Falling dynamics of flexible plates with concentrated weight

We investigate the free fall of a flexible plate with a concentrated central weight using numerical simulations to explore the coupling effects of structural flexibility and concentrated loading. Two-dimensional and three-dimensional (3-D) simulations, combining lattice Boltzmann and immersed boundary methods, identify four distinct falling modes: swing, stable falling, fluttering and tumbling. The swing and stable falling modes are reported for the first time in this context. By analysing the balance between bending stiffness upper K <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> mml:miK</mml:mi> </mml:math> $K$ and concentrated weight upper G <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> mml:miG</mml:mi> </mml:math> $G$ , we introduce a new dimensionless effective stiffness italic Kg equals upper K divided by upper G <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> mml:mrow <mml:mtext mathvariant="italic" class="MJX-tex-mathit">Kg</mml:mtext> </mml:mrow> mml:mo=</mml:mo> mml:miK</mml:mi> mml:mrow mml:mo/</mml:mo> </mml:mrow> mml:miG</mml:mi> </mml:math> $\textit{Kg} = K/G$ , which governs mode transitions and successfully collapses the kinematic data. A bistable transition zone between fluttering and tumbling is observed, with the mode selection dependent on the initial inclination angle, which scales as a power law of italic Kg <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> mml:mrow <mml:mtext mathvariant="italic" class="MJX-tex-mathit">Kg</mml:mtext> </mml:mrow> </mml:math> $\textit{Kg}$ . Additionally, within the swing mode, we observe a transition in wake topology from a 2P to a 2S vortex-shedding pattern, modulated by the mass ratio. In 3-D simulations, the finite aspect ratio upper A Subscript r <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> mml:msub mml:miA</mml:mi> mml:mir</mml:mi> </mml:msub> </mml:math> $A_r$ significantly influences mode selection due to pressure leakage at the side edges. While the italic Kg <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> mml:mrow <mml:mtext mathvariant="italic" class="MJX-tex-mathit">Kg</mml:mtext> </mml:mrow> </mml:math> $\textit{Kg}$ scaling remains valid for the global dynamics, 3-D effects suppress tumbling in narrow plates and induce spanwise oscillations in the stable falling mode at higher aspect ratios.