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Nonlinear vibration of a deploying laminated Rayleigh beam with a spinning motion in hygrothermal environment

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Abstract

The free vibration of a deploying laminated beam with spinning motion in hygrothermal environment is studied. The model of this system is given within the framework of the Rayleigh beam theory and the von Karman nonlinear strain theory. The nonlinear dynamic equilibrium equation of the cantilever boundary condition is established based on Hamilton's principle with considering the combined effects of the axial motion, transverse vibration, spinning motion, and hygrothermal environment. The frequencies of the system are numerically determined by using the Newmark method. The detailed parameter analyses are provided to investigate the effects of the deploying speed, the spinning speed, the temperature variation, the moisture concentration, and the fiber direction angle on frequency and the corresponding sensitivity.

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Abbreviations

L :

Total length of the beam

\(l\left( t \right)\) :

Length outside of the hub is a function of time

\(V_{x}\) :

Axial deploying speed in x-direction

\(\Omega\) :

Spinning speed

\(l_{0}\) :

Initial length outside of the hub

\(r_{n}\) , \(r_{w}\) :

Inner and outer diameters of the annulus

\(n\) :

Number of the composite material layers

\(\eta\) :

Fiber orientation angle

\(\Delta T\), \(\Delta C\) :

Variation of the temperature and moisture concentration

\(\overline{Q}_{11i}\) :

Reduced rigidity of the i-th layer

\(\alpha_{1}\) , \(\alpha_{2}\) :

Thermal expansion coefficients

\(\beta_{1}\) , \(\beta_{2}\) :

Moisture expansion coefficients

\(\rho\) :

Material density

\(r_{i}\) , \(r_{i + 1}\) :

Inner and outer radii of the i-th layer

\(I_{m}\) :

Mass per unit length of the beam

\(I_{d}\) :

Diametrical mass of inertia

\(I_{p}\) :

Polar mass of inertia

N :

Galerkin truncation order

\(\beta_{m}\) , \(\beta_{p}\) :

m/p-Th solutions of the frequency equation

\(\theta\) :

Rotation angle

\(\lambda\) :

Eigenvalues

\({\mathbf{M}}_{u}\) , \({\mathbf{M}}\) :

Mass matrix in axial and transverse direction

\({\mathbf{C}}_{u}\) , \({\mathbf{C}}\) :

Damping matrix in axial and transverse direction

\({\mathbf{K}}_{u}\) , \({\mathbf{K}}^{{\mathbf{L}}}\) :

Linear stiffness matrix in axial and transverse direction

\({\mathbf{K}}^{{{\mathbf{NL}}}}\) :

Nonlinear stiffness matrix in transverse direction

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 11872319). The support from Doctoral Innovation Fund Program of Southwest Jiaotong University (No. DCX201739) is gratified as well.

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  1. School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu, 610031, People’s Republic of China

    L. Wang, J. Yang & Y. H. Li

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  1. L. Wang
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Correspondence to Y. H. Li.

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Wang, L., Yang, J. & Li, Y.H. Nonlinear vibration of a deploying laminated Rayleigh beam with a spinning motion in hygrothermal environment. Engineering with Computers 37, 3825–3841 (2021). https://doi.org/10.1007/s00366-020-01035-6

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