1. 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

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

3. 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

4. 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.”

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

7. 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

9. 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

10. Linear Solution
***

11. 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)

12. Nonlinear Solution
***

13. Hysteresis Curve

14. Hysteresis Split

15. 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
******