Midterm-Flight Dynamics and Control

Student Name: _______________________ Student ID:

Course Title: Flight Dynamics and Control Section: A

Course Code: EE519 Semester: Spring

Instructor: JAMEEL AHMAD Academic Year: 2012

Date: April 17, 2012 Total Marks: 100

Exam: Midterm Time Allowed: 150

min.

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Provide your answers in space provided. You may also use back sides of the

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Formula sheet is given at the end.

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Good Luck!

Question Q1 Q2 Q3 Q4 Q5 Total

Marks 20 20 20 28 12 100

Marks Obtained

University of Management & Technology

Student Name: _______________________ Student ID: _______________________

Question 1 (20points)

Part-A (5 Points)

Explain the terms Coefficient of lift, Coefficient of Drag, Pitching Moment Coefficient and

write down the corresponding mathematical expressions for each. Which coefficient is

used for longitudinal static stability of aircraft and missiles?

Solution

Part-B (2 points)

Consider the following equations:

ααLL CC = (1)

2

min LDD CKCC += . (2)

Figure 1: Lift and drag for subsonic and supersonic speeds.

Qualitatively identify the regions in Figure 1 that correspond to subsonic speeds.

Identify the region of attack angles, such that the coefficients are linearly related to α.

LC DC

minDC

αα

Student Name:________________________ Student ID:_________________________________

Part-C (3 Points)

Figure 2: Lift coefficient versus angle of attack for a cambered airfoil

Explain what is the reason of using cambered airfoils in aircraft? From Figure-2, give a

quantitative mathematical expression for CL as a function of α.

Solution


Part-D (10 points)

Consider a simple glider as shown in the

figure with its center of gravity xcg

Figure 3: Glider (top and side view)

It consists of a wing, tail and body. Each item is rectangular, and each has the same

thickness, t. Denote the length and width of the wing as wl and

ww , respectively. Use

similar notation for the dimensions of the tail and the body. Assume center of gravity is

given with

tot

ttbww

cgW

wWWwWx

)2/()2/()2/( −++=

ll

You should assume that the speed is sufficiently slow that the backwash onto the tail

region is negligible. Also, assume that drag moments are negligible. In addition to the

stated assumptions, we will make the following assumptions:

(A1): The body contributes zero lift.

(A2): The wing lift force is αγρ )(

2VSL ww =

and the tail lift force is αγρ )(

2VSL tt =

γ is flight path angle, V is air speed, ρ is air density, S is wing span and α is angle of attack.

a) Determine angle of attack α for un-accelerated level flight. For un-accelerated

level flight we must have tottw WLL =+

0 lx

0 lx

cgx 2/l

bW

tW

wW wwtw


b) The pitch moment about the cg is:

)2/()2/( tcgtwcgwm wxLwxLC −−−−= l .

Show that

)]2/()2/([/2

tcgtwcgwm wxSwxSVC −−−−=∂∂ lγρα .

[If the pitch moment, mC, equals zero for some angle of attack, oα

, the airplane is said to

be in the condition of longitudinal balance. This condition may also be termed

longitudinal equilibrium.]


Question-2[20 points] Answer the following questions in the space provided. Be

brief.

a) What are the factors which decide the flying path of an airplane as a rigid

body?

b) Why the airplane is considered as a dynamic system in six degrees of

freedom? What are the conditions to be satisfied for equilibrium along a straight

un-accelerated flight path?

c) What is meant by control of an airplane, how longitudinal, roll and

directional controls are provided in airplane?


d) What is meant by static and dynamic stability of an airplane?

e) What are the characteristic modes of longitudinal motion of airplane?


f) What is the purpose of wind tunnel test?

Question-3 (5X4=20 Points)

A dynamic system is represented by a set of differential equations as given below.

1. Represent the system in state-space form

2. Determine A,B,C and D matrices of state-space representation.

3. Determine State Transition Matrix

4. Determine C(SI-A)-1 B+D


Question-4 (7X4=28 Points)

Part-A

The body axes fixed to an aircraft are oriented at ψ= -90 deg, pitch Ѳ= -45 deg and roll

ф= 45 deg, with respect to the inertial reference frame. The aircraft is flying at 300 ft/s,

with an angle of attack of 5 degrees and an angle of side-slip at 3 degrees.

Calculate

a) The body axes velocity components;

b) The direction cosine matrix from the body axes to the stability axes,


c) The direction cosine matrix from the body axes to the wind axes,

d) The direction cosine matrix from the inertial to the body axes

Hint: Inter-conversion and Rotation between two axes systems might look like this


Part-B

I. Determine the missing components (marked x) from the following Direction

Cosine Matrix

C=

−

7195.04963.0

1218.08595.0

4858.01587.0

x

x

x


II. After determining the missing components in C matrix in part (I) , determine the

corresponding Euler Angles: Ѳ , ф and ψ

III. Determine the corresponding Quaternions from Euler Angles.


Question-5 (6x2=12 Points) [6-DOF Equation of Motion of aircraft]

An aircraft is in trim condition (e.g. equilibrium state).

Translational Velocity Vector= [U V W] T such that U = U0, V =0, W = 0;

Angular Position Vector = [ф Ѳ ψ] T such that 0=ψ ; Ѳ= Ѳo; ф= 0;

Angular Velocity Vector= [P Q R] T

Note-1: A cheat sheet on last page is provided to help you solve Question-5

For this case,

(a) Using equations for Translational Dynamics (TD), Rotational Dynamics (RD),

Rotational Kinematics (RK) and Translational Kinematics (KT), determine all

the body axis angular velocity components [P Q R]


(b) Using Translational Dynamics (TD), Rotational Dynamics (RD), Rotational

Kinematics (RK) and Translational Kinematics (KT), Find the forces (FX FY FZ[ and

moments[L M N] (aerodynamics propulsion/Thrust) in the body frame components in

terms of the given flight condition variables, the aircraft mass, m and the acceleration

due to gravity, g.


Summary of 6-DOF Equations Governing a Rigid Body (BodyFrame Equations)

Assumptions: Flat Earth

Translational Dynamics

U = RV −QW − g sin θ + (FXA + FXT )/m

V = −RU + PW + g sinφ cos θ + (FY A + FY T )/m

W = QU − PV + g cosφ cos θ + (FZA + FZT )/m

Rotational Dynamics

Γ P = Ixz [Ixx − Iyy + Izz] PQ−[Izz(Izz − Iyy) + I2

xz

]QR + Izz L+ Ixz N

Iyy Q = (Izz − Ixx) PR− Ixz (P 2 −R2) +MΓ R = −Ixz [Ixx − Iyy + Izz] QR +

[Ixx(Ixx − Iyy) + I2

xz

]PQ+ Ixz L+ Ixx N

Γ = IxxIzz − I2xz

Rotational Kinematics Euler angles

φ = P + tan θ (Q sinφ+R cosφ)

θ = Q cosφ−R sinφ

ψ = (Q sinφ+R cosφ) / cos θ

Translational Kinematics

XI = cos θ cosφ U + (− cosφ sinψ + sinφ sin θ cosψ) V + (sinφ sinψ + cosφ sin θ cosψ) W

YI = cos θ sinφ U + (cosφ cosψ + sinφ sin θ sinψ) V + (− sinφ cosψ + cosφ sin θ sinψ) W

ZI = sin θ U − sinφ cos θ V − cosφ cos θ W

Note:

• L, M, N are the roll, pitch and yaw moments (aero/propulsion) in the body frameabout the vehicle CG .

• FXA, FY A, FZA represent the components of the sum of all the aerodynamic forces inthe body frame.

• FXT , FY T , FZT represent the components of the sum of all the propulsive forces inthe body frame.

• m is the mass of the vehicle, I is the moment of inertia matrix of the vehicle with anXB − ZB plane of symmetry.

• Angle of attack, α = tan−1

(W

U

)and Angle of sideslip, β = sin−1

(V

VT

)and VT =

√U2 + V 2 +W 2.

2

Midterm-Flight Dynamics and Control

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