Exam Experimental Research and Data Analysis (AE2223-II)

15 April 2014 14:00 - 17:00

Please note:

• This exam consists of 5 problems divided in a number of items

• For each problem: start with a new piece of paper and clearly write your name andstudent number on the top of each page. Pages without name and student number willnot be graded !

1. Strategies for experimental research (20 points)

Please provide only one option for the multiple choice questions a-e and only the numericalvalue for the open questions f-h.

a. 3pt

In designing an experiment, blocking is used:1. to reduce bias.2. to reduce variation.3. as a substitute for a control group.4. as a first step in randomization.5. to control the level of the experiment.

b. 3pt

Which of the following are important in the design of experiments?

I. Control of confounding variablesII. Randomization in assigning subjects to different treatmentsIII. Replication of the experiment using sufficient numbers of subjects

1. I and II2. I and III3. II and III4. I, II, and III5. None of the above gives the complete set of true responses.

c. 3pt

Consider a test assessing the validity of the null-hypothesis H0. The power of thistest is defined as:

1. P(reject H0 — H0 is false)2. P(fail to reject H0 — H0 is false)3. P(reject H0 — H0 is true)4. P(fail to reject H0 — H0 is true)

d. 3pt

A factorial design is one in which ....

1. Only one independent variable is studied to determine its effect on the dependentvariable

2. Only two independent variables are simultaneously studied to determine their in-dependent and interactive effects on the dependent variable

3. Two or more independent variables are simultaneously studied to determine theirindependent and interactive effects on the dependent variable

4. Two dependent variables are studied to determine their interactive effects

e. 2pt

Which one of the following statements is true?

1. the greater the difference in means, the greater the chance of a type II error2. the sample size does not affect the chance of a type II error3. the larger the sample size, the greater the chance of a type II error4. the larger the sample size, the smaller the chance of a type II error

f. 2pt

Replication.

If the standard deviation of the response variable of one test is equal to 3.0, then how bigis the standard deviation for the mean of the response variable of 100 such tests assuming aGaussian distribution?

g. 2pt

You plan to do an experiment with 4 factors, each with 3 possible levels, and 6 replications,using the Factorial strategy. How many different runs do we have in the experiment?

h. 2pt

You plan to do an experiment with 6 factors, each with 4 possible levels, and 3 replica-tions, using the One-factor-at-a-time strategy. What is the minimum number of differenttreatments (not runs) that we have in the experiment?

2. Space topic (20 points)

Please provide only the numerical value for questions a-c.

The position of a low earth orbiting satellite is obtained from GPS observations by the pointpositioning method.

a. 5pt

The weighted least-squares solution is given by the following equations for a certaintime instance:

x = (AT WA)−1AT Wy

where x represents the vector of estimated parameters, namely the position coordinates x,y and z, and the clock correction multiplied with the speed of light cτ :

x =

x

y

z

cτ

For this time instance, the following numerical values are found for part of the weightedleast-squares equation (using SI units):

(AT WA) =

3.0 2.0 0.0 0.02.0 3.0 0.0 0.00.0 0.0 4.0 0.00.0 0.0 0.0 1.0

Compute the predicted precision of the y coordinate in SI units.

b. 5pt

The GPS receiver provides observations with 1-sec time interval. If the total obser-vation period is half an hour (30 min) and if on the average 8 satellites are in view, howmany falsely detected outliers do you expect if a level of significance equal to 0.001 is used.Please round off to the nearest integer.

c. 5pt

Consider the following picture for an earth orbiting satellite moving from 800 km to 5000km altitude orbit, which is tracked by GPS:

12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:0010

15

20

25

30

35

40

45

C/N

0 [dB

−H

z]

Signal to Noise Ratio (SNR) for GPS observations

allvisibletracked

What is the minimum required signal-to-noise (SNR) ratio for GPS signals to be acquiredby the GPS receiver? Provide the SNR as the real ratio (i.e. convert from dB).

d. 5pt

For the point positioning solutions, the Global Overall Model (GOM) value is found to beequal to 1.8.

Specify/mention two possible causes for the fact that this GOM value is significantly devi-ating from ONE. Please do not use more than ONE line of text for describing these causes.

P.S.: Please note that it is no use to specify more than two causes. For each wrong cause,half of the points will be deducted.

Question 3 Aerodynamics Topic (20 points)

a. (2pts) If performing an experiment on the propulsion properties of an engine, which type of wind tunnel is considered best and why:

1. Open return wind tunnel – Low operating cost 2. Closed return wind tunnel – Quiet operation conditions 3. Closed return wind tunnel – Good inflow quality 4. Open return wind tunnel – No accumulation of exhaust products

b. (2pts) Consider a wind tunnel experiment where the pressure distribution is measured on

the airfoil. Which of the following airfoil characteristics is hardest to estimate from the experimental data.

1. Maximum lift coefficient 2. Minimum drag coefficient 3. Lift curve slope 4. Stall angle of attack

c. (3pts) Suppose that, at a certain angle of attack, the lift and drag coefficients of an airfoil

are 1.02 and 0.013 respectively with uncertainty of 5% for both of them. The absolute uncertainty on the aerodynamic efficiency equals (use the error propagation formula

L D

2 2

2 2

E C C

L D

E E

C C

∂ ∂σ = σ + σ ∂ ∂ , where σE, σCL and σCD are the absolute uncertainties of

efficiency, lift coefficient and drag coefficient respectively): 1. 3.92 2. 5.55 3. 1.98 4. 301.8

d. (2pts) For an airfoil, the lift curve slope is computed using the least-squares regression

with a linear fitting of n = 11 points; the sum of the squares of the residuals between the data CL,i and the linear fitting y = a1 x + a0 is Sr = 0.055. The standard deviation sy/x from the regression line equals:

1. 0.074 2. 0.078 3. 0.021 4. 0.071

e. (2pts) The model of a wing, having area S = 830 cm2, is tested in a wind tunnel in standard atmospheric conditions. At free-stream velocity 45.3 m/s and angle of attack α = 5 deg, the generated lift is 33 N. The actual wing area Sreal = 11 m2 and is flying at velocity of 240 km/h. What is the weight of the aircraft in straight static flight at the same angle of attack, in kN?

1. 8.33 2. 9.44 3. 9.47 4. 10.10

f. (2pts) A regional aircraft flies at cruise speed of 800 km/h at an altitude of 10,000 m. A wind tunnel needs to be designed in sea-level conditions to study such an aircraft. Which ones of the following experimental parameters should be selected to get the best approximation of flow similarity? Assume the following formula for the speed of sound in m/s: a = 165.97 + 0.606*T

1. V∞ = 196.6 m/s and 1:3.2 scale model 2. V∞ = 251.2 m/s and 1:4.1 scale model 3. V∞ = 251.2 m/s and 1:2.7 scale model 4. V∞ = 196.6 m/s and 1:2.1 scale model

NOTE Complete the figures on the separate aerodynamics answer sheet to answer

questions 3g and 3h.

g. (3pts) During the design of a wind tunnel experiment the location of the pressure sensors need to be determined. Indicate on the aerodynamics answer sheet (figure i) the location of the pressure sensors. Elaborate on the number of sensors and their distribution from leading edge to trailing edge. Consider measurement objectives, sensor accuracy and sensor installation.

h. (4pts) Sketch on four plots on the aerodynamics answer sheet (figures ii, iii, iv, v) the typical results of a wind tunnel experiment with pressure sensors on the airfoil surface:

1. Cp vs x/c at moderate angle of attack, 2. Cl vs α 3. Cd vs α 4. Cl vs Cd

Indicate in the plots the following characteristics: minimum drag, maximum lift, maximum aerodynamic efficiency, and stagnation point.

4. Materials topic (20 points)

a.Differentiate the log-likelihood function of the Weibull distribution with respect to λ.(nf represents the number of specimens in set I1, which were tested to failure) (3 points)

1.∑

iεI1∪I0

β(xi−λδ

)β−1δ

+ (β − 1)∑iεI1

(1

λ− xi

)

2.∑

iεI1∪I0

β(xi−λδ

)β−1δ2

+ β∑iεI1

(1

λ− xi

)

3.∑

iεI1∪I0

β (xi − λδ

)β−1+ (β − 1)nfλ

4.∑

iεI1∪I0

β(xi−λδ

)β−1δ2

+ (β − 1)nfλ

b. Which of the Weibull functions in the figure has the lowest value of β? (2 points)

c. Given the following design points,

σa (MPa) N (cycles)120.5 46,156102.8 93,828

use the Basquin relation (σkaN = C) to find the stress amplitude at a life of 78,000 cycles.(3 points)

1. 107.1 MPa2. 115.4 MPa3. 127.8 MPa4. 114.0 MPa

d. Which of the following best summarizes the method of obtaining a maximum likelihoodestimate? (1 points)

1. Finding the maximum value of the partial derivatives of the likelihood function2. Setting the likelihood function equal to zero3. Setting the partial derivatives of the likelihood function equal to zero4. Taking the logarithm of the likelihood function and solving for its zeros

e. Given a likelihood function:

∆ =n∑i=1

λe−λxi

Which term represents the test data? (1 points)

1. xi2. e3. n4. λ

f. A given log10(life) distribution has the following Normal parameters: µ = 4.047 andσ = 0.22. What proportion (percentage) of the samples would you expect to fail at a lifeof 81,500 cycles? (2 points)

Remember that:

f(x) =1√

2πσ2e−

(x−µ)2

2σ2

F (x) =1

2

[1 + erf

(x− µσ√

2

)]Knowing that: erfinv(0.999914) = 2.77751

1. 1.8895 %2. 0.0050 %3. 0.0043 %4. 0.0011 %

g. A given log10(life) distribution has the following Weibull parameters: β = 3.844,δ = 0.75 and λ = 3.58. At what life would you expect 0.2% of the samples to fail? (2points)

Remember that:

f(x) =β

δ

(x− λδ

)β−1e−(x−λδ )

β

F (x) = 1− e−(x−λδ )β

1. 7,109 cycles2. 3,804 cycles3. 12,237 cycles4. 5,155 cycles5. 5,357 cycles

h. Consider the below figure from a stress concentration factor handbook, showing thestress concentration factors for a flat tension bar with opposite U-shaped notches.

The following relations are given:

Ktn =σmaxσnom

σnom =P

hd

What is the Kt with d = 100 mm, H = 160 mm, h = 3 mm, r = 5 mm and P = 25 kN?(2 points)

1. 3.22. 3.73. 7.44. 10.6

i. From the previous question, what is the peak (maximum) stress at the edge? (3 points)

1. 193 MPa2. 83 MPa3. 308 MPa4. 617 MPa

5. Image Processing (20 points)

Please provide only one option for the multiple choice questions a-e and a written answer for open

question 5f.

(5a) A greyscale image consists of 831 by 768 pixels and has a bit depth of 16. How many bytes

are needed to store the data without compression? (2 points)

a. 638208

b. 1276416

c. 1914624

d. 3829248

(5b) What scaling factor is needed for the filter shown below? (2 points)

1 2 3 2 1

2 4 6 4 2

3 6 9 6 3

2 4 6 4 2

1 2 3 2 1

a. 1

b. 25

c. 81

d. 273

(5c) What is the maximum intensity that can be stored in an 14-bit image? (2 points)

a. 225

b. 16383

c. 32768

d. 65535

(5d) What is the improvement in signal to noise ratio when using a 3x3 averaging filter? (2 points)

a. 1

b. 3

c. 9

d. 27

(5e) What type of filter is this? (2 points)

-1 0 1

-2 0 2

-1 0 1

a. Sobel

b. Roberts

c. Median

d. Gaussian

(5f) Sketch a test plan (e.g. as a flowchart) for assessing image processing algorithms. As a

minimum several algorithms must be used in series. (10 points)