An Improved Preliminary Design Tool for Turbomachinery Blades Using Van der Pol Based Reduced-Order Model for Non-Synchronous Vibrations

Abstract

Abstract Starting from an oscillating cylinder, this paper navigates the shift of a Van der Pol based Reduced-Order Model from that simple case to airfoil and turbomachinery blade. The structure will be free to vibrate under the influence of a von Karman vortex street. The first degree of freedom tracks the oscillatory motion of the vortex shedding utilizing a combined Van der Pol and a Duffing equation, previously shown to better match the flow physics. The other degree of freedom is a cylinder or pitching blade with stiffness, mass, and damping, making the coupled system one of fluid-structure interaction. A third degree of freedom is added in a separate configuration to provide the airfoil with both pitching and plunging motion. Using this model to study the time-history of the fluid and the structure oscillation, additional parameters are extracted to understand the underlying mechanisms of frequency lock-in and limit cycle oscillation. The coefficients in the model are tuned to match limit cycle oscillation data captured in experiments. Then, the phase shift between the vortex shedding and the structural motion is calculated when the former locks-on to the latter, and then unlocks. Also, the work done per cycle of vibration is analyzed from the contributions of the mass, spring, and damping forces to determine the dominant contributor when locking-in versus unlocking. This model is a more accurate case for turbomachinery applications compared to the previous models. This model, especially with the third degree of freedom, will serve as the preliminary design tool for engine manufacturers to use for preventing Non-Synchronous Vibrations in turbomachinery.