Combination resonances of spinning composite shafts considering geometric nonlinearity

Abstract

In this paper, the combination resonance of a spinning composite shaft with geometric nonlinearity is studied by the method of harmonic balance. The full equations of motion containing the flexural–flexural–extensional–torsional coupling are employed for the analysis. The equations were discretized by both one and two modes, so two different forms of combination resonances can be analyzed. The shear deformation is neglected due to the shaft slenderness, whereas the rotary inertia and the gyroscopic effects are considered. The effects of different parameters such as external damping and the eccentricity on the response bifurcation of the shaft are investigated. The effect of the fiber orientation angle was also investigated. The results obtained for the composite shaft are compared to those of its metallic counterpart. It is shown that two geometrically identical shafts, one metallic and one composite, have different behaviors under the condition of combination resonance. It is observed that the vibration of the composite shaft occurs with smaller amplitude. This confirms the superiority of the composite shafts over metallic ones. Finally, the results were validated by the numerical simulations and there was a good agreement between the results.

Appendix

• 1. Stress–strain relationship for a composite shaft in the Cartesian and cylindrical coordinate system is [20]

  $$ \left\{ \begin{aligned} \sigma_{11} \hfill \\ \sigma_{22} \hfill \\ \sigma_{33} \hfill \\ \tau_{23} \hfill \\ \tau_{31} \hfill \\ \tau_{12} \hfill \\ \end{aligned} \right\} = \left[ {\begin{array}{*{20}c} {Q_{11} } & {Q_{12} } & {Q_{13} } & 0 & 0 & 0 \\ {Q_{12} } & {Q_{22} } & {Q_{23} } & 0 & 0 & 0 \\ {Q_{13} } & {Q_{23} } & {Q_{33} } & 0 & 0 & 0 \\ 0 & 0 & 0 & {Q_{44} } & 0 & 0 \\ 0 & 0 & 0 & 0 & {Q_{55} } & 0 \\ 0 & 0 & 0 & 0 & 0 & {Q_{66} } \\ \end{array} } \right]\left\{ \begin{aligned} \varepsilon_{11} \hfill \\ \varepsilon_{22} \hfill \\ \varepsilon_{33} \hfill \\ \gamma_{23} \hfill \\ \gamma_{31} \hfill \\ \gamma_{12} \hfill \\ \end{aligned} \right\} $$
  $$ \left\{ \begin{aligned} \sigma_{xx} \hfill \\ \sigma_{\theta \theta } \hfill \\ \sigma_{rr} \hfill \\ \tau_{r\theta } \hfill \\ \tau_{xr} \hfill \\ \tau_{x\theta } \hfill \\ \end{aligned} \right\} = \left[ {\begin{array}{*{20}c} {\overline{Q}_{11} } & {\overline{Q}_{12} } & {\overline{Q}_{13} } & 0 & 0 & {\overline{Q}_{16} } \\ {\overline{Q}_{12} } & {\overline{Q}_{22} } & {\overline{Q}_{23} } & 0 & 0 & {\overline{Q}_{26} } \\ {\overline{Q}_{13} } & {\overline{Q}_{23} } & {\overline{Q}_{33} } & 0 & 0 & {\overline{Q}_{36} } \\ 0 & 0 & 0 & {\overline{Q}_{44} } & {\overline{Q}_{45} } & 0 \\ 0 & 0 & 0 & {\overline{Q}_{45} } & {\overline{Q}_{55} } & 0 \\ {\overline{Q}_{16} } & {\overline{Q}_{26} } & {\overline{Q}_{36} } & 0 & 0 & {\overline{Q}_{66} } \\ \end{array} } \right]\left\{ \begin{aligned} \varepsilon_{xx} \hfill \\ \varepsilon_{\theta \theta } \hfill \\ \varepsilon_{rr} \hfill \\ \gamma_{r\theta } \hfill \\ \gamma_{xr} \hfill \\ \gamma_{x\theta } \hfill \\ \end{aligned} \right\} $$

  where σ and τ are normal and shear stresses, respectively. The subscript defines the coordinate system in which they are presented, and ε and γ are also normal and shear strains. They are related by

  $$ \left[ {\bar{Q}} \right] = \left[ T \right]^{ - 1} \left[ Q \right]\left[ T \right]^{ - T} $$

  In the above equations

  $$ \begin{aligned} & \left[ T \right] = \left[ {\begin{array}{*{20}c} {m^{2} } & {n^{2} } & 0 & 0 & 0 & {2mn} \\ {n^{2} } & {m^{2} } & 0 & 0 & 0 & { - 2mn} \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & m & { - n} & 0 \\ 0 & 0 & 0 & n & m & 0 \\ { - mn} & {mn} & 0 & 0 & 0 & {m^{2} - n^{2} } \\ \end{array} } \right] \\ & m = \cos \eta ,\quad n = \sin \eta \\ \end{aligned} $$

  and η is the fiber orientation angle in each composite layer. In addition,

  $$ \begin{aligned} & Q_{11} = \frac{{E_{11} }}{{1 - \frac{{\nu_{12}^{2} E_{22} }}{{E_{11} }}}},\quad Q_{12} = \frac{{E_{22} \nu_{12} }}{{1 - \frac{{\nu_{12}^{2} E_{22} }}{{E_{11} }}}},\quad Q_{22} = \frac{{E_{22} }}{{1 - \frac{{\nu_{12}^{2} E_{22} }}{{E_{11} }}}},\quad\\& Q_{44} = G_{23} ,\quad Q_{55} = G_{13} ,\quad Q_{66} = G_{12} , \\ & Q_{13} = Q_{23} = Q_{33} = 0 \\ \end{aligned}$$

• 2. Natural frequencies of the composite shaft for one-mode discretization are [20]

  $$ \begin{aligned} & \omega_{\text{f}} = - \frac{1}{2}I\sqrt { - 2{\mkern 1mu} \pi^{4} \sqrt {I_{p}^{2}\Omega ^{2} \left( {I_{\text{p}}^{2}\Omega ^{2} + 4{\mkern 1mu} D_{11} } \right)} + \left( { - 2{\mkern 1mu} I_{\text{p}}^{2}\Omega ^{2} - 4{\mkern 1mu} D_{11} } \right)\pi^{4} } \\ & \omega_{\text{b}} = - \frac{1}{2}I\sqrt {2{\mkern 1mu} \pi^{4} \sqrt {I_{\text{p}}^{2}\Omega ^{2} \left( {I_{\text{p}}^{2}\Omega ^{2} + 4{\mkern 1mu} D_{11} } \right)} + \left( { - 2{\mkern 1mu} I_{\text{p}}^{2}\Omega ^{2} - 4{\mkern 1mu} D_{11} } \right)\pi^{4} } \\ & \omega_{u} = \pi \sqrt {A_{11} } \\ & \omega_{\varphi } = \pi \sqrt {D_{66} } \\ \end{aligned} $$

• 3. Equations of motion obtained for two-mode discretization are [20]

  $$ \begin{aligned} & \pi^{4} A_{11} [ - 8{\mkern 1mu} V_{1} \left( t \right)V_{2} \left( t \right)U_{2} \left( t \right) - 8{\mkern 1mu} W_{1} \left( t \right)W_{2} \left( t \right)U_{2} \left( t \right) - \frac{3}{2}{\mkern 1mu} \left( {V_{1} \left( t \right)} \right)^{2} U_{1} \left( t \right) \\ & \quad - 4{\mkern 1mu} \left( {V_{2} \left( t \right)} \right)^{2} U_{1} \left( t \right) - \frac{3}{2}{\mkern 1mu} \left( {W_{1} \left( t \right)} \right)^{2} U_{1} \left( t \right) - 4{\mkern 1mu} \left( {W_{2} \left( t \right)} \right)^{2} U_{1} \left( t \right)]\varepsilon^{3} \\ & \quad +\, \sqrt 2 \pi^{3} A_{11} \left( {V_{1} \left( t \right)V_{2} \left( t \right) + W_{1} \left( t \right)W_{2} \left( t \right)} \right)\varepsilon^{2} - \left[ {\frac{{32{\mkern 1mu} }}{3}\pi B_{16} \varphi_{2} \left( t \right) + \pi^{2} A_{11} U_{1} \left( t \right) + \frac{{{\text{d}}^{2} }}{{{\text{d}}t^{2} }}U_{1} \left( t \right)} \right]\varepsilon = 0 \\ & \pi^{4} {\kern 1pt} A_{11} [ - 8V_{1} \left( t \right)V_{2} \left( t \right)U_{1} \left( t \right) - 8{\kern 1pt} W_{1} \left( t \right)W_{2} \left( t \right)U_{1} \left( t \right) - 4{\kern 1pt} \left( {V_{1} \left( t \right)} \right)^{2} U_{2} \left( t \right) \\ & \quad - 24{\kern 1pt} \left( {V_{2} \left( t \right)} \right)^{2} U_{2} \left( t \right) - 4{\kern 1pt} \left( {W_{1} \left( t \right)} \right)^{2} U_{2} \left( t \right) - 24{\kern 1pt} \left( {W_{2} \left( t \right)} \right)^{2} U_{2} \left( t \right)]\varepsilon^{3} \\ & \quad +\, \frac{1}{2}\sqrt 2 \pi^{3} A_{11} \left( {\left( {V_{1} \left( t \right)} \right)^{2} + {\kern 1pt} \left( {W_{1} \left( t \right)} \right)^{2} } \right)\varepsilon^{2} + \left[ {4{\kern 1pt} \pi^{2} A_{11} U_{2} \left( t \right) + \frac{16}{3}{\kern 1pt} \pi B_{16} \varphi_{1} \left( t \right) + \frac{{{\text{d}}^{2} }}{{{\text{d}}t^{2} }}U_{2} \left( t \right)} \right]\varepsilon = 0 \\ & \quad \left[ {\frac{3}{4}{\kern 1pt} \left( {V_{1} \left( t \right)} \right)^{3} - \frac{3}{2}V_{1} \left( t \right)\left( {U_{1} \left( t \right)} \right)^{2} - 8{\kern 1pt} V_{2} \left( t \right)U_{2} \left( t \right)U_{1} \left( t \right) + 6{\kern 1pt} V_{1} \left( t \right)\left( {V_{2} \left( t \right)} \right)^{2} } \right. \\ & \quad \left. { + 4{\kern 1pt} W_{2} \left( t \right)V_{2} \left( t \right)W_{1} \left( t \right) + 2{\kern 1pt} \left( {W_{2} \left( t \right)} \right)^{2} V_{1} \left( t \right) + \frac{3}{4}{\kern 1pt} \left( {W_{1} \left( t \right)} \right)^{2} V_{1} \left( t \right) - 4{\kern 1pt} V_{1} \left( t \right)\left( {U_{2} \left( t \right)} \right)^{2} } \right]\pi^{4} A_{11} \varepsilon^{3} \\ & \quad +\, \left( {\frac{\text{d}}{{{\text{d}}t}}V_{1} \left( t \right)} \right)c\varepsilon^{3} + \pi^{3} \sqrt 2 A_{11} \varepsilon^{2} \left( {U_{1} \left( t \right)V_{2} \left( t \right) + V_{1} \left( t \right)U_{2} \left( t \right)} \right) + \left[ {D_{11} \pi^{4} V_{1} \left( t \right) + I_{p} \varOmega \pi^{2} {\kern 1pt} \left( {\frac{\text{d}}{{{\text{d}}t}}W_{1} \left( t \right)} \right)} \right. \\ & \quad +\, \left. {\Omega ^{2} \sin \left( {\Omega {\kern 1pt} t} \right)e_{1,1} -\Omega ^{2} \cos \left( {\Omega {\kern 1pt} t} \right)e_{2,1} + \frac{{{\text{d}}^{2} }}{{{\text{d}}t^{2} }}V_{1} \left( t \right)} \right]\varepsilon = 0 \\ & \quad \left[ { - 4{\kern 1pt} \left( {U_{1} \left( t \right)} \right)^{2} V_{2} \left( t \right) - 8{\kern 1pt} U_{1} \left( t \right)V_{1} \left( t \right)U_{2} \left( t \right) + 6{\kern 1pt} \left( {V_{1} \left( t \right)} \right)^{2} V_{2} \left( t \right) + 4{\kern 1pt} V_{1} \left( t \right)W_{1} \left( t \right)W_{2} \left( t \right)} \right. \\ & \quad \left. { + 12{\kern 1pt} \left( {V_{2} \left( t \right)} \right)^{3} + 2{\kern 1pt} V_{2} \left( t \right)\left( {W_{1} \left( t \right)} \right)^{2} + 12{\kern 1pt} V_{2} \left( t \right)\left( {W_{2} \left( t \right)} \right)^{2} - 24{\kern 1pt} V_{2} \left( t \right)\left( {U_{2} \left( t \right)} \right)^{2} } \right]\pi^{4} A_{11} \varepsilon^{3} \\ & \quad +\, \varepsilon^{3} c{\kern 1pt} \frac{\text{d}}{{{\text{d}}t}}V_{2} \left( t \right) + \varepsilon^{2} \sqrt 2 \pi^{3} A_{11} V_{1} \left( t \right)U_{1} \left( t \right) + \left[ {16{\kern 1pt} D_{11} \pi^{4} V_{2} \left( t \right) + 4{\kern 1pt} I_{\text{p}}\Omega {\kern 1pt} \pi^{2} \left( {\frac{\text{d}}{{{\text{d}}t}}W_{2} \left( t \right)} \right)} \right. \\ & \quad \left. { +\Omega ^{2} \sin \left( {\Omega {\kern 1pt} t} \right)e_{1,2} -\Omega ^{2} \cos \left( {\Omega {\kern 1pt} t} \right)e_{2,2} + \frac{{{\text{d}}^{2} }}{{{\text{d}}t^{2} }}V_{2} \left( t \right)} \right]\varepsilon = 0 \\ & \quad \left[ {4{\kern 1pt} W_{2} \left( t \right)V_{2} \left( t \right)V_{1} \left( t \right) - 8{\kern 1pt} W_{2} \left( t \right)U_{2} \left( t \right)U_{1} \left( t \right) + \frac{3}{4}{\kern 1pt} {\kern 1pt} \left( {W_{1} \left( t \right)} \right)^{3} + \frac{3}{4}{\kern 1pt} W_{1} \left( t \right)\left( {V_{1} \left( t \right)} \right)^{2} } \right. \\ & \quad \left. { + 2{\kern 1pt} W_{1} \left( t \right)\left( {V_{2} \left( t \right)} \right)^{2} + 6{\kern 1pt} W_{1} \left( t \right)\left( {W_{2} \left( t \right)} \right)^{2} - 4{\kern 1pt} W_{1} \left( t \right)\left( {U_{2} \left( t \right)} \right)^{2} - \frac{3}{2}{\kern 1pt} W_{1} \left( t \right)\left( {U_{1} \left( t \right)} \right)^{2} } \right]\pi^{4} A_{11} \varepsilon^{3} \\ & \quad +\, \varepsilon^{3} c{\kern 1pt} \frac{\text{d}}{{{\text{d}}t}}W_{1} \left( t \right) + \varepsilon^{2} A_{11} \pi^{3} \sqrt 2 \left( {W_{2} \left( t \right)U_{1} \left( t \right) + U_{2} \left( t \right)W_{1} \left( t \right)} \right) + \left[ {D_{11} \pi^{4} W_{1} \left( t \right)} \right. \\ & \quad \left. { - I_{\text{p}}\Omega \pi^{2} {\kern 1pt} \left( {\frac{\text{d}}{{{\text{d}}t}}V_{1} \left( t \right)} \right) -\Omega ^{2} \cos \left( {\Omega {\kern 1pt} t} \right)e_{1,1} -\Omega ^{2} \sin \left( {\Omega {\kern 1pt} t} \right)e_{1,1} + \frac{{{\text{d}}^{2} }}{{{\text{d}}t^{2} }}W_{1} \left( t \right)} \right]\varepsilon = 0 \\ & \quad \left[ {12{\kern 1pt} \left( {W_{2} \left( t \right)} \right)^{3} + 12{\kern 1pt} W_{2} \left( t \right)\left( {V_{2} \left( t \right)} \right)^{2} + 2{\kern 1pt} W_{2} \left( t \right)\left( {V_{1} \left( t \right)} \right)^{2} - 24{\kern 1pt} W_{2} \left( t \right)\left( {U_{2} \left( t \right)} \right)^{2} } \right. \\ & \quad \left. { - 4{\kern 1pt} W_{2} \left( t \right)\left( {U_{1} \left( t \right)} \right)^{2} + 6{\kern 1pt} W_{2} \left( t \right)\left( {W_{1} \left( t \right)} \right)^{2} + 4{\kern 1pt} V_{2} \left( t \right)V_{1} \left( t \right)W_{1} \left( t \right) - 8{\kern 1pt} U_{2} \left( t \right)U_{1} \left( t \right)W_{1} \left( t \right)} \right]\pi^{4} A_{11} \varepsilon^{3} \\ & \quad +\, \varepsilon^{3} c{\kern 1pt} \frac{\text{d}}{{{\text{d}}t}}W_{2} \left( t \right) + \varepsilon^{2} \sqrt 2 \pi^{3} A_{11} W_{1} \left( t \right)U_{1} \left( t \right) + \left[ {16{\kern 1pt} \pi^{4} D_{11} W_{2} \left( t \right) - 4{\kern 1pt} \pi^{2} I_{\text{p}}\Omega {\kern 1pt} \left( {\frac{\text{d}}{{{\text{d}}t}}V_{2} \left( t \right)} \right)} \right. \\ & \quad \left. { -\Omega ^{2} \cos \left( {\Omega {\kern 1pt} t} \right)e_{1,2} -\Omega ^{2} \sin \left( {\Omega {\kern 1pt} t} \right)e_{2,2} + \frac{{{\text{d}}^{2} }}{{{\text{d}}t^{2} }}W_{2} \left( t \right)} \right]\varepsilon = 0 \\ & \quad \left( { - 8{\kern 1pt} U_{2} \left( t \right)V_{1} \left( t \right)W_{2} \left( t \right) - \frac{3}{2}{\kern 1pt} U_{1} \left( t \right)V_{1} \left( t \right)W_{1} \left( t \right) - 4{\kern 1pt} U_{1} \left( t \right)V_{2} \left( t \right)W_{2} \left( t \right)} \right)\pi^{5} D_{66} \varepsilon^{3} \\ & \quad +\, \pi^{4} \sqrt 2 \varepsilon^{2} D_{66} \left( { - W_{2} \left( t \right)V_{1} \left( t \right) + 2{\kern 1pt} V_{2} \left( t \right)W_{1} \left( t \right)} \right) + \left[ {\pi^{2} D_{66} \varphi_{1} \left( t \right) + \frac{{{\text{d}}^{2} }}{{{\text{d}}t^{2} }}\varphi_{1} \left( t \right) + \frac{{64{\kern 1pt} }}{3}B_{16} U_{2} \left( t \right)\pi } \right]\varepsilon = 0 \\ & \quad \left( { - 48{\kern 1pt} W_{2} \left( t \right)V_{2} \left( t \right)U_{2} \left( t \right) - 16{\kern 1pt} V_{2} \left( t \right)U_{1} \left( t \right)W_{1} \left( t \right) - 8{\kern 1pt} V_{1} \left( t \right)U_{2} \left( t \right)W_{1} \left( t \right)} \right)\pi^{5} D_{66} \varepsilon^{3} \\ & \quad +\, \pi^{4} \varepsilon^{2} \sqrt 2 D_{66} V_{1} \left( t \right)W_{1} \left( t \right) + \left[ {4{\kern 1pt} \pi^{2} D_{66} \varphi_{2} \left( t \right) - \frac{8}{3}\pi {\kern 1pt} B_{16} U_{1} \left( t \right) + \frac{{{\text{d}}^{2} }}{{{\text{d}}t^{2} }}\varphi_{2} \left( t \right)} \right]\varepsilon = 0 \\ \end{aligned} $$

  where

  $$ \begin{aligned} & e_{1,i} = \int_{0}^{1} {[\sqrt 2 e_{y} (x)\sin (i\pi x)]{\text{d}}x} \\ & e_{2,j} = \int_{0}^{1} {[\sqrt 2 e_{z} (x)\sin (j\pi x)]{\text{d}}x} \\ \end{aligned} $$

• 4. Natural frequencies obtained for two-mode discretization are [20]

  $$ \begin{aligned} & \omega_{f,1} = - \frac{1}{2}i\sqrt { - 2{\mkern 1mu} \sqrt {\pi^{8} I_{\text{p}}^{2}\Omega ^{2} \left( {I_{p}^{2}\Omega ^{2} + 4{\mkern 1mu} D_{11} } \right)} + \left( { - 4{\mkern 1mu} D_{11} - 2{\mkern 1mu} I_{p}^{2}\Omega ^{2} } \right)\pi^{4} } \\ & \omega_{b,1} = - \frac{1}{2}i\sqrt {2{\mkern 1mu} \sqrt {\pi^{8} I_{\text{p}}^{2}\Omega ^{2} \left( {I_{\text{p}}^{2}\Omega ^{2} + 4{\mkern 1mu} D_{11} } \right)} + \left( { - 4{\mkern 1mu} D_{11} - 2{\mkern 1mu} I_{\text{p}}^{2}\Omega ^{2} } \right)\pi^{4} } \\ & \omega_{f,2} = - 2i\sqrt { - 2{\mkern 1mu} \sqrt {\pi^{8} I_{\text{p}}^{2}\Omega ^{2} \left( {I_{\text{p}}^{2}\Omega ^{2} + 4{\mkern 1mu} D_{11} } \right)} + \left( { - 4{\mkern 1mu} D_{11} - 2{\mkern 1mu} I_{\text{p}}^{2}\Omega ^{2} } \right)\pi^{4} } \\ & \omega_{b,2} = - 2i\sqrt {2{\mkern 1mu} \sqrt {\pi^{8} I_{\text{p}}^{2}\Omega ^{2} \left( {I_{\text{p}}^{2}\Omega ^{2} + 4{\mkern 1mu} D_{11} } \right)} + \left( { - 4{\mkern 1mu} D_{11} - 2{\mkern 1mu} I_{\text{p}}^{2}\Omega ^{2} } \right)\pi^{4} } \\ & \omega_{u1} = - \frac{1}{6}i\sqrt { - 6{\mkern 1mu} \sqrt 9 \sqrt {\left( {\left( {A_{11} - 4{\mkern 1mu} D_{66} } \right)^{2} \pi^{2} + \frac{1024}{9}{\mkern 1mu} B_{16}^{2} } \right)\pi^{2} } + \left( { - 72{\mkern 1mu} D_{66} - 18{\mkern 1mu} A_{11} } \right)\pi^{2} } \\ & \omega_{u2} = - \frac{1}{6}i\sqrt { - 6{\mkern 1mu} \sqrt {144} \sqrt {\left( {\left( {A_{11} - \frac{1}{4}{\mkern 1mu} D_{66} } \right)^{2} \pi^{2} + \frac{256}{9}{\mkern 1mu} B_{16}^{2} } \right)\pi^{2} } + \left( { - 18{\mkern 1mu} D_{66} - 72{\mkern 1mu} A_{11} } \right)\pi^{2} } \\ & \omega_{\varphi 1} = - \frac{1}{6}{\mkern 1mu} i\sqrt {6{\mkern 1mu} \sqrt {144} \sqrt {\left( {\left( {A1 - \frac{1}{4}{\mkern 1mu} D_{66} } \right)^{2} \pi^{2} + \frac{256}{9}{\mkern 1mu} B_{16}^{2} } \right)\pi^{2} } + \left( { - 18{\mkern 1mu} D_{66} - 72{\mkern 1mu} A_{11} } \right)\pi^{2} } \\ & \omega_{\varphi 2} = \, - \frac{1}{6}{\mkern 1mu} i\sqrt {6{\mkern 1mu} \sqrt 9 \sqrt {\left( {\left( {A_{11} - 4{\mkern 1mu} D_{66} } \right)^{2} \pi^{2} + \frac{1024}{9}{\mkern 1mu} B_{16}^{2} } \right)\pi^{2} } + \left( { - 72{\mkern 1mu} D_{66} - 18{\mkern 1mu} A_{11} } \right)\pi^{2} } \\ \end{aligned} $$

• 5. Coefficients introduced in Eq. (46) are defined

  $$ \begin{aligned}\Gamma _{1} & = \pi^{4} A_{11} \left( {P_{{{\text{f}},1}} } \right)\left( t \right)\left( {4{\kern 1pt} \cos \left( {\beta_{2} \left( t \right) - \beta_{1} \left( t \right)} \right)P_{1} \left( t \right)^{2} - G_{1} \left( t \right)^{2} - H_{2} \left( t \right)^{2} + \frac{11}{4}{\kern 1pt} P_{1} \left( t \right)^{2} + \frac{3}{4}{\kern 1pt} P_{{{\text{b}},1}} \left( t \right)^{2} + 2{\kern 1pt} P_{{{\text{f}},2}} \left( t \right)^{2} - \frac{3}{8}H_{1} \left( t \right)^{2} - \frac{3}{8}{\kern 1pt} G_{2} \left( t \right)^{2} } \right) \\ & \quad + \pi^{4} A_{11} \left( {\frac{3}{8}{\kern 1pt} P_{{{\text{f}},1}} \left( t \right)^{3} + 2{\kern 1pt} \cos \left( {\beta_{1} \left( t \right) - \psi \left( t \right)} \right)P_{{{\text{f}},2}} \left( t \right)P_{1} \left( t \right)P_{{{\text{b}},1}} \left( t \right) + 2{\kern 1pt} \cos \left( {\beta_{2} \left( t \right) - \psi \left( t \right)} \right)P_{{{\text{f}},2}} \left( t \right)P_{1} \left( t \right)P_{{{\text{b}},1}} \left( t \right)} \right) \\ & \quad + \frac{1}{2}P_{{{\text{f}},1}} \left( t \right)\left( {\pi^{2} I_{\text{p}} {\kern 1pt} \left[ {\omega_{{{\text{f}},1}} \sigma {\kern 1pt} + {\kern 1pt} \omega_{{{\text{f}},1}} \omega_{{{\text{f}},2}} - {\kern 1pt} \omega_{{{\text{f}},1}} \omega_{{{\text{b}},1}} + \omega_{{{\text{f}},1}}^{2} } \right] + {\kern 1pt} D_{11} \pi^{4} - {\kern 1pt} \omega_{{{\text{f}},1}}^{2} } \right) \\\Gamma _{2} & = \pi^{4} A_{11} P_{{{\text{f}},2}} \left( t \right)\left( {4{\kern 1pt} \cos \left( {\beta_{2} \left( t \right) - \beta_{1} \left( t \right)} \right)P_{1} \left( t \right)^{2} + 14{\kern 1pt} P_{1} \left( t \right)^{2} + 2{\kern 1pt} P_{{{\text{f}},1}} \left( t \right)^{2} - 6{\kern 1pt} G_{1} \left( t \right)^{2} - 6{\kern 1pt} H_{2} \left( t \right)^{2} - G_{2} \left( t \right)^{2} - H_{1} \left( t \right)^{2} + 2{\kern 1pt} P_{{{\text{b}},1}} \left( t \right)^{2} } \right) \\ & \quad + \pi^{4} A_{11} \left( {6{\kern 1pt} P_{{{\text{f}},2}} \left( t \right)^{3} + \frac{3}{4}\cos \left( {\beta_{1} \left( t \right) - \psi \left( t \right)} \right)P_{{{\text{f}},1}} \left( t \right)P_{1} \left( t \right)P_{{{\text{b}},1}} \left( t \right) + 2{\kern 1pt} \cos \left( {\beta_{2} \left( t \right) - \psi \left( t \right)} \right)P_{1} \left( t \right)P_{{{\text{f}},1}} \left( t \right)P_{{{\text{b}},1}} \left( t \right)} \right) \\ & \quad + 2\pi^{2} I_{\text{p}} P_{{{\text{f}},2}} \left( t \right)\left( {\omega_{{{\text{f}},2}}^{2} + \omega_{{{\text{f}},1}} \omega_{{{\text{f}},2}} - \omega_{{{\text{f}},2}} \omega_{{{\text{b}},1}} + \omega_{{{\text{f}},2}} \sigma {\kern 1pt} } \right) + P_{{{\text{f}},2}} \left( t \right)\left( {8{\kern 1pt} D_{11} \pi^{4} - \frac{1}{2}{\kern 1pt} \omega_{{{\text{f}},2}}^{2} } \right) \\\Gamma _{3} & = \pi^{4} A_{11} P_{{{\text{b}},1}} \left( t \right)\left( {4{\kern 1pt} \cos \left( {\beta_{2} \left( t \right) - \beta_{1} \left( t \right)} \right)P_{1} \left( t \right)^{2} - G_{1} \left( t \right)^{2} - H_{2} \left( t \right)^{2} + \frac{3}{4}P_{{{\text{f}},1}} \left( t \right)^{2} + \frac{11}{4}{\kern 1pt} P_{1} \left( t \right)^{2} - \frac{3}{8}{\kern 1pt} G_{2} \left( t \right)^{2} + 2{\kern 1pt} P_{{{\text{f}},2}} \left( t \right)^{2} - \frac{3}{8}{\kern 1pt} H_{1} \left( t \right)^{2} } \right) \\ & \quad + \pi^{4} A_{11} \left( {\frac{3}{8}{\kern 1pt} P_{{{\text{b}},1}} \left( t \right)^{3} + 2{\kern 1pt} \cos \left( {\beta_{1} \left( t \right) - \psi \left( t \right)} \right)P_{{{\text{f}},2}} \left( t \right)P_{{{\text{f}},1}} \left( t \right)P_{1} \left( t \right) + 2{\kern 1pt} \cos \left( {\beta_{2} \left( t \right) - \psi \left( t \right)} \right)P_{{{\text{f}},2}} \left( t \right)P_{1} \left( t \right)P_{{{\text{f}},1}} \left( t \right)} \right) \\ & \quad + \frac{1}{2}P_{{{\text{b}},1}} \left( t \right)\left( {\pi^{2} I_{\text{p}} \left[ {\omega_{{{\text{f}},2}} \omega_{{{\text{b}},1}} + {\kern 1pt} \sigma {\kern 1pt} \omega_{{{\text{b}},1}} + \omega_{{{\text{f}},1}} \omega_{{{\text{b}},1}} - {\kern 1pt} \omega_{{{\text{b}},1}}^{2} } \right]{\kern 1pt} + {\kern 1pt} D_{11} \pi^{4} - {\kern 1pt} \omega_{{{\text{b}},1}}^{2} } \right) \\ \end{aligned} $$