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Nonlinear vibration, stability, and bifurcation of rotating axially moving conical shells

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Abstract

The nonlinear vibration characteristics of rotating axially moving conical shells are investigated in the current paper. The nonlinear equations of motion and strain compatibility equation based on Donnell’s nonlinear shell theory are obtained. Three nonlinear equations of motion are reduced to a radial equation by applying the appropriate Airy stress function, forming a set of equations with the compatibility equation. The compatibility equation is solved by employing the seven degrees of freedom with respect to the system’s flexural mode shape. By substituting the flexural mode shape into the equation of motion and applying the Galerkin method, seven nonlinear coupled nonhomogeneous ODEs are achieved, then the set of equations is transformed into the normal form where it has been solved by the numerical method. The effects of the axial and rotational velocity on bifurcation diagrams, frequency response curves, time history, and the phase portraits of the system are discussed. The results of the present paper are validated against available data, and good agreements are achieved.

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Acknowledgements

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Authors and Affiliations

  1. Faculty of Mechanical Engineering, Shahid Rajaee Teacher Training University, Tehran, Iran

    Hadi Vahidi, Majid Shahgholi, Ali Rahmani Hanzaki & Arash Mohamadi

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  1. Hadi Vahidi
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Correspondence to Majid Shahgholi.

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Appendices

Appendix 1

$$L_{11} = S\frac{{e^{x} }}{\cot \left( \alpha \right)}\left( {\frac{{\partial^{2} }}{{\partial x^{2} }} + 3\frac{\partial }{\partial x} + 2} \right) - \cot \left( \alpha \right) \times \int gd\varphi,$$
(54)
$$\begin{aligned} L_{12} = & \frac{1}{{e^{2x} }}\left( { - A_{3} \left( {\frac{{\partial^{4} }}{{\partial \varphi^{4} }} + 2\frac{{\partial^{2} }}{{\partial \varphi^{2} }} + \frac{{\partial^{4} }}{{\partial x^{4} }} - 4\frac{{\partial^{3} }}{{\partial x^{3} }} + 4\frac{{\partial^{2} }}{{\partial x^{2} }}} \right)} \right. \\ & - 2\left( {A_{4} + A_{6} } \right)\left( {\frac{{\partial^{4} }}{{\partial x^{2} \partial \varphi^{2} }} - 2\frac{{\partial^{3} }}{{\partial x\partial \varphi^{2} }} + \frac{{\partial^{2} }}{{\partial \varphi^{2} }}} \right) \\ & - S^{4} e^{4x} \left( {\rho h\left( {\frac{{\partial^{2} }}{{\partial t^{2} }} + \frac{{2V_{c} \left( t \right)}}{{Se^{x} }}\frac{\partial }{\partial t} + \frac{{V_{c} \left( t \right)^{2} }}{{S^{2} e^{2x} }}\frac{{\partial^{2} }}{{\partial x^{2} }}} \right)} \right) \\ & + \left. {C\left( {\frac{\partial }{\partial t} + \frac{{V_{c} \left( t \right)}}{{Se^{x} }}\frac{\partial }{\partial t}} \right)} \right) + \int gd\varphi \times (\frac{1}{{se^{x} }}\left( {\frac{{\partial^{2} }}{{\partial \varphi^{2} }} + \frac{\partial }{\partial x}} \right) \\ & + \rho h\omega_{r}^{2} \left( {se^{x} \frac{\partial }{\partial x}\cos \left( \alpha \right)\cot \left( \alpha \right) + \frac{{\partial^{2} }}{{\partial \varphi^{2} }} + \cos \left( \alpha \right)^{2} } \right), \\ \end{aligned}$$
(55)
$$\begin{aligned} L_{13} = & e^{2x} \left( {\frac{{\partial^{2} }}{{\partial \varphi^{2} }} + \frac{\partial }{\partial x} + 2} \right)\left( {\frac{{\partial^{2} }}{{\partial x^{2} }} - \frac{\partial }{\partial x}} \right) \\ & + \left( {\frac{{\partial^{2} }}{{\partial x^{2} }} + 3\frac{\partial }{\partial x} + 2} \right)\left( {\frac{{\partial^{2} }}{{\partial \varphi^{2} }} + \frac{\partial }{\partial x}} \right) \\ & - 2\left( {\frac{{\partial^{2} }}{\partial x\partial \varphi } + \frac{\partial }{\partial \varphi }} \right)\left( {\frac{{\partial^{2} }}{\partial x\partial \varphi } - \frac{\partial }{\partial \varphi }} \right), \\ \end{aligned}$$
(56)
$$\begin{aligned} L_{21} = & B_{1} \cdot e^{2x} \frac{{\partial^{4} }}{{\partial \varphi^{4} }} + 2\left( {\frac{1 + \mu }{{Eh}} - \frac{\mu }{Eh}} \right)e^{2x} \frac{{\partial^{4} }}{{\partial x^{2} \partial \varphi^{2} }} + 4\left( {\frac{1 + \mu }{{Eh}} - \frac{\mu }{Eh}} \right)e^{2x} \frac{{\partial^{3} }}{{\partial x\partial \varphi^{2} }} \\ & + 2\left( {\frac{1 + \mu }{{Eh}} - \frac{\mu }{Eh} + \frac{1}{Eh}} \right)e^{2x} \frac{{\partial^{2} }}{{\partial \varphi^{2} }} + B_{1} \cdot e^{2x} \frac{{\partial^{4} }}{{\partial x^{4} }} + 4B_{1} .e^{2x} \frac{{\partial^{3} }}{{\partial x^{3} }} + 4B_{1} .e^{2x} \frac{{\partial^{2} }}{{\partial x^{2} }}, \\ \end{aligned}$$
(57)
$$L_{22} = - B_{4} \frac{{\partial^{4} }}{{\partial \varphi^{4} }} - \left( {\frac{{Se^{x} }}{\cot \left( \alpha \right)}} \right)\frac{{\partial^{2} }}{{\partial x^{2} }},$$
(58)
$$\begin{aligned} L_{23} = & - \left( {\frac{\partial }{\partial \varphi }} \right)^{2} + 2\left( {\frac{\partial }{\partial \varphi }} \right)\left( {\frac{{\partial^{2} }}{\partial x\partial \varphi }} \right) \\ & - \left( {\frac{\partial }{\partial x} - \frac{{\partial^{2} }}{{\partial x^{2} }}} \right)\left( {\frac{{\partial^{2} }}{{\partial \varphi^{2} }}} \right) - \left( {\frac{{\partial^{2} }}{\partial x\partial \varphi }} \right)^{2} \\ & - \left( {\frac{\partial }{\partial x} - \frac{{\partial^{2} }}{{\partial x^{2} }}} \right)\left( {\frac{\partial }{\partial x}} \right). \\ \end{aligned}$$
(59)

Appendix 2

$$\begin{aligned} & A_{29} \left( {A_{1,n} } \right)^{3} + A_{37} \left( {A_{1,n} } \right)^{2} A_{2,n} + \left( {A_{48} \left( {A_{2,n} } \right)^{2} + A_{60} \left( {A_{1,n} } \right)^{2} + (A_{5,0} A_{90} + A_{3,0} A_{94} + A_{93} } \right)A_{10} + A_{66} \left( {A_{3,0} } \right)^{2} \\ & \quad + \left( {A_{5,0} A_{97} + A_{3,0} A_{94} + A_{96} } \right)A_{30} + A_{42} \left( {B_{1,n} } \right)^{2} + A_{54} \left( {B_{2,n} } \right)^{2} + A_{72} \left( {A_{5,0} } \right)^{2} + A_{85} B_{1,n} B_{2,n} + \left( {A_{5,0} A_{98} } \right) \\ & \quad + \left( {A_{1,5} } \right)A_{1,n} + A_{31} \left( {A_{2,n} } \right)^{3} + \left( {A_{62} \left( {A_{1,0} } \right)^{2} } \right. + \left( {A_{109} A_{3,0} + A_{110} A_{5,0} + A_{108} } \right)A_{1,0} + A_{68} \left( {A_{3,0} } \right)^{2} \\ & \quad + A_{3,0} \left( {A_{112} A_{5,0} + A_{111} } \right) + A_{43} \left( {B_{1,n} } \right)^{2} + A_{56} \left( {B_{2,n} } \right)^{2} + A_{74} \left( {A_{5,0} } \right)^{2} + A_{100} B_{1,n} B_{2,n} + A_{113} A_{5,0} + \left. {A_{17} } \right)\\ & A_{2,n} - F\cos \left( {\omega t} \right), \\ \end{aligned}$$
(60)
$$\begin{aligned} & B_{2} \frac{{d^{2} }}{{dt^{2} }}\left( {B_{1,n} } \right) + B_{4} \frac{{d^{2} }}{{dt^{2} }}\left( {B_{2,n} } \right) + B_{9} \frac{d}{dt}\left( {B_{1,n} } \right) + B_{11} \frac{d}{dt}\left( {B_{2,n} } \right) + B_{6} B_{1,n} + B_{18} B_{2,n} + B_{30} \left( {B_{1,n} } \right)^{3} + B_{32} \left( {B_{2,n} } \right)^{3} \\ & \quad + \left( {A_{1,n} } \right)^{2} \left( {B_{36} B_{1,n} + B_{38} B_{2,n} } \right) + B_{44} \left( {B_{1,n} } \right)^{2} B_{2,n} + B_{49} \left( {A_{2,n} } \right)^{2} B_{1,n} + B_{50} \left( {A_{2,n} } \right)^{2} B_{2,n} + B_{55} \left( {B_{2,n} } \right)^{2} B_{1,n} \\ & \quad + A_{1,0}^{2} \left( {B_{61} B_{1,n} + B_{63} B_{2,n} } \right) + A_{3,0}^{2} \left( {B_{67} B_{1,n} + B_{69} B_{2,n} } \right) + A_{5,0}^{2} \left( {B_{73} B_{1,n} + B_{75} B_{2,n} } \right)\\ &\quad + A_{1,n} A_{2,n} \left( {B_{79} B_{1,n} + B_{80} B_{2,n} } \right) \\ & \quad + B_{1,n} A_{1,0} \left( {B_{119} A_{3,0} + B_{118} } \right) + \left( {A_{5,0} B_{120} A_{1,0} + A_{3,0} B_{122} + B_{123} } \right)A_{5,0} + A_{3,0} B_{121} )B_{1,n} \\ & \quad + B_{2,n} A_{1,0} \left( {A_{3,0} B_{125} + A_{5,0} B_{126} + B_{124} } \right) + B_{2,n} \left( {A_{5,0} B_{128} + B_{127} } \right)A_{3,0}, \\ \end{aligned}$$
(61)
$$\begin{aligned} & E_{29} \left( {A_{1,n} } \right)^{3} + E_{37} \left( {A_{1,n} } \right)^{2} A_{2,n} + \left( {E_{48} \left( {A_{2,n} } \right)^{2} + E_{60} \left( {A_{1,n} } \right)^{2} + \left( {A_{5,0} E_{95} + A_{3,0} E_{94} + E_{93} } \right)} \right.A_{10} \\ & \quad + E_{66} \left( {A_{3,0} } \right)^{2} + \left( {A_{5,0} E_{97} + E_{96} } \right)A_{30} + E_{42} \left( {B_{1,n} } \right)^{2} + E_{56} \left( {B_{2,n} } \right)^{2} + E_{72} \left( {A_{5,0} } \right)^{2} \\ & \quad + \left. {E_{85} B_{1,n} \left( t \right)B_{2,n} + A_{5,0} E_{98} + E_{1,5} } \right)A_{1,n} + \left( {A_{2,n} } \right)^{2} (E_{31} \left( {A_{2,n} } \right)^{2} + (E_{62} \left( {A_{1,0} } \right)^{2} \\ & \quad + \left( {E_{109} A_{3,0} + E_{110} A_{5,0} + E_{108} } \right)A_{1,0} + E_{68} \left( {A_{3,0} } \right)^{2} + A_{3,0} \left( {E_{112} A_{5,0} + EA_{111} } \right) \\ & \quad + E_{43} \left( {B_{1,n} } \right)^{2} + E_{56} \left( {B_{2,n} } \right)^{2} + E_{74} \left( {A_{5,0} } \right)^{2} \left. { + E_{100} B_{1,n} B_{2,n} + E_{113} A_{5,0} + E_{17} } \right)A_{2,n}, \\ \end{aligned}$$
(62)
$$\begin{aligned} & G_{30} \left( {B_{1,n} } \right)^{3} + G_{44} \left( {B_{1,n} } \right)^{2} B_{2,n} + B_{49} \left( {A_{2,n} } \right)^{2} B_{1,n} + \left( G_{55} \left( {B_{2,n} } \right)^{2} + G_{67} \left( {A_{3,0} } \right)^{2}\right.\\ &\quad + \left( {E_{122} A_{5,0} + G_{119} A_{1,0} + E_{112} } \right) A_{3,0} + G_{73} \left( {A_{5,0} } \right)^{2} \\ & \quad + \left( {E_{120} A_{1,0} + E_{123} } \right)A_{5,0} + \left( {A_{1,n} } \right)^{2} G_{36} + \left( {A_{2,n} } \right)^{2} G_{49}\\ &\quad + A_{1,0}^{2} G_{61} + G_{79} A_{2,n} A_{1,n} + A_{1,0} \left( t \right)G_{118} \left. { + G_{16} } \right)B_{1,n} + \left( {A_{3,0} } \right)^{2} G_{69} \\ & \quad + \left. {\left( {E_{125} A_{1,0} + E_{128} A_{5,0} + E_{127} } \right)A_{3,0} } \right) + \left( {A_{5,0} } \right)^{2} G_{75} + \left( {\left. {E_{126} A_{1,0} + E_{129} } \right)A_{5,0} + G_{38} \left( {A_{1,n} } \right)^{2} + G_{50} \left( {A_{2,n} } \right)^{2} } \right. \\ & \quad + \left. {G_{63} \left( {A_{1,0} } \right)^{2} + G_{90} A_{2,n} A_{1,n} + E_{124} A_{1,0} + G_{18} } \right)B_{2,n} + E_{126} A_{3,0} A_{5,0}, \\ \end{aligned}$$
(63)
$$\begin{aligned} & K_{33} \left( {A_{1,0} } \right)^{3} + \left( {K_{64} \left( {A_{3,0} } \right)^{2} + \left( {A_{5,0} K_{65} + A_{3,0} K_{64} + K_{26} } \right)} \right)\left( {A_{1,0} } \right)^{2}\\ &\quad + \left( {K_{70} \left( {A_{3,0} } \right)^{2} + \left( {K_{131} A_{5,0} + K_{130} } \right)A_{5,0} + \left( {A_{1,n} } \right)^{2} K_{39} }K_{45} \left( {B_{1,n} } \right)^{2} \right. \\ & \quad \left. + { + K_{51} \left( {A_{2,n} } \right)^{2} + K_{57} \left( {B_{1,n} } \right)^{2} + K_{76} \left( {A_{5,0} } \right)^{2} + K_{81} A_{2,n} A_{1,n} + K_{115} B_{1,n} B_{2,n} + A_{5,0} K_{132} + K_{19} } \right)A_{1,0} \\ & \quad + K_{34} \left( {A_{3,0} } \right)^{3} + \left( {A_{5,0} K_{71} + K_{27} } \right)\left( {A_{3,0} } \right)^{2} + (K_{40} \left( {A_{1,n} } \right)^{2} + \left( {B_{1,n} } \right)^{2} K_{46} + K_{52} \left( {A_{2,n} } \right)^{2} + K_{58} \left( {B_{2,n} } \right)^{2} \\ & \quad + \left( {A_{5,0} } \right)^{2} K_{77} + K_{82} A_{2,n} A_{1,n} + K_{116} B_{2,n} B_{1,n} + K_{133} A_{5,0} + K_{20} ) A_{3,0} + K_{35} \left( {A_{3,0} } \right)^{3} + K_{28} \left( {A_{5,0} } \right)^{2} \\ & \quad + \left( {\left( {A_{1,n} } \right)^{2} K_{41} + K_{47} \left( {B_{1,n} } \right)^{2} + K_{53} \left( {A_{2,n} } \right)^{2} + K_{59} \left( {B_{2,n} } \right)^{2} + K_{83} A_{2,n} A_{1,n} + K_{21} } \right)A_{5,0} + \left( {A_{1,n} } \right)^{2} K_{22} \\ & \quad + \left( {B_{1,n} } \right)^{2} K_{23} + K_{24} \left( {A_{2,n} } \right)^{2} + \left( {B_{1,n} } \right)^{2} K_{25} + K_{79} A_{2,n} A_{1,n} + K_{116} B_{2,n} B_{1,n}, \\ \end{aligned}$$
(64)
$$\begin{aligned} & J_{33} \left( {A_{1,0} } \right)^{3} + (J_{64} \left( {A_{3,0} } \right)^{2} + A_{5,0} J_{65} + A_{3,0} K_{64} + J_{26} )\left( {A_{1,0} } \right)^{2}\\ &\quad + \left( {J_{70} \left( {A_{3,0} } \right)^{2} + J_{131} A_{5,0} + J_{130} A_{5,0} + \left( {A_{1,n} } \right)^{2} J_{39} } \right. \\ & \quad + \left. {J_{45} \left( {B_{1,n} } \right)^{2} + J_{51} \left( {A_{2,n} } \right)^{2} + J_{57} \left( {B_{1,n} } \right)^{2} + J_{76} \left( {A_{5,0} } \right)^{2} + J_{81} A_{2,n} A_{1,n} + J_{115} B_{1,n} B_{2,n} + A_{5,0} J_{132} } \right)A_{1,0} \\ & \quad + J_{34} \left( {A_{3,0} } \right)^{3} + \left( {A_{5,0} J_{71} + J_{27} } \right)\left( {A_{3,0} } \right)^{2} + (J_{40} \left( {A_{1,n} } \right)^{2} + \left( {B_{1,n} } \right)^{2} J_{46} + J_{52} \left( {A_{2,n} } \right)^{2} \\ & \quad + J_{58} \left( {B_{2,n} } \right)^{2} + \left( {A_{5,0} } \right)^{2} J_{77} + J_{82} A_{2,n} A_{1,n} + J_{116} B_{2,n} B_{1,n} + J_{133} A_{5,0} + J_{20} )A_{3,0} + J_{35} \left( {A_{3,0} } \right)^{3} \\ & \quad + J_{28} \left( {A_{5,0} } \right)^{2} + \left( {\left( {A_{1,n} } \right)^{2} J_{41} + J_{47} \left( {B_{1,n} } \right)^{2} + J_{53} \left( {A_{2,n} } \right)^{2} + J_{59} \left( {B_{2,n} } \right)^{2} + J_{83} A_{2,n} A_{1,n} + J_{21} } \right)A_{5,0} \\ & \quad + \left( {A_{1,n} } \right)^{2} J_{22} + \left( {B_{1,n} } \right)^{2} J_{23} + J_{24} \left( {A_{2,n} } \right)^{2} + \left( {B_{1,n} } \right)^{2} J_{25} + J_{79} A_{2,n} A_{1,n} + J_{114} B_{2,n} B_{1,n}, \\ \end{aligned}$$
(65)
$$\begin{aligned} & K_{33} \left( {A_{1,0} } \right)^{3} + (K_{64} \left( {A_{3,0} } \right)^{2} + \left. {\left( {A_{5,0} K_{65} + A_{3,0} K_{64} + K_{26} } \right)} \right)\left( {A_{1,0} } \right)^{2}\\ & \quad + \left( {K_{70} \left( {A_{3,0} } \right)^{2} + \left( {K_{131} A_{5,0} + K_{130} } \right)A_{5,0} + \left( {A_{1,n} } \right)^{2} K_{39} + K_{45} \left( {B_{1,n} } \right)^{2} } \right. \\ & \quad + \left. {K_{51} \left( {A_{2,n} } \right)^{2} + K_{57} \left( {B_{1,n} } \right)^{2} + K_{76} \left( {A_{5,0} } \right)^{2} + K_{81} A_{2,n} A_{1,n} + K_{115} B_{1,n} B_{2,n} + A_{5,0} K_{132} + K_{19} } \right)A_{1,0} \\ & \quad + K_{34} \left( {A_{3,0} } \right)^{3} + \left( {(A_{5,0} K_{71} + K_{27} } \right)\left( {A_{3,0} } \right)^{2} + (K_{40} \left( {A_{1,n} } \right)^{2} + \left( {B_{1,n} } \right)^{2} K_{46} + K_{52} \left( {A_{2,n} } \right)^{2} + K_{58} \left( {B_{2,n} } \right)^{2} \\ & \quad + \left( {A_{5,0} } \right)^{2} K_{77} + K_{82} A_{2,n} A_{1,n} + K_{116} B_{2,n} B_{1,n} + K_{133} A_{5,0} + K_{20} )A_{3,0} + K_{35} \left( {A_{3,0} } \right)^{3} + K_{28} \left( {A_{5,0} } \right)^{2} \\ & \quad + \left( {\left( {A_{1,n} } \right)^{2} K_{41} + K_{47} \left( {B_{1,n} } \right)^{2} + K_{53} \left( {A_{2,n} } \right)^{2} + K_{59} \left( {B_{2,n} } \right)^{2} + K_{83} A_{2,n} A_{1,n} + K_{21} } \right)A_{5,0} \\ & \quad + \left( {A_{1,n} } \right)^{2} K_{22} + \left( {B_{1,n} } \right)^{2} K_{23} + K_{24} \left( {A_{2,n} } \right)^{2} + \left( {B_{1,n} } \right)^{2} K_{25} + K_{79} A_{2,n} A_{1,n} + K_{116} B_{2,n} B_{1,n}. \\ \end{aligned}$$
(66)

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Vahidi, H., Shahgholi, M., Hanzaki, A.R. et al. Nonlinear vibration, stability, and bifurcation of rotating axially moving conical shells. Acta Mech 233, 3175–3196 (2022). https://doi.org/10.1007/s00707-022-03255-x

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