13.4 to "wutter' Substituting these values in Eq. (13.57) we obtain

Transcription

1 568 Elementary aeroelasticity where the origin for z is taken at the built-in end and a is a constant term which includes the tip load and the flexural rigidity of the beam. From Eq. (i) d2 V V(L) = 2aL3 and - dz2-6a(l - '1 Substituting these values in Eq. (13.57) we obtain 2 w1 = 36EIa2 $(L - z)~ dz paa2 Jk(3L - z )~z~ dz + 2m(2ai3)2 Evaluating Eq. (ii) and expressing pa in terms of m we obtain w1 = dZ EI (ii) (iii) which value is only 0.02 per cent higher than the true value given above. The estimation of higher natural frequencies requires the assumption of further, more complex, shapes for V(z). It is clear from the previous elementary examples of normal mode and natural frequency calculation that the estimation of such modes and frequencies for a complete aircraft is a complex process. However, the aircraft designer is not restricted to calculation for the solution of such problems, although the advent of the digital computer has widened the scope and accuracy of this approach. Other possible methods are to obtain the natural frequencies and modes by direct measurement from the results of a resonance test on the actual aircraft or to carry out a similar test on a simplified scale model. Details of resonance tests are discussed in Section Usually a resonance test is impracticable since the designer requires the information before the aircraft is built, although this type of test is carried out on the completed aircraft as a design check. The alternative of building a scale model has found favour for many years. Such models are usually designed to be as light as possible and to represent the stiffness characteristics of the full-scale aircraft. The inertia properties are simulated by a suitable distribution of added masses. A full description of model construction, testing techniques and the estimation of normal modes and frequencies is given in Ref. 3. The calculation of normal modes and frequencies is also treated in Refs 3 and to "wutter' We have previously defined flutter as the dynamic instability of an elastic body in an airstream. It is found most frequently in aircraft structures subjected to large aerodynamic loads such as wings, tail units and control surfaces. Flutter occurs at a critical or flutter speed Vf which in turn is defined as the lowest airspeed at which a given structure will oscillate with sustained simple harmonic motion. Flight at speeds below and above the flutter speed represents conditions of stable and unstable (that is divergent) structural oscillation respectively. Generally, an elastic system having just one degree of freedom cannot be unstable unless some peculiar mechanical characteristic exists such as a negative spring force or

2 13.4 Introduction to 'flutter' 569 a negative damping force. However, it is possible for systems with two or more degrees of freedom to be unstable without possessing unusual characteristics. The forces associated with each individual degree of freedom can interact, causing divergent oscillations for certain phase differences. The flutter of a wing in which the flexural and torsional modes are coupled is an important example of this type of instability. Some indication of the physical nature of wing bending-torsion flutter may be had from an examination of aerodynamic and inertia forces during a combined bending and torsional oscillation in which the individual motions are 90' out of phase. In a pure bending or pure torsional oscillation the aerodynamic forces produced by the effective wing incidence oppose the motion; the geometric incidence in pure bending remains constant and therefore does not affect the aerodynamic damping force, while in pure torsion the geometric incidence produces aerodynamic forces which oppose the motion during one half of the cycle but assist it during the other half so that the overall effect is nil. Thus, pure bending or pure torsional oscillations are quickly damped out. This is not the case in the combined oscillation when the maximum twist occurs at zero bending and vice versa; that is, a 90" phase difference. Consider the wing shown in Fig in various stages of a bending-torsion osciiiation. At the position of zero bending the twisting of the wing causes a positive geometric incidence and therefore an aerodynamic force in the same direction as the motion of the wing. A similar but reversed situation exists as the wing moves in a downward direction; the negative geometric incidence due to wing twist causes a downward aerodynamic force. It follows that, although the effective wing incidence produces aerodynamic forces which oppose the motion at all stages, the aerodynamic forces associated with the geometric incidence have a destabilizing effect. At a certain speed - the flutter speed V, - this destabilization action becomes greater than the stabilizing forces and the oscillations diverge. In practical cases the bending and torsional oscillations would not be as much as 90" out of phase; however, the same basic principles apply. t ' geometric %% Positive incidence producing positive lift 1 - Motion Flexural axis Negative geometric incidence producing negative lift of wing Fig Coupling of bending and torsional oscillations and destabilizing effect of geometric incidence.

3 570 Elementary aeroelasticity The type of flutter described above, in which two distinctly different types of oscillating motion interact such that the resultant motion is divergent, is known as classical flutter. Other types of flutter, non-classical flutter, may involve only one type of motion. For example, stallingflutter of a wing occurs at a high incidence where, for particular positions of the spanwise axis of twist, self-excited twisting oscillations occur which, above a critical speed, diverge. Another non-classical form of flutter, aileron buzz, occurs at high subsonic speeds and is associated with the shock wave on the wing forward of the aileron. If the aileron oscillates downwards the flow over the upper surface of the wing accelerates, intensifying the shock and resulting in a reduction in pressure in the boundary layer behind the shock. The aileron, therefore, tends to be sucked back to its neutral position. When the aileron rises the shock intensity reduces and the pressure in the boundary layer increases, tending to push the aileron back to its neutral position. At low frequencies these pressure changes are approximately 180" out of phase with the aileron deflection and therefore become aerodynamic damping forces. At higher frequencies a component of pressure appears in phase with the aileron velocity which excites the oscillation. If this is greater than all other damping actions on the aileron a high frequency oscillation results in which only one type of motion, rotation of the aileron about its hinge, is present, i.e. aileron buzz. Aileron buzz may be prevented by employing control jacks of sufficient stiffness to ensure that the natural frequency of aileron rotation is high. Bufeting is produced most commonly in a tailplane by eddies caused by poor airflow in the wing wake striking the tailplane at a frequency equal to its natural frequency; a resonant oscillation having one degree of freedom could then occur. The problem may be alleviated by proper positioning of the tailplane and clean aerodynamic design Coupling We have seen that the classical flutter of an aircraft wing involves the interaction of flexural and torsional motions. Separately neither motion will cause flutter but together, at critical values of amplitude and phase angle, the forces produced by one motion excite the other; the two types of motion are then said to be coupled. Various forms of coupling occur: inertial, aerodynamic and elastic. The cross-section of a small length of wing is shown in Fig Its centre of gravity is a distance gc ahead of its flexural axis, c is the wing section chord and the mass of the small length of wing is m. If the length of wing is subjected to an upward acceleration j; an accompanying inertia force my acts at its centre of gravity in a downward direction, thereby producing a nose down torque about the flexural axis of mygc, causing the wing to twist. The vertical motion therefore induces a twisting motion by virtue of the inertia forces present, i.e. inertial coupling. Conversely, an angular acceleration ti about the flexural axis causes a linear acceleration of gc& at the centre of gravity with a corresponding inertia force of mgcii. Thus, angular acceleration generates a force producing translation, again inertial coupling. Note that the inertia torque due to unit linear acceleration (mgc) is equal to the inertia force due to unit angular acceleration (mgc); the inertial coupling therefore possesses symmetry. -

4 13.4 Introduction to 'flutter' 57 1 Centre of gravity \ I Y I Flexural axis Fig Inertial coupling of a wing. Aerodynamic coupling is associated with changes of lift produced by wing rotation or translation. A change of wing incidence, that is a rotation of the wing, induces a change of lift which causes translation while a translation of velocity 3, say, results in an effective change in incidence, thereby yielding a lift which causes rotation. These aerodynamic forces, which oscillate in a flutter condition, act through a centre analogous to the aerodynamic centre of a wing in steady motion; this centre is known as the centre of independence. Consider now the wing section shown in Fig and suppose that the wing stiffness is represented by a spring of stiffness k positioned at its flexural axis. Suppose also that the displacement of the wing is defined by the vertical deflection y of an arbitrary point 0 (Fig (a)) and a rotation a about 0 (Fig (b)). In Fig (a) the vertical displacement produces a spring force which causes a clockwise torque (kyd) on the wing section about 0, resulting in an increase in wing incidence a. In Fig (b) the clockwise rotation a about 0 results in a spring force kda acting in an upward direction on the wing section, thereby producing translations in the positive y direction. Thus, translation and rotation are coupled by virtue of the elastic stieness of the wing, hence elastic coupling. We note that, as in the case of inertial coupling, elastic coupling possesses symmetry since the moment due to 1- Flexural axis L* (a) Fig Elastic coupling of a wing.

5 572 Elementary aeroelasticity unit displacement (kd) is equal to the force produced by the unit rotation (kd). Also, if the arbitrarily chosen point 0 is made to coincide with the flexural axis, d = 0 and the coupling disappears. From the above it can be seen that flutter will be prevented by uncoupling the two constituent motions. Thus, inertial coupling is prevented if the centre of gravity coincides with the flexural axis, while aerodynamic coupling is eliminated when the centre of independence coincides with the flexural axis. This, in fact, would also eliminate elastic coupling since 0 in Fig would generally be the centre of independence. Unfortunately, in practical situations, the centre of independence is usually forward of the flexural axis, while the centre of gravity is behind it giving conditions which promote flutter Determination of critical flutter speed Consider a wing section of chord c oscillating harmonically in an airflow of velocity V and density p and having instantaneous displacements, velocities and accelerations of, rotationally, a, d!, &, and, translationally, y, j, 9. The oscillation causes a reduction in lift from the steady state lift4 so that, in effect, the lift due to the oscillation acts downwards. The downward lift corresponding to a, d! and & is, respectively 1,pcv2a = L,a lbpc2vd! = L&& I.. c3& = L&& CYP in which I,, I&, lii! are non-dimensional coefficients analogous to the lift-curve slopes in steady motion. Similarly, downward forces due to the translation of the wing section occur and are IypcV2y/c = Lyy Ijpc2vj/c = Lyj I.. c3 " c - L *- yp Yl - yy Thus, the total aerodynamic lift on the wing section due to the oscillating motion is given by L = Lyy + LyJj + Lyy + L,a + L&d! + L&& (1 3.60) We have previously seen that rotational and translational displacements produce moments about any chosen centre. Thus, the total nose up moment on the wing section is where M = Myy + Myj + Mjj + M,a + Mbd! + Mcii (13.61) Myy = lypc2 V2y/c ~~j = iypc3 v$lc

6 13.4 Introduction to 'flutter' 573 Centre of gravity 1 Mean position Flexural axis -\ Fig Flutter of a wing section. Myy = 1ypc4y/c 2 2 M,a = m,pc V a/c 3 M&ci = m&pc Vcilc ~ ~ = m6pc4ii/c i i in which m, etc. re analogous to the steady motion local pitching moment coefficients. Now consider the wing section shown in Fig The wing section is oscillating about a mean position and its flexural and torsional stiffnesses are represented by springs of stiffness k and ke respectively. Suppose that its instantaneous displacement from the mean position is y, which is now taken as positive downwards. In additior, to the aerodynamic lift and moment forces of Eqs (13.60) and (13.61) the wing section experiences inertial and elastic forces and moments. Thus, if the mass of the wing section is m and Io is its moment of inertia about 0, instantaneous equations of vertical force and moment equilibrium may be written as follows. For vertical force equilibrium and for moment equilibrium about 0 L - my i- mgcii - ky = 0 (1 3.62) M - I0&+mgcy - kea = 0 (13.63) Substituting for L and M from Eqs (13.60) and (13.61) we obtain (m - Ly)j - LFj + (k - L,)y - (mgc + Lh)2i! - L&& - L,a = 0 (-mgc - Mj)j - Mjj - M,y + (Io - M&)ii - M&ci + (ke - M,)a = 0 ( ) (13.65) The terms involving y in the force equation and a in the moment equation are known as direct terms, while those containing a in the force equation and y in the moment equation are known as coupling terms. The critical flutter speed Vf is contained in Eqs (13.64) and (13.65) within the terms L,, Lj, L,, L&, My, My, M, and Mb. Its value corresponds to the condition that these

7 574 Elementary aeroelasticity equations represent simple harmonic motion. Above this critical value the equations represent divergent oscillatory motion, while at lower speeds they represent damped oscillatory motion. For simple harmonic motion y = yo e'"', a = a. e Substituting in Eqs (13.64) and (13.65) and rewriting in matrix form we obtain -w (m- L?) - idj, + k - Ly J(mgc + Lii) - iwl& - La [ ' E} =O (13.66) w2(mgc + Mj) - iwml - M), -J(Io - M6) - iwm& + ke - M, I{ The solution of Eq. (13.66) is most readily obtained by computer4 for which several methods are available. One method represents the motion of the system at a general speed V by Y'Yoe iwt (6+w)z (6+iw)t, a=ao in which S + iw is one of the complex roots of the determinant of Eq. (13.66). For any speed V the imaginary part w gives the frequency.of the oscillating system while S represents the exponential growth rate. At low speeds the oscillation decays (6 is negative) and at high speeds it diverges (6 is positive). Zero growth rate corresponds to the critical flutter speed V,, which may therefore be obtained by calculating 6 for a range of speeds and determining the value of Vf for S = Prevention of flutter We have previously seen that flutter can be prevented by eliminating inertial, aerodynamic and elastic coupling by arranging for the centre of gravity, the centre of independence and the flexural axis of the wing section to coincide. The means by which this may be achieved are indicated in the coupling terms in Eqs (13.64) and (1 3.65). In Eq. (13.65) the inertial coupling term is mgc + My in which My is usually very much smaller than mgc. Thus, inertial coupling may be virtually eliminated by adjusting the position of the centre of gravity of the wing section through mass balancing so that it coincides with the flexural axis, i.e. gc = 0. The aerodynamic coupling term Mjj vanishes, as we have seen, when the centre of independence coincides with the flexural axis. Further, the terms M,J and L&d! are very small and may be neglected so that Eqs (13.64) and (13.65) now reduce to and (m - Lji)ji - Lpj + (k - Ly)y - L,a = 0 (13.67) (Io - Me)& - Mbd! + (ke - Ma)a = 0 (1 3.68) The remaining coupling term L,a cannot be eliminated since the vertical force required to maintain flight is produced by wing incidence. Equation (13.68) governs the torsional motion of the wing section and contains no coupling terms so that, since all the coefficients are positive at speeds below the wing

8 13.4 Introduction to 'flutter' 575 section torsional divergence speed, any torsional oscillation produced, say, by a gust will decay. Also, from Eq. (13.67), it would appear that a vertical oscillation could be maintained by the incidence term Loo. However, rotational oscillations, as we have seen from Eq. (13.68), decay so that the lift force L,a is a decaying force and cannot maintain any vertical oscillation. In practice it is not always possible to prevent flutter by eliminating coupling terms. However, increasing structural stiffness, although carrying the penalty of increased weight, can raise the value of V, above the operating speed range. Further, arranging for the centre of gravity of the wing section to be as close as possible to and forward of the flexural axis is beneficial. Thus, wing mounted jet engines are housed in pods well ahead of the flexural axis of the wing. The previous analysis has been concerned with the flutter of a simple two degrees of freedom model. In practice the structure of an aircraft can oscillate in many different ways. For example, a wing has fundamental bending and torsional modes of oscillation on which secondary or overtone modes of oscillation are superimposed. Also it is possible for fuselage bending oscillations to produce changes in wing camber thereby affecting wing lift and for control surfaces oscillating about their hinges to produce aerodynamic forces on the main surfaces. The equations of motion for an actual aircraft are therefore complex with a number N, say, of different motions being represented (N can be as high as 12). There are, therefore, N equations of motion which are aerodynamically coupled. At a given speed, solution of these N equations yields N different values of S + iw corresponding to the N modes of oscillation. Again, as in the simple two degrees of freedom case, the critical flutter speed for each mode may be found by calculating S for a range of speeds and determining the value of speed at which S = 0. A similar approach is used experimentally on actual aircraft. The aircraft is flown at a given steady speed and caused to oscillate either by exploding a small detonator on the wing or control surface or by a sudden control jerk. The resulting oscillations I Decay ra: Fig Experimental determination of flutter speed.

9 576 Elementary aeroelasticity are recorded and analysed to determine the decay rate. The procedure is repeated at increasing speeds with smaller increments being used at higher speeds. The measured decay rates are plotted against speed, producing a curve such as that shown in Fig This curve is then extrapolated to the zero decay point which corresponds to Vf. Clearly this approach requires as accurate as possible a preliminary estimation of flutter speed since induced oscillations above the flutter speed diverge leading to possibly catastrophic results. Other experimental work involves wind tunnel tests on flutter models, the results being used to check theoretical calculations Control surface flutter If a control surface oscillates about its hinge, oscillating forces are induced on the main surface. For example, if a wing oscillates in bending at the same time as the aileron oscillates about its hinge, flutter can occur provided there is a phase difference between the two motions. In similar ways elevator and rudder flutter can occur as the fuselage oscillates in bending. Other forms of control surface flutter involve more than two different types of motion. Included in this category are wing bending/aileron rotation/tab rotation and elevator rotation/fuselage bending/rigid body pitching and translation of the complete aircraft. It can be shown4 that control surface flutter can be prevented by eliminating the inertial coupling between the control rotation and the motion of the main surface. This may be achieved by mass balancing the control surface whereby weights are attached to the control surface forward of the hinge line. All newly designed aircraft are subjected early in the life of a prototype to a ground resonance test to determine actual normal modes and frequencies. The primary objects of such tests are to check the accuracy of the calculated normal modes on which the flutter predictions are based and to show up any unanticipated peculiarities in the vibrational behaviour of the aircraft. Usually the aircraft rests on some low frequency support system or even on its deflated tyres. Electrodynamic exciters are mounted in pairs on the wings and tail with accelerometers as the measuring devices. The test procedure is generally first to discover the resonant frequencies by recording amplitude and phase of a selected number of accelerometers over a given frequency range. Having obtained the resonant frequencies the aircraft is then excited at each of these frequencies in turn and all accelerometer records taken simultaneously. Babister, A. W., Aircraft Stability and Control, Pergamon Press, London, Duncan, W. J., The Principles of the Control and Stability of Aircraft, Cambridge University Press, Cambridge, Bisplinghoff, R. L., Ashley, H., and Halfman, R. L., Aeroelasticity, Addison-Wesley Publishing Co. Inc., Cambridge, Mass., Dowell, E. H. et al., A Modern Course in Aeroelasticity, Sijthoff and Noordhoff, Alphen am den Rijn, Netherlands, 1978.