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Stability and bifurcations in oscillations of composite laminates with curvilinear fibres under a supersonic airflow

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Abstract

Nonlinear flutter of variable stiffness composite plates—using a reduced-order model benefiting from a large-enough number of modes or degrees of freedom—is the subject of this research. Dynamic of such plates is enriched by different scenarios including period-n (un)stable limit cycle oscillations, quasi-periodic or chaotic solutions, along with Hopf, symmetry breaking and period doubling bifurcations. The fibre path orientation, in any individual ply of these rectangular plates, changes linearly from the left edge to the right one. The plates are subjected to a supersonic airflow. The plate’s displacement field is modelled using a third-order shear deformation theory; a p-version finite element is used to discretize the displacement components. Aerodynamic pressure due to the airflow is approximated using linear piston theory. The self-exciting vibrational full-order model (FOM) is formed by employing the principle of virtual work, and then, the ROM is extracted from the FOM using two reduction techniques, namely static condensation and modal summation method. The inclusion of a large-enough number of modes of vibration in vacuum in the latter technique is critical for the observation of, often overlooked, complex dynamic behaviour of these plates. The equations of motion representing the ROM are solved using the shooting method (to obtain stable and unstable LCOs) and using Newmark method (for stable or non-oscillatory solutions). The rich dynamics of these plates are explored with the help of vibration and flutter mode shapes, phase-plane plots, bifurcation diagrams, fast Fourier transform diagrams and Poincaré sections.

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Abbreviations

a, b :

Length and width of the plate, respectively (m)

c :

\(4/3h^2\) (\(\hbox {m}^{-2}\))

D :

\(\frac{E_1 h^3}{12\left( 1-\nu ^2\right) }\) flexural rigidity of the plate (N m)

\(E_1,E_2\) :

Major and minor Young’s moduli, respectively, in material coordinates 1 and 2 (GPa)

\(G_{12},G_{13},G_{23}\) :

Shear moduli, in material coordinates 1, 2 and 3 (GPa)

h :

Thickness of the plate (m)

M :

Mach number

\({{\mathbf {N}} ^{\mathbf {i}}\left( x,y\right) }\) :

Shape function vectors (\({\mathbf {i}}= {\mathbf {u}}\), \({\mathbf {v}}\), \({\mathbf {w}}\), \({\phi }_{\mathbf {x}}\) and \({\phi }_{\mathbf {y}}\))

p :

Aerodynamic pressure (Pa)

q :

\(\frac{\rho _\infty U^2_\infty }{2}\) dynamic pressure (Pa)

\({\mathbf {q}}_{\mathbf {i}}\left( t\right) \) :

Generalized displacement vectors (\({\mathbf {i}}= {\mathbf {u}}\), \({\mathbf {v}}\), \({\mathbf {w}}\), \({\phi }_{\mathbf {x}}\) and \({\phi }_{\mathbf {y}}\))

t :

Time (s)

\(T_0,T_1\) :

Characteristic angles of the fibre path (\(^{\circ }\))

\(T,{\overline{T}}\) :

Period and its initial guess (s)

\(U_\infty \) :

Free-stream velocity (m/s)

u, v, w :

Displacement components in x, y, z directions (m)

\(u^0, v^0, w^0\) :

Displacement components at the mid-plane (m)

W :

LCO amplitude (m)

x, y, z :

Cartesian coordinate system

\(\beta \) :

\(\sqrt{M^2-1}\)

\(\lambda \) :

\(\frac{2qa^3}{\beta D}\) non-dimensional dynamic pressure

\(\mu \) :

\(\frac{a \rho _\infty }{h \rho }\)

\(\nu \) :

Poisson’s ratio

\(\phi {}_x, \phi {}_y\) :

Independent rotations of normal to the middle surface about y and x axes

\(\rho _\infty \) :

Ambient density of air (kg/\(\hbox {m}^3\))

\(\rho \) :

Density of the plate (kg/\(\hbox {m}^3\))

\(\theta \left( x\right) \) :

Angle of the reference fibre path with respect to x axis

\(\varOmega \) :

Surface of the plate

T :

Transpose of matrix or vector

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Funding

This research was carried out in the framework of project No 030348, POCI-01-0145-FEDER-030348, “Laminated composite panels reinforced with carbon nanotubes and curvilinear carbon fibres for enhanced vibration and flutter characteristics”, funded by FEDER, through Programa Operacional Competitividade e Internacionalização—COMPETE 2020, and by National Funds (PIDDAC), through FCT/MCTES. The authors gratefully acknowledge the funding provided.

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  1. INEGI/DEMec, Universidade do Porto, 4200-465, Porto, Portugal

    Hamed Akhavan & Pedro Ribeiro

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  1. Hamed Akhavan
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Akhavan, H., Ribeiro, P. Stability and bifurcations in oscillations of composite laminates with curvilinear fibres under a supersonic airflow. Nonlinear Dyn 103, 3037–3058 (2021). https://doi.org/10.1007/s11071-020-05838-6

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