SECTION A Question 1 (a) In the usua notation derive the governing differentia equation of motion in free vibration for the singe degree of freedom system shown in Figure Q1(a) by using Newton's second aw, agrange's equation and Hamiton s principe, respectivey. [9 marks] k θ a m Figure 1(a) (b) The undercarriage of the aircraft shown in Figure Q1(b) has been ideaised by two vertica springs. The nose undercarriage has a stiffness of k whie the main undercarriage has a tota stiffness of 7k. The centre of gravity of the aircraft is at G, its mass is M, and its pitching moment of inertia about its centre of gravity is 6Ma. Treating the aircraft structure as rigid, cacuate the two natura frequencies and the corresponding mode shapes of natura vibration. [16 marks] G k 7k a a Figure Q1(b) Figure not to scae Page 1 of 7
Question (a) Derive the governing differentia equation of motion of a bar with etensiona rigidity EA, mass per unit ength ρa and ength undergoing free natura vibration in aia motion and then obtain its genera soution. [6 marks] (b) Determine the frequency equation of the bar when its one end is fied and a mass M is attached to the other, as shown in Figure Q(b). [4 marks] M Figure Q(b) (c) Using the norma mode method of anaysis, obtain an epression for the steady state feura response of a pinned-pinned beam subjected to a harmonic force f(,t) = f 0 sin Ωt appied at = a as shown in Figure Q(c). [15 marks] In the usua notation, the natura frequencies and mode shapes of a simpy supported beam in feura vibration are respectivey given by and n π ω n = W n EI ρa nπ ( ) = sin f 0 sinωt a Figure Q(c) Page of 7
Question (a) The stiffness and mass matrices of a -D bar eement in the usua notation are given by EA 1 1 ρa 1 ρa 1 0 [ k ] =, [ m] c = 1 1, [ m] = 6 1 0 1 where the subscripts c and to the mass matri denote the consistent and umped matrices, respectivey. Determine the fundamenta natura frequency of the fied-fied bar using two eements (see Figure Q(a)) and both consistent and umped matri representations E and compare with the eact vaue π, where is the tota ength of the bar. ρ Comment briefy on the differences amongst the resuts. [10 marks] 1 Figure Q(a) (b) The stiffness and mass matrices of a -D beam eement in feure are given by 1 6 1 6 156 6 4 6 EI = ρa [ k ] and [ m] = 1 6 1 6 40 54 6 6 4 1 4 1 54 1 156 1 4 Determine the natura frequencies of a simpy supported beam using ony one finite eement and by comparing them with the ones given by the epression for ω n in Q(c), estimate the errors. [15 marks] Page of 7
SECTION B Question 4 Figure Q4 shows the essentia detais of an aircraft wing. The wing itsef is considered cantievered with a ength = 4.5 m and its structura strength and stiffness in torsion may be considered to derive entirey from a torsion bo made of auminium. The feura ais runs aiay aong the centre of the torsion bo and the wing has a centre of ift ais ec = 0.075 m forward of the feura ais as shown in Figure Q4(a). The wing chord tapers ineary from 1.5 m at the root to 0.75 m at the tip. The torsion bo chord aso tapers ineary from 0.6 m at the root to 0. m at the tip. The skin thickness of the torsion bo is t = mm. A cross section of the wing at the root is shown in Figure Q4(b). Assume two-dimensiona ift curve sope a 1 = 5.7 and subsonic strip theory is appicabe. Obtain the divergence speed for the wing using an assumed twist mode θ ( ) = q, where is the distance measured aong the span from the wing root and q is the generaised coordinate. For air ρ =1. 5 kg/m and auminium G =.6 10 10 N/m. Aso J = 4A t / s for a cosed section, A being the encosed area, s the periphera ength and t the skin thickness of the torsion bo. [5 marks] Aero ais V Section of wing torsion bo at wing root ec 0.6 m c =1.5 m Feura ais = 4.5 m 0.5 m Figure Q4(a) Figure Q4(b) Page 4 of 7
Question 5 For the purpose of investigating the dynamic behaviour of the wing of a high-wing aircraft, the wing is modeed as a uniform beam of ength s, as shown in Figure Q5. The wing is pin-jointed at point O on the fuseage which is considered to be fied. It is supported by a strut at mid-span which is represented by a spring from earth norma to the wing. The wing has feura rigidity EI and mass per unit ength ρ and the spring stiffness is 8EI/s. The transverse defection z(y) of the wing is given in terms of assumed modes q1 and q where y y z( y) = q1 + q s s and y is distance aong the wing from O. Obtain the equations of motion of q1 and q and hence find the two natura frequencies. [19 marks] Find aso the corresponding norma modes and sketch their mode shapes, indicating any nodes. [6 marks] The strain energy of a beam in bending is given as 1 d z V = EI 0 dy where the symbos have been defined above. s dy O y z(y) 8EI s s/ Figure Q5 Page 5 of 7
Question 6 Describe briefy the eperimenta apparatus and procedures required to carry out a simpe resonance test on a sma engineering structure in the aboratory. [5 marks] An aircraft empennage (fin and taipane) is attached to a massive and rigid fiture for the purpose of conducting a resonance test, as shown in Figure Q6. The mass distribution is modeed as three umped masses of 10, 0 and 10 kg at points 1, and respectivey. The measured moda modes for the dispacements 1, and at points 1, and as indicated in the figure and their corresponding moda frequencies are found to be 1 st Mode: {.6, 1.0, 0.60} nd Mode: {.0, 0.05, 0.90} rd Mode: { 1.0, 0.61, 0.95} 0 at 6 Hz 1 at 14 Hz at 16Hz Cacuate the generaised mass matri in the measured norma modes and check if it is reasonabe to discard the off-diagona terms. [8 marks] Aso, obtain the generaised forcing vector if a sinusoida force of 00sin Ωt N acts as shown at point, where Ω is the circuar forcing frequency and t is time. [4 marks] Additionay, formuate a set of uncouped equations of motion for vibration of the empennage in the measured norma coordinates, given that the measured damping ratio in each norma mode is 1.5% and that the damping may be assumed viscous. [8 marks] 00sin Ωt N 1 1 Figure Q6 Eaminers: Prof. J.R. Banerjee Dr C.W. Cheung Eterna Eaminer: Prof. D.I.A. Po Page 6 of 7
Information Sheet The strain energy V of a thin-wa section beam of ength and constant wa thickness t in torsion θ () is given as where the torsiona constant d V = 1 G J ( ) θ d d 0 [ A( ) ] s( ) 4t J ( ) =. A() being the encosed area of crosssection and s() the section periphera ength. The strain energy V of the same beam in bending z() is given as 1 d z V = E I ( ) d 0 d where I() is the section moment of area of cross-section. The angrange s equations of motion with usua symbos may be epressed as d dt T T V q i q + & i qi = Q i where Q i is the vector of generaised forces incuding non-conservative forces. The transformation from the structura mass and stiffness matrices, M and K, to the generaised mass and stiffness matrices, M and K can be defined as where R is caed the moda matri. M = R K = R T T M R K R Page 7 of 7