Analysis of Non-linear Vibrations of a Fractionally Damped Cylindrical Shell Under the Conditions of Combinational Internal Resonance

Abstract

Non-linear damped vibrations of a cylindrical shell subjected to the different conditions of the combinational internal resonance are investigated. Its viscous properties are described by Riemann-Liouville fractional derivative. The displacement functions are determined in terms of eigenfunctions of linear vibrations. The procedure resulting in decoupling linear parts of equations is proposed with the further utilization of the method of multiple time scales for solving nonlinear governing equations of motion, in so doing the amplitude functions are expanded into power series in terms of the small parameter and depend on different time scales. It is shown that the phenomenon of internal resonance can be very critical, since in a circular cylindrical shell the internal additive and difference combinational resonances are always present. The influence of viscosity on the energy exchange mechanism is analyzed. It is shown that each mode is characterized by its damping coefficient connected with the natural frequency by the exponential relationship with a negative fractional exponent.

Appendices

Appendix A

$$ \displaystyle\begin{array}{rcl} a_{1\mathit{mn}}^{m_{1}n_{1}m_{2}n_{2} }\! =\!\int _{ 0}^{1}\!\int _{ 0}^{2\pi }\sin \pi m_{ 1}x\sin n_{1}\varphi \cos \pi m_{2}x\sin n_{2}\varphi \cos \pi \mathit{mx}\sin n\varphi \;\mathit{dxd}\varphi,& & {}\\ a_{2\mathit{mn}}^{m_{1}n_{1}m_{2}n_{2} }\! =\!\int _{ 0}^{1}\!\int _{ 0}^{2\pi }\cos \pi m_{ 1}x\cos n_{1}\varphi \sin \pi m_{2}x\cos n_{2}\varphi \cos \pi \mathit{mx}\sin n\varphi \;\mathit{dxd}\varphi,& & {}\\ a_{3\mathit{mn}}^{m_{1}n_{1}m_{2}n_{2} }\! =\!\int _{ 0}^{1}\!\int _{ 0}^{2\pi }\sin \pi m_{ 1}x\cos n_{1}\varphi \sin \pi m_{2}x\sin n_{2}\varphi \sin \pi \mathit{mx}\cos n\varphi \;\mathit{dxd}\varphi,& & {}\\ a_{4\mathit{mn}}^{m_{1}n_{1}m_{2}n_{2} }\! =\!\int _{ 0}^{1}\!\int _{ 0}^{2\pi }\cos \pi m_{ 1}x\sin n_{1}\varphi \cos \pi m_{2}x\cos n_{2}\varphi \sin \pi \mathit{mx}\cos n\varphi \;\mathit{dxd}\varphi,& & {}\\ a_{5\mathit{mn}}^{m_{1}n_{1}m_{2}n_{2} }\! =\!\int _{ 0}^{1}\!\int _{ 0}^{2\pi }\sin \pi m_{ 1}x\sin n_{1}\varphi \sin \pi m_{2}x\sin n_{2}\varphi \sin \pi \mathit{mx}\sin n\varphi \;\mathit{dxd}\varphi,& & {}\\ a_{6\mathit{mn}}^{m_{1}n_{1}m_{2}n_{2} }\! =\!\int _{ 0}^{1}\!\int _{ 0}^{2\pi }\cos \pi m_{ 1}x\cos n_{1}\varphi \cos \pi m_{2}x\cos n_{2}\varphi \sin \pi \mathit{mx}\sin n\varphi \;\mathit{dxd}\varphi,& & {}\\ a_{7\mathit{mn}}^{m_{1}n_{1}m_{2}n_{2} }\! =\!\int _{ 0}^{1}\!\int _{ 0}^{2\pi }\cos \pi m_{ 1}x\sin n_{1}\varphi \cos \pi m_{2}x\sin n_{2}\varphi \sin \pi \mathit{mx}\sin n\varphi \;\mathit{dxd}\varphi,& & {}\\ a_{8\mathit{mn}}^{m_{1}n_{1}m_{2}n_{2} }\! =\!\int _{ 0}^{1}\!\int _{ 0}^{2\pi }\sin \pi m_{ 1}x\cos n_{1}\varphi \sin \pi m_{2}x\cos n_{2}\varphi \sin \pi \mathit{mx}\sin n\varphi \;\mathit{dxd}\varphi,& & {}\\ a_{9\mathit{mn}}^{m_{1}n_{1}m_{2}n_{2} }\! =\!\int _{ 0}^{1}\!\int _{ 0}^{2\pi }\cos \pi m_{ 1}x\sin n_{1}\varphi \cos \pi m_{2}x\sin n_{2}\varphi \sin \pi \mathit{mx}\sin n\varphi \;\mathit{dxd}\varphi,& & {}\\ a_{10\mathit{mn}}^{m_{1}n_{1}m_{2}n_{2} }\! =\!\int _{ 0}^{1}\!\int _{ 0}^{2\pi }\sin \pi m_{ 1}x\cos n_{1}\varphi \sin \pi m_{2}x\cos n_{2}\varphi \sin \pi \mathit{mx}\sin n\varphi \;\mathit{dxd}\varphi \;.& & {}\\ \end{array} $$

The above integrals could be easily calculated using the following formulas [36]:

$$\displaystyle\begin{array}{rcl} & & \sin \alpha \sin \beta \sin \gamma = \frac{1} {4}{\Bigl [\sin (\alpha +\beta -\gamma ) +\sin (\beta +\gamma -\alpha ) +\sin (\gamma +\alpha -\beta ) -\sin (\alpha +\beta +\gamma )\Bigr ]}, {}\\ & & \sin \alpha \cos \beta \cos \gamma = \frac{1} {4}{\Bigl [\sin (\alpha +\beta -\gamma ) -\sin (\beta +\gamma -\alpha ) +\sin (\gamma +\alpha -\beta ) -\sin (\alpha +\beta +\gamma )\Bigr ]}, {}\\ & & \sin \alpha \sin \beta \cos \gamma = \frac{1} {4}{\Bigl [ -\cos (\alpha +\beta -\gamma ) +\cos (\beta +\gamma -\alpha ) +\cos (\gamma +\alpha -\beta ) +\cos (\alpha +\beta +\gamma )\Bigr ]}, {}\\ & & \cos \alpha \cos \beta \cos \gamma = \frac{1} {4}{\Bigl [\cos (\alpha +\beta -\gamma ) +\cos (\beta +\gamma -\alpha ) +\cos (\gamma +\alpha -\beta ) +\cos (\alpha +\beta +\gamma )\Bigr ]}. {}\\ \end{array}$$

Appendix B

$$\displaystyle{k_{1} = \frac{a_{11}^{I}} {3\varOmega _{1}^{2}},\quad k_{2} = -\frac{a_{11}^{I}} {\varOmega _{1}^{2}},\quad k_{3} = \frac{a_{22}^{I}} {4\varOmega _{2}^{2} -\varOmega _{1}^{2}},\quad k_{4} = -\,\frac{a_{22}^{I}} {\varOmega _{1}^{2}},\quad k_{5} = \frac{a_{33}^{I}} {4\varOmega _{3}^{2} -\varOmega _{1}^{2}},}$$

$$\displaystyle{k_{6} = -\,\frac{a_{33}^{I}} {\varOmega _{1}^{2}},\quad k_{7} = \frac{a_{12}^{I}} {(\varOmega _{1} +\varOmega _{2})^{2} -\varOmega _{1}^{2}},\quad k_{8} = \frac{a_{12}^{I}} {(\varOmega _{1} -\varOmega _{2})^{2} -\varOmega _{1}^{2}},}$$

$$\displaystyle{k_{9} = \frac{a_{13}^{I}} {(\varOmega _{1} +\varOmega _{3})^{2} -\varOmega _{1}^{2}},\quad k_{10} = \frac{a_{13}^{I}} {(\varOmega _{1} -\varOmega _{3})^{2} -\varOmega _{1}^{2}},}$$

$$\displaystyle{k_{11} = \frac{a_{23}^{I}} {(\varOmega _{2} +\varOmega _{3})^{2} -\varOmega _{1}^{2}},\quad k_{12} = \frac{a_{23}^{I}} {(\varOmega _{2} -\varOmega _{3})^{2} -\varOmega _{1}^{2}}\;;}$$

$$\displaystyle{g_{1} = \frac{a_{11}^{\mathit{II}}} {4\varOmega _{1}^{2} -\varOmega _{2}^{2}},\quad g_{2} = -\frac{a_{11}^{\mathit{II}}} {\varOmega _{2}^{2}},\quad g_{3} = \frac{a_{22}^{\mathit{II}}} {3\varOmega _{2}^{2}},\quad g_{4} = -\,\frac{a_{22}^{\mathit{II}}} {\varOmega _{2}^{2}},\quad g_{5} = \frac{a_{33}^{\mathit{II}}} {4\varOmega _{3}^{2} -\varOmega _{2}^{2}},}$$

$$\displaystyle{g_{6} = -\,\frac{a_{33}^{\mathit{II}}} {\varOmega _{2}^{2}},\quad g_{7} = \frac{a_{12}^{\mathit{II}}} {(\varOmega _{1} +\varOmega _{2})^{2} -\varOmega _{2}^{2}},\quad g_{8} = \frac{a_{12}^{\mathit{II}}} {(\varOmega _{1} -\varOmega _{2})^{2} -\varOmega _{2}^{2}},}$$

$$\displaystyle{g_{9} = \frac{a_{13}^{\mathit{II}}} {(\varOmega _{1} +\varOmega _{3})^{2} -\varOmega _{2}^{2}}\;,\quad g_{10} = \frac{a_{13}^{\mathit{II}}} {(\varOmega _{1} -\varOmega _{3})^{2} -\varOmega _{2}^{2}},}$$

$$\displaystyle{g_{11} = \frac{a_{23}^{\mathit{II}}} {(\varOmega _{2} +\varOmega _{3})^{2} -\varOmega _{2}^{2}},\quad g_{12} = \frac{a_{23}^{\mathit{II}}} {(\varOmega _{2} -\varOmega _{3})^{2} -\varOmega _{2}^{2}}\;;}$$

$$\displaystyle{q_{1} = \frac{a_{11}^{\mathit{III}}} {4\varOmega _{3}^{2} -\varOmega _{1}^{2}},\quad q_{2} = -\frac{a_{11}^{\mathit{III}}} {\varOmega _{3}^{2}},\quad q_{3} = \frac{a_{22}^{\mathit{III}}} {4\varOmega _{2}^{2} -\varOmega _{3}^{2}},\quad q_{4} = -\frac{a_{22}^{\mathit{III}}} {\varOmega _{3}^{2}},}$$

$$\displaystyle{q_{5} = \frac{a_{33}^{\mathit{III}}} {3\varOmega _{3}^{2}},\quad q_{6} = -\frac{a_{33}^{\mathit{III}}} {\varOmega _{3}^{2}},\quad q_{7} = \frac{a_{12}^{\mathit{III}}} {(\varOmega _{1} +\varOmega _{2})^{2} -\varOmega _{3}^{2}},\quad q_{8} = \frac{a_{12}^{\mathit{III}}} {(\varOmega _{1} -\varOmega _{2})^{2} -\varOmega _{3}^{2}},}$$

$$\displaystyle{q_{9} = \frac{a_{13}^{\mathit{III}}} {(\varOmega _{1} +\varOmega _{3})^{2} -\varOmega _{3}^{2}},\quad q_{10} = \frac{a_{13}^{\mathit{III}}} {(\varOmega _{1} -\varOmega _{3})^{2} -\varOmega _{3}^{2}},}$$

$$\displaystyle{q_{11} = \frac{a_{23}^{\mathit{III}}} {(\varOmega _{2} +\varOmega _{3})^{2} -\varOmega _{3}^{2}},\quad q_{12} = \frac{a_{23}^{\mathit{III}}} {(\varOmega _{2} -\varOmega _{3})^{2} -\varOmega _{3}^{2}}\;;}$$

$$\displaystyle{d_{1}^{J} = 2a_{ 11}^{J}(k_{ 1} + 2k_{2}) + a_{12}^{J}(g_{ 1} + 2g_{2}) + a_{13}^{J}(q_{ 1} + 2q_{2}),}$$

$$\displaystyle{d_{2}^{J} = 4a_{ 11}^{J}k_{ 4}+2a_{22}^{J}(g_{ 7}+g_{8})+a_{12}^{J}(2g_{ 4}+k_{7}+k_{8})+2a_{13}^{J}q_{ 4}+a_{23}^{J}(q_{ 7}+q_{8}),}$$

$$\displaystyle{d_{3}^{J} = 4a_{ 11}^{J}k_{ 6}+2a_{33}^{J}(q_{ 9}+q_{10})+2a_{12}^{J}g_{ 6}+a_{13}^{J}(2q_{ 6}+k_{9}+k_{10})+a_{23}^{J}(g_{ 9}+g_{10}),}$$

$$\displaystyle{d_{4}^{J} = 2a_{ 22}^{J}(g_{ 3} + 2g_{4}) + a_{12}^{J}(k_{ 3} + 2k_{4}) + a_{23}^{J}(q_{ 3} + 2q_{4}),}$$

$$\displaystyle{d_{5}^{J} = 2a_{ 11}^{J}(k_{ 7}+k_{8})+4a_{22}^{J}g_{ 2}+a_{12}^{J}(2k_{ 2}+g_{7}+g_{8})+a_{12}^{J}(q_{ 7}+q_{8})+2a_{23}^{J}q_{ 2},}$$

$$\displaystyle{d_{6}^{J} = 4a_{ 22}^{J}g_{ 6}+2a_{33}^{J}(q_{ 11}+q_{12})+2a_{12}^{J}k_{ 6}+a_{13}^{J}(k_{ 11}+k_{12})+a_{23}^{J}(2q_{ 6}+g_{11}+g_{12}),}$$

$$\displaystyle{d_{7}^{J} = 2a_{ 33}^{J}(q_{ 5} + 2q_{6}) + a_{13}^{J}(k_{ 5} + 2k_{6}) + a_{23}^{J}(g_{ 5} + 2g_{6}),}$$

$$\displaystyle{d_{8}^{J} = 2a_{ 11}^{J}(k_{ 9}+k_{10})+4a_{33}^{J}q_{ 2}+a_{12}^{J}(g_{ 9}+g_{10})+a_{13}^{J}(2k_{ 2}+q_{9}+q_{10})+2a_{23}^{J}g_{ 2},}$$

$$\displaystyle{d_{9}^{J} = 2a_{ 22}^{J}(g_{ 11}+g_{12})+4a_{33}^{J}q_{ 4}+a_{12}^{J}(k_{ 11}+k_{12})+2a_{13}^{J}k_{ 4}+a_{23}^{J}(2g_{ 4}+q_{11}+q_{12}),}$$

$$\displaystyle{d_{10}^{J} = 2a_{ 11}^{J}k_{ 1} + a_{12}^{J}g_{ 1} + a_{13}^{J}q_{ 1},\quad d_{11}^{J} = 2a_{ 22}^{J}g_{ 3} + a_{12}^{J}k_{ 3} + a_{23}^{J}q_{ 3},}$$

$$\displaystyle{d_{12}^{J} = 2a_{ 33}^{J}q_{ 5} + a_{13}^{J}k_{ 5} + a_{23}^{J}g_{ 5}\;;}$$

$$\displaystyle{e_{1}^{J} = 2a_{ 11}^{J}k_{ 11}+2a_{22}^{J}g_{ 9}+2a_{33}^{J}q_{ 7}+a_{12}^{J}(g_{ 11}+k_{9})+a_{13}^{J}(q_{ 11}+k_{7})+a_{23}^{J}(q_{ 9}+g_{7}),}$$

$$\displaystyle{e_{2}^{J} = 2a_{ 11}^{J}k_{ 12}+2a_{22}^{J}g_{ 10}+2a_{33}^{J}q_{ 7}+a_{12}^{J}(g_{ 12}+k_{10})+a_{13}^{J}(q_{ 12}+k_{7})+a_{23}^{J}(q_{ 10}+g_{7}),}$$

$$\displaystyle{e_{3}^{J} = 2a_{ 11}^{J}k_{ 12}+2a_{22}^{J}g_{ 9}+2a_{33}^{J}q_{ 8}+a_{12}^{J}(g_{ 12}+k_{9})+a_{13}^{J}(q_{ 12}+k_{8})+a_{23}^{J}(q_{ 9}+g_{8}),}$$

$$\displaystyle{e_{4}^{J} = 2a_{ 11}^{J}k_{ 11}+2a_{22}^{J}g_{ 10}+2a_{33}^{J}q_{ 8}+a_{12}^{J}(g_{ 11}+k_{10})+a_{13}^{J}(q_{ 11}+k_{8})+a_{23}^{J}(q_{ 9}+g_{8})\;;}$$

$$\displaystyle{c_{1}^{J} = 2a_{ 11}^{J}k_{ 3} + 2a_{22}^{J}g_{ 7} + a_{12}^{J}(g_{ 3} + k_{7}) + a_{13}^{J}q_{ 3} + a_{23}^{J}q_{ 7},}$$

$$\displaystyle{c_{2}^{J} = 2a_{ 11}^{J}k_{ 5} + 2a_{33}^{J}q_{ 9} + a_{12}^{J}g_{ 5} + a_{13}^{J}(q_{ 5} + k_{9}) + a_{23}^{J}g_{ 9},}$$

$$\displaystyle{c_{3}^{J} = 2a_{ 11}^{J}k_{ 7} + 2a_{22}^{J}g_{ 1} + a_{12}^{J}(g_{ 7} + k_{1}) + a_{13}^{J}q_{ 7} + a_{23}^{J}q_{ 1},}$$

$$\displaystyle{c_{4}^{J} = 2a_{ 11}^{J}k_{ 9} + 2a_{33}^{J}q_{ 1} + a_{12}^{J}g_{ 9} + a_{13}^{J}(q_{ 9} + k_{1}) + a_{23}^{J}g_{ 1},}$$

$$\displaystyle{c_{5}^{J} = 2a_{ 22}^{J}g_{ 5} + 2a_{33}^{J}q_{ 11} + a_{12}^{J}k_{ 5} + a_{13}^{J}k_{ 11} + a_{23}^{J}(q_{ 5} + g_{11}),}$$

$$\displaystyle{c_{6}^{J} = 2a_{ 22}^{J}g_{ 11} + 2a_{33}^{J}q_{ 3} + a_{12}^{J}k_{ 11} + a_{13}^{J}k_{ 3} + a_{23}^{J}(q_{ 11} + g_{3}),}$$

$$\displaystyle{c_{7}^{J} = 2a_{ 11}^{J}k_{ 8} + 2a_{22}^{J}g_{ 1} + a_{12}^{J}(g_{ 8} + k_{1}) + a_{13}^{J}q_{ 8} + a_{23}^{J}q_{ 1},}$$

$$\displaystyle{c_{8}^{J} = 2a_{ 11}^{J}k_{ 10} + 2a_{33}^{J}q_{ 1} + a_{12}^{J}g_{ 10} + a_{13}^{J}(q_{ 10} + k_{1}) + a_{23}^{J}g_{ 1},}$$

$$\displaystyle{c_{9}^{J} = 2a_{ 11}^{J}k_{ 3} + 2a_{22}^{J}g_{ 8} + a_{12}^{J}(g_{ 3} + k_{8}) + a_{13}^{J}q_{ 3} + a_{23}^{J}q_{ 8},}$$

$$\displaystyle{c_{10}^{J} = 2a_{ 11}^{J}k_{ 5} + 2a_{33}^{J}q_{ 10} + a_{12}^{J}g_{ 5} + a_{13}^{J}(q_{ 5} + k_{10}) + a_{23}^{J}g_{ 10},}$$

$$\displaystyle{c_{11}^{J} = 2a_{ 22}^{J}g_{ 12} + 2a_{33}^{J}q_{ 3} + a_{12}^{J}k_{ 12} + a_{13}^{J}k_{ 3} + a_{23}^{J}(q_{ 12} + g_{3}),}$$

$$\displaystyle{c_{12}^{J} = 2a_{ 22}^{J}g_{ 5} + 2a_{33}^{J}q_{ 12} + a_{12}^{J}k_{ 5} + a_{13}^{J}k_{ 12} + a_{23}^{J}(q_{ 5} + g_{12}).}$$