Exercise 02 Part2 Aileron Deflection solutions PDF

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Solutions of the exercise 2, part 2 about aileron deflection...

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Wind Energy Institute Prof. Dr. Carlo L. Bottasso

Technische Universität München

Aileron Deflection and Reversal Part 2 Marta Bertelè, Carlo R. Sucameli E-Mail: [email protected] [email protected] The assignment aims at helping the student fully understand the concept of aileron deflection and reversal and program with Matlab: the analysis of the static behaviour of the given system will be performed with the aid of this software.

Task 1: Matlab implementation Let’s perform a critical analysis of the static behaviour of an elastic system with the help of Matlab. From Moodle you can download: -

-

The .dat files containing the airfoil properties. Each file represents the values for the lift and moment coefficients (about the AC) respectively as a function of the angle of attack (first column) and of the aileron deflection angle (first row); A pre-written MainGuided.mat file that can be used as backbone for your script.

Zero lift line

𝛼𝑟

Structural deformation

𝜃

𝐾

Aerodynamic center

𝑒𝑐

Elastic axis

𝛿

𝛼𝑟 Figure 1: Elastic system

Study the behavior of a system as in Fig. 1 (able to twist about the elastic axis EA and provided with an aileron) with the following properties: 

airfoil GA(W)-1: the aerodynamic properties can be found in GA(W)-1_CL.dat and GA(W)1_Cm.dat;

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Wind Energy Institute Prof. Dr. Carlo L. Bottasso



structural properties: o o o o

Technische Universität München

stiffness of the torsional spring: 𝐾 = 218 000 [

𝑁𝑚

𝑚 𝑟𝑎𝑑

]

airfoil chord: 𝑐 = 1.7 [𝑚] distance between aerodynamic center and elastic axis: 𝑒 = 0.145 [−] geometric angle of attack: 𝛼𝐺 = 3.5°.

Task 1.1: Linear regime Let’s assume now a linear relation between lift coefficient and of attack and aileron deflection. Using Matlab: 1) Plot the aerodynamic properties as a function both of angle of attack and aileron deflection and approximatively identify the linear region. Solution:

The linearity region can be identified looking for the range of angle of attack and deflection for which both 𝐶𝐿 and 𝐶𝑀𝐴𝐶 are constant. The linearity region is here considered to be 𝛼 ∈ [−6 ,6] and 𝛿 ∈ [−16 ,16]. Remark: the following results are computed based on this choice.

2) Compute the following parameters of the simplified linear aerodynamic model: a. 𝛼0 b. 𝐶𝐿𝛼 c. 𝐶𝐿0 d. 𝐶𝐿𝛿 Page 2 of 7

Wind Energy Institute Prof. Dr. Carlo L. Bottasso

e. 𝐶𝑀𝐴𝐶 f.

𝐶𝑀𝐴𝐶

Technische Universität München

𝛿

Solution: 𝐶𝐿𝛼 = 5.7014 [𝑟𝑎𝑑 −1 ] 𝐶𝐿0 = 0.2868 [– ]

𝐶𝐿𝛿 = 2.3048 [𝑟𝑎𝑑 −1 ] 𝛼0 = −2.8824 [𝑑𝑒𝑔] 𝐶𝑀𝐴𝐶 = −0.0761 [−] 𝐶𝑀𝐴𝐶 = −0.4381 [𝑟𝑎𝑑 −1 ] 𝛿

3) Compute divergence and reversal dynamic pressure, 𝑞𝐷 𝑎𝑛𝑑 𝑞𝑅 . Solution: 𝑞𝐷 =

𝐾

𝑒𝑐 2 𝐶𝐿𝛼

= 91 244.892 [𝑃𝑎], 𝑞𝑅 =

𝐾𝐶𝐿

𝛿

𝑐 2 𝐶𝑀𝐴𝐶 𝐶𝐿𝛼 𝛿

= 69 603.924 [𝑃𝑎]

4) Assuming 𝛿 = [−5 0 5]°, i.e. for three different deflection configurations, compute and plot the structural torsional deflection and the elastic lift as a function of the dynamic pressure. Solution:

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Wind Energy Institute Prof. Dr. Carlo L. Bottasso

Technische Universität München

𝑞 1− ⁄ 𝑞𝑅

5) Plot now the Aileron Effectiveness, i.e. 𝐴𝐸 =1−𝑞 Solution:

⁄𝑞𝐷

From the plot, it can be easily noticed that when 𝑞 = 𝑞𝑅 , AE is equal to 0. This parameter shows therefore whether we are incurring in aileron reversal or not: when 𝐴𝐸 < 0 aileron reversal occurs.

Tips:

1) To plot the coefficients both with respect to 𝛿 and 𝛼, you need to create a 3D plot. To do so, you can use the function mesh. mesh(X,Y,Z): where Z is a matrix representing a 2D function of X and Y, which in turn are the matrices containing the coordinates for which Z is known. Other options such as ‘FaceColor’, ‘EdgeColor’ etc can be added to the input to modify the colors of the plot. 2) To extract the airfoil properties, you can either use the least-squares method or the function fit. In both cases, you have to express the relation between angles and coefficients as follows: 𝑦 = 𝑎 ∙ 𝑥 + 𝑏 ∙ 𝑧 + 𝑐, even if your coefficient y is expressed in matrix form as a function of x and z. How can we rewrite our matrix in order to obtain the previous formulation? Here an example. The coefficient matrix Y is scheduled with respect to x row-wise and to z column-wise as follows

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Wind Energy Institute Prof. Dr. Carlo L. Bottasso

Technische Universität München

𝑥1 ⋮

𝑥𝑛

[

𝑧1

𝑌𝑥1 𝑧1 ⋮ 𝑌𝑥𝑛 𝑧1

… 𝑧𝑚 … 𝑌𝑥1 𝑧𝑚 ⋮ ] ⋯ 𝑌𝑥𝑛 𝑧𝑚

To rearrange the matrix system into simple vectors we can put each column of Y below the previous one, such that 𝑌𝑥1 𝑧1 ⋮ 𝑌𝑥𝑛 𝑧1 𝑌𝑥1 𝑧2 𝑦= ⋮ 𝑌𝑥𝑛 𝑧2 𝑌𝑥1 𝑧𝑚 ⋮ [𝑌𝑥𝑛 𝑧𝑚 ] Now we coherently have to rearrange both x and z, creating vectors in which each entry is coherent with the values in y : 𝑥1 ⋮ 𝑥𝑛 𝑥1 𝑥 = ⋮ and 𝑧 = 𝑥𝑛 𝑥1 ⋮ [𝑥𝑛 ]

𝑧1 ⋮ 𝑧1 𝑧2 ⋮ 𝑧2 𝑧𝑛 ⋮ [𝑧𝑛 ]

𝑥1 the vector [ ⋮ ] will be repeated 𝑚-times, while each element of the vector [𝑧1 … 𝑧𝑚 ] 𝑥𝑛 will be repeated 𝑛-times. As a quick check, assume 𝑛 = 4 and 𝑚 = 3. The 7𝑡ℎ element of y will be 𝑌𝑥3 𝑧2 : if we

have a look at the 7𝑡ℎ element of both x and z we will find 𝑥 = 𝑥 3 , 𝑧 = 𝑧2 , which is indeed correct. To replicate the vectors you can use the function repmat(…).

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Wind Energy Institute Prof. Dr. Carlo L. Bottasso

Technische Universität München

Task 1.2: Non-linear regime Assume now that the lift coefficient non-linearly depends on the angle of attack. The lift (i.e. 𝐶𝐿 ) will now be expressed as a non-linear function of the angles 𝛼 and 𝛿 such as 𝐿 = 𝑞𝑐𝐶𝐿 〈𝛼, 𝛿 〉.

(1)

Since no linearity is now assumed, the stall behavior will be automatically included in our modelling. Using Matlab:

6) Assuming 𝛿 = [−5 0 5]°, i.e. for three different deflection configurations, compute and plot the structural torsional deflection as a function of 𝑞. Solution:

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Wind Energy Institute Prof. Dr. Carlo L. Bottasso

Technische Universität München

7) Assuming 𝛿 = [−5 0 5]°, i.e. for three different deflection configurations, compute and plot the elastic lift as a function of 𝑞. Solution:

Tips:

To interpolate the coefficients with respect to both 𝛿 and 𝛼 you can use the provided tables and interpolate the coefficients with the function interp2. Vq=interp2(X,Y,V,Xq,Yq ): “interpolates to find Vq, the values of the underlying 2-D function V

at the query points in matrices Xq and Yq. Matrices X and Y specify the points at which the data V is given. “ Therefore V is the matrix of the given coefficient, X and Y the matrices containing the values of 𝛿 and 𝛼 for which each entry in V is known. Xq and Yq represent the values of 𝛿 and 𝛼 for which we want to interpolate, i.e. compute the values of the coefficient.

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