Fluid-structure interactions of flexible structures in different fluid flow regimes

The research presented in this dissertation focuses on fluid-structure interactions of thin flexible structures for three different problems, namely, the aeroelastic flutter of webs and ribbons, the nonlinear aerodynamic damping of slender beams, and the gas damping of microcantilevers. All these problems involve flexible structures at different length scales resulting in different physical regimes for the surrounding fluid flows, which are modeled using simplified fluid flow models. The first part of this dissertation presents a theoretical investigation of the aeroelastic flutter of tensioned wide webs and narrow ribbons commonly used in the paper-handling, textile, and sheet-metal industries. The web or ribbon is modeled as a uni-axially tensioned Kirchhoff plate with vanishingly small bending stiffness and is submerged in an incompressible inviscid fluid flow across its free edges resulting in a coupled non-conservative dynamical system with gyroscopic and circulatory terms. Wide webs mainly destabilize through a divergence instability due to cross-flow-induced, conservative centrifugal effects, and for certain values of applied tension, destabilize via a weak flutter instability, due to the wake-induced non-conservative effects. Contrarily, narrow ribbons in cross flow exhibit either flutter or divergence instability depending on the value of applied tension. Wind tunnel experiments, conducted to qualitatively corroborate these theoretical results, were inconclusive due to the lack of sufficient control over the important physical parameters. Nonetheless, the experiments show interesting dynamical phenomena such as simultaneous occurrence of oscillatory and zero frequency response of ribbons. The second part of this dissertation focuses on the nonlinear aerodynamic damping of slender, sharp-edged beams commonly found in flapping wings of micro-air-vehicles (MAVs), piezoelectric fans and insect wings. When such structures oscillate at moderate to large non-dimensional frequencies with large amplitudes comparable to their widths, vortex-shedding from the beam\u27s sharp edges gives rise to nonlinear aerodynamic damping. In this work, a general theory is developed to predict the amplitude and frequency dependence of this aerodynamic damping by coupling the structural motions to an inviscid, incompressible fluid. The fluid-structure interaction model developed here accounts for separation of flow and studies vortex-shedding-induced aerodynamic damping in slender, sharp-edged beams for different values of the Keulegan-Carpenter number and the non-dimensional frequency parameter. The theoretical predictions are validated against carefully performed experiments with piezoelectric fans under vacuum and atmospheric conditions. The third part of this dissertation studies the gas damping of microcantilevers oscillating in different vibration modes in an unbounded gas at low ambient pressures varying over 6 orders of magnitude. The accurate prediction of gas damping of microcantilevers at low ambient pressures is essential for improving the sensitivity of microcantilever-based sensors and improving the efficiency of microresonators. In this work, solutions of a sub-continuum, quasisteady Boltzmann equation with a simplified ellipsoidal statistical Bhatnagar-Gross-Krook (ES-BGK) collision operator are used to provide a closed-form fit, which can be used to predict the gas damping of different microcantilever vibration modes. The fit is uniformly valid over 5 orders of magnitude of the Knudsen number and spans the free-molecular, the transition, and the lower pressure side of the slip flow regime. For the higher pressure side of the slip flow regime, this work proposes a boundary-integral-method-based approach for including the slip boundary condition in existing continuum regime models. Detailed experimental data on gas damping of silicon microcantilevers obtained from research collaborators shows excellent agreement with the predictions of the ES-BGK-model-based fit