With Andrew Magstadt and Dr. David Thayer Computational Analysis of a Nonlinear Driven Damped Harmonic Oscillator Kristy Katein-Taylor With Andrew Magstadt and Dr. David Thayer University of Wyoming

Overview Brief Overview of Project Definition of Key Aerodynamic Principles Description of Research Mathematical Models Results and Future Research Opportunities

Project Description Effect of Blade Design on Efficiency of Wind Turbines Sub-project to analyze model of harmonic oscillator, e.g. a mass on a spring

Definition of Key Aerodynamic Principles Aerodynamic Forces: Lift vs. Drag Angle of Attack α Normal Stall vs. Dynamic Stall Rapid variations in α Hysteresis Lift: force normal to velocity vector; drag: force parallel to velocity vector. Reynolds number: “named for Osborne Reynolds, who showed in the 1880s that this group governs the role of viscous forces in the flow of liquids or gases. In particular, it controls the transition between laminar and turbulent.” (Basic Aero book, page 29) Mach number: “…was identified as a key parameter in describing the effects of compressibility in aerodynamic flows.”

Original Wind Tunnel Model Ongoing project with Composite Materials Research Group (CMRG) Research regarding dynamic stall, effect of airfoil design ***See above***

Experimental Problem How physical can we make this model? The windmill blades will experience torsional vibration in addition to the change in α Need to find a way to model this vibration Solution: nonlinear spring coefficient Better explanation of why the spring is added

Mathematical Models Φ +2𝜉𝜔 Φ + 𝜔 2 Φ= 𝐹 0 𝑠𝑖𝑛 𝜔 𝑓 𝑡 +𝑄(𝑡) 𝜔= 𝑘 𝐼 One-dimensional harmonic oscillator in α = Φ Φ +2𝜉𝜔 Φ + 𝜔 2 Φ= 𝐹 0 𝑠𝑖𝑛 𝜔 𝑓 𝑡 +𝑄(𝑡) 𝜔= 𝑘 𝐼 Φ = 𝜕Φ 𝜕 𝑡

Linear Solution ***

Nonlinear k term 𝑘 𝜃 = 𝑟 1 𝑘 sin⁡(𝜃) 𝜃 𝑥 ***see above*** 𝑥= 𝑟 2 − 𝑟 1 + 𝑟 1 𝑟 2 𝑠𝑖𝑛 2 (𝜃)

Nonlinear Solution ***

Hysteresis Curve

Hysteresis Split

Future Research Opportunities Comparison to hysteresis curves from unsteady model or quasi-steady model with Theodorsen’s function applied Improvement of future dynamic stall models, and wind foil analysis and design Potential for further analysis into the aerodynamics of the windfoil Calculation of variables of interest, such as power generation of the turbine ******