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Active vibration control of rotating laminated composite truncated conical shells through magnetostrictive layers based on first-order shear deformation theory

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Abstract

In this paper, for the first time active vibration control of rotating laminated composite truncated conical shells containing magnetostrictive layers by employing first-order shear deformation theory is investigated. The active vibration control task is done through magnetostrictive layers employing velocity feedback control law. The effects of initial hoop tension and centrifugal and Coriolis forces are considered in extraction of the partial differential equations through Hamilton principle. The ordinary differential equations are derived by employing modified Galerkin method. This study agrees with the mentioned results of the literature. Finally, the effects of several parameters on the vibration suppression are investigated.

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Authors and Affiliations

  1. Faculty of Mechanical Engineering, K. N. Toosi University of Technology, Tehran, Iran

    Shahin Mohammadrezazadeh & Ali Asghar Jafari

Authors
  1. Shahin Mohammadrezazadeh
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  2. Ali Asghar Jafari
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Correspondence to Shahin Mohammadrezazadeh.

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Technical Editor: Wallace Moreira Bessa, D.Sc..

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Appendix

Appendix

The differential operators of matrix L which is introduced in Eq. (29) are defined as the following for symmetric cross-ply lamination scheme:

$$\begin{aligned} L_{11} & = A_{11} R(x)\frac{{\partial^{2} }}{{\partial x^{2} }} + A_{11} \sin \alpha \frac{\partial }{\partial x} - \frac{{A_{22} \sin^{2} \alpha }}{R(x)} + \frac{{A_{66} }}{R(x)}\frac{{\partial^{2} }}{{\partial \theta^{2} }} - I_{1} R(x)\frac{{\partial^{2} }}{{\partial t^{2} }} \\ & \quad + I_{1} \varOmega^{2} R(x)\sin^{2} \alpha + I_{1} \varOmega^{2} R(x)\frac{{\partial^{2} }}{{\partial \theta^{2} }} \\ \end{aligned}$$
(46)
$$L_{12} = A_{12} \frac{{\partial^{2} }}{\partial x\partial \theta } - \frac{{A_{22} \sin \alpha }}{R(x)}\frac{\partial }{\partial \theta } + A_{66} \frac{{\partial^{2} }}{\partial x\partial \theta } - \frac{{A_{66} \sin \alpha }}{R(x)}\frac{\partial }{\partial \theta } + 2I_{1} \varOmega R(x)\sin \alpha \frac{\partial }{\partial t}$$
(47)
$$\begin{aligned} L_{13} & = A_{12} \cos \alpha \frac{\partial }{\partial x} - A_{31} R(x)\frac{{\partial^{2} }}{\partial x\partial t} - A_{31} \sin \alpha \frac{\partial }{\partial t} - \frac{{A_{22} \sin \alpha \cos \alpha }}{R(x)} + A_{32} \sin \alpha \frac{\partial }{\partial t} \\ & \quad + I_{1} \varOmega^{2} R(x)\sin \alpha \cos \alpha - I_{1} (R(x))^{2} \varOmega^{2} \cos \alpha \frac{\partial }{\partial x} \\ \end{aligned}$$
(48)
$$L_{14} = 0$$
(49)
$$L_{15} = 0$$
(50)
$$\begin{aligned} L_{21} & = A_{12} \frac{{\partial^{2} }}{\partial x\partial \theta } + \frac{{A_{22} \sin \alpha }}{R(x)}\frac{\partial }{\partial \theta } + A_{66} \frac{{\partial^{2} }}{\partial x\partial \theta } + \frac{{A_{66} \sin \alpha }}{R(x)}\frac{\partial }{\partial \theta } - 2I_{1} \varOmega R(x)\sin \alpha \frac{\partial }{\partial t} \\ & \quad + I_{1} \varOmega^{2} (R(x))^{2} \frac{{\partial^{2} }}{\partial x\partial \theta } + I_{1} \varOmega^{2} R(x)\sin \alpha \frac{\partial }{\partial \theta } \\ \end{aligned}$$
(51)
$$\begin{aligned} L_{22} & = \frac{{A_{22} }}{R(x)}\frac{{\partial^{2} }}{{\partial \theta^{2} }} + A_{66} R(x)\frac{{\partial^{2} }}{{\partial x^{2} }} + A_{66} \sin \alpha \frac{\partial }{\partial x} - \frac{{A_{66} }}{R(x)}\sin^{2} \\ & \quad\alpha - \frac{{K_{s} A_{44} \cos^{2} \alpha }}{R(x)} - I_{1} R(x)\frac{{\partial^{2} }}{{\partial t^{2} }} + I_{1} \varOmega^{2} R(x)\\ & \quad + I_{1} \varOmega^{2} (R(x))^{2} \sin \alpha \frac{\partial }{\partial x} - I_{1} \varOmega^{2} R(x)\sin^{2} \alpha \\ \end{aligned}$$
(52)
$$L_{23} = \frac{{A_{22} \cos \alpha }}{R(x)}\frac{\partial }{\partial \theta } - A_{32} \frac{{\partial^{2} }}{\partial \theta \partial t} + \frac{{K_{s} A_{44} \cos \alpha }}{R(x)}\frac{\partial }{\partial \theta } - 2I_{1} \varOmega R(x)\cos \alpha \frac{\partial }{\partial t}$$
(53)
$$L_{24} = 0$$
(54)
$$L_{25} = K_{\text{s}} A_{44} \cos \alpha$$
(55)
$$\begin{aligned} L_{31} & = - A_{12} \cos \alpha \frac{\partial }{\partial x} - \frac{{A_{22} \sin \alpha \cos \alpha }}{R(x)} + I_{1} \varOmega^{2} R(x)\sin \alpha \cos \alpha \\ & \quad - m_{kh} I_{1} \varOmega^{2} (R(x))^{2} \cos \alpha \frac{\partial }{\partial x} \\ \end{aligned}$$
(56)
$$ \begin{aligned}L_{32} &= - \frac{{A_{22} \cos \alpha }}{R(x)}\frac{\partial }{\partial \theta } - \frac{{K_{\text{s}} A_{44} \cos \alpha }}{R(x)}\frac{\partial }{\partial \theta } \\ &\quad+ 2I_{1} \varOmega R(x)\cos \alpha \frac{\partial }{\partial t} - I_{1} os\varOmega^{2} R(x)\frac{\partial }{\partial \theta }\end{aligned} $$
(57)
$$\begin{aligned} L_{33} & = - \frac{{A_{22} \cos^{2} \alpha }}{R(x)} + A_{32} \cos \alpha \frac{\partial }{\partial t} + K_{\text{s}} A_{55} R(x)\frac{{\partial^{2} }}{{\partial x^{2} }} + K_{s} A_{55} \sin \alpha \frac{\partial }{\partial x} \\ & \quad + \frac{{K_{\text{s}} A_{44} }}{R(x)}\frac{{\partial^{2} }}{{\partial \theta^{2} }} - I_{1} R(x)\frac{{\partial^{2} }}{{\partial t^{2} }} + I_{1} \varOmega^{2} R(x)\cos^{2} \alpha + I_{1} \varOmega^{2} R(x)\frac{{\partial^{2} }}{{\partial \theta^{2} }} \\ \end{aligned}$$
(58)
$$L_{34} = K_{\text{s}} A_{55} R(x)\frac{\partial }{\partial x} + K_{\text{s}} A_{55} \sin \alpha$$
(59)
$$L_{35} = K_{\text{s}} A_{44} \frac{\partial }{\partial \theta }$$
(60)
$$L_{41} = 0$$
(61)
$$L_{42} = 0$$
(62)
$$L_{43} = - K_{\text{s}} A_{55} R(x)\frac{\partial }{\partial x}$$
(63)
$$\begin{aligned} L_{44} & = D_{11} R(x)\frac{{\partial^{2} }}{{\partial x^{2} }} + D_{11} \sin \alpha \frac{\partial }{\partial x} - \frac{{D_{22} \sin^{2} \alpha }}{R(x)} + \frac{{D_{66} }}{R(x)}\frac{{\partial^{2} }}{{\partial \theta^{2} }} - K_{\text{s}} A_{55} R(x) \\ & \quad - I_{3} R(x)\frac{{\partial^{2} }}{{\partial t^{2} }} + I_{3} \varOmega^{2} R(x)\sin^{2} \alpha \\ \end{aligned}$$
(64)
$$ \begin{aligned}L_{45} &= D_{12} \frac{{\partial^{2} }}{\partial x\partial \theta } - \frac{{D_{22} \sin \alpha }}{R(x)}\frac{\partial }{\partial \theta } + D_{66} \frac{{\partial^{2} }}{\partial x\partial \theta } \\ &\quad- \frac{{D_{66} \sin \alpha }}{R(x)}\frac{\partial }{\partial \theta } + 2I_{3} \varOmega R(x)\sin \alpha \frac{\partial }{\partial t}\end{aligned} $$
(65)
$$L_{51} = 0$$
(66)
$$L_{52} = K_{\text{s}} A_{44} \cos \alpha$$
(67)
$$L_{53} = - K_{\text{s}} A_{44} \frac{\partial }{\partial \theta }$$
(68)
$$ \begin{aligned}L_{54} &= D_{12} \frac{{\partial^{2} }}{\partial x\partial \theta } + \frac{{D_{22} \sin \alpha }}{R(x)}\frac{\partial }{\partial \theta } + D_{66} \frac{{\partial^{2} }}{\partial x\partial \theta } \\ &\quad+ \frac{{D_{66} \sin \alpha }}{R(x)}\frac{\partial }{\partial \theta } - 2I_{3} \varOmega R(x)\sin \alpha \frac{\partial }{\partial t}\end{aligned} $$
(69)
$$\begin{aligned} L_{55} & = \frac{{D_{22} }}{R(x)}\frac{{\partial^{2} }}{{\partial \theta^{2} }} + D_{66} R(x)\frac{{\partial^{2} }}{{\partial x^{2} }} + D_{66} \sin \alpha \frac{\partial }{\partial x} \\ & \quad - \frac{{D_{66} \sin^{2} \alpha }}{R(x)} - K_{\text{s}} A_{44} R(x) - I_{3} R(x)\frac{{\partial^{2} }}{{\partial t^{2} }} + I_{3} \varOmega^{2} R(x) \\ \end{aligned}$$
(70)

The elements of matrix P which is presented in Eq. (32) can be obtained in the following types for cross-ply symmetric laminate scheme:

$$P_{11} = - A_{11} R(x)\frac{\partial }{\partial x} - A_{12} \sin \alpha$$
(71)
$$P_{12} = - A_{12} \frac{\partial }{\partial \theta }$$
(72)
$$P_{13} = - A_{12} \cos \alpha + A_{31} R(x)\frac{\partial }{\partial t}$$
(73)
$$P_{44} = - D_{11} R(x)\frac{\partial }{\partial x} - D_{12} \sin \alpha$$
(74)
$$P_{45} = - D_{12} \frac{\partial }{\partial \theta }$$
(75)

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Mohammadrezazadeh, S., Jafari, A.A. Active vibration control of rotating laminated composite truncated conical shells through magnetostrictive layers based on first-order shear deformation theory. J Braz. Soc. Mech. Sci. Eng. 42, 304 (2020). https://doi.org/10.1007/s40430-020-02363-w

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