Stability and bifurcations in oscillations of composite laminates with curvilinear fibres under a supersonic airflow
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ORIGINAL PAPER
Stability and bifurcations in oscillations of composite laminates with curvilinear fibres under a supersonic airflow Hamed Akhavan
· Pedro Ribeiro
Received: 28 January 2020 / Accepted: 21 July 2020 © Springer Nature B.V. 2020
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, quasiperiodic 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 largeenough 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 H. Akhavan (B) · P. Ribeiro INEGI/DEMec, Universidade do Porto, 4200-465 Porto, Portugal e-mail: [email protected] P. Ribeiro e-mail: [email protected]
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. Keywords Bifurcations · Stability · Nonlinear flutter · Composite laminates · Curvilinear fibres List of symbols a, b c D E1, E2 G 12 , G 13 , G 23 h M Ni (x, y) p q qi (t)
Length and width of the plate, respectively (m) 4/3h 2 (m−2 ) E1 h 3 flexural rigidity of the plate 12(1−ν 2 ) (N m) Major and minor Young’s moduli, respectively, in material coordinates 1 and 2 (GPa) Shear moduli, in material coordinates 1, 2 and 3 (GPa) Thickness of the plate (m) Mach number Shape function vectors (i = u, v, w, φx and φy ) Aerodynamic pressure (Pa) 2 ρ∞ U ∞ 2
dynamic pressure (Pa) Generalized displacement vectors (i = u, v, w, φx and φy )
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H. Akhavan, P. Ribeiro
t T0 , T1 T, T U∞ u, v, w u 0 , v 0 , w0 W x, y, z β λ
Time (s) Characteristic angles of the fibre path (◦ ) Period and its initial guess (s) Free-stream velocity (m/s) Displacement components in x, y, z directions (m) Displacement components at the midplane (m) LCO amplitude (m) Cartesian coordinate system √ M2 − 1 2qa 3 β D non-dimensional dyna
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