{\rho^{fk} \omega^{2} n^{fk} }}{{(i\omega s_{vk} - \rho^{fk} \omega^{2} )}}} \right]}}{{\rho^{sk} \omega^{2} + \frac{{\rho^{fk} \omega^{2} i\omega s_{vk} }}{{(i\omega s_{vk} - \rho^{fk} \omega^{2} )}}}}. $$
Appendix 2 Determined coefficients in the pile–soil solutions
Coefficients in Eqs. (28 and 29):
For the outer soil part,
$$ F_{O1}^{n} (r) = T_{2O} K_{1} (b_{1On} r);F_{O2}^{n} (r) = T_{3O} K_{1} (b_{n} r);F_{O3}^{n} (r) = K_{1} (b_{4On} r); $$
$$ F_{O4}^{n} (r) = T_{4O} K_{0} (b_{1On} r);F_{O5}^{n} (r) = T_{5O} K_{0} (b_{n} r);F_{O6}^{n} (r) = \frac{{b_{4On} }}{{b_{n} }}K_{0} (b_{4On} r); $$
$$ \begin{gathered} F_{O7}^{n} (r) = \left\{ {\frac{{i\omega s_{vO} T_{2O} \alpha_{O}^{2} + n^{fO} a_{3O} b_{1On} }}{{(i\omega s_{vO} - \rho^{fO} \omega^{2} )\alpha_{O}^{2} }}} \right\}K_{1} (b_{1On} r); \hfill \\ F_{O8}^{n} (r) = \left\{ {\frac{{i\omega s_{vO} T_{3O} - n^{fO} b_{n} }}{{(i\omega s_{vO} - \rho^{fO} \omega^{2} )}}} \right\}K_{1} (b_{n} r);F_{O9}^{n} (r) = \frac{{i\omega s_{vO} }}{{(i\omega s_{vO} - \rho^{fO} \omega^{2} )}}K_{1} (b_{4On} r); \hfill \\ \end{gathered} $$
$$ F_{O10}^{n} (r) = \mu^{sO} (T_{2O} b_{n} + T_{4O} b_{1On} )K_{1} (b_{1On} r);F_{O11}^{n} (r) = \mu^{sO} b_{n} (T_{3O} + T_{5O} )K_{1} (b_{n} r);F_{O12}^{n} (r) = \mu^{sO} (b_{n} + \frac{{b_{4On}^{2} }}{{b_{n} }})K_{1} (b_{4On} r); $$
$$ \begin{gathered} F_{O13}^{n} (r) = (\lambda^{sO} + 2\mu^{sO} T_{2O} b_{1On} + \frac{{a_{3O} }}{{\alpha_{O}^{2} }})K_{0} (b_{1On} r) + 2\mu^{sO} T_{2O} \frac{1}{r}K_{1} (b_{1On} r); \hfill \\ F_{O14}^{n} (r) = 2\mu^{sO} T_{3O} \left[ {(b_{n} - \frac{1}{{2\mu^{sO} T_{3O} }})K_{0} (b_{n} r) + \frac{1}{r}K_{1} (b_{n} r)} \right]; \hfill \\ F_{O15}^{n} (r) = 2\mu^{sO} \left[ {b_{4On} K_{0} (b_{4On} r) + \frac{1}{r}K_{1} (b_{4On} r)} \right]; \hfill \\ \end{gathered} $$
For the inner soil part,
$$ F_{I1}^{n} (r) = T_{2I} I_{1} (b_{1In} r);F_{I2}^{n} (r) = T_{3I} I_{1} (b_{n} r);F_{I3}^{n} (r) = I_{1} (b_{4In} r); $$
$$ F_{I4}^{n} (r) = T_{4I} I_{0} (b_{1In} r);F_{I5}^{n} (r) = T_{5I} I_{0} (b_{n} r);F_{I6}^{n} (r) = \frac{{b_{4In} }}{{b_{n} }}I_{0} (b_{4In} r); $$
$$ \begin{gathered} F_{I7}^{n} (r) = \left\{ {\frac{{i\omega s_{vI} T_{2I} \alpha_{I}^{2} + n^{fI} a_{3I} b_{1In} }}{{(i\omega s_{vI} - \rho^{fI} \omega^{2} )\alpha_{I}^{2} }}} \right\}I_{1} (b_{1In} r); \hfill \\ F_{I8}^{n} (r) = \left\{ {\frac{{i\omega s_{vI} T_{3I} - n^{fI} b_{n} }}{{(i\omega s_{vI} - \rho^{fI} \omega^{2} )}}} \right\}I_{1} (b_{n} r);F_{I9}^{n} (r) = \frac{{i\omega s_{vI} }}{{(i\omega s_{vI} - \rho^{fI} \omega^{2} )}}I_{1} (b_{4In} r); \hfill \\ \end{gathered} $$
$$ F_{I10}^{n} (r) = \mu^{sI} (T_{2I} b_{n} + T_{4I} b_{1In} )I_{1} (b_{1In} r);F_{I11}^{n} (r) = \mu^{sI} b_{n} (T_{3I} + T_{5I} )I_{1} (b_{n} r);F_{I12}^{n} (r) = \mu^{sI} (b_{n} + \frac{{b_{4In}^{2} }}{{b_{n} }})I_{1} (b_{4In} r); $$
$$ \begin{gathered} F_{I13}^{n} (r) = (\lambda^{sI} + 2\mu^{sI} T_{2I} b_{1In} + \frac{{a_{3I} }}{{\alpha_{I}^{2} }})I_{0} (b_{1In} r) - 2\mu^{sI} T_{2I} \frac{1}{r}I_{1} (b_{1In} r); \hfill \\ F_{I14}^{n} (r) = 2\mu^{sI} T_{3I} \left[ {(b_{n} - \frac{1}{{2\mu^{sI} T_{3I} }})I_{0} (b_{n} r) - \frac{1}{r}I_{1} (b_{n} r)} \right]; \hfill \\ F_{I15}^{n} (r) = 2\mu^{sI} \left[ {b_{4In} I_{0} (b_{4In} r) - \frac{1}{r}I_{1} (b_{4In} r)} \right], \hfill \\ \end{gathered} $$
$$ b_{4kn} = \sqrt {b_{n}^{2} - \frac{1}{{\mu^{sk} }}\left[ {\rho^{sk} \omega^{2} + \frac{{\rho^{fk} \omega^{2} i\omega s_{vk} }}{{(i\omega s_{vk} - \rho^{fk} \omega^{2} )}}} \right]} ,\;k = O\;{\text{or}}\;I. $$
Coefficients in Eq. (41):
$$ P_{1} = - \frac{{w_{g} \frac{{2\pi r_{O} }}{{\mu^{p} J_{p} }}(r_{O}^{2} m_{O} - r_{I}^{2} m_{I} )}}{{\left[ {\frac{4}{{(1 - 2\upsilon_{p} )}}\frac{{A_{p} }}{{J_{p} }} - \frac{{\rho_{p} }}{{\mu^{p} }}\omega^{2} } \right]}};\;P_{2n} = \alpha_{1n} C_{1n} + \alpha_{2n} C_{2n} - \alpha_{3n} C_{3n} + \alpha_{4n} C_{4n} + \alpha_{5n} C_{5n} + \alpha_{6n} C_{6n} + w_{g} \alpha_{gn} ; $$
$$ P_{3n} = \beta_{1n} C_{1n} + \beta_{2n} C_{2n} - \beta_{3n} C_{3n} + \beta_{4n} C_{4n} + \beta_{5n} C_{5n} + \beta_{6n} C_{6n} + w_{g} \beta_{gn} ; $$
$$ \Delta_{1n} = \frac{{2\upsilon_{p} }}{{(1 - \upsilon_{p} )}}\frac{{4\upsilon_{p} }}{{(1 - 2\upsilon_{p} )}}\frac{{A_{p} }}{{J_{p} }}b_{n}^{2} - \left[ {b_{n}^{2} - \frac{{(1 - 2\upsilon_{p} )}}{{2\mu^{p} (1 - \upsilon_{p} )}}\rho_{p} \omega^{2} } \right]\left\{ {b_{n}^{2} + \left[ {\frac{4}{{(1 - 2\upsilon_{p} )}}\frac{{A_{p} }}{{J_{p} }} - \frac{{\rho_{p} }}{{\mu^{p} }}\omega^{2} } \right]} \right\}; $$
$$ \alpha_{1n} = \frac{1}{{\Delta_{1n} }}\frac{{\pi r_{O}^{2} }}{{\mu^{p} J_{p} }}\left\{ {\frac{{4\upsilon_{p} }}{{(1 - \upsilon_{p} )}}b_{n} F_{O10}^{n} (r_{O} ) - \left[ {b_{n}^{2} - \frac{{(1 - 2\upsilon_{p} )}}{{2\mu^{p} (1 - \upsilon_{p} )}}\rho_{p} \omega^{2} } \right]2r_{O} F_{O13}^{n} (r_{O} )} \right\}; $$
$$ \alpha_{2n} = \frac{1}{{\Delta_{1n} }}\frac{{\pi r_{O}^{2} }}{{\mu^{p} J_{p} }}\left\{ {\frac{{4\upsilon_{p} }}{{(1 - \upsilon_{p} )}}b_{n} F_{O11}^{n} (r_{O} ) - \left[ {b_{n}^{2} - \frac{{(1 - 2\upsilon_{p} )}}{{2\mu^{p} (1 - \upsilon_{p} )}}\rho_{p} \omega^{2} } \right]2r_{O} F_{O14}^{n} (r_{O} )} \right\}; $$
$$ \alpha_{3n} = \frac{1}{{\Delta_{1n} }}\frac{{\pi r_{O}^{2} }}{{\mu^{p} J_{p} }}\left\{ {\frac{{4\upsilon_{p} }}{{(1 - \upsilon_{p} )}}b_{n} F_{O12}^{n} (r_{O} ) - \left[ {b_{n}^{2} - \frac{{(1 - 2\upsilon_{p} )}}{{2\mu^{p} (1 - \upsilon_{p} )}}\rho_{p} \omega^{2} } \right]2r_{O} F_{O15}^{n} (r_{O} )} \right\}; $$
$$ \alpha_{4n} = \frac{1}{{\Delta_{1n} }}\frac{{2\pi r_{I} r_{O} }}{{\mu^{p} J_{p} }}\left\{ {\frac{{2\upsilon_{p} }}{{(1 - \upsilon_{p} )}}b_{n} F_{I10}^{n} (r_{I} ) + \left[ {b_{n}^{2} - \frac{{(1 - 2\upsilon_{p} )}}{{2\mu^{p} (1 - \upsilon_{p} )}}\rho_{p} \omega^{2} } \right]r_{I} F_{I13}^{n} (r_{I} )} \right\}; $$
$$ \alpha_{5n} = \frac{1}{{\Delta_{1n} }}\frac{{2\pi r_{I} r_{O} }}{{\mu^{p} J_{p} }}\left\{ {\frac{{2\upsilon_{p} }}{{(1 - \upsilon_{p} )}}b_{n} F_{I11}^{n} (r_{I} ) + \left[ {b_{n}^{2} - \frac{{(1 - 2\upsilon_{p} )}}{{2\mu^{p} (1 - \upsilon_{p} )}}\rho_{p} \omega^{2} } \right]r_{I} F_{I14}^{n} (r_{I} )} \right\}; $$
$$ \alpha_{6n} = \frac{1}{{\Delta_{1n} }}\frac{{2\pi r_{I} r_{O} }}{{\mu^{p} J_{p} }}\left\{ {\frac{{2\upsilon_{p} }}{{(1 - \upsilon_{p} )}}b_{n} F_{I12}^{n} (r_{I} ) + \left[ {b_{n}^{2} - \frac{{(1 - 2\upsilon_{p} )}}{{2\mu^{p} (1 - \upsilon_{p} )}}\rho_{p} \omega^{2} } \right]r_{I} F_{I15}^{n} (r_{I} )} \right\}; $$
$$ \alpha_{gn} = \frac{1}{{\Delta_{1n} }}\frac{{2\pi r_{O} }}{{\mu^{p} J_{p} }}\left\{ \begin{gathered} \left[ {b_{n}^{2} - \frac{{(1 - 2\upsilon_{p} )}}{{2\mu^{p} (1 - \upsilon_{p} )}}\rho_{p} \omega^{2} } \right](r_{O}^{2} F_{O16}^{n} P_{On} - r_{I}^{2} F_{I16}^{n} P_{In} ) \hfill \\ + \frac{{(r_{O}^{2} m_{O} - r_{I}^{2} m_{I} )}}{{\left[ {\frac{4}{{(1 - 2\upsilon_{p} )}}\frac{{A_{p} }}{{J_{p} }} - \frac{{\rho_{p} }}{{\mu^{p} }}\omega^{2} } \right]}}\frac{{2\upsilon_{p} }}{{(1 - \upsilon_{p} )}}\frac{{4\upsilon_{p} }}{{(1 - 2\upsilon_{p} )}}\frac{{A_{p} }}{{J_{p} }}\frac{2}{L}\sin (b_{n} L) \hfill \\ \end{gathered} \right\}; $$
$$ \beta_{1n} = \frac{1}{{\Delta_{1n} }}\frac{{(1 - 2\upsilon_{p} )}}{{(1 - \upsilon_{p} )}}\frac{{\pi r_{O} }}{{\mu^{p} A_{p} }}\left\{ {\left\{ {b_{n}^{2} + \left[ {\frac{4}{{(1 - 2\upsilon_{p} )}}\frac{{A_{p} }}{{J_{p} }} - \frac{{\rho_{p} }}{{\mu^{p} }}\omega^{2} } \right]} \right\}F_{O10}^{n} (r_{O} ) - \frac{{2\upsilon_{p} }}{{(1 - 2\upsilon_{p} )}}\frac{{A_{p} }}{{J_{p} }}b_{n} 2r_{O} F_{O13}^{n} (r_{O} )} \right\}; $$
$$ \beta_{2n} = \frac{1}{{\Delta_{1n} }}\frac{{(1 - 2\upsilon_{p} )}}{{(1 - \upsilon_{p} )}}\frac{{\pi r_{O} }}{{\mu^{p} A_{p} }}\left\{ {\left\{ {b_{n}^{2} + \left[ {\frac{4}{{(1 - 2\upsilon_{p} )}}\frac{{A_{p} }}{{J_{p} }} - \frac{{\rho_{p} }}{{\mu^{p} }}\omega^{2} } \right]} \right\}F_{O11}^{n} (r_{O} ) - \frac{{2\upsilon_{p} }}{{(1 - 2\upsilon_{p} )}}\frac{{A_{p} }}{{J_{p} }}b_{n} 2r_{O} F_{O14}^{n} (r_{O} )} \right\}; $$
$$ \beta_{3n} = \frac{1}{{\Delta_{1n} }}\frac{{(1 - 2\upsilon_{p} )}}{{(1 - \upsilon_{p} )}}\frac{{\pi r_{O} }}{{\mu^{p} A_{p} }}\left\{ {\left\{ {b_{n}^{2} + \left[ {\frac{4}{{(1 - 2\upsilon_{p} )}}\frac{{A_{p} }}{{J_{p} }} - \frac{{\rho_{p} }}{{\mu^{p} }}\omega^{2} } \right]} \right\}F_{O12}^{n} (r_{O} ) - \frac{{2\upsilon_{p} }}{{(1 - 2\upsilon_{p} )}}\frac{{A_{p} }}{{J_{p} }}b_{n} 2r_{O} F_{O15}^{n} (r_{O} )} \right\}; $$
$$ \beta_{4n} = \frac{1}{{\Delta_{1n} }}\frac{{(1 - 2\upsilon_{p} )}}{{(1 - \upsilon_{p} )}}\frac{{\pi r_{I} }}{{\mu^{p} A_{p} }}\left\{ {\left\{ {b_{n}^{2} + \left[ {\frac{4}{{(1 - 2\upsilon_{p} )}}\frac{{A_{p} }}{{J_{p} }} - \frac{{\rho_{p} }}{{\mu^{p} }}\omega^{2} } \right]} \right\}F_{I10}^{n} (r_{I} ) + \frac{{2\upsilon_{p} }}{{(1 - 2\upsilon_{p} )}}\frac{{A_{p} }}{{J_{p} }}b_{n} 2r_{I} F_{I13}^{n} (r_{I} )} \right\}; $$
$$ \beta_{5n} = \frac{1}{{\Delta_{1n} }}\frac{{(1 - 2\upsilon_{p} )}}{{(1 - \upsilon_{p} )}}\frac{{\pi r_{I} }}{{\mu^{p} A_{p} }}\left\{ {\left\{ {b_{n}^{2} + \left[ {\frac{4}{{(1 - 2\upsilon_{p} )}}\frac{{A_{p} }}{{J_{p} }} - \frac{{\rho_{p} }}{{\mu^{p} }}\omega^{2} } \right]} \right\}F_{I11}^{n} (r_{I} ) + \frac{{2\upsilon_{p} }}{{(1 - 2\upsilon_{p} )}}\frac{{A_{p} }}{{J_{p} }}b_{n} 2r_{I} F_{I14}^{n} (r_{I} )} \right\}; $$
$$ \beta_{6n} = \frac{1}{{\Delta_{1n} }}\frac{{(1 - 2\upsilon_{p} )}}{{(1 - \upsilon_{p} )}}\frac{{\pi r_{I} }}{{\mu^{p} A_{p} }}\left\{ {\left\{ {b_{n}^{2} + \left[ {\frac{4}{{(1 - 2\upsilon_{p} )}}\frac{{A_{p} }}{{J_{p} }} - \frac{{\rho_{p} }}{{\mu^{p} }}\omega^{2} } \right]} \right\}F_{I12}^{n} (r_{I} ) + \frac{{2\upsilon_{p} }}{{(1 - 2\upsilon_{p} )}}\frac{{A_{p} }}{{J_{p} }}b_{n} 2r_{I} F_{I15}^{n} (r_{I} )} \right\}; $$
$$ \beta_{gn} = \frac{1}{{\Delta_{1n} }}\frac{{2\upsilon_{p} }}{{(1 - \upsilon_{p} )}}\frac{2\pi }{{\mu^{p} J_{p} }}\left\{ \begin{gathered} b_{n} (r_{O}^{2} F_{O16}^{n} P_{On} - r_{I}^{2} F_{I16}^{n} P_{In} ) \hfill \\ + \frac{1}{{b_{n} }}\left\{ {b_{n}^{2} + \left[ {\frac{4}{{(1 - 2\upsilon_{p} )}}\frac{{A_{p} }}{{J_{p} }} - \frac{{\rho_{p} }}{{\mu^{p} }}\omega^{2} } \right]} \right\}\frac{{(r_{O}^{2} m_{O} - r_{I}^{2} m_{I} )}}{{\left[ {\frac{4}{{(1 - 2\upsilon_{p} )}}\frac{{A_{p} }}{{J_{p} }} - \frac{{\rho_{p} }}{{\mu^{p} }}\omega^{2} } \right]}}\frac{2}{L}\sin (b_{n} L) \hfill \\ \end{gathered} \right\}. $$
Determined coefficients \(C_{1n}\) ~ \(C_{6n}\) and \(U_{1}\) ~ \(U_{4}\):
$$ C_{1n} = \sum\limits_{j = 1}^{4} {U_{j} \frac{1}{{\gamma_{28n} }}\xi_{jn} } + w_{g} \frac{1}{{\gamma_{28n} }}\xi_{gn} , $$
(52)
$$ C_{2n} = - \sum\limits_{j = 1}^{4} {U_{j} \frac{1}{{\gamma_{28n} }}\delta_{jn} } - w_{g} \frac{1}{{\gamma_{28n} }}\delta_{gn} , $$
(53)
$$ C_{3n} = \sum\limits_{j = 1}^{4} {U_{j} \frac{1}{{\gamma_{28n} }}\frac{1}{{c_{3n} }}\left( {c_{1n} \xi_{jn} - c_{2n} \delta_{jn} } \right)} + w_{g} \frac{1}{{\gamma_{28n} }}\frac{1}{{c_{3n} }}\left( {c_{1n} \xi_{gn} - c_{2n} \delta_{gn} } \right), $$
(54)
$$ C_{4n} = \sum\limits_{j = 1}^{4} {U_{j} \frac{1}{{\gamma_{28n} }}\frac{1}{{\gamma_{21n} }}\left\{ {\gamma_{29n} \xi_{jn} - \gamma_{30n} \delta_{jn} } \right\}} + w_{g} \frac{1}{{\gamma_{28n} }}\frac{1}{{\gamma_{21n} }}\left\{ {\gamma_{29n} \xi_{gn} - \gamma_{30n} \delta_{gn} + \gamma_{28n} (P_{On} - P_{In} )} \right\}, $$
(55)
$$ C_{5n} = - \sum\limits_{j = 1}^{4} {U_{j} \frac{1}{{\gamma_{28n} }}\frac{1}{{\gamma_{20n} }}\left\{ {\gamma_{31n} \xi_{jn} - \gamma_{32n} \delta_{jn} } \right\}} - w_{g} \frac{1}{{\gamma_{28n} }}\frac{1}{{\gamma_{20n} }}\left\{ {\gamma_{31n} \xi_{gn} - \gamma_{32n} \delta_{gn} + \frac{1}{{\gamma_{21n} }}\gamma_{19n} \gamma_{28n} (P_{On} - P_{In} )} \right\}, $$
(56)
$$ C_{6n} = - \sum\limits_{j = 1}^{4} {U_{j} \frac{1}{{\gamma_{28n} }}\frac{1}{{F_{I9}^{n} (r_{I} )}}\left\{ {\gamma_{33n} \xi_{jn} - \gamma_{34n} \delta_{jn} } \right\}} - w_{g} \frac{1}{{\gamma_{28n} }}\frac{1}{{F_{I9}^{n} (r_{I} )}}\left\{ {\gamma_{33n} \xi_{gn} - \gamma_{34n} \delta_{gn} + \frac{1}{{\gamma_{21n} }}\gamma_{28n} \gamma_{g3n} (P_{On} - P_{In} )} \right\}, $$
(57)
$$ U_{1} = - \tilde{p}\frac{1}{R}\frac{1}{{k_{1} e^{{\eta_{1} L}} }}m_{p3} - w_{g} \frac{1}{{k_{1} e^{{\eta_{1} L}} }}m_{g6} , $$
(58)
$$ U_{2} = \tilde{p}\frac{1}{R}\frac{1}{{m_{1} }}(m_{2} m_{p2} - m_{3} m_{p1} ) + w_{g} \frac{1}{{m_{1} }}m_{g5} , $$
(59)
$$ U_{3} = - \tilde{p}\frac{1}{R}m_{p2} - w_{g} \frac{1}{R}m_{g4} , $$
(60)
$$ U_{4} = \tilde{p}\frac{1}{R}m_{p1} + w_{g} \frac{1}{R}m_{g3} , $$
(61)
where
$$ \xi_{jn} = h_{j} \left\{ {\gamma_{26n} - \gamma_{27n} \frac{{\left[ {F_{O2}^{n} (r_{O} ) - F_{O8}^{n} (r_{O} )} \right]}}{{\left[ {F_{O3}^{n} (r_{O} ) - F_{O9}^{n} (r_{O} )} \right]}}} \right\}I_{jn}^{s} + k_{j} \left\{ {\gamma_{23n} - \gamma_{24n} \frac{{\left[ {F_{O2}^{n} (r_{O} ) - F_{O8}^{n} (r_{O} )} \right]}}{{\left[ {F_{O3}^{n} (r_{O} ) - F_{O9}^{n} (r_{O} )} \right]}}} \right\}I_{jn}^{c} ; $$
$$ \delta_{jn} = h_{j} \left\{ {\gamma_{25n} - \gamma_{27n} \frac{{\left[ {F_{O1}^{n} (r_{O} ) - F_{O7}^{n} (r_{O} )} \right]}}{{\left[ {F_{O3}^{n} (r_{O} ) - F_{O9}^{n} (r_{O} )} \right]}}} \right\}I_{jn}^{s} + k_{j} \left\{ {\gamma_{22n} - \gamma_{24n} \frac{{\left[ {F_{O1}^{n} (r_{O} ) - F_{O7}^{n} (r_{O} )} \right]}}{{\left[ {F_{O3}^{n} (r_{O} ) - F_{O9}^{n} (r_{O} )} \right]}}} \right\}I_{jn}^{c} ; $$
$$ I_{jn}^{s} = \frac{2}{L}\int_{0}^{L} {e^{{\eta_{j} z}} \sin (b_{n} z)dz} ;\;I_{jn}^{c} = \frac{2}{L}\int_{0}^{L} {e^{{\eta_{j} z}} \cos (b_{n} z)dz} ;\;j = 1,2,3,4; $$
$$ \xi_{gn} = \left\{ {\gamma_{26n} - \gamma_{27n} \frac{{\left[ {F_{O2}^{n} (r_{O} ) - F_{O8}^{n} (r_{O} )} \right]}}{{\left[ {F_{O3}^{n} (r_{O} ) - F_{O9}^{n} (r_{O} )} \right]}}} \right\}I_{gn}^{s} + \left\{ {\gamma_{23n} - \gamma_{24n} \frac{{\left[ {F_{O2}^{n} (r_{O} ) - F_{O8}^{n} (r_{O} )} \right]}}{{\left[ {F_{O3}^{n} (r_{O} ) - F_{O9}^{n} (r_{O} )} \right]}}} \right\}I_{gn}^{c} + \gamma_{g1n} (P_{On} - P_{In} ); $$
$$ \delta_{gn} = \left\{ {\gamma_{25n} - \gamma_{27n} \frac{{\left[ {F_{O1}^{n} (r_{O} ) - F_{O7}^{n} (r_{O} )} \right]}}{{\left[ {F_{O3}^{n} (r_{O} ) - F_{O9}^{n} (r_{O} )} \right]}}} \right\}I_{gn}^{s} + \left\{ {\gamma_{22n} - \gamma_{24n} \frac{{\left[ {F_{O1}^{n} (r_{O} ) - F_{O7}^{n} (r_{O} )} \right]}}{{\left[ {F_{O3}^{n} (r_{O} ) - F_{O9}^{n} (r_{O} )} \right]}}} \right\}I_{gn}^{c} + \gamma_{g2n} (P_{On} - P_{In} ); $$
$$ I_{gn}^{s} = \alpha_{gn} - \frac{{\frac{{2\pi r_{O} }}{{\mu^{p} J_{p} }}(r_{O}^{2} m_{O} - r_{I}^{2} m_{I} )}}{{\left[ {\frac{4}{{(1 - 2\upsilon_{p} )}}\frac{{A_{p} }}{{J_{p} }} - \frac{{\rho_{p} }}{{\mu^{p} }}\omega^{2} } \right]}}\frac{2}{L}\int_{0}^{L} {z\sin (b_{n} z)dz} ;\;I_{gn}^{c} = \beta_{gn} - \frac{2}{L}\int_{0}^{L} {\cos (b_{n} z)dz} - P_{On} ; $$
$$ \begin{aligned} \gamma_{g1n} = & \left\{ {\gamma_{26n} - \gamma_{27n} \frac{{\left[ {F_{O2}^{n} (r_{O} ) - F_{O8}^{n} (r_{O} )} \right]}}{{\left[ {F_{O3}^{n} (r_{O} ) - F_{O9}^{n} (r_{O} )} \right]}}} \right\}\frac{1}{{\gamma_{21n} }}\left( {\gamma_{4n} - \frac{1}{{\gamma_{20n} }}\gamma_{5n} \gamma_{19n} } \right) \\ {\kern 1pt} & + \left\{ {\gamma_{23n} - \gamma_{24n} \frac{{\left[ {F_{O2}^{n} (r_{O} ) - F_{O8}^{n} (r_{O} )} \right]}}{{\left[ {F_{O3}^{n} (r_{O} ) - F_{O9}^{n} (r_{O} )} \right]}}} \right\}\frac{1}{{\gamma_{21n} }}\left( {\gamma_{9n} - \frac{1}{{\gamma_{20n} }}\gamma_{10n} \gamma_{19n} } \right); \\ \end{aligned} $$
$$ \begin{aligned} \gamma_{g2n} = & \left\{ {\gamma_{25n} - \gamma_{27n} \frac{{\left[ {F_{O1}^{n} (r_{O} ) - F_{O7}^{n} (r_{O} )} \right]}}{{\left[ {F_{O3}^{n} (r_{O} ) - F_{O9}^{n} (r_{O} )} \right]}}} \right\}\frac{1}{{\gamma_{21n} }}\left( {\gamma_{4n} - \frac{1}{{\gamma_{20n} }}\gamma_{5n} \gamma_{19n} } \right) \\ {\kern 1pt} & + \left\{ {\gamma_{22n} - \gamma_{24n} \frac{{\left[ {F_{O1}^{n} (r_{O} ) - F_{O7}^{n} (r_{O} )} \right]}}{{\left[ {F_{O3}^{n} (r_{O} ) - F_{O9}^{n} (r_{O} )} \right]}}} \right\}\frac{1}{{\gamma_{21n} }}\left( {\gamma_{9n} - \frac{1}{{\gamma_{20n} }}\gamma_{10n} \gamma_{19n} } \right); \\ \end{aligned} $$
$$ \gamma_{g3n} = F_{I7}^{n} (r_{I} ) - \frac{1}{{\gamma_{20n} }}\gamma_{19n} F_{I8}^{n} (r_{I} );\;\gamma_{g4n} = \alpha_{4n} - \frac{1}{{\gamma_{20n} }}\alpha_{5n} \gamma_{19n} - \frac{1}{{F_{I9}^{n} (r_{I} )}}\alpha_{6n} \gamma_{g3n} ; $$
$$ \gamma_{1n} = \alpha_{7n} - \frac{{r_{I} }}{{r_{O} }}\alpha_{6n} \frac{1}{{F_{I9}^{n} (r_{I} )}}F_{O7}^{n} (r_{O} );\;\gamma_{2n} = \alpha_{8n} - \frac{{r_{I} }}{{r_{O} }}\alpha_{6n} \frac{1}{{F_{I9}^{n} (r_{I} )}}F_{O8}^{n} (r_{O} );\;\gamma_{3n} = \alpha_{9n} - \frac{{r_{I} }}{{r_{O} }}\alpha_{6n} \frac{1}{{F_{I9}^{n} (r_{I} )}}F_{O9}^{n} (r_{O} ); $$
$$ \gamma_{4n} = \alpha_{4n} - \alpha_{6n} \frac{1}{{F_{I9}^{n} (r_{I} )}}F_{I7}^{n} (r_{I} );\;\gamma_{5n} = \alpha_{5n} - \alpha_{6n} \frac{1}{{F_{I9}^{n} (r_{I} )}}F_{I8}^{n} (r_{I} );\;\gamma_{6n} = \beta_{7n} + \frac{{r_{I} }}{{r_{O} }}\beta_{6n} \frac{1}{{F_{I9}^{n} (r_{I} )}}F_{O7}^{n} (r_{O} ); $$
$$ \gamma_{7n} = \beta_{8n} + \frac{{r_{I} }}{{r_{O} }}\beta_{6n} \frac{1}{{F_{I9}^{n} (r_{I} )}}F_{O8}^{n} (r_{O} );\;\gamma_{8n} = \beta_{9n} + \frac{{r_{I} }}{{r_{O} }}\beta_{6n} \frac{1}{{F_{I9}^{n} (r_{I} )}}F_{O9}^{n} (r_{O} );\;\gamma_{9n} = \beta_{4n} - \beta_{6n} \frac{1}{{F_{I9}^{n} (r_{I} )}}F_{I7}^{n} (r_{I} ); $$
$$ \gamma_{10n} = \beta_{5n} - \beta_{6n} \frac{1}{{F_{I9}^{n} (r_{I} )}}F_{I8}^{n} (r_{I} );\;\gamma_{11n} = F_{O4}^{n} (r_{O} ) + \frac{{r_{I} }}{{r_{O} }}\frac{1}{{F_{I9}^{n} (r_{I} )}}F_{I6}^{n} (r_{I} )F_{O7}^{n} (r_{O} ); $$
$$ \gamma_{12n} = F_{O5}^{n} (r_{O} ) + \frac{{r_{I} }}{{r_{O} }}\frac{1}{{F_{I9}^{n} (r_{I} )}}F_{I6}^{n} (r_{I} )F_{O8}^{n} (r_{O} );\;\gamma_{13n} = F_{O6}^{n} (r_{O} ) + \frac{{r_{I} }}{{r_{O} }}\frac{1}{{F_{I9}^{n} (r_{I} )}}F_{I6}^{n} (r_{I} )F_{O9}^{n} (r_{O} ); $$
$$ \gamma_{14n} = F_{I4}^{n} (r_{I} ) - \frac{1}{{F_{I9}^{n} (r_{I} )}}F_{I6}^{n} (r_{I} )F_{I7}^{n} (r_{I} );\;\gamma_{15n} = F_{I5}^{n} (r_{I} ) - \frac{1}{{F_{I9}^{n} (r_{I} )}}F_{I6}^{n} (r_{I} )F_{I8}^{n} (r_{I} ); $$
$$ \gamma_{16n} = \frac{{r_{I} }}{{r_{O} }}F_{O1}^{n} (r_{O} ) - \frac{{r_{I} }}{{r_{O} }}\frac{1}{{F_{I9}^{n} (r_{I} )}}F_{I3}^{n} (r_{I} )F_{O7}^{n} (r_{O} );\;\gamma_{17n} = \frac{{r_{I} }}{{r_{O} }}F_{O2}^{n} (r_{O} ) - \frac{{r_{I} }}{{r_{O} }}\frac{1}{{F_{I9}^{n} (r_{I} )}}F_{I3}^{n} (r_{I} )F_{O8}^{n} (r_{O} ); $$
$$ \gamma_{18n} = \frac{{r_{I} }}{{r_{O} }}F_{O3}^{n} (r_{O} ) - \frac{{r_{I} }}{{r_{O} }}\frac{1}{{F_{I9}^{n} (r_{I} )}}F_{I3}^{n} (r_{I} )F_{O9}^{n} (r_{O} );\;\gamma_{19n} = F_{I1}^{n} (r_{I} ) - \frac{1}{{F_{I9}^{n} (r_{I} )}}F_{I3}^{n} (r_{I} )F_{I7}^{n} (r_{I} ); $$
$$ \gamma_{20n} = F_{I2}^{n} (r_{I} ) - \frac{1}{{F_{I9}^{n} (r_{I} )}}F_{I3}^{n} (r_{I} )F_{I8}^{n} (r_{I} );\;\gamma_{21n} = \gamma_{14n} - \frac{1}{{\gamma_{20n} }}\gamma_{15n} \gamma_{19n} ; $$
$$ \gamma_{22n} = \left( {\gamma_{1n} - \frac{1}{{\gamma_{20n} }}\gamma_{5n} \gamma_{16n} } \right) + \frac{1}{{\gamma_{21n} }}\left( {\gamma_{4n} - \frac{1}{{\gamma_{20n} }}\gamma_{5n} \gamma_{19n} } \right)\left( {\gamma_{11n} + \frac{1}{{\gamma_{20n} }}\gamma_{15n} \gamma_{16n} } \right); $$
$$ \gamma_{23n} = \left( {\gamma_{2n} - \frac{1}{{\gamma_{20n} }}\gamma_{5n} \gamma_{17n} } \right) + \frac{1}{{\gamma_{21n} }}\left( {\gamma_{4n} - \frac{1}{{\gamma_{20n} }}\gamma_{5n} \gamma_{19n} } \right)\left( {\gamma_{12n} + \frac{1}{{\gamma_{20n} }}\gamma_{15n} \gamma_{17n} } \right); $$
$$ \gamma_{24n} = (\gamma_{3n} - \frac{1}{{\gamma_{20n} }}\gamma_{5n} \gamma_{18n} ) + \frac{1}{{\gamma_{21n} }}(\gamma_{4n} - \frac{1}{{\gamma_{20n} }}\gamma_{5n} \gamma_{19n} )(\gamma_{13n} + \frac{1}{{\gamma_{20n} }}\gamma_{15n} \gamma_{18n} ); $$
$$ \gamma_{25n} = \left( {\gamma_{6n} + \frac{1}{{\gamma_{20n} }}\gamma_{10n} \gamma_{16n} } \right) - \frac{1}{{\gamma_{21n} }}\left( {\gamma_{9n} - \frac{1}{{\gamma_{20n} }}\gamma_{10n} \gamma_{19n} } \right)\left( {\gamma_{11n} + \frac{1}{{\gamma_{20n} }}\gamma_{15n} \gamma_{16n} } \right); $$
$$ \gamma_{26n} = \left( {\gamma_{7n} + \frac{1}{{\gamma_{20n} }}\gamma_{10n} \gamma_{17n} } \right) - \frac{1}{{\gamma_{21n} }}\left( {\gamma_{9n} - \frac{1}{{\gamma_{20n} }}\gamma_{10n} \gamma_{19n} } \right)\left( {\gamma_{12n} + \frac{1}{{\gamma_{20n} }}\gamma_{15n} \gamma_{17n} } \right); $$
$$ \gamma_{27n} = \left( {\gamma_{8n} + \frac{1}{{\gamma_{20n} }}\gamma_{10n} \gamma_{18n} } \right) - \frac{1}{{\gamma_{21n} }}\left( {\gamma_{9n} - \frac{1}{{\gamma_{20n} }}\gamma_{10n} \gamma_{19n} } \right)\left( {\gamma_{13n} + \frac{1}{{\gamma_{20n} }}\gamma_{15n} \gamma_{18n} } \right); $$
$$ \begin{aligned} \gamma_{28n} = & \left\{ {\gamma_{23n} - \gamma_{24n} \frac{{\left[ {F_{O2}^{n} (r_{O} ) - F_{O8}^{n} (r_{O} )} \right]}}{{\left[ {F_{O3}^{n} (r_{O} ) - F_{O9}^{n} (r_{O} )} \right]}}} \right\}\left\{ {\gamma_{25n} - \gamma_{27n} \frac{{\left[ {F_{O1}^{n} (r_{O} ) - F_{O7}^{n} (r_{O} )} \right]}}{{\left[ {F_{O3}^{n} (r_{O} ) - F_{O9}^{n} (r_{O} )} \right]}}} \right\} \\ & - \left\{ {\gamma_{22n} - \gamma_{24n} \frac{{\left[ {F_{O1}^{n} (r_{O} ) - F_{O7}^{n} (r_{O} )} \right]}}{{\left[ {F_{O3}^{n} (r_{O} ) - F_{O9}^{n} (r_{O} )} \right]}}} \right\}\left\{ {\gamma_{26n} - \gamma_{27n} \frac{{\left[ {F_{O2}^{n} (r_{O} ) - F_{O8}^{n} (r_{O} )} \right]}}{{\left[ {F_{O3}^{n} (r_{O} ) - F_{O9}^{n} (r_{O} )} \right]}}} \right\}; \\ \end{aligned} $$
$$ \gamma_{29n} = \left( {\gamma_{11n} + \frac{1}{{\gamma_{20n} }}\gamma_{15n} \gamma_{16n} } \right) - \frac{1}{{c_{3n} }}\left( {\gamma_{13n} + \frac{1}{{\gamma_{20n} }}\gamma_{15n} \gamma_{18n} } \right)c_{1n} ; $$
$$ \gamma_{30n} = \left( {\gamma_{12n} + \frac{1}{{\gamma_{20n} }}\gamma_{15n} \gamma_{17n} } \right) - \frac{1}{{c_{3n} }}\left( {\gamma_{13n} + \frac{1}{{\gamma_{20n} }}\gamma_{15n} \gamma_{18n} } \right)c_{2n} ; $$
$$ \gamma_{31n} = \gamma_{16n} - \gamma_{18n} \frac{1}{{c_{3n} }}c_{1n} + \frac{1}{{\gamma_{21n} }}\gamma_{19n} \gamma_{29n} ;\;\gamma_{32n} = \gamma_{17n} - \gamma_{18n} \frac{1}{{c_{3n} }}c_{2n} + \frac{1}{{\gamma_{21n} }}\gamma_{19n} \gamma_{30n} ; $$
$$ \gamma_{33n} = \frac{{r_{I} }}{{r_{O} }}F_{O7}^{n} (r_{O} ) - \frac{1}{{c_{3n} }}\frac{{r_{I} }}{{r_{O} }}F_{O9}^{n} (r_{O} )c_{1n} + \frac{1}{{\gamma_{21n} }}F_{I7}^{n} (r_{I} )\gamma_{29n} - \frac{1}{{\gamma_{20n} }}F_{I8}^{n} (r_{I} )\gamma_{31n} ; $$
$$ \gamma_{34n} = \frac{{r_{I} }}{{r_{O} }}F_{O8}^{n} (r_{O} ) - \frac{1}{{c_{3n} }}\frac{{r_{I} }}{{r_{O} }}F_{O9}^{n} (r_{O} )c_{2n} + \frac{1}{{\gamma_{21n} }}F_{I7}^{n} (r_{I} )\gamma_{30n} - \frac{1}{{\gamma_{20n} }}F_{I8}^{n} (r_{I} )\gamma_{32n} ; $$
$$ \gamma_{35n} = \alpha_{1n} - \frac{1}{{c_{3n} }}\alpha_{3n} c_{1n} + \frac{1}{{\gamma_{21n} }}\alpha_{4n} \gamma_{29n} - \frac{1}{{\gamma_{20n} }}\alpha_{5n} \gamma_{31n} - \frac{1}{{F_{I9}^{n} (r_{I} )}}\alpha_{6n} \gamma_{33n} ; $$
$$ \gamma_{36n} = \alpha_{2n} - \frac{1}{{c_{3n} }}\alpha_{3n} c_{2n} + \frac{1}{{\gamma_{21n} }}\alpha_{4n} \gamma_{30n} - \frac{1}{{\gamma_{20n} }}\alpha_{5n} \gamma_{32n} - \frac{1}{{F_{I9}^{n} (r_{I} )}}\alpha_{6n} \gamma_{34n} ; $$
$$ \alpha_{7n} = \alpha_{1n} + F_{O1}^{n} (r_{O} );\;\alpha_{8n} = \alpha_{2n} + F_{O2}^{n} (r_{O} );\;\alpha_{9n} = \alpha_{3n} + F_{O3}^{n} (r_{O} ); $$
$$ \beta_{7n} = F_{O4}^{n} (r_{O} ) - \beta_{1n} ;\;\beta_{8n} = F_{O5}^{n} (r_{O} ) - \beta_{2n} ;\;\beta_{9n} = F_{O6}^{n} (r_{O} ) - \beta_{3n} ; $$
$$ c_{1n} = F_{O1}^{n} (r_{O} ) - F_{O7}^{n} (r_{O} );\;c_{2n} = F_{O2}^{n} (r_{O} ) - F_{O8}^{n} (r_{O} );\;c_{3n} = F_{O3}^{n} (r_{O} ) - F_{O9}^{n} (r_{O} ); $$
$$ m_{1} = \left( {h_{2} \eta_{2} - k_{2} h_{1} \eta_{1} \frac{1}{{k_{1} }}} \right)e^{{\eta_{2} L}} ;\;m_{2} = \left( {h_{3} \eta_{3} - k_{3} h_{1} \eta_{1} \frac{1}{{k_{1} }}} \right)e^{{\eta_{3} L}} ;\;m_{3} = \left( {h_{4} \eta_{4} - k_{4} h_{1} \eta_{1} \frac{1}{{k_{1} }}} \right)e^{{\eta_{4} L}} ; $$
$$ m_{4} = X_{2} - X_{1} \frac{1}{{k_{1} e^{{\eta_{1} L}} }}k_{2} e^{{\eta_{2} L}} ;\;m_{5} = X_{3} - X_{1} \frac{1}{{k_{1} e^{{\eta_{1} L}} }}k_{3} e^{{\eta_{3} L}} ;\;m_{6} = X_{4} - X_{1} \frac{1}{{k_{1} e^{{\eta_{1} L}} }}k_{4} e^{{\eta_{4} L}} ; $$
$$ m_{7} = \lambda_{2} - \lambda_{1} \frac{1}{{k_{1} e^{{\eta_{1} L}} }}k_{2} e^{{\eta_{2} L}} ;\;m_{8} = \lambda_{3} - \lambda_{1} \frac{1}{{k_{1} e^{{\eta_{1} L}} }}k_{3} e^{{\eta_{3} L}} ;\;m_{9} = \lambda_{4} - \lambda_{1} \frac{1}{{k_{1} e^{{\eta_{1} L}} }}k_{4} e^{{\eta_{4} L}} ; $$
$$ X_{j} = h_{j} \eta_{j} + \sum\limits_{n = 1}^{\infty } {\frac{1}{{\gamma_{28n} }}b_{n} \left\{ {\gamma_{35n} \xi_{jn} - \gamma_{36n} \delta_{jn} } \right\}} ;\;j = 1,2,3,4; $$
$$ m_{p1} = m_{1} (m_{1} m_{5} - m_{2} m_{4} )\frac{{(1 - 2\upsilon_{p} )}}{{2\mu^{p} A_{p} }};\;m_{p2} = m_{1} (m_{1} m_{6} - m_{3} m_{4} )\frac{{(1 - 2\upsilon_{p} )}}{{2\mu^{p} A_{p} }}; $$
$$ m_{p3} = \frac{1}{{m_{1} }}k_{2} e^{{\eta_{2} L}} (m_{2} m_{p2} - m_{3} m_{p1} ) - k_{3} e^{{\eta_{3} L}} m_{p2} + k_{4} e^{{\eta_{4} L}} m_{p1} ; $$
$$ m_{g1} = m_{1} \left( {X_{g} - X_{1} \frac{1}{{k_{1} e^{{\eta_{1} L}} }}} \right) - m_{4} \left( {Y_{g} - h_{1} \eta_{1} \frac{1}{{k_{1} }}} \right);\;m_{g2} = m_{1} \lambda_{1} \frac{1}{{k_{1} e^{{\eta_{1} L}} }} + m_{7} \left( {Y_{g} - h_{1} \eta_{1} \frac{1}{{k_{1} }}} \right); $$
$$ m_{g3} = m_{g2} (m_{1} m_{5} - m_{2} m_{4} ) + m_{g1} (m_{1} m_{8} - m_{2} m_{7} );\;m_{g4} = m_{g2} (m_{1} m_{6} - m_{3} m_{4} ) + m_{g1} (m_{1} m_{9} - m_{3} m_{7} ); $$
$$ m_{g5} = \frac{1}{R}m_{2} m_{g4} - \frac{1}{R}m_{3} m_{g3} + \left( {Y_{g} - h_{1} \eta_{1} \frac{1}{{k_{1} }}} \right);\;m_{g6} = \frac{1}{{m_{1} }}k_{2} e^{{\eta_{2} L}} m_{g5} - \frac{1}{R}k_{3} e^{{\eta_{3} L}} m_{g4} + \frac{1}{R}k_{4} e^{{\eta_{4} L}} m_{g3} - 1; $$
$$ R = (m_{1} m_{8} - m_{2} m_{7} )(m_{1} m_{6} - m_{3} m_{4} ) - (m_{1} m_{9} - m_{3} m_{7} )(m_{1} m_{5} - m_{2} m_{4} ); $$
$$ X_{g} = \frac{{\frac{{2\pi r_{O} }}{{\mu^{p} J_{p} }}(r_{O}^{2} m_{O} - r_{I}^{2} m_{I} )}}{{\left[ {\frac{4}{{(1 - 2\upsilon_{p} )}}\frac{{A_{p} }}{{J_{p} }} - \frac{{\rho_{p} }}{{\mu^{p} }}\omega^{2} } \right]}} - \sum\limits_{n = 1}^{\infty } {b_{n} \left\{ {\frac{1}{{\gamma_{28n} }}\left\{ {\gamma_{35n} \xi_{gn} - \gamma_{36n} \delta_{gn} } \right\} + \frac{1}{{\gamma_{21n} }}\gamma_{g4n} (P_{On} - P_{In} ) + \alpha_{gn} } \right\}} ; $$
\(Y_{g} = \frac{{\frac{{2\pi r_{O} }}{{\mu^{p} J_{p} }}(r_{O}^{2} m_{O} - r_{I}^{2} m_{I} )}}{{\left[ {\frac{4}{{(1 - 2\upsilon_{p} )}}\frac{{A_{p} }}{{J_{p} }} - \frac{{\rho_{p} }}{{\mu^{p} }}\omega^{2} } \right]}};\) \(\tilde{p}\) represents the external load amplitude of the pile top.
