EMCH721 Test 1 Solution Aeroelasticity Dr.
SOLUTION
Test 1
EIGEN ANALYSIS OF FLUTTER
A. AEROELASTIC MODEL .................................................................................................................................................3
B. GROUND VIBRATION TEST (GVT) ANALYSIS ................................................................................................................4
C. FLUTTER ANALYSIS......................................................................................................................................................5
D. DAMPED GVT AND FLUTTER ..................................................................................................................................... 14
E. EXTRA CREDIT CHALLENGE PROBLEMS ..................................................................................................................... 19
LIST OF MATLAB CODES .................................................................................................. ERROR! BOOKMARK NOT DEFINED.
Consider an airfoil in wind tunnel testing with airspeed U (Figure 1). The airfoil is supported by springs
at P. The center of mass is at C. The distance from P to C is the static offset xCP. The aerodynamic lift L
acts at Q. The distance from P to Q is the aerodynamic offset xQP. The airfoil undergoes oscillatory motion
with plunge displacement h(t) and pitch displacement (t). The positive plunge direction is downward.
The positive pitch direction is “nose up” (clockwise).
Figure 1: Airfoil in wind tunnel testing
Input data
Air density: =1.225 kg/m3
Chord: c=0.45 m
Mass, m= 4.5 kg
Mass moment of inertia, I0=0.06 kg.m2
Plunge frequency, fh=2.5 Hz
Pitch frequency, f=5.5 Hz
Static offset, xCP= –15% of c
Aerodynamic offset, xQP= 25% of c
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EMCH721 Test 1 Solution Aeroelasticity Dr. G
NOTES:
• Always display input data!
• Show work and relevant comments
• Partial credit might be given based on work done and MATLAB codes if they are attached; without
them, no partial credit can be given.
• Display numerical results to three significant digits (four if the first digit is 1).
• Problems solved beyond the minimum requirements will be given bonus points.
• Challenge problems are optional. They may be solved for extra credit
Notations:
• EOM = equation of motion
• CL = closed loop
• FBD = free body diagram
• FB = feedback
• GVT = ground vibration test
• LHS = left hand side
• LT = Laplace transform
• MIMO = multi-input multi-output
• ODE = ordinary differential equation
• PDE = partial differential equation
• RHS = right hand side
• SISO = single-input single-output
• SS = state space
• TF = transfer function
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EMCH721 Test 1 Solution Aeroelasticity Dr. G
A. AEROELASTIC MODEL
nothing here
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EMCH721 Test 1 Solution Aeroelasticity Dr. G
B. GROUND VIBRATION TEST (GVT) ANALYSIS
nothing here
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EMCH721 Test 1 Solution Aeroelasticity Dr. G
C. FLUTTER ANALYSIS
C.1 EIGEN ANALYSIS OF FLUTTER
nothing here
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EMCH721 Test 1 Solution Aeroelasticity Dr. G
C.2 FLUTTER DIAGRAM
Let xCP= –15% of c, xQP= 25% of c. Let U=0,…,14 m/s with 1001 steps.
(a) Display input data
(b) Plot frequencies and damping vs. airspeed. Use these plots to find the flutter speed UF and
insert datatips appropriately.
(c) State the value of UF in m/s and knots.
Answer:
(a)
(b)
(c)
UF ≈11.102 m/s = 21.58 knots
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EMCH721 Test 1 Solution Aeroelasticity Dr. G
C.3 FREQUENCIES AND MODESHAPES NEAR FLUTTER SPEED
Let U= 0, 6m/s, (1–)UF, UF, (1+)UF, with =1%
(a) find frequencies and modeshapes at each of these speeds
(b) discuss your results
Answer:
(a)
(b) Students should insert here discussion of results
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EMCH721 Test 1 Solution Aeroelasticity Dr. G
C.4 STATIC OFFSET EFFECT ON FLUTTER SPEED
Let U=10,…,17 m/s with 1001 steps. Keep xQP= 25% of c.
Let xCP = –20%, –15%, –10%, –5%, –1%, –0.1% of c. Do the following:
(a) Display input data
(b) Plot overlapped (hold on) frequencies and damping vs. airspeed for each value of xCP. Insert
datatips to locate the flutter speed UF. Present the UF results in table format.
(c) Plot the variation of flutter speed UF with xCP%. Make sure the plot covers the whole U
range.
(d) Discuss how xCP influences the flutter speed.
(e) Repeat for xCP = 0, 0.1%, 0.5%, 10%, 15%, 20% of c.
Answer:
(a)
(b)
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EMCH721 Test 1 Solution Aeroelasticity Dr. G
(c)
(d) Every student should insert here discussion of results
(e) There is no flutter for xCP≥0
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EMCH721 Test 1 Solution Aeroelasticity Dr. G
C.5 AERODYNAMIC OFFSET EFFECT ON FLUTTER SPEED
Let U=10.5,…,12.5 m/s with 1001 steps. Keep xCP= –15% of c.
Let xQP = 20%, 25%, 30% of c. Do the following:
(a) Display input data
(b) Plot overlapped (hold on) frequencies and damping vs. airspeed for each value of xCP. Insert
datatips to locate the flutter speed UF. Present the UF results in table form.
(c) Plot the variation of flutter speed UF with xQP%. Make sure the plot covers the whole U
range.
(d) Discuss how xQP influences the flutter speed.
Answer:
(a)
(b)
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EMCH721 Test 1 Solution Aeroelasticity Dr. G
(c)
(d) Every student should insert here discussion of results
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EMCH721 Test 1 Solution Aeroelasticity Dr. G
C.6 DIVERGENCE SPEED
(a) Let xCP= –15% of c, xQP= 25% of c. Let U=0,…, 25 m/s with 1001 steps.
(i) Display input data
(ii) Plot extended flutter diagram
(iii)Identify with datatips the location of UD on the extended flutter diagram
(b) Let xQP = 20%, 25%, 30% of c. Calculate and plot divergence speed UD vs. xQP%. Make
sure the plot covers the whole range of UD.
(c) Discuss the comparison between the results of items (a) and (b)
Answer:
(a)
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EMCH721 Test 1 Solution Aeroelasticity Dr. G
(b)
(c) Every student should insert here discussion of results
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EMCH721 Test 1 Solution Aeroelasticity Dr. G
D. DAMPED GVT AND FLUTTER
D.1 ANALYSIS OF DAMPED GVT AND FLUTTER
nothing here
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EMCH721 Test 1 Solution Aeroelasticity Dr. G
D.2 SETUP FOR DAMPED GVT AND FLUTTER CALCULATIONS
Augment input data with the following modal damping values:
• Plunge damping ratio: h=3%
• Pitch damping ratio: =2%
Display updated input data
Answer
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EMCH721 Test 1 Solution Aeroelasticity Dr. G
D.3 DAMPED GVT ANALYSIS
Let U=0 and study the effect of damping on GVT:
(a) Calculate damped frequencies and compare them with the undamped frequencies
(b) Calculate modal damping and compare with the uncoupled modal damping
(c) Calculate damped modeshapes and compare them with the undamped modeshapes
Answer:
Every student should insert here discussion of results
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EMCH721 Test 1 Solution Aeroelasticity Dr. G
D.4 DAMPED FLUTTER ANALYSIS
Let U=0…14 m/s with 1001 steps and study the effect of damping on flutter speed UF
(a) Plot frequencies and damping vs. airspeed; find the damped flutter speed UF on these plots.
(b) State the value of UF in m/s and knots
(c) Plot the undamped diagram on top of the damped diagram and identify the undamped flutter speed
U Fundamped and the damped flutter speed UF
(d) Discuss the difference between the damped and undamped flutter speeds
Answer:
(a)
(b) UF =11.088 m/s = 21.55 knots
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EMCH721 Test 1 Solution Aeroelasticity Dr. G
(c)
UF=11.088 m/s
U Fundamped =11.102 m/s
(d) Every student should insert here discussion of results
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EMCH721 Test 1 Solution Aeroelasticity Dr. G
E. EXTRA CREDIT CHALLENGE PROBLEMS
Challenge problems are optional.
E.1 GVT MODELING
Consider the airfoil shown in Figure 1 but with zero airspeed U=0. This condition corresponds to ground
vibration testing (GVT). Do the following:
(a) Deduce equations of motion. Clearly state your assumptions.
ˆ st , = ˆe st and cast equations of motion in matrix form as a polynomial eigenvalue
(b) Assume h = he
hˆ
( )
problem s 2M S + K S x = 0 where x =
ˆ
Answer:
Students should write their own derivation using the class notes as a guide
E.2 DIVERGENCE SPEED MODELING
Derive the expression of the divergence speed UD
Answer:
Students should write their own derivation using the class notes as a guide
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