University of Management & TechnologyStudent Name: _______________________ Student ID: Section: A Course Title: Flight Dynamics and Control Course Code: EE519 Instructor: JAMEEL AHMAD Semester: Spring Academic Year: 2012 Total Marks: 100 Time Allowed: 150 Date: April 17, 2012 Exam: min. DO NOT OPEN THIS EXAM UNTIL TOLD TO DO SO Midterm The instructions below must be followed strictly. Failure to do so can result in serious grade loss. ⇒ You may not • Talk to anyone once the exam begins. • Exchange calculator, book, stationary etc. • Use Mobile phone, PDA, Digital Diary. ⇒ Specific instructions Calculator and laptop Allowed. Please write within space provided. Open Notes. Provide your answers in space provided. You may also use back sides of the paper. Use a ball point or pen. Please do not use lead pencils. Show all work, it helps me give partial credit. I will not grade your examination if you fail to 1) put your name and ID or 2) sign the blank below acknowledging the terms of this test and the honor code policy. Formula sheet is given at the end. Pledge Signature: ____________________________________________________ I acknowledge the above terms for taking this examination. I have neither given nor received unauthorized help on this exam. I have followed the UMT honor code in preparing and submitting the test. Good Luck! Question Marks Marks Obtained Q1 20 Q2 20 Q3 20 Q4 28 Q5 12 Total 100 Which coefficient is used for longitudinal static stability of aircraft and missiles? Solution Part-B (2 points) Consider the following equations: CL CD = CLα α 2 = C Dmin + K C L . Qualitatively identify the regions in Figure 1 that correspond to subsonic speeds. Coefficient of Drag. Identify the region of attack angles.Student Name: _______________________ Student ID: _______________________ Question 1 (20points) Part-A (5 Points) Explain the terms Coefficient of lift. . such that the coefficients are linearly related to α. (1) (2) CL CD CDmin α α Figure 1: Lift and drag for subsonic and supersonic speeds. Pitching Moment Coefficient and write down the corresponding mathematical expressions for each. give a quantitative mathematical expression for CL as a function of α. Solution .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. Denote the length and width of the wing as lw and ww . (A2): The wing lift force is and the tail lift force is Lw = ( ρ S w V 2 γ )α Lt = ( ρ S t V 2 γ )α γ is flight path angle. tail and body. respectively.Student Name:________________________ Student ID:_________________________________ Part-D (10 points) Consider a simple glider as shown in the figure with its center of gravity xcg Ww ww Wb wt Wt 0 l x Figure 3: Glider (top and side view) 0 xcg l / 2 l x It consists of a wing. In addition to the stated assumptions. assume that drag moments are negligible. Also. t. Assume center of gravity is given with xcg = Ww ( ww / 2) + Wb (l / 2) + Wt (l − wt / 2) Wtot You should assume that the speed is sufficiently slow that the backwash onto the tail region is negligible. we will make the following assumptions: (A1): The body contributes zero lift. S is wing span and α is angle of attack. ρ is air density. For un-accelerated level flight we must have Lw + Lt = Wtot . and each has the same thickness. Use similar notation for the dimensions of the tail and the body. Each item is rectangular. a) Determine angle of attack α for un-accelerated level flight. V is air speed. This condition may also be termed longitudinal equilibrium. α o .Student Name:________________________ Student ID:_________________________________ b) The pitch moment about the cg is: C m = Lw ( x cg − ww / 2) − Lt (l − x cg − wt / 2) . Cm . [If the pitch moment.] . equals zero for some angle of attack. Show that ∂C m / ∂α = ρV 2 γ [ S w ( xcg − ww / 2) − S t (l − xcg − wt / 2)] . the airplane is said to be in the condition of longitudinal balance. Answer the following questions in the space provided.Student Name:________________________ Student ID:_________________________________ Question-2[20 points] brief. roll and directional controls are provided in airplane? . Be 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. Student Name:________________________ Student ID:_________________________________ d) What is meant by static and dynamic stability of an airplane? e) What are the characteristic modes of longitudinal motion of airplane? . 1. Represent the system in state-space form .Student Name:________________________ Student ID:_________________________________ 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. Determine A.B. 3.2. Determine C(SI-A)-1 B+D . Determine State Transition Matrix 4.C and D matrices of state-space representation. with an angle of attack of 5 degrees and an angle of side-slip at 3 degrees. The aircraft is flying at 300 ft/s. Calculate a) The body axes velocity components. with respect to the inertial reference frame.Student Name:________________________ Student ID:_________________________________ Question-4 (7X4=28 Points) Part-A The body axes fixed to an aircraft are oriented at ψ= -90 deg. b) The direction cosine matrix from the body axes to the stability axes. pitch Ѳ= -45 deg and roll ф= 45 deg. . Student Name:________________________ Student ID:_________________________________ 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 . 4963 0.1218 0.1587 C= 0. Determine the missing components (marked x) from the following Direction Cosine Matrix 0.7195 .8595 x x − 0.4858 x 0.Student Name:________________________ Student ID:_________________________________ Part-B I. determine the corresponding Euler Angles: Ѳ . . Determine the corresponding Quaternions from Euler Angles. ф and ψ III.Student Name:________________________ Student ID:_________________________________ II. After determining the missing components in C matrix in part (I) . equilibrium state). determine all the body axis angular velocity components [P Q R] Student Name:________________________ Student ID:_________________________________ . 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.g. V =0. Rotational Kinematics (RK) and Translational Kinematics (KT). (a) Using equations for Translational Dynamics (TD).Student Name:________________________ Student ID:_________________________________ Question-5 (6x2=12 Points) [6-DOF Equation of Motion of aircraft] An aircraft is in trim condition (e. ф= 0. Angular Position Vector = [ф Ѳ ψ] T such thatψ = 0 . W = 0. Translational Velocity Vector= [U V W] T such that U = U0. Ѳ= Ѳo. Rotational Dynamics (RD). m and the acceleration due to gravity. 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. Rotational Kinematics (RK) and Translational Kinematics (KT). the aircraft mass.(b) Using Translational Dynamics (TD). g. Rotational Dynamics (RD). Student Name:________________________ Student ID:_________________________________ . . FY T . FZT represent the components of the sum of all the propulsive forces in the body frame. • FXA . β = sin−1 V VT and VT = 2 . W U and Angle of sideslip. • m is the mass of the vehicle. FZA represent the components of the sum of all the aerodynamic forces in the body frame.Summary of 6-DOF Equations Governing a Rigid Body (Body Frame Equations) Assumptions: Flat Earth Translational Dynamics ˙ U = RV − QW − g sin θ + (FXA + FXT )/m ˙ V = −RU + P W + g sin φ cos θ + (FY A + FY T )/m ˙ W = QU − P V + g cos φ cos θ + (FZA + FZT )/m Rotational Dynamics 2 ˙ Γ P = Ixz [Ixx − Iyy + Izz ] P Q − Izz (Izz − Iyy ) + Ixz QR + Izz L + Ixz N ˙ Iyy Q = (Izz − Ixx ) P R − Ixz (P 2 − R2 ) + M 2 ˙ Γ R = −Ixz [Ixx − Iyy + Izz ] QR + Ixx (Ixx − Iyy ) + Ixz P Q + Ixz L + Ixx N 2 Γ = Ixx Izz − Ixz Rotational Kinematics Euler angles ˙ φ = P + tan θ (Q sin φ + R cos φ) ˙ θ = Q cos φ − R sin φ ˙ ψ = (Q sin φ + R cos φ) / cos θ Translational Kinematics ˙ XI = cos θ cos φ U + (− cos φ sin ψ + sin φ sin θ cos ψ) V + (sin φ sin ψ + cos φ sin θ cos ψ) W ˙ YI = cos θ sin φ U + (cos φ cos ψ + sin φ sin θ sin ψ) V + (− sin φ cos ψ + cos φ sin θ sin ψ) W ˙ ZI = sin θ U − sin φ cos θ V − cos φ cos θ W Note: • L. α = tan−1 √ U 2 + V 2 + W 2. • FXT . I is the moment of inertia matrix of the vehicle with an XB − ZB plane of symmetry. • Angle of attack. M. N are the roll. pitch and yaw moments (aero/propulsion) in the body frame about the vehicle CG . FY A .