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Stability behavior of rotating axially moving conical shell made of shape memory alloy

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Abstract

The current study investigates the nonlinear vibration characteristic of rotating axially moving conical shell made of shape memory alloy (SMA). For this purpose, the material behavior of SMA is simulated via Boyd-Lagoudas and Brinson models, and three nonlinear governing equations are derived by employing Hamilton principle, Donnell’s nonlinear theory assumptions, and SMA constitutive relations. By applying a suitable parametric airy stress function, three nonlinear equations of motion are reduced to one in radial direction, which must be solved with the help of the compatibility equation. By the aid of Jordan conical form and applying the Galerkin method on the equilibrium equation in the radial direction, seven nonlinear nonhomogeneous ODEs are resulted. Then, the set of nonlinear equations is solved using the fourth-order Runge–Kutta method and pseudo-arc length continuation. Furthermore, the bifurcation analysis based on the different parameters especially frequency responses along with the curves of the time histories and phase portraits mention the influence of different phases of the material, axial motion and spinning on the conical shells made of SMA. The results of the present work are validated with available approved data, which shows good agreements.

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This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

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Authors and Affiliations

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

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

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  1. Hadi Vahidi
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  2. Ali Rahmani Hanzaki
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  3. Majid Shahgholi
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  4. Arash Mohamadi
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Correspondence to Ali Rahmani Hanzaki.

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Appendices

Appendix A

In this portion, the relation for equation of motion which is mentioned in Sect. 3 is presented.

$$L_{11} = S\frac{{e^{x} }}{\cot \left( \alpha \right)}\left( {\frac{{\partial^{2} }}{{\partial x^{2} }} + 3\frac{\partial }{\partial x} + 2} \right) - \cot \left( \alpha \right) \times \int gd\varphi$$
(A1)
$$\begin{gathered} L_{12} = \frac{1}{{e^{2x} }}\left( { - A3\left( {\frac{{\partial^{4} }}{{\partial \varphi^{4} }} + 2\frac{{\partial^{2} }}{{\partial \varphi^{2} }} + \frac{{\partial^{4} }}{{\partial x^{4} }} - 4\frac{{\partial^{3} }}{{\partial x^{3} }} + 4\frac{{\partial^{2} }}{{\partial x^{2} }}} \right) - 2\left( {A4 + A6} \right)\left( {\frac{{\partial^{4} }}{{\partial x^{2} \partial \varphi^{2} }} - 2\frac{{\partial^{3} }}{{\partial x\partial \varphi^{2} }} + \frac{{\partial^{2} }}{{\partial \varphi^{2} }}} \right)} \right. \hfill \\ \left. {\quad \quad - S^{4} e^{4x} \left( {\rho h\left( {\frac{{\partial^{2} }}{{\partial t^{2} }} + \frac{{2V_{c} \left( t \right)}}{{Se^{x} }}\frac{\partial }{\partial t} + \frac{{V_{c} \left( t \right)^{2} }}{{S^{2} e^{2x} }}\frac{{\partial^{2} }}{{\partial x^{2} }}} \right)} \right) + C\left( {\frac{\partial }{\partial t} + \frac{{V_{c} \left( t \right)}}{{Se^{x} }}\frac{\partial }{\partial t}} \right)} \right) \hfill \\ \quad \quad + \int gd\varphi \times (\frac{1}{{se^{x} }}\left( {\frac{{\partial^{2} }}{{\partial \varphi^{2} }} + \frac{\partial }{\partial x}} \right) + \rho h\omega_{r}^{2} \left( {se^{x} \frac{\partial }{\partial x}\cos \left( \alpha \right)\cot \left( \alpha \right) + \frac{{\partial^{2} }}{{\partial \varphi^{2} }} + \cos \left( \alpha \right)^{2} } \right) \hfill \\ \end{gathered}$$
(A2)
$$L_{13} = e^{2x} (\left( {\frac{{\partial^{2} }}{{\partial \varphi^{2} }} + \frac{\partial }{\partial x} + 2} \right)\left( {\frac{{\partial^{2} }}{{\partial x^{2} }} - \frac{\partial }{\partial x}} \right) + \left( {\frac{{\partial^{2} }}{{\partial x^{2} }} + 3\frac{\partial }{\partial x} + 2} \right)\left( {\frac{{\partial^{2} }}{{\partial \varphi^{2} }} + \frac{\partial }{\partial x}} \right) - 2\left( {\frac{{\partial^{2} }}{\partial x\partial \varphi } + \frac{\partial }{\partial \varphi }} \right)\left( {\frac{{\partial^{2} }}{\partial x\partial \varphi } - \frac{\partial }{\partial \varphi }} \right)$$
(A3)
$$\begin{gathered} L_{21} = B1.e^{2x} \frac{{\partial^{4} }}{{\partial \varphi^{4} }} + 2\left( {\frac{1 + \mu }{{Eh}} - \frac{\mu }{Eh}} \right)e^{2x} \frac{{\partial^{4} }}{{\partial x^{2} \partial \varphi^{2} }} + 4\left( {\frac{1 + \mu }{{Eh}} - \frac{\mu }{Eh}} \right)e^{2x} \frac{{\partial^{3} }}{{\partial x\partial \varphi^{2} }} \hfill \\ \quad \quad + 2\left( {\frac{1 + \mu }{{Eh}} - \frac{\mu }{Eh} + \frac{1}{Eh}} \right)e^{2x} \frac{{\partial^{2} }}{{\partial \varphi^{2} }} + B1.e^{2x} \frac{{\partial^{4} }}{{\partial x^{4} }} + 4B1.e^{2x} \frac{{\partial^{3} }}{{\partial x^{3} }} + 4B1.e^{2x} \frac{{\partial^{2} }}{{\partial x^{2} }} \hfill \\ \end{gathered}$$
(A4)
$$L_{22} = - B4\frac{{\partial^{4} }}{{\partial \varphi^{4} }} - \left( {\frac{{Se^{x} }}{\cot \left( \alpha \right)}} \right)\frac{{\partial^{2} }}{{\partial x^{2} }}$$
(A5)
$$L_{23} = - \left( {\frac{\partial }{\partial \varphi }} \right)^{2} + 2\left( {\frac{\partial }{\partial \varphi }} \right)\left( {\frac{{\partial^{2} }}{\partial x\partial \varphi }} \right) - \left( {\frac{\partial }{\partial x} - \frac{{\partial^{2} }}{{\partial x^{2} }}} \right)\left( {\frac{{\partial^{2} }}{{\partial \varphi^{2} }}} \right) - \left( {\frac{{\partial^{2} }}{\partial x\partial \varphi }} \right)^{2} - \left( {\frac{\partial }{\partial x} - \frac{{\partial^{2} }}{{\partial x^{2} }}} \right)\left( {\frac{\partial }{\partial x}} \right)$$
(A6)

Appendix B

$$\begin{aligned} & A_{29} \left( {\left( {A_{1,n} } \right)} \right)^{3} + A_{37} \left( {\left( {A_{1,n} } \right)} \right)^{2} A_{2,n} + \left( {A_{48} \left( {\left( {A_{2,n} } \right)} \right)^{2} + A_{60} \left( {\left( {A_{1,n} } \right)} \right)^{2} + \left( {A_{5,0} A_{90} + A_{3,0} A_{94} + A_{93} } \right)} \right.A_{10} \hfill \\ &\quad + A_{66} \left( {\left( {A_{3,0} } \right)} \right)^{2} + \left( {A_{5,0} A_{97} + A_{3,0} A_{94} + A_{96} } \right)A_{30} + A_{42} \left( {\left( {B_{1,n} } \right)} \right)^{2} + A_{54} \left( {\left( {B_{2,n} } \right)} \right)^{2} + A_{72} \left( {\left( {A_{5,0} } \right)} \right)^{2} \hfill \\ &\quad \left. { + A_{85} B_{1,n} B_{2,n} + \left( {A_{5,0} A_{98} } \right) + A_{1,5} } \right)A_{1,n} + A_{31} \left( {\left( {A_{2,n} } \right)} \right)^{3} + \left( {A_{62} \left( {\left( {A_{1,0} } \right)} \right)^{2} } \right. \hfill \\ &\quad + \left( {A_{109} \left( {A_{3,0} } \right) + A_{110} \left( {A_{5,0} } \right) + A_{108} } \right)A_{1,0} + A_{68} \left( {\left( {A_{3,0} } \right)} \right)^{2} + A_{3,0} \left( {A_{112} A_{5,0} + A_{111} } \right) + A_{43} \left( {\left( {B_{1,n} } \right)} \right)^{2} \hfill \\ &\quad \left. { + A_{56} \left( {\left( {B_{2,n} } \right)} \right)^{2} + A_{74} \left( {\left( {A_{5,0} } \right)} \right)^{2} + \left( {A_{100} B_{1,n} B_{2,n} } \right) + A_{113} \left( {A_{5,0} } \right) + A_{17} } \right)A_{2,n} - F\cos \left( {\omega t} \right) \hfill \\ \end{aligned}$$
(B1)
$$\begin{aligned} & B_{2} \frac{{d^{2} }}{{dt^{2} }}\left( {B_{1,n} } \right) + B_{4} \frac{{d^{2} }}{{dt^{2} }}\left( {B_{2,n} } \right) + B_{9} \frac{d}{dt}\left( {B_{1,n} } \right) + B_{11} \frac{d}{dt}\left( {B_{2,n} } \right) + B_{6} B_{1,n} \hfill \\ &\quad + B_{18} B_{2,n} + B_{30} \left( {\left( {B_{1,n} } \right)} \right)^{3} + B_{32} \left( {\left( {B_{2,n} } \right)} \right)^{3} + \left( {\left( {A_{1,n} } \right)} \right)^{2} \left( {B_{36} \left( {B_{1,n} } \right) + B_{38} \left( {B_{2,n} } \right)} \right) \hfill \\ &\quad + B_{44} \left( {\left( {B_{1,n} } \right)} \right)^{2} B_{2,n} + B_{49} \left( {\left( {A_{2,n} } \right)} \right)^{2} B_{1,n} + B_{50} \left( {\left( {A_{2,n} } \right)} \right)^{2} B_{2,n} + B_{55} \left( {\left( {B_{2,n} } \right)} \right)^{2} B_{1,n} \hfill \\ &\quad + A_{1,0}^{2} \left( {B_{61} B_{1,n} + B_{63} B_{2,n} } \right) + A_{3,0}^{2} \left( {B_{67} B_{1,n} + B_{69} B_{2,n} } \right) + A_{5,0}^{2} \left( {B_{73} B_{1,n} + B_{75} B_{2,n} } \right) \hfill \\ &\quad + A_{1,n} A_{2,n} \left( {B_{79} B_{1,n} + B_{80} B_{2,n} } \right) + B_{1,n} A_{1,0} \left( {B_{119} A_{3,0} + B_{118} } \right) \hfill \\ &\quad + \left( {A_{5,0} B_{120} A_{1,0} + A_{3,0} B_{122} + B_{123} } \right)A_{5,0} + A_{3,0} B_{121} )B_{1,n} \hfill \\ &\quad + B_{2,n} A_{1,0} \left( {A_{3,0} B_{125} + A_{5,0} B_{126} + B_{124} } \right) + B_{2,n} \left( {A_{5,0} B_{128} + B_{127} } \right)A_{3,0} + A_{5,0} B_{129} \hfill \\ \end{aligned}$$
(B2)
$$\begin{aligned} & E_{29} \left( {\left( {A_{1,n} } \right)} \right)^{3} + E_{37} \left( {\left( {A_{1,n} } \right)} \right)^{2} A_{2,n} \hfill \\ &\quad + \left( {E_{48} \left( {\left( {A_{2,n} } \right)} \right)^{2} + E_{60} \left( {\left( {A_{1,n} } \right)} \right)^{2} + (A_{5,0} E_{95} + A_{3,0} E_{94} + E_{93} } \right)A_{10} \hfill \\ &\quad + E_{66} \left( {\left( {A_{3,0} } \right)\left( t \right)} \right)^{2} + \left( {A_{5,0} E_{97} + E_{96} } \right)A_{30} + E_{42} \left( {\left( {B_{1,n} } \right)} \right)^{2} + E_{56} \left( {\left( {B_{2,n} } \right)} \right)^{2} \hfill \\ &\quad + E_{72} \left( {\left( {A_{5,0} } \right)} \right)^{2} + E_{85} B_{1,n} \left( t \right)B_{2,n} + \left( {A_{5,0} E_{98} ) + E_{1,5} } \right)A_{1,n} \hfill \\ &\quad + \left( {A_{2,n} } \right)^{2} (E_{31} \left( {\left( {A_{2,n} } \right)} \right)^{2} + (E_{62} \left( {\left( {A_{1,0} } \right)} \right)^{2} + \left( {E_{109} \left( {A_{3,0} } \right) + E_{110} \left( {A_{5,0} } \right)} \right. \hfill \\ &\quad \left. { + E_{108} } \right)A_{1,0} + E_{68} \left( {\left( {A_{3,0} } \right)} \right)^{2} + A_{3,0} \left( {E_{112} A_{5,0} + EA_{111} } \right) + E_{43} \left( {\left( {B_{1,n} } \right)} \right)^{2} \hfill \\ &\quad + E_{56} \left( {\left( {B_{2,n} } \right)} \right)^{2} + E_{74} \left( {\left( {A_{5,0} } \right)} \right)^{2} + \left( {E_{100} B_{1,n} B_{2,n} } \right) + E_{113} \left( {A_{5,0} } \right) + E_{17} )A_{2,n} \hfill \\ \end{aligned}$$
(B3)
$$\begin{aligned} & G_{30} \left( {\left( {B_{1,n} } \right)} \right)^{3} + G_{44} \left( {\left( {B_{1,n} } \right)} \right)^{2} B_{2,n} + B_{49} \left( {\left( {A_{2,n} } \right)} \right)^{2} B_{1,n} + (G_{55} \left( {\left( {B_{2,n} } \right)} \right)^{2} + G_{67} \left( {\left( {A_{3,0} } \right)} \right)^{2} \hfill \\ &\quad + (E_{122} A_{5,0} + G_{119} A_{1,0} + E_{112} )A_{3,0} + G_{73} \left( {\left( {A_{5,0} } \right)} \right)^{2} + (E_{120} A_{1,0} + E_{123} )A_{5,0} + \left( {\left( {A_{1,n} } \right)} \right)^{2} G_{36} + \left( {\left( {A_{2,n} } \right)} \right)^{2} G_{49} \hfill \\ &\quad + A_{1,0}^{2} \left( {G_{61} } \right) + G_{79} \left( {A_{2,n} A_{1,n} } \right) + A_{1,0} \left( t \right)\left( {G_{118} } \right) + G_{16} )B_{1,n} + \left( {\left( {A_{3,0} } \right)} \right)^{2} G_{69} \hfill \\ &\quad + \left( {E_{125} A_{1,0} + E_{128} A_{5,0} + E_{127} } \right)A_{3,0} ) + \left( {\left( {A_{5,0} } \right)} \right)^{2} G_{75} + (E_{126} A_{1,0} + E_{129} )A_{5,0} + G_{38} \left( {\left( {A_{1,n} } \right)} \right)^{2} \hfill \\ &\quad + G_{50} \left( {\left( {A_{2,n} } \right)} \right)^{2} + G_{63} \left( {\left( {A_{1,0} } \right)} \right)^{2} + G_{90} \left( {A_{2,n} A_{1,n} } \right) + E_{124} A_{1,0} + G_{18} )B_{2,n} + E_{126} \left( {A_{3,0} } \right)A_{5,0} \hfill \\ \end{aligned}$$
(B4)
$$\begin{aligned} & K_{33} \left( {\left( {A_{1,0} } \right)} \right)^{3} + (K_{64} \left( {\left( {A_{3,0} } \right)} \right)^{2} + (A_{5,0} K_{65} + A_{3,0} K_{64} + K_{26} ))\left( {\left( {A_{1,0} } \right)} \right)^{2} \hfill \\ &\quad + \left( {K_{70} \left( {\left( {A_{3,0} } \right)} \right)^{2} + \left( {K_{131} A_{5,0} + K_{130} } \right)A_{5,0} + \left( {\left( {A_{1,n} } \right)} \right)^{2} K_{39} + K_{45} \left( {\left( {B_{1,n} } \right)} \right)^{2} } \right. \hfill \\ &\quad \left. { + K_{51} \left( {\left( {A_{2,n} } \right)} \right)^{2} + K_{57} \left( {\left( {B_{1,n} } \right)} \right)^{2} + K_{76} \left( {\left( {A_{5,0} } \right)} \right)^{2} + K_{81} \left( {A_{2,n} A_{1,n} } \right) + K_{115} B_{1,n} \left( t \right)B_{2,n} + \left( {A_{5,0} } \right)K_{132} + K_{19} } \right) \hfill \\ &\quad + A_{1,0} + K_{34} \left( {\left( {A_{3,0} } \right)} \right)^{3} + \left( {(A_{5,0} K_{71} + K_{27} } \right)\left( {\left( {A_{3,0} } \right)} \right)^{2} + (K_{40} \left( {\left( {A_{1,n} } \right)} \right)^{2} + \left( {\left( {B_{1,n} } \right)} \right)^{2} K_{46} + K_{52} \left( {\left( {A_{2,n} } \right)} \right)^{2} \hfill \\ &\quad + K_{58} \left( {\left( {B_{2,n} } \right)} \right)^{2} + \left( {\left( {A_{5,0} } \right)} \right)^{2} K_{77} + K_{82} \left( {A_{2,n} A_{1,n} } \right) + K_{116} \left( {B_{2,n} B_{1,n} } \right) + K_{133} A_{5,0} + K_{20} )A_{3,0} + K_{35} \left( {\left( {A_{3,0} } \right)} \right)^{3} \hfill \\ &\quad + K_{28} \left( {\left( {A_{5,0} } \right)} \right)^{2} + \left( {\left( {\left( {A_{1,n} } \right)} \right)^{2} K_{41} + K_{47} \left( {\left( {B_{1,n} } \right)} \right)^{2} + K_{53} \left( {\left( {A_{2,n} } \right)} \right)^{2} + K_{59} \left( {\left( {B_{2,n} } \right)} \right)^{2} + K_{83} \left( {A_{2,n} A_{1,n} } \right) + K_{21} } \right)\left( {A_{5,0} } \right) \hfill \\ &\quad + \left( {\left( {A_{1,n} } \right)} \right)^{2} K_{22} + \left( {\left( {B_{1,n} } \right)} \right)^{2} K_{23} + K_{24} \left( {\left( {A_{2,n} } \right)} \right)^{2} + \left( {\left( {B_{1,n} } \right)} \right)^{2} K_{25} + K_{79} \left( {A_{2,n} A_{1,n} } \right) + K_{116} \left( {B_{2,n} B_{1,n} } \right) \hfill \\ \end{aligned}$$
(B5)
$$\begin{aligned} & J_{33} \left( {\left( {A_{1,0} } \right)} \right)^{3} + (J_{64} \left( {\left( {A_{3,0} } \right)} \right)^{2} + (A_{5,0} J_{65} + A_{3,0} K_{64} + J_{26} ))\left( {\left( {A_{1,0} } \right)} \right)^{2} \hfill \\ &\quad + \left( {J_{70} \left( {\left( {A_{3,0} } \right)} \right)^{2} + \left( {J_{131} A_{5,0} + J_{130} } \right)A_{5,0} + \left( {\left( {A_{1,n} } \right)} \right)^{2} J_{39} + J_{45} \left( {\left( {B_{1,n} } \right)} \right)^{2} } \right. \hfill \\ &\quad \left. { + J_{51} \left( {\left( {A_{2,n} } \right)} \right)^{2} + J_{57} \left( {\left( {B_{1,n} } \right)} \right)^{2} + J_{76} \left( {\left( {A_{5,0} } \right)} \right)^{2} + J_{81} \left( {A_{2,n} A_{1,n} } \right) + J_{115} B_{1,n} \left( t \right)B_{2,n} + \left( {A_{5,0} } \right)J_{132} + J_{19} } \right)A_{1,0} \hfill \\ &\quad + J_{34} \left( {\left( {A_{3,0} } \right)} \right)^{3} + \left( {(A_{5,0} J_{71} + J_{27} } \right)\left( {\left( {A_{3,0} } \right)} \right)^{2} + (J_{40} \left( {\left( {A_{1,n} } \right)} \right)^{2} + \left( {\left( {B_{1,n} } \right)} \right)^{2} J_{46} + J_{52} \left( {\left( {A_{2,n} } \right)} \right)^{2} \hfill \\ &\quad + J_{58} \left( {\left( {B_{2,n} } \right)} \right)^{2} + \left( {\left( {A_{5,0} } \right)} \right)^{2} J_{77} + J_{82} \left( {A_{2,n} A_{1,n} } \right) + J_{116} \left( {B_{2,n} B_{1,n} } \right) + J_{133} A_{5,0} + J_{20} )A_{3,0} \hfill \\ &\quad + J_{35} \left( {\left( {A_{3,0} } \right)} \right)^{3} + J_{28} \left( {\left( {A_{5,0} } \right)} \right)^{2} + \left( {\left( {\left( {A_{1,n} } \right)} \right)^{2} J_{41} + J_{47} \left( {\left( {B_{1,n} } \right)} \right)^{2} + J_{53} \left( {\left( {A_{2,n} } \right)} \right)^{2} + J_{59} \left( {\left( {B_{2,n} } \right)} \right)^{2} + J_{83} \left( {A_{2,n} A_{1,n} } \right) + J_{21} } \right) \hfill \\ &\quad + \left( {A_{5,0} } \right) + \left( {\left( {A_{1,n} } \right)} \right)^{2} J_{22} + \left( {\left( {B_{1,n} } \right)} \right)^{2} J_{23} + J_{24} \left( {\left( {A_{2,n} } \right)} \right)^{2} + \left( {\left( {B_{1,n} } \right)} \right)^{2} J_{25} + J_{79} \left( {A_{2,n} A_{1,n} } \right) + J_{114} \left( {B_{2,n} B_{1,n} } \right) \hfill \\ \hfill \\ \end{aligned}$$
(B6)
$$\begin{aligned} & K_{33} \left( {\left( {A_{1,0} } \right)} \right)^{3} + (K_{64} \left( {\left( {A_{3,0} } \right)} \right)^{2} + (A_{5,0} K_{65} + A_{3,0} K_{64} + K_{26} ))\left( {\left( {A_{1,0} } \right)} \right)^{2} \hfill \\ &\quad + \left( {K_{70} \left( {\left( {A_{3,0} } \right)} \right)^{2} + \left( {K_{131} A_{5,0} + K_{130} } \right)A_{5,0} + \left( {\left( {A_{1,n} } \right)} \right)^{2} K_{39} + K_{45} \left( {\left( {B_{1,n} } \right)} \right)^{2} + K_{51} \left( {\left( {A_{2,n} } \right)} \right)^{2} } \right. \hfill \\ &\quad \left. { + K_{57} \left( {\left( {B_{1,n} } \right)} \right)^{2} + K_{76} \left( {\left( {A_{5,0} } \right)} \right)^{2} + K_{81} \left( {A_{2,n} A_{1,n} } \right) + K_{115} B_{1,n} \left( t \right)B_{2,n} + \left( {A_{5,0} } \right)K_{132} + K_{19} } \right)A_{1,0} \hfill \\ &\quad + K_{34} \left( {\left( {A_{3,0} } \right)} \right)^{3} + \left( {(A_{5,0} K_{71} + K_{27} } \right)\left( {\left( {A_{3,0} } \right)} \right)^{2} + (K_{40} \left( {\left( {A_{1,n} } \right)} \right)^{2} + \left( {\left( {B_{1,n} } \right)} \right)^{2} K_{46} \hfill \\ &\quad + K_{52} \left( {\left( {A_{2,n} } \right)} \right)^{2} + K_{58} \left( {\left( {B_{2,n} } \right)} \right)^{2} + \left( {\left( {A_{5,0} } \right)} \right)^{2} K_{77} + K_{82} \left( {A_{2,n} A_{1,n} } \right) + K_{116} \left( {B_{2,n} B_{1,n} } \right) \hfill \\ &\quad + K_{133} A_{5,0} + K_{20} )A_{3,0} + K_{35} \left( {\left( {A_{3,0} } \right)} \right)^{3} + K_{28} \left( {\left( {A_{5,0} } \right)} \right)^{2} \hfill \\ &\quad + \left( {\left( {\left( {A_{1,n} } \right)} \right)^{2} K_{41} + K_{47} \left( {\left( {B_{1,n} } \right)} \right)^{2} + K_{53} \left( {\left( {A_{2,n} } \right)} \right)^{2} + K_{59} \left( {\left( {B_{2,n} } \right)} \right)^{2} + K_{83} \left( {A_{2,n} A_{1,n} } \right) + K_{21} } \right)\left( {A_{5,0} } \right) \hfill \\ &\quad + \left( {\left( {A_{1,n} } \right)} \right)^{2} K_{22} + \left( {\left( {B_{1,n} } \right)} \right)^{2} K_{23} + K_{24} \left( {\left( {A_{2,n} } \right)} \right)^{2} + \left( {\left( {B_{1,n} } \right)} \right)^{2} K_{25} + K_{79} \left( {A_{2,n} A_{1,n} } \right) + K_{116} \left( {B_{2,n} B_{1,n} } \right) \hfill \\ \end{aligned}$$
(B7)

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Vahidi, H., Rahmani Hanzaki, A., Shahgholi, M. et al. Stability behavior of rotating axially moving conical shell made of shape memory alloy. Acta Mech 234, 5725–5748 (2023). https://doi.org/10.1007/s00707-023-03674-4

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