On the direct determination of the eigenmodes of finite flow–structure systems

Abstract

A new method for directly determining the eigenmodes of finite flow-structure systems is presented using the classical problem of the interaction of a uniform incompressible flow with a flexible panel, held at both ends, as an exemplar. The method is a hybrid of theoretical analysis and computational modelling. This method is contrasted with Galerkin and travelling-wave methods, which are most often used to study the hydroelasticity of such systems. The new method does not require an a priori approximation of perturbations via a finite sum of modes. Instead, the coupled equations for the wall-flow system are used to derive a single matrix equation for the system that is a second-order differential equation for the panel-displacement variable. This is achieved in this exemplar by applying a combination of boundary-element and finite-element methods to the discretized system. Standard statespace methods are then used to extract the eigenmodes of the system directly. We present the results for the stability of the case of an unsupported flexible plate, elucidating its divergence and flutter characteristics, and the effect of energy dissipation in the structure. We then present the results for the case of a spring-backed flexible plate, showing that its motion is dominated by travelling waves. Finally, we illustrate the versatility of the approach by extracting the stability diagrams and modes for a panel with spatially varying properties and a panel with non-standard boundary conditions. In doing so, we show how spatial inhomogeneity can modify the energy exchanges between flow and structure, thereby introducing a single-mode flutter instability at pre-divergence flow speeds.

Interaction between a cantilevered-free flexible plate and ideal flow

“…Pitman and Lucey (2009) showed that the introduction of structural damping prevents exact coalescence in the panel-flutter problem but the system continues to yield what remains essentially a singlefrequency response comprising a highly amplifying and a highly attenuated pair of roots. Of course, only the amplifying root would have significance in a physical system.…”

“…Global predictions of the infinite-time system behaviour are generated using a standard state-space method, implemented in a similar way to that detailed by Pitman and Lucey (2009). As applied in this paper, we do not incorporate the effects of the downstream wake nor of an unsteady free-stream.…”

“…The assembled system is then used to conduct a variety of numerical simulations, the results of which map out the response space of the system. However, we also use the computational model to extract directly the eigenmodes of the fluid-structure system using the state-space methods developed by Pitman and Lucey (2009). An equivalent approach was adopted by Argentina and Mahadevan (2005), although they made simplifying assumptions in their flow model in order to develop a tractable system equation.…”

Controlling aero-elastic instability of curtain wall systems in high-rise buildings

“…It is well known, e.g. Dugundji et al (1963), Weaver & Unny (1971), Ellen (1973), Lucey & Carpenter (1993), Pitman & Lucey (2009), that, for an initially flat plate, low-amplitude deformations generate a pressure field that, at critically high water or wind speeds, applies a force that amplifies the plate deformation. This is best known as the problem of panel flutter.…”

Global instabilities and transient growth in Blasius boundary-layer flow over a compliant panel

“…However the methods developed could readily be used to analyse interactions between Falkner-Skan type boundary-layer profiles when a non-zero pressure gradient determines the mean flow. The approach builds from the global stability analysis of Pitman and Lucey (2009) ;Burke et al (2014) for external and channel potential flow and Pitman and Lucey (2010) for Poiseuille flow interacting with a flexible panel. In the present development, we extend the velocity-vorticity formulation of the flow equations and combine it with a generalized Helmholtz decomposition (Wu and Thompson (1973); Kempka et al (1995)) to investigate the global asymptotic and transient behavior of the FSI system.…”