Exercises for the Course AE2-914: Vibrations of Aerospace Structures Principal lecturer: dr.ir. A.S.J. Suiker Case 3: Vibration behaviour of an aircraft wing REMARKS - All exercises must be elaborated analytically, unless it is stated explicitly that the elaboration may be done numerically. - For the plotting of results, small numerical programs need to be written, for the purpose of which MATLAB, FORTRAN, or any other programming language of a similar type may be used. - Provide the (numerical) answers with the corresponding units. - After the exercises are made, a report must be prepared. The report must contain a copy of the exercises, the analytical elaborations of the exercises with the required figures, and a hard copy of all numerical programs developed (in an appendix). - The exercises (and the report) should be made together with a colleague student. The names of the students and the student identification numbers must be written on the cover of the report. - The exercises are not compulsory; however, they provide a good preparation for the exam. The result of the exercises will be combined with the result of the exam as follows: if the percentage of correct answers to the exercises is (i) less than 40% : 0.0 point will be added to the exam grade, (ii) between 40% and 70% : 0.5 point will be added to the exam grade, (iii) above 70% : 1.0 point will be added to the exam grade. Final grades higher than 10.0 will be rounded off to 10.0.
- The exercises are divided into four categories. The number of a category relates to the chapter in Engineering Vibrations (second edition) by D.J. Inman that contains the necessary background theory. - The main purpose of the exercises is to develop a good insight into vibration theory, rather than to learn how to apply well-known formulas to solve specific vibration problems. In correspondence with this philosophy, the exercises are made such that one should be careful with straightforwardly applying formulas given Inman s textbook; an adaptation of these formulas to the actual problem might be necessary for computing the correct answer. - Questions regarding the exercises will be dealt by dr.ir. A.S.J. Suiker. During the course period the Friday afternoon shall be reserved for support. Every two collaborating students can register for a maximum of ten minutes support per week; it is required to arrive wellprepared. Register forms will be hung up at the 2nd years announcements board. - The report should be turned in to dr.ir. A.S.J. Suiker (room NB 1.54 in the building of Aerospace Engineering) at last three days before the day of the exam. Reports turned in after this deadline will not be considered. - To be able to work out all exercises within the available amount of time, it is recommended to do the exercises in parallel with the corresponding AE2-914 lectures.
Fuselage Wing elastic axis L c.g. e.a. e lch Cross-section of wing e.a. = elastic axis (shear centre) c.g. = centre of gravity c.g. θ 1 e.a. M1 x 1 e l ch Figure 1: Sketch of a wing. EXERCISES Figure 1 illustrates the cross-section of a wing of a light, two-engine aircraft. For design purposes, the vibration behaviour of the wing needs to be studied under various loading conditions during a flight. The parameter set necessary for carrying out the analyses is given below. Parameter set Wing cross-section Mass per unit depth: M = 1.0 10 2 kg/m 1 Mass moment of inertia per unit depth (about the centre of gravity): J = 2.0 10 3 kgm 2 /m 1 Elasticity modulus: E = 7.0 10 10 N/m 2 Shear modulus: G = 2.7 10 10 N/m 2 Bending moment of inertia (about the elastic axis): I b = 1.0 10 5 m 4 Torsional moment of inertia (about the elastic axis): I t = 3.0 10 5 m 4 Eccentricity of the elastic axis with respect to the centre of gravity: e = 0.6 m Wing distance at which the cross-section is considered: L = 3.0 m Reference chord: l ch = 1.225 m Viscous damping per unit depth: c = 1.0 10 3 Ns/m 1 /m 1
Aeroelastic loading characteristics Velocity of aircraft, (i) subsonic: v = 100 m/s (ii) transsonic: v = 300 m/s Mass density of air: ρ = 0.5 kg/m 3 Aeroelastic parameters: A (1) = 5 10 2 s A (2) = 6.2 [ ] Loading amplitude: ˆM = 1.0 10 3 Nm/m 1 Loading frequency: ω = 2.0 rad/s Response End of time domain: t end = 10 s Initial conditions Initial rotation: θ 0 1 = 0.008 rad Initial rotation rate: θ0 1 = 0.02 rad/s
1 Free response of a mass-spring system Initially the wing cross-section is modelled as a single mass-spring system by only accounting for the rotation at the centre of gravity, θ 1 (t), and ignoring the vertical displacement, at the centre of gravity, x 1 (t). The torsional moment of inertia of the wing cross-section about the elastic axis is I t, and the bending moment of inertia about the elastic axis is I b. The shear modulus and the elasticity modulus of the wing are G and E, respectively, and the mass moment of inertia about the centre of gravity is J. The wing cross-section is taken at a distance L from the wing-fuselage connection, see Figure 1. a) Derive the translational stiffness due to bending, k b, and the rotational stiffness due to torsion, k t, at the elastic axis (= shear centre). b) Schematise the wing cross-section as a mass-spring system. c) Draw the free-body diagram for the mass-spring system and derive the corresponding equation of motion at the centre of gravity, c.g.. It is noted that the overall stiffness, k, consists of two parts: (i) a part due to bending of the wing and (ii) a part due to torsion of the wing. d) The wing cross-section has a structural viscous damping, c, that works in parallel with its overall stiffness, k. Extend the equation of motion derived under b) with the effect of structural damping. As a result of aerodynamic lift forces, the centre of gravity of the wing cross-section is subjected to a moment per unit depth (see Figure 1) M (1) 1 (t) = ( A (1) θ1 + A (2) θ 1 ) 1 2 ρv2 l 2 ch [Nm/m 1 ] where ρ is the mass density of air, v is the velocity of the aircraft, l ch is the reference chord of the wing cross-section (see Figure 1) and A (1) and A (2) are aeroelastic parameters. e) Add the above loading to the equation of motion and reformulate the expression to a homogeneous differential equation. Compute the eigen frequency, ω n, of the mass-spring system at (i) subsonic speed and (ii) transsonic speed. f) Calculate the free response of the wing at subsonic speed and transsonic speed, as caused by the initial conditions: θ 1 (0) = θ 0 1 and θ 1 (0) = θ 0 1. Plot the responses for 0 t t end. Explain why the flutter instability that appears at transsonic speed can be considered as critical to the wing structure. g) What is the critical velocity of the aircraft at which the flutter instability is initiated?
shock loading M (3) 1 M 1 0 π ω time t Figure 2: Half-sine pulse. 2 Harmonic forced response of a mass-spring system Now, consider a case at subsonic speed where the wing is loaded by the aerodynamic lift, M (1) 1 (t), given under 1d), and by a harmonic wind load, M (2) 1 (t) = ˆM 1 sin(ωt), with ˆM 1 the amplitude and ω the angular frequency of the harmonic wind load. a) Compute the steady-state response of the wing, θ1 ss (t), to the wind loading, and plot the result for 0 t 2t end. b) Use the result under a) for the numerical determination of the total response, θ 1 (t) = θ1 tr (t) + θ1 ss (t). The transient part of the response can be formulated as (Inman, 2001) θ tr 1 (t) = A exp( ζω n t) sin(ω d t + φ), where the amplitude A and the phase angle φ must be determined from the the initial conditions θ 1 (0) = θ 0 1 and θ 1 (0) = θ 0 1. Plot the total response for 0 t 2tend and indicate how long it approximately takes to reach a steady-state. 3 General forced response of a mass-spring system Next, consider a case at subsonic speed with the wing being loaded by the aerodynamic lift, M (1) 1 (t), and a shock loading, M (3) 1 (t). The shock loading is represented by a half-sine pulse, and is caused by a gust of wind. The amplitude of the half-sine pulse is ˆM 1 and its angular frequency is ω, see Figure 2. a) Give the mathematical expression for the half-sine pulse loading (make use of the principle of superposition). b) Compute the response, θ 1 (t), of the wing to the half-sine pulse loading 1. It may be assumed that the the initial conditions are zero, θ 1 (0) = θ 1 (0) = 0 and that the effect of viscous damping is negligible (i.e. an undamped system). Plot the response for 0 t 1 2 tend. 1 Hint: A convenient way to solve this problem is to use the Laplace transform method in combination with the method of partial fractions, see Section 6.2 of Elementary Differential Equations and Boundary Value Problems (5th edition) by W.E. Boyce and R.C. DiPrima.
The plot of the peak response of the mass-spring system, θ max 1, as a function of the (normalised) frequency of the half-sine pulse loading, ω n /ω, with ω n the eigen frequency of the mass-spring system, is known as the shock spectrum. c) Compute the shock spectrum numerically and plot it for 0 ω n /ω 20. Give a physical explanation for the asymptotes ω n /ω 0 and ω n /ω. 4 Response of a multiple-degree-of-freedom system In order to obtain a more realistic model for the wing, the vertical displacement at the centre of gravity, x 1 (t), needs to be taken into account. a) Schematise the wing cross-section as a two degree-of-freedom system (by neglecting the effect of damping and external forces), draw the free-body diagram and derive the equations of motion. b) Add the aerodynamic lift forces F (1) 1 (t) = [ ] (1) F 1 (t) [N] M (1) 1 (t) [Nm] = [ 0 A (2) θ 1 1 2 ρv2 l 2 ch ] to the above equations of motion and reformulate the set of equations to a set of homogeneous differential equations (in matrix-vector format). Calculate the eigen frequencies ω n of this system. c) Derive the modal vectors (eigen vectors) u n of the two-degree-of-freedom system. Similar to the example 2), at subsonic speed the wing is subjected to an aerodynamic lift, F (1) 1 (t), and to a harmonic wind loading, F (2) 1 (t), given by F (2) 1 (t) = [ ] (2) F 1 (t) [N] M (2) 1 (t) [Nm] = [ 0 ˆM 1 sin(ωt) d) Compute the steady-state response, x ss 1 = [x ss 1, θ1 ss ] T, to the wind loading and plot the result for 0 t t end. Compare the result for θ1 ss (t) with that derived under 2a) for the simpler, one-degree-of-freedom system. e) Generally wind loading is not characterised by a singular angular frequency ω, but by a spectrum of frequencies. To examine the frequency effect on the response, construct the frequency-response curves (transfer functions) for the two-degree of freedom system: i.e. the curves that show the the amplitudes of the steady-state response, ˆx ss 1 = [ˆxss ss 1, ˆθ 1 ]T, as a function of the loading frequency ω. Take the frequency domain as 0 ω 40 rad/s. ].