Mech481 Wind Energy: Aerodynamics Assignment Answers

Assignment Title: Structural Response to Unsteady Loads

The goal of this assignment is to explore some of the concepts that we learned in the lectures in more depth. There are two parts to the assignment. The first is fairly simple: you will add the mass element matrices to the bar code and analyze the natural frequencies. In the second part, you will do several more simulations of the cantilever-beam airfoil to better understand how the aerodynamic loads affect the structural dynamics of the beam.

Problem 1: Calculate natural frequencies of a 2-bar linkage

Objective: Implement the theory presented in the “Modeling Structures” lecture

What to hand in:

• A write-up summarizing your resulting natural frequencies and the requested observations. Thiswrite-up must be sufficient resolution/font size that we can read all relevant text (including figure labels).
• The Matlab code you wrote to generate the solutions.

Problem:

Part 1. In the problem_1.m file provided with the assignment, add in the mass matrices. The body/element mass matrix can be found in the lecture slides.

Part 2. Calculate the natural frequencies (in Hz) of a 2-bar linkage with the following structural properties:

	E [Pa]	A [m2]	L [m]	rho [kg/m3]
Bar 1	200	0.5	10	1
Bar 2	170	0.7	8	0.7

 

Part 3. If you double the stiffness of Bar 1, how do you expect the natural frequencies to change and why? Verify your hypothesis using the code.

Part 4. If you were instead to take the parameters in Part 2 and then double the density of Bar 2, how would you expect the natural frequencies to change? Verify your hypothesis using the code.

Problem 2: Exploration of aeroelastic responses

Objective: A deeper understanding of the cantilever-beam airfoil problem

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Description of the beam-airfoil:

• Same as in “Simulating Unsteady Aerodynamic Loads” lecture

• Cantilever, Euler-Bernoulli beam with two degrees of freedom: deflection and rotation at tip of blade

• FFA-W3-241 airfoil with optional S. Øye dynamic stall model

• Aerodynamic force applied as a tip load (very simplistic assumption)

What to hand in:

• A write-up with pictures and words that answers the questions in the text below. Someone who is not familiar with the question should be able to follow your analysis/observations purely from your writing. This write-up must be sufficient resolution/font size that we can read allrelevant text (including figure labels).

• Any code you modified for this assignment.

Problem:

Part 1. Changing the angle at which the beam is mounted to the wall (Sims. A and B)

The airfoil beam is mounted to the wall with a certain angle ????_geom, which is not shown in the diagram above. Using problem_2.m, simulate two different mounting angles (????_geom = 20 degrees and ????_geom = 5 degrees) with the following options: steady wind at 2 m/s, an initial tip displacement/angle of 0.06 m/0 degrees, and no dynamic stall. Describe the difference in simulated tip deflections for the two mounting angles. Can you explain why? Provide plots to support your explanations.

Part 2. Comparing no dynamic stall with dynamic stall (Sims. A and C)

Set ????_geom to 20 degrees, but now turn on dynamic stall. Simulate the response to steady wind with 2 m/s and plot the tip deflection. Extract C_L(t ) and ???? (t ) for [0.2 s, 2 s]. Plot versus for that range and compare to Simulation A. Describe the plot. Whatcharacteristics of this plot explain the differences in time-domain behavior? (Think in terms of what we learned in the previous part.)

Part 3. Dynamic stall (and none) in unsteady wind (Sims. D, E, F, G, H, and I)

Using the parameters for Simulations D through G, simulate the system behaviour for 15 seconds, then throw out the first 5 seconds to get rid of transience.

Plot the time traces of the tip displacements with and without dynamic stall on top of one another, and then plot the time traces of C_Lversus ???? with and without dynamic stall. What do you see? How different are the time traces with and without dynamic stall? Give your theories on why the differences are so little.

Propose a new set of simulation parameters that should result in a more significant difference between simulations with and without dynamic stall. Run these simulations and verify your hypothesis.

Aerodynamics and aeroelasticity of wind turbines

Structural Response to Unsteady Response

General Objectives

• To explore the concepts on structural response to unsteady response learnt in lectures in more depth.

Problem 1: Calculating Natural Frequencies of A 2-Bar Linkage

Objectives

• To Implement the theory presented in the “Modeling Structures” lecture

Procedure

1. The mass matrices were added to the problem_1.m file provided with the assignment.
2. We saved the changes and run the program to obtain the natural frequencies (in Hz) of a 2-bar linkage with the following structural properties:

      Figure 1: Data set for problem 1

Matlab code:

Results:

Natural frequencies for bar 1 and bar 2 are obtained as 0.2041Hz and 0.8141Hz respectively.

3. We doubled the stiffness for bar 1 and run the code.

Matlab code:

Results and discussion:

When we double the stiffness of bar 1 we would expect the natural frequency to increase. The natural frequencies for bar 1 and bar 2 were obtained as 0.2761 Hz and 0.8508 Hz respectively. This is because the system oscillates more.

4. We doubled the density of bar two and ran the code:

Matlab code:

Results and discussion:

When we double the density of bar 2 we would expect the natural frequency to decrease. The natural frequencies for bar 1 and bar 2 were obtained as 0.1541 Hz and 0.6512 Hz respectively.

Problem 2: Exploration of Aeroelastic Responses

Objectives

• To obtain a deeper understanding of the cantilever-beam airfoil problem.

Procedure

1. Changing the angle at which the beam is mounted to the wall (Sims. A and B)

The airfoil beam is mounted to the wall with a certain angle alfageon=20, which is not shown in the diagram above. We used the mfile-problem_2.m, to simulate two different mounting angles (alfageon =20 degrees and alfageon =5 degrees) with the following options: steady wind at 2 m/s, an initial tip displacement/angle of 0.06 m/0 degrees, and no dynamic stall. We were to describe the difference in simulated tip deflections for the two mounting angles.

Matlab code:

Results and Discussion:

Figure 2: Plot deflection for the air foil with two different displacement angles 20 and 5 for A and B respectively

From the plots shown it can be observed that the deflection of A is huge compared to that B this is because the air-foil beam in A is subjected to a lot of wind turbulence hence resulting to deflections as shown above.

2. Comparing no dynamic stall with dynamic stall (Sims. A and C). Set alfa-Geon to 20 degrees, but now turn on dynamic stall. Simulate the response to steady wind with 2 m/s and plot the tip deflection. Extract C_L (t) and alpha for [0.2 s, 2 s]. Plot C_Lversus alpha for that range and compare to Simulation A. Describe the plot. What characteristics of this plot explain the differences in time-domain behavior?

Matlab code:

Results and discussion:

Figure 3: Matlab plot of coefficient versus alpha

From the plots shown above it can be observed that the plot for coefficient of lift against alpha for dynamic stall section air-foil beam is quite steady compared to that of a static air-foil beam. This is because the former is able to adjust the angle of attack hence smoothening the coefficient of lift.

3. Dynamic stall (and none) in unsteady wind (Sims. D, E, F, G, H, and I). Using the parameters for Simulations D through G, simulate the system behavior for 15 seconds, then throw out the first 5 seconds to get rid of transience. Plot the time traces of the tip displacements with and without dynamic stall on top of one another, and then plot the time traces of C_Lversus alpha with and without dynamic stall. What do you see? How different are the time traces with and without dynamic stall? Give your theories on why the differences are so little. Propose a new set of simulation parameters that should result in a more significant difference between simulations with and without dynamic stall. Run these simulations and verify your hypothesis.

We proposed that angle of mounting of the airfoil beam should also be varied to obtain the variation between the two sets.

References.

Fortmann, J. (2015). Modeling of Wind Turbines with Doubly Fed Generator System. Wiesbaden, Springer Fachmedien Wiesbaden. Available from: https://link.springer.com/book/10.1007/978-3-658-06882-0. [Accessed Date: 2^nd May 2018].

Hansen, M. O. L. (2015). Aerodynamics of wind turbines. London, [England], Earthscan.

Ho?Lling, M., Peinke, J., & Ivanell, S. (2014). Wind energy-- impact of turbulence. Available from: https://site.ebrary.com/id/10845476. [Accessed Date: 2^nd May 2018].

Matha, D. (2011). Challenges in simulation of aerodynamics, hydrodynamics, and mooring-line dynamics of floating offshore wind turbines. Golden, CO, National Renewable Energy Laboratory, U.S. Dept. of Energy, Office of Energy Efficiency and Renewable Energy. Available from: https://purl.fdlp.gov/GPO/gpo16404. [Accessed Date: 2^nd May 2018].