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Nonlinear characteristics of randomly excited transonic flutter
Mathematics and Computers in Simulation 58 (2002) 385–405
Nonlinear characteristics of randomly excited transonic flutter L.E. Christiansen a , T. Lehn-Schiøler a , E. Mosekilde a,∗ , P. Gránásy b , H. Matsushita c a
Department of Physics, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark b G.E. Hungary Ltd., 1340 Budapest, Vací ut 77, Hungary c National Aerospace Laboratory, 16-13-1 Ohsawa, Mitaka-shi, Tokyo 181-0015, Japan
Abstract The paper describes the effects of random external excitations on the onset and dynamical characteristics of transonic flutter (i.e. large-amplitude, self-sustained oscillations) for a high aspect ratio wing. Wind tunnel experiments performed at the National Aerospace Laboratory (NAL) in Japan have shown that the self-sustained oscillations arise in a subcritical Hopf bifurcation. However, analysis of the experimental data also reveals that this bifurcation is modified in various ways. We present an outline of the construction of a 6 DOF model of the aeroelastic behavior of the wing structure. When this model is extended by the introduction of nonlinear terms, it can reproduce the subcritical Hopf bifurcation. We hereafter consider the effects of subjecting simplified versions of the model to random external excitations representing the fluctuations present in the airflow. These models can reproduce several of the experimentally observed modifications of the flutter transition. In particular, the models display the characteristic phenomena of coherence resonance. © 2002 IMACS. Published by Elsevier Science B.V. All rights reserved. Keywords: Transonic flutter; Subcritical Hopf bifurcation; Noise excitation; Coherence resonance; Wind tunnel experiments
1. Introduction Flutter denotes a characteristic form of self-excited oscillations that can arise through the interaction of an aerodynamic flow with the elastic modes of a mechanical structure, e.g. the bending and torsion modes of an aircraft wing [1,2]. Limit cycle oscillations (LCO) of this form typically arise in the transonic region (i.e. when the aircraft operates close to the velocity of sound). They are thought to be related with a separation of the airflow associated with shock waves propagating along the surface of the wing. Emergence of flutter compromises not only the long term durability of the wing structure, but also the operational safety, flight performance and energy efficiency of the aircraft. Effective means of flutter prevention are, therefore, mandatory in the certification of new flight vehicles, and considerable effort, theoretically as well as experimentally, is devoted to the study of methods for active flutter control and of the interaction between structural dynamics and unsteady airflows. ∗
Corresponding author. Tel.: +45-45-88-16-11; fax: +45-45-93-16-69. E-mail address: [email protected] (E. Mosekilde). 0378-4754/02/$ – see front matter © 2002 IMACS. Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 4 7 5 4 ( 0 1 ) 0 0 3 7 9 - 2
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A number of investigators have recently succeeded in producing limit cycle oscillations in numerical simulations of various aeroelastic systems using Euler code [3,4] or Navier–Stokes code [5]. Experimental studies of transonic LCO in fighter aircraft and two-dimensional supercritical wing structures have also been reported [6–8], and a few papers have compared test results with numerical simulations and obtained reasonable agreement [4,9,10]. In studies of this type the concepts and methods of nonlinear dynamics and bifurcation theory are indispensable in explaining the onset and dynamics of the self-excited oscillations. Namachchivaya and Van Roessel [11,12], for example, have provided an early analysis of the stability and bifurcation of the single degree of freedom pitching motion for a double-wedge airfoil in a supersonic flow, emphasizing in particular the possible occurrence of a degenerate Hopf bifurcation at certain parameter values. Wind tunnel experiments performed at the National Aerospace Laboratory (NAL) in Japan over a period of 4–5 years with a high aspect ratio wing model have consistently shown that the onset of flutter occurs abruptly as the dynamical pressure reaches a certain critical value [13,14]. Moreover, as the dynamic pressure is subsequently reduced, the flutter oscillations occur all the way down to pressures that are about 10% below the critical onset pressure, again to disappear suddenly. With its hysteresis and abrupt transitions the above behavior is consistent with the occurrence of a subcritical Hopf bifurcation. The equilibrium point of the aeroelastic system loses its stability as a pair of complex conjugated eigenvalues cross the imaginary axis. However, by contrast to the supercritical Hopf bifurcation, the transition is not associated with the formation of a stable limit cycle of smoothly growing amplitude. Rather, for aerodynamic pressures below the bifurcation point, an unstable (saddle) cycle coexist with the stable equilibrium point, and the loss of stability is associated with the saddle cycle closing in on this point. At aerodynamic pressures somewhat below the Hopf bifurcation point, the saddle cycle stabilizes in a saddle-node bifurcation, and the finite amplitude stable limit cycle (the node) continues to exist for increasing values of the aerodynamic pressure (this bifurcation structure is illustrated in Fig. 3). In a particular experiment, a transition with a smoothly decreasing flutter amplitude was observed. Originally, this was interpreted in terms of a distortion of the deterministic bifurcation diagram, caused by the presence of higher nonlinear interactions and leading to a supercritical transition for certain parameter values [15,16]. One of the purposes of the present work is to discuss an alternative explanation according to which modifications in the deterministic bifurcation structure, caused by random fluctuations associated with the unsteady airflow, produce phenomena that appear similar to a supercritical transition. In particular, the presence of random excitations may produce the so-called noisy precursors, i.e. a coherent oscillatory response from the wing structure that grows in amplitude as the bifurcation point is approached. This behavior is related to the phenomenon of coherence resonance [17,18]. Using a Fokker–Planck equation approach, the influence of noise on the characteristics of a supercritical Hopf bifurcation has been studied in considerable detail by Arnold et al. [19]. They conclude that the interaction between noise and bifurcation structure can produce interesting distortions in the invariant density and split the pair of two complex conjugated eigenvalues into different Lyapunov exponents. The subcritical Hopf bifurcation appears not to have been studied in similar detail. Besides the above mentioned phenomena, including the emergence of noisy precursors, we expect a couple of additional modifications: the presence of noise may cause a transition to large amplitude limit cycle oscillations (across the saddle solution) before the Hopf bifurcation point is reached, producing a shift in the apparent bifurcation point that depends on the noise amplitude. At larger noise amplitudes, it is also possible that
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the system can start to jump forth and back between the equilibrium state and the oscillatory state, and this switching may develop into a pseudo-coherent phenomenon [17,18]. Finally, it is worth mentioning that Leng and Namachchivaya [20] have considered the influence of random external excitations on the behavior of systems with several (i.e. two) marginally stable or unstable modes, a condition which may very well apply to our aeroelastic wing problem. For such systems, the bifurcation structure is not robust. In particular, while one mode may be stabilized by the presence of noise, another may be destabilized, the actual result depending on details of the interaction between the critical modes. Only under very special conditions can one expect the deterministic bifurcation structure to be preserved.
2. Wind tunnel experiments Experiments were performed with the transonic wind tunnel at the NAL in Japan [13,14,21]. This tunnel can operate at relatively low noise levels up to air speeds of 1.4 Mach. Fig. 1 shows a sketch of the wing model (a) together with a photograph of the model as mounted in the 2 m × 2 m test section of the tunnel (b). The model simulates a high aspect ratio wing span of an advanced, energy efficient aircraft. The length of the wing is 1.043 m. Except for the middle part, where the cross-section is inflated
Fig. 1. Sketch of the wing model (a) with its leading edge and trailing edge control surfaces crosshatched. The model simulates a high aspect ratio wing of an advanced, energy efficient aircraft. Positions of the accelerometers and strain gauges are indicated. Photograph of the model as mounted in the 2 m × 2 m test section of the transonic wind tunnel (b). Note the flutter stopping device.
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and symmetric to make room for a couple of servo motors, the wing has a supercritical cross-section. To prevent the wing from being destructed during the experiments, a flutter stopping device has been installed in the test section. This device has the form of a triangular plate that normally lies along the floor of the tunnel. However, when the wing goes into self-excited oscillations the plate can immediately be raised to protrude into the airflow and lower the dynamic pressure [13]. The servo motors are used for experiments with active flutter control [22]. For this purpose the model is also equipped with leading edge and trailing edge control surfaces (shown crosshatched in Fig. 1a) that can be activated by the servo motors. During the experiments the behavior of the wing is monitored by means of four accelerometers and seven pairs of strain gauges fixed at wing spars and measuring various bending and torsion components of the strain (Fig. 1a). Tests performed in the transonic regime at different Mach numbers and different pressures have consistently led to the appearance of large amplitude limit cycle oscillations. These experiments have also confirmed the existence of the transonic dip, characteristic of a high aspect ratio wing [23], i.e. of a minimum in the dynamic pressure required to excite flutter oscillations near 0.8 Mach [13]. In this connection it was noted that the flutter amplitude is relatively modest near the minimum of the transonic dip to grow for both higher and lower Mach numbers. The flutter frequency for the refurbished wing (i.e. after installation of the electric motors) was found to be around 22 Hz. The results to be reported here were performed with air speeds around M = 0.8 and with an incidence angle (angle of attack) α in the interval from α = −0.2 to +0.75◦ . Under these circumstances, the onset of flutter typically occurred at a dynamical pressure of pd = 27.9 kPa. This will be referred to as the nominal flutter pressure. To examine the nature of the flutter transition, a series of wind tunnel experiments were performed in which the leading edge control surface was perturbed by a sinusoidal forcing signal. The stationary state reached by the wing after turning off the sinusoidal forcing was determined as a function of the air speed. These experiments showed that the transition to flutter can occur below the nominal flutter pressure, and that the position of the transition point depends both on the frequency and the amplitude of the forcing signal. Most efficient are forcing frequencies around the flutter frequency of about 22 Hz, and the effect weakens significantly as the forcing frequency is either reduced to 10 Hz or increased to 30 Hz. The transition to flutter does not necessarily occur instantaneously when the forcing signal is applied, but may require a build-up time of several seconds. On the other hand, when first excited, the large amplitude oscillations tend to persist even when the forcing is removed. For a range of dynamical pressures, the above results established the coexistence of a stable equilibrium state and a state of large amplitude LCO, and hence, pointed to a subcritical nature of the flutter bifurcation [15]. In a subsequent series of experiments performed at a forcing frequency of 21.5 Hz, an attempt was made to determine the separatrix between the forcing amplitudes that lead to large amplitude oscillations and the amplitudes for which the wing remains close to its stable equilibrium state. Fig. 2 reproduces the results of this experiment. Here, the limiting state has been plotted as a function of the amplitude of the leading edge excitation and of the dynamic pressure in the wind tunnel. Inspection of Fig. 2 shows that the nominal flutter transition point in these experiments (α = +0.75◦ ) occurs around pd = 27.9 kPa. Above this point all experiments produce large amplitude limit cycle oscillations even without forcing. As the dynamic pressure is reduced below the critical value, larger and larger forcing amplitudes are required to elicit LCO until, at a dynamic pressure of about 24.8 kPa, the experiments fail to produce self-sustained oscillations, even with a forcing amplitude of the leading edge surface as large as 6◦ . We assume that the curve that delineates the two types of behavior for 24.8 kPa < pd < 27.9 kPa represents the separatrix between the stable equilibrium state and the flutter
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Fig. 2. Experimental results obtained with a sinusoidal forcing of the leading edge control surface. In the depicted flap amplitude vs. dynamical pressure plane, crosses represent conditions that lead to self-sustained oscillations, and circles represent conditions that produce stable equilibrium behavior. The nominal flutter transition pressure is pd ∼ = 27.9 kPa.
∼ 24.8 kPa represents the position of the saddle-node bifurcation in state, and that the threshold at pd = which the unstable LCO involved in the subcritical transition are stabilized. The results of the above experiments have subsequently been reanalyzed in order to construct a proper bifurcation diagram. The accelerometer signal was numerically integrated twice to obtain the velocity and deflection signals for the wing model, and phase plots were prepared for different initial conditions and various dynamical pressures. This has allowed us to determine the variation of the amplitudes for both the stable and the unstable limit cycle as a function of the dynamic pressure. As depicted in Fig. 3, the bifurcation diagram clearly illustrates the subcritical nature of the flutter transition. The crosshatched area represents the significant uncertainty in locating the separatrix (unstable limit cycle). Fig. 4 displays the results of a particular wind tunnel experiment. Here, the dynamic pressure has been initiated at a value beyond the nominal flutter pressure, the flutter control has been switched off, and we follow the changes in the accelerometer signal as the dynamic pressure is gradually reduced. At time t ∼ = 11.5 s the dynamic pressure reaches the saddle-node bifurcation, stable self-sustained oscillations cease to occur, and the accelerometer signal drops from an amplitude of about 6 V to less than 1 V. This latter value represents the stable equilibrium state excited by the random fluctuations in the air flow. Fig. 5a shows the power spectrum of the signal recorded from the accelerometer during the initial period of self-excited oscillations in Figs. 4 and 5b displays the spectrum for the noise excited equilibrium state attained for t > 12 s. The spectrum in Fig. 5a clearly reveals not only the frequency of the flutter mode (f ∼ = 22 Hz), but also its higher harmonics up to the order of seven or more. Once the transition to the noise excited equilibrium state has occurred, one no longer observes the narrow peaks in the spectral distribution. The power spectrum displayed in Fig. 6 applies to a state of noise excited equilibrium dynamics for a dynamical pressure pd ∼ = 27.0 kPa, i.e. below but relatively close to the nominal flutter pressure. This spectrum exhibits two narrow and relatively close peaks in the frequency range of 20–25 Hz characteristic
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Fig. 3. Experimentally obtained bifurcation diagram illustrating the subcritical nature of the flutter transition. The diagram was constructed from phase plots obtained with different initial conditions and six different values of the dynamic pressure. Note that the self-sustained oscillations in the 1 m long wing structure reach amplitudes of the order of 12 mm.
Fig. 4. Variation of the accelerometer signal as the dynamic pressure is gradually reduced. At t ∼ = 11.5 s, the amplitude abruptly drops from about 6 V to less than 1 V as the aeroelastic system passes the saddle-node bifurcation in Fig. 3 and stable self-sustained oscillations cease to exist.
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Fig. 5. Power spectrum for the flutter oscillation observed in Fig. 4 for t < 10 s (a). Here, one can observe the build up of lines in the spectrum corresponding to the flutter frequency and its harmonics. Spectrum observed in the noise excited equilibrium state attained in Fig. 4 for t > 12 s (b).
of flutter oscillations in the considered wing model. This result, which seems to be reproduced fairly consistently, indicates that two different modes may be involved in the flutter transition. The beating phenomenon, i.e. the periodic modulation of the flutter amplitude observed in Fig. 4 points to the same conclusion. We consider this spectrum as an example of the emergence of noisy precursors, i.e. of relatively strong
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Fig. 6. Power spectrum for a noise excited equilibrium state immediately below the point of nominal flutter transition. Note the appearance of two nearby lines. Such noisy precursors may be used in the development of active flutter control.
components of the main flutter frequency and its harmonics even before the Hopf bifurcation has occurred [18]. Such signals may play an important role in the development of methods for active flutter control [22]. A particular test with active flutter control has produced a very different behavior [16]. In this experiment the dynamical pressure was held constant at a value pd ∼ = 28.3 kPa, i.e. slightly above the nominal flutter ∼ pressure (pd = 27.9 kPa). When the flutter control was switched off, large amplitude flutter oscillations were observed to build up with an incubation time of the order of 5 s. To stop the oscillations the flutter control was activated and then again released. This time a low amplitude beating oscillation appeared to finally again give way to large amplitude flutter oscillations. After a new activation and release of the flutter control, the low amplitude beating oscillation reappeared. However, this time it seemed to be stable. Moreover, upon reduction of the dynamic pressure the oscillation amplitude decreased smoothly as in the presence of a supercritical (or soft) bifurcation. The above results indicate the possibility that two different flutter modes may occur for the same dynamical pressure, and that one of the modes arises via a supercritical Hopf bifurcation [16]. Unfortunately, in spite of much work we have never been able to reproduce these results or to find anything resembling them. However, the results again seem to indicate that two different aerodynamic modes can become critical almost simultaneously, and that cooperative effects of mode-to-mode interactions and noise excitation might influence the dynamics near the bifurcation point.
3. Finite state aeroelastic model The purpose of this section is to sketch the development of a finite state aeroelastic model for our experimental wing system, and to present the main results of a linear stability analyses for the onset of
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flutter [13,14]. In Section 4, we shall consider a reduced nonlinear model in order to examine some of the phenomena that can arise via the interaction of noise with a subcritical Hopf bifurcation. The structural dynamics of the wing are represented by its six basic modes: four bending modes and two torsion modes, and each of these modes is described by its generalized coordinate q and velocity q. ˙ In the linearized analysis, the phase shift associated with the unsteady airflow is represented by a set of six delay variables [24], and in this way we arrive at an equation of motion of the following standard form: x˙ = Ax with x = [q, q, ˙ z]T ∈ R18 where x denotes the 18-dimensional state vector. The system matrix A = {aij } ∈ R18×18 takes the form 0 I −1 A = −(M − A0 ) (K − A2 ) −(M − A0 )−1 (C − A1 ) Ba 0
(3.1)
0 −(M − A0 )−1 Λ
(3.2)
where M, C, and K are the mass, damping coefficient and stiffness matrices for the considered structural modes, and I is the unit matrix. The matrices Ai , i = 0, 1, and 2 and the matrix Ba are coefficient matrices in the finite state expression of the aerodynamic forces, and Λ is a diagonal matrix of reciprocal delay times. Methods to obtain these matrices have been discussed by several authors. In the present analyses we have adopted the method described by Roger [24]. To determine the coefficients of the various matrices in the structural model a number of vibration tests were performed. Using both sinusoidal and random forcing, the wing model was vibrated at selected points, and the deflections at 39 points over the wing surface were measured by means of a laser deflection meter. The results were treated with the help of a modal analysis software (FEM analysis) to determine the natural frequencies, damping coefficients and shapes (nodal lines) for the six basic modes. The mass distribution across the wing and the variation of rigidity were adjusted to obtain the best possible fit between the linear model and the vibration results. Table 1 lists the corresponding natural frequencies. For all six modes, the analytically calculated frequencies agree with the experimentally determined frequencies to within an error of 3%, and the nodal lines also show good agreement. Table 1 Natural frequencies in Hz for the basic modes of the wing modela Mode
First bending Second bending First torsion Third bending Second torsion Fourth bending a
Excitation
Model
Random
Sinusoidal
12.5 37.0 44.0 86.5 134.0 173.0
12.22 37.23 45.11 87.20 134.64 177.33
12.52 36.23 44.01 90.10 138.06 172.45
The frequencies of all six modes agree within an error of 3%. The nodal lines also show good agreement between experiments and the linear model [13].
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Fig. 7. Root locus analysis for the eigenvalues of the six basic modes as functions of the air speed. Only the lowest mode undergoes a Hopf bifurcation. The corresponding frequency is 22 Hz.
The aerodynamic forces depend on the air speed and are distributed across the wing surface. In order to perform a linear, finite-mode analysis these forces must be approximated by lumped parameters for the individual modes, and the generalized forces calculated in this way must be expanded with respect to the position, velocity and phase shift variables. Fig. 7 presents the results of an eigenvalue analysis for the coefficient matrix A in terms of a root locus plot with the air speed as a parameter. Inspection of the figure shows how a pair of complex conjugated eigenvalues for the lowest mode (the first bending mode) start very close to the imaginary axis and with increasing air speed first move in the direction of increasing damping to subsequently move to the right and cross the imaginary axis for air speeds of the order of 280 m/s. The point of crossing corresponds to a frequency of approximately 26 Hz. Several of the other modes also have fairly low damping coefficients at low air speeds. However, as the air speed increases the damping becomes stronger, and the eigenvalues move away from the imaginary axis. We must conclude that the analytical results do not provide support to the idea that two modes of nearly the same frequency become unstable at approximately the same wind speed. A similar conclusion may be drawn from a reduced two-mode analysis of the problem [13]. When comparing the experimental results with the analytic calculations it is also observed that the root locus analysis predicts a flutter dynamic pressure that is only about one-third of the nominal flutter pressure determined in the experiments. To account for this disagreement, a factor of 0.368 has been applied to the generalized aerodynamic forces. We suppose that a better agreement could be achieved if the description of the dynamics of the unsteady air flow was improved, e.g. by replacing the six first order differential equations for the delay variables by six equations of motion of second order.
4. Noise response The above discussion has left us with a number of questions. In particular, the linear analyses shows that only a single structural mode is under bifurcation, whereas some of the experimental results (beating
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Fig. 8. Bifurcation diagram for the extended normal-form model (4.1). The diagram illustrates the subcritical Hopf bifurcation and the saddle-node bifurcation in which the unstable limit cycle oscillations are stabilized. Here, α = −1 and β = 0.2.
of the flutter amplitude and double-peaked spectra) indicate that two critical modes of nearly the same frequency might be involved. Another problem is the observation (in a single experiment) of a flutter transition that could appear to be supercritical. We cannot resolve these problems at present. However, in view of the relatively strong noise level in the air flow (presumably caused by a weak separation on the wing surface associated with shock waves that exist even at these relatively low wind speeds) it appears worthwhile to try to understand some of the phenomena that can arise when a subcritical Hopf bifurcation is excited by a random external signal. The wind tunnel experiments (see, e.g. Fig. 4) show an amplitude for the noise response of the equilibrium state of nearly 20% of the fully developed flutter amplitude. As a first approach to this problem we have considered a simple model representing the normal-form of a subcritical Hopf bifurcation, but extended with higher order terms to account for the stabilization of the unstable LCO in a saddle-node bifurcation. Neglecting some of the possible cross terms and assuming a symmetrical form for the nonlinear terms, our simplified model may be expressed as x˙ = µx − ωy + αx(x 2 + y 2 ) + βx(x 4 + y 4 ) and y˙ = ωx + µy + αy(x 2 + y 2 ) + βy(x 4 + y 4 )
(4.1)
Here, ω represents the angular frequency of the oscillations at the bifurcation point µ = 0 where the damping vanishes. The Hopf bifurcation is subcritical for α < 0 and supercritical for α > 0. A typical bifurcation diagram is illustrated in Fig. 8. Here, α = −1 and β = 0.2. Fig. 9a illustrates the dynamics of the extended normal-form model when the stable equilibrium point is perturbed by a noise signal of relatively low amplitude, and Fig. 9b shows the corresponding power spectrum. Here, µ = −1. As expected, the trajectory fluctuates around the stable equilibrium point, and the spectrum shows a peak near the natural frequency of the system (22.4 Hz). With higher amplitude it becomes possible for the noise to excite the system forth and back across the unstable limit cycle, i.e. between the noise perturbed equilibrium state and fully developed flutter oscillations. This is illustrated in Fig. 10. A similar behavior cannot be observed in the wind tunnel experiments because the wing model is not allowed to operate in the flutter state for extended periods of
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Fig. 9. Temporal behavior of the extended normal-form model subjected to a low amplitude noise signal (a). Corresponding power spectrum showing a peak near the resonance frequency of the model (b). Here, µ = −1.
time. However, the phenomenon is characteristic of coherence resonance in a bistable system and will be discussed in more detail in Section 5. When applying a low level noise signal to the extended normal-form model below but relatively close to the Hopf bifurcation point, a relatively strong response is elicited at the resonant flutter frequency. In many ways this response may appear like a low amplitude limit cycle. As the distance to the bifurcation point
Fig. 10. In the presence of a noise signal of larger amplitude the system may jump forth and back between a state of noise-perturbed equilibrium dynamics and fully developed flutter oscillations µ = −1.
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Fig. 11. A gradual shift of the operation point away from the Hopf bifurcation causes the noise response of the equilibrium state to decrease. The phenomenon may explain the observation in a particular experiment of an apparent supercritical flutter transition.
is increased by changing the bifurcation parameter, the damping of the resonant mode also increases, and one can observe something that looks like a smooth transition. This is illustrated in Fig. 11. It is possible that this phenomenon provides an explanation to the result obtained in a single wind tunnel experiment and interpreted as a supercritical Hopf bifurcation. A particular problem concerns the influence of noise on the presence of higher harmonics of the flutter frequency in the power spectra. Such harmonics are quite conspicuous in some of the experimental results, but nearly absent in others. Fig. 12 shows a sequence of power spectra calculated for the extended normal-form model with increasing noise amplitude. Here, µ = 0.2, and we are operating just above the Hopf bifurcation point. Fig. 12a was calculated for the noise free system, and we clearly observe the higher harmonics of the limit cycle oscillations. As the noise amplitude is increased, the higher harmonics are gradually suppressed (Fig. 12b) to finally disappear almost completely (Fig. 12c). After this introductory investigation, we have considered the effects of noise on a nonlinear 2 DOF structural model representing the dynamics of the first bending and the first torsion mode, i.e. x˙ = A(v)x + b(x, v) with x = (q1 , q2 , q˙1 , q˙2 , z1 , z2 )T ∈ R 6
(4.2)
Here, v denotes the air speed. The coefficients of the matrix A(v) were determined as described in Section 3 for the full 6 DOF version of the model. The nonlinear contribution b(x, v) contains second and fourth order terms in the bending mode coordinate q1 and is added to the acceleration term for this mode, i.e.
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Fig. 12. Power spectra obtained for the extended normal-form model in the presence of noise signals of increasing amplitude. Here, µ = +0.2, and the model is operating immediately above the Hopf bifurcation point.
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Fig. 12. (Continued).
0 0 a (v)(αq 2 + βq 4 ) 33 1 1 b(x, v) = 0 0 0
(4.3)
There is no detailed physical argument for this expression, and other authors have used a formulation in which the nonlinear terms depend on the velocity q˙1 rather than on the coordinate q1 itself [25]. With a proper choice of the parameters, the two approaches appear to give similar results. The aerodynamic forces were adjusted such that the model reproduced the experimentally observed flutter transition point, and the nonlinear parameters α and β were chosen to produce a proper position of the saddle-node bifurcation point and a reasonable amplitude of the flutter oscillations. With α ∼ = −1 and β∼ = 0.3, the nonlinear 2 DOF model can reproduce the experimentally observed bifurcation diagram in Fig. 3. A root locus analysis for the (linearized) 2 DOF model shows that for this model it is the second mode (i.e. the first torsion mode) that becomes unstable. At the Hopf bifurcation point the frequency of this mode is approximately 22 Hz. An exchange of the active modes must, of course, be considered with a certain caution. However, none of the modes are pure, and a mixing of the two modes has occurred well before the Hopf bifurcation takes place.
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Fig. 13. A series of power spectra obtained for the nonlinear 2 DOF structural model immediately above the Hopf bifurcation point. As in Fig. 12 we find that increasing noise intensity suppresses the higher harmonics of the flutter oscillations.
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Fig. 13. (Continued).
Application of noise to the nonlinear 2 DOF structural model reproduces all the same results as we have obtained for the extended normal-form model. In particular, when operating in the subcritical regime (i.e. at dynamic pressures below the Hopf bifurcation point), with the proper initial conditions the trajectory stays close to the stable equilibrium point at low noise levels to jump into the state of fully developed flutter oscillations at larger noise amplitudes. At even larger noise amplitudes, the system starts to jump forth and back between the two states in a manner similar to the behavior depicted in Fig. 10. The 2 DOF model can also reproduce the smooth reduction of the noise response as the point of operation is shifted away from the bifurcation point. This is similar to the results displayed in Fig. 11. As a further example of the similarity between the results obtained with the two models, Fig. 13 shows a sequence of power spectra calculated for the supercritical regime immediately above the Hopf bifurcation point. Again we see how increasing the noise intensity gradually suppresses the higher harmonics in the spectrum while leaving the peak at the flutter frequency relatively unaffected.
5. Conclusion We presented the results of a series of transonic flutter experiments performed at the NAL in Japan and outlined the basic aeroelastic theory used to analyze the results. In order to understand better some of the observed phenomena, particularly the significance of random fluctuations in the airflow in the presence of a subcritical flutter transition, we considered two nonlinear models. First an extended normal-form
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model and secondly a nonlinear version of a 2 DOF structural model. For both models the findings were the same. • When operating in the subcritical regime (i.e. at dynamic pressures below the nominal flutter pressure), one can observe a gradual increase in the noise excited fluctuations of the wing as the bifurcation point is approached. Particularly, the resonant components of the spectrum tend to grow. Under proper conditions, this phenomenon may appear like a supercritical transition. • At higher noise amplitudes, the system may jump from the noise perturbed equilibrium state into the state of fully developed flutter (i.e. across the separatrix provided by the unstable limit cycle oscillations). This causes a reduction of the effective flutter transition point relative to the deterministically calculated bifurcation point. • At even higher noise amplitudes the system may start to jump forth and back between the noise perturbed equilibrium state and the flutter state. This produces the phenomenon of coherence resonance [17,18]. • When operating in the supercritical regime immediately above the Hopf bifurcation, the presence of noise is found to suppress the higher harmonics of the flutter frequency. Coherence resonance is a phenomenon by which the application of noise to a nonlinear system can elicit a coherent component in the response. This phenomenon is related to some extent to the more well-known phenomenon of stochastic resonance [26] by which the application of noise to a nonlinear system in the presence of a (weak) periodic signal can enhance the periodic component. Stochastic resonance has been applied, for instance, to explain the pseudo-regular transitions of the earth’s climatic system between ice ages and more normal conditions [27] or the ability of crayfish to detect their preys [28]. Experimental observation of coherence resonance has recently been reported for cascaded excitable systems [29]. In order to characterize the switching behavior between the noise perturbed equilibrium state and the flutter state in the subcritical regime (as displayed, for instance, in Fig. 10), one can consider the degree of regularity in the jumps for a very long time series. At the lower end of the interval of noise amplitudes that can elicit this phenomenon, jumps from one of the states will be quite rare, and a spectral analysis of the temporal behavior will reveal a very broadband spectrum. In the other end of the interval, i.e. for very large noise amplitudes, jumps will occur quite frequently and will again be random in nature. However, there may be an optimal choice of the noise amplitude (typically when the jump intervals are nearly the same for the two directions) when the spectral analyses reveals a clear peak. When this is the case, we observe a noise induced time scale in the switching behavior, and a time scale which is not related to noisy precursors of the deterministic dynamics or to any other inherent deterministic periodicity. A more quantitative description of the degree of regularity may be obtained by calculating the entropy S(s) for the spectral distribution of the switching dynamics relative to the entropy S(n) of a completely random switching process or, alternatively, the so-called regularity parameter γ =1−
S(s) S(n)
(5.1)
As applied here, the spectral entropy is defined by N S(s) = − si ln si i=1
(5.2)
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Fig. 14. Variation of the regularity parameter γ with the noise level and the air speed for the nonlinear 2 DOF structural model. When scanning the model vertically between the two bifurcations, one can observe a maximum in the regularity of the switching process. The scale to the right defines the size of the regularity parameter. This parameter is found to vary between 0.02 and 0.20.
Fig. 15. Variation of the lifetime in the noise-perturbed equilibrium state as a function of the noise level and the air speed. For a noise level of 1 m/s, transition to flutter oscillations occurs nearly 5% below the nominal flutter speed. The scale to the right defines the lifetime in the noise-perturbed equilibrium state in seconds.
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where si represents the relative contribution to the power spectrum from the frequency interval #Ωi = Ω/N, N being the total number of spectral bins and Ω the frequency range of the considered spectrum. For a random process (with a flat spectrum), S(s) = S(n) = ln N, and the regularity parameter is γ = 0. For a completely monochromatic signal, on the other hand, S(s) = 0 and γ = 1. Fig. 14 shows the variation of the regularity parameter with the noise level and the air speed for the nonlinear 2 DOF structural model discussed in Section 4. The subcritical Hopf bifurcation occurs at v ∼ = 266 m/s, and ∼ the saddle-node bifurcation at v = 248 m/s. In the interval between these two bifurcations, the unstable limit cycle represents a separatrix between the equilibrium state and the flutter state. When scanning the diagram vertically in this interval one can clearly observe that there is a maximum in the regularity of the switching dynamics. Fig. 15 shows the lifetime for the system in the noise perturbed equilibrium state as a function of the noise level and the air speed, now at somewhat lower noise levels. Inspection of the figure shows how the effective threshold for excitation of flutter oscillations moves to lower and lower air speeds as the noise level is increased. Moreover, this shift appears to be nearly linear. For a noise level of 1 m/s, transition to flutter oscillations occurs within 20 s at an air speed of v ∼ = 252 m/s, or nearly 5% below the nominal flutter speed. References [1] A.M. Cunningham Jr., Practical problem: air-planes, in: D. Nixon (Ed.), Unsteady Transonic Aerodynamics, Progress in Astronautics and Aeronautics, Vol. 120, AIAA, 1989, p. 75. [2] E.H. Dowell, Nonlinear Aeroelasticity, Flight-Vehicle Materials, Structures and Dynamics, Vol. 5, ASME, 1993, p. 213. [3] O.O. Bendiksen, G.-Y. Hwang, Nonlinear flutter calculations for transonic wings, in: Proceedings of the CEAS International Forum on Aeroelasticity and Structural Dynamics, Rome, Italy, 1997, p. 105. [4] S. Schulze, Transonic aeroelastic simulation of a flexible wing section, in: Proceedings of the AGARD Structures and Materials Panel Workshop on Numerical Simulation of Unsteady Aerodynamics, AGARD R-822, Aalborg, Denmark, 1997, pp. 10–11. [5] H. Kheirandish, G. Beppu, J. Nakamichi, Flutter simulation by Navier–Stokes equations, in: Proceedings of the 34th Aircraft Symposium, FSASS, 1996, p. 545. [6] J.J. Meijer, A.M. Cunningham Jr., Applications of a semi-empirical method for predicting transonic limit cycle oscillation characteristics of fighter aircraft, in: Proceedings of the International Forum on Aeroelasticity and Structural Dynamics, 1995, p. 75.1. [7] H. Timmermann, L. Tichy, Application of the Hilbert transform to flight vibration testing, in: Proceedings of the International Forum on Aeroelasticity and Structural Dynamics, 1995, p. 78.1. [8] G. Schewe, H. Deyhle, Experiments on transonic flutter of a two-dimensional supercritical wing with emphasis on the nonlinear effects, in: Proceedings of the Royal Aeronautical Society Conference on Unsteady Aerodynamics, 1996. [9] A. Knipfer, G. Schewe, Investigations of an oscillating supercritical two-dimensional wing section in a transonic flow, in: Proceedings of the 37th Aerospace Sciences Meeting and Exhibition, AIAA 99-0653, Reno, USA, 1999. [10] D.R. Dreim, S.B. Jacobson, R.T. Britt, Simulation of nonlinear transonic aeroelastic behavior on the B-2, in: Proceedings of the CEAS/AIAA/ICASE/NASA Longley International Forum on Aeroelasticity and Structural Dynamics, 1999, p. 511. [11] N.S. Namachchivaya, H.J. Van Roessel, Unfolding of degenerate Hopf bifurcation for supersonic flow past a pitching wedge, J. Guidance Control Dynamics 9 (1986) 413. [12] N.S. Namachchivaya, H.J. Van Roessel, Unfolding of double-zero eigenvalue bifurcations for supersonic flow past a pitching wedge, J. Guidance Control Dynamics 13 (1990) 343. [13] H. Matsushita, M. Hashidate, K. Saitoh, Y. Ando, K. Fujii, K. Suzuki, D.H. Baldelli, Transonic flutter control of a high aspect ratio wing: mathematical modeling, control law design and wind tunnel tests, in: Proceedings of the 19th ICAS Congress, ICAS-94-5.6.2, Anaheim, CA, 1994.
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[14] H. Matsushita, K. Saitoh, M. Hashidate, Robust stabilization of transonic flutter based on a linearized math model, in: Proceedings of the 20th ICAS Congress, ICAS-96, Sorronto, Italy, 1996. [15] P. Gránásy, H. Matsushita, K. Saitoh, Nonlinear aeroelastic phenomena at transonic region, in: Proceedings of the CEAS International Forum on Aeroelasticity and Structural Dynamics, Rome, Italy, 1997, p. 379. [16] H. Matsushita, K. Saitoh, P. Gránásy, Nonlinear characteristics of transonic flutter of a high aspect ratio wing, in: Proceedings of the 21st ICAS Congress, ICAS-98-4.1.2, Melbourne, Australia, 1998. [17] A.P. Pikovsky, J. Kurths, Coherence resonance in a noise-driven excitable system, Phys. Rev. Lett. 78 (1997) 775. [18] A. Neiman, P.I. Saparin, L. Stone, Coherence resonance at noisy precursors of bifurcations in nonlinear dynamical systems, Phys. Rev. E 56 (1997) 270. [19] L. Arnold, N.S. Namachchivaya, K.R. Schenk-Hoppé, Toward an understanding of stochastic Hopf bifurcation: a case study, Int. J. Bifurcation Chaos 6 (1996) 1947. [20] G. Leng, N.S. Namachchivaya, Critical mode interaction in the presence of external random excitation, J. Guidance Control Dynamics 14 (1991) 770. [21] K. Saitoh, D.H. Baldelli, H. Matsushita, M. Hashidate, Robust controller design and its experimental validation of active transonic flutter suppression, in: Proceedings of the CEAS International Forum on Aeroelasticity and Structural Dynamics, Rome, Italy, 1997, p. 393. [22] H. Matsushita, K. Saitoh, P Gránásy, Research on active control of aeroelastic systems at National Aerospace Laboratory in Japan, Int. J. Intelligent Mechatron.: Design Prod. 3 (1998) 34. [23] K. Isogai, Transonic-dip mechanism of flutter of a sweptback wing: part II, AIAA J. 19 (1981) 1240. [24] K.L. Roger, Airplane Math Modeling for Active Control Design, AGARD-CP-228, 1997, p. 4.1. [25] J.M.T. Thompson, H.B. Stewart, Nonlinear Dynamics and Chaos, Wiley, Chichester, 1986. [26] A. Neiman, A. Silchenko, V. Anishchenko, L. Schimansky-Geier, Stochastic resonance: noise-enhanced phase coherence, Phys. Rev. E 58 (1998) 7118. [27] R. Benzi, A. Sutera, A. Vulpiani, The mechanism of stochastic resonance, J. Phys. A 14 (1981) L453. [28] J.K. Douglass, L. Wilkens, E. Pantazelou, F. Moss, Noise-enhancement of information transfer in crayfish mechanoreceptors by stochastic resonance, Nature (London) 365 (1993) 337. [29] D.E. Postnov, S.K. Han, T.G. Yim, O.V. Sosnovtseva, Experimental observation of coherence resonance in cascaded excitable systems, Phys. Rev. E (1999) R3791.
Nonlinear characteristics of randomly excited transonic flutter L.E. Christiansen a , T. Lehn-Schiøler a , E. Mosekilde a,∗ , P. Gránásy b , H. Matsushita c a
Department of Physics, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark b G.E. Hungary Ltd., 1340 Budapest, Vací ut 77, Hungary c National Aerospace Laboratory, 16-13-1 Ohsawa, Mitaka-shi, Tokyo 181-0015, Japan
Abstract The paper describes the effects of random external excitations on the onset and dynamical characteristics of transonic flutter (i.e. large-amplitude, self-sustained oscillations) for a high aspect ratio wing. Wind tunnel experiments performed at the National Aerospace Laboratory (NAL) in Japan have shown that the self-sustained oscillations arise in a subcritical Hopf bifurcation. However, analysis of the experimental data also reveals that this bifurcation is modified in various ways. We present an outline of the construction of a 6 DOF model of the aeroelastic behavior of the wing structure. When this model is extended by the introduction of nonlinear terms, it can reproduce the subcritical Hopf bifurcation. We hereafter consider the effects of subjecting simplified versions of the model to random external excitations representing the fluctuations present in the airflow. These models can reproduce several of the experimentally observed modifications of the flutter transition. In particular, the models display the characteristic phenomena of coherence resonance. © 2002 IMACS. Published by Elsevier Science B.V. All rights reserved. Keywords: Transonic flutter; Subcritical Hopf bifurcation; Noise excitation; Coherence resonance; Wind tunnel experiments
1. Introduction Flutter denotes a characteristic form of self-excited oscillations that can arise through the interaction of an aerodynamic flow with the elastic modes of a mechanical structure, e.g. the bending and torsion modes of an aircraft wing [1,2]. Limit cycle oscillations (LCO) of this form typically arise in the transonic region (i.e. when the aircraft operates close to the velocity of sound). They are thought to be related with a separation of the airflow associated with shock waves propagating along the surface of the wing. Emergence of flutter compromises not only the long term durability of the wing structure, but also the operational safety, flight performance and energy efficiency of the aircraft. Effective means of flutter prevention are, therefore, mandatory in the certification of new flight vehicles, and considerable effort, theoretically as well as experimentally, is devoted to the study of methods for active flutter control and of the interaction between structural dynamics and unsteady airflows. ∗
Corresponding author. Tel.: +45-45-88-16-11; fax: +45-45-93-16-69. E-mail address: [email protected] (E. Mosekilde). 0378-4754/02/$ – see front matter © 2002 IMACS. Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 4 7 5 4 ( 0 1 ) 0 0 3 7 9 - 2
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A number of investigators have recently succeeded in producing limit cycle oscillations in numerical simulations of various aeroelastic systems using Euler code [3,4] or Navier–Stokes code [5]. Experimental studies of transonic LCO in fighter aircraft and two-dimensional supercritical wing structures have also been reported [6–8], and a few papers have compared test results with numerical simulations and obtained reasonable agreement [4,9,10]. In studies of this type the concepts and methods of nonlinear dynamics and bifurcation theory are indispensable in explaining the onset and dynamics of the self-excited oscillations. Namachchivaya and Van Roessel [11,12], for example, have provided an early analysis of the stability and bifurcation of the single degree of freedom pitching motion for a double-wedge airfoil in a supersonic flow, emphasizing in particular the possible occurrence of a degenerate Hopf bifurcation at certain parameter values. Wind tunnel experiments performed at the National Aerospace Laboratory (NAL) in Japan over a period of 4–5 years with a high aspect ratio wing model have consistently shown that the onset of flutter occurs abruptly as the dynamical pressure reaches a certain critical value [13,14]. Moreover, as the dynamic pressure is subsequently reduced, the flutter oscillations occur all the way down to pressures that are about 10% below the critical onset pressure, again to disappear suddenly. With its hysteresis and abrupt transitions the above behavior is consistent with the occurrence of a subcritical Hopf bifurcation. The equilibrium point of the aeroelastic system loses its stability as a pair of complex conjugated eigenvalues cross the imaginary axis. However, by contrast to the supercritical Hopf bifurcation, the transition is not associated with the formation of a stable limit cycle of smoothly growing amplitude. Rather, for aerodynamic pressures below the bifurcation point, an unstable (saddle) cycle coexist with the stable equilibrium point, and the loss of stability is associated with the saddle cycle closing in on this point. At aerodynamic pressures somewhat below the Hopf bifurcation point, the saddle cycle stabilizes in a saddle-node bifurcation, and the finite amplitude stable limit cycle (the node) continues to exist for increasing values of the aerodynamic pressure (this bifurcation structure is illustrated in Fig. 3). In a particular experiment, a transition with a smoothly decreasing flutter amplitude was observed. Originally, this was interpreted in terms of a distortion of the deterministic bifurcation diagram, caused by the presence of higher nonlinear interactions and leading to a supercritical transition for certain parameter values [15,16]. One of the purposes of the present work is to discuss an alternative explanation according to which modifications in the deterministic bifurcation structure, caused by random fluctuations associated with the unsteady airflow, produce phenomena that appear similar to a supercritical transition. In particular, the presence of random excitations may produce the so-called noisy precursors, i.e. a coherent oscillatory response from the wing structure that grows in amplitude as the bifurcation point is approached. This behavior is related to the phenomenon of coherence resonance [17,18]. Using a Fokker–Planck equation approach, the influence of noise on the characteristics of a supercritical Hopf bifurcation has been studied in considerable detail by Arnold et al. [19]. They conclude that the interaction between noise and bifurcation structure can produce interesting distortions in the invariant density and split the pair of two complex conjugated eigenvalues into different Lyapunov exponents. The subcritical Hopf bifurcation appears not to have been studied in similar detail. Besides the above mentioned phenomena, including the emergence of noisy precursors, we expect a couple of additional modifications: the presence of noise may cause a transition to large amplitude limit cycle oscillations (across the saddle solution) before the Hopf bifurcation point is reached, producing a shift in the apparent bifurcation point that depends on the noise amplitude. At larger noise amplitudes, it is also possible that
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the system can start to jump forth and back between the equilibrium state and the oscillatory state, and this switching may develop into a pseudo-coherent phenomenon [17,18]. Finally, it is worth mentioning that Leng and Namachchivaya [20] have considered the influence of random external excitations on the behavior of systems with several (i.e. two) marginally stable or unstable modes, a condition which may very well apply to our aeroelastic wing problem. For such systems, the bifurcation structure is not robust. In particular, while one mode may be stabilized by the presence of noise, another may be destabilized, the actual result depending on details of the interaction between the critical modes. Only under very special conditions can one expect the deterministic bifurcation structure to be preserved.
2. Wind tunnel experiments Experiments were performed with the transonic wind tunnel at the NAL in Japan [13,14,21]. This tunnel can operate at relatively low noise levels up to air speeds of 1.4 Mach. Fig. 1 shows a sketch of the wing model (a) together with a photograph of the model as mounted in the 2 m × 2 m test section of the tunnel (b). The model simulates a high aspect ratio wing span of an advanced, energy efficient aircraft. The length of the wing is 1.043 m. Except for the middle part, where the cross-section is inflated
Fig. 1. Sketch of the wing model (a) with its leading edge and trailing edge control surfaces crosshatched. The model simulates a high aspect ratio wing of an advanced, energy efficient aircraft. Positions of the accelerometers and strain gauges are indicated. Photograph of the model as mounted in the 2 m × 2 m test section of the transonic wind tunnel (b). Note the flutter stopping device.
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and symmetric to make room for a couple of servo motors, the wing has a supercritical cross-section. To prevent the wing from being destructed during the experiments, a flutter stopping device has been installed in the test section. This device has the form of a triangular plate that normally lies along the floor of the tunnel. However, when the wing goes into self-excited oscillations the plate can immediately be raised to protrude into the airflow and lower the dynamic pressure [13]. The servo motors are used for experiments with active flutter control [22]. For this purpose the model is also equipped with leading edge and trailing edge control surfaces (shown crosshatched in Fig. 1a) that can be activated by the servo motors. During the experiments the behavior of the wing is monitored by means of four accelerometers and seven pairs of strain gauges fixed at wing spars and measuring various bending and torsion components of the strain (Fig. 1a). Tests performed in the transonic regime at different Mach numbers and different pressures have consistently led to the appearance of large amplitude limit cycle oscillations. These experiments have also confirmed the existence of the transonic dip, characteristic of a high aspect ratio wing [23], i.e. of a minimum in the dynamic pressure required to excite flutter oscillations near 0.8 Mach [13]. In this connection it was noted that the flutter amplitude is relatively modest near the minimum of the transonic dip to grow for both higher and lower Mach numbers. The flutter frequency for the refurbished wing (i.e. after installation of the electric motors) was found to be around 22 Hz. The results to be reported here were performed with air speeds around M = 0.8 and with an incidence angle (angle of attack) α in the interval from α = −0.2 to +0.75◦ . Under these circumstances, the onset of flutter typically occurred at a dynamical pressure of pd = 27.9 kPa. This will be referred to as the nominal flutter pressure. To examine the nature of the flutter transition, a series of wind tunnel experiments were performed in which the leading edge control surface was perturbed by a sinusoidal forcing signal. The stationary state reached by the wing after turning off the sinusoidal forcing was determined as a function of the air speed. These experiments showed that the transition to flutter can occur below the nominal flutter pressure, and that the position of the transition point depends both on the frequency and the amplitude of the forcing signal. Most efficient are forcing frequencies around the flutter frequency of about 22 Hz, and the effect weakens significantly as the forcing frequency is either reduced to 10 Hz or increased to 30 Hz. The transition to flutter does not necessarily occur instantaneously when the forcing signal is applied, but may require a build-up time of several seconds. On the other hand, when first excited, the large amplitude oscillations tend to persist even when the forcing is removed. For a range of dynamical pressures, the above results established the coexistence of a stable equilibrium state and a state of large amplitude LCO, and hence, pointed to a subcritical nature of the flutter bifurcation [15]. In a subsequent series of experiments performed at a forcing frequency of 21.5 Hz, an attempt was made to determine the separatrix between the forcing amplitudes that lead to large amplitude oscillations and the amplitudes for which the wing remains close to its stable equilibrium state. Fig. 2 reproduces the results of this experiment. Here, the limiting state has been plotted as a function of the amplitude of the leading edge excitation and of the dynamic pressure in the wind tunnel. Inspection of Fig. 2 shows that the nominal flutter transition point in these experiments (α = +0.75◦ ) occurs around pd = 27.9 kPa. Above this point all experiments produce large amplitude limit cycle oscillations even without forcing. As the dynamic pressure is reduced below the critical value, larger and larger forcing amplitudes are required to elicit LCO until, at a dynamic pressure of about 24.8 kPa, the experiments fail to produce self-sustained oscillations, even with a forcing amplitude of the leading edge surface as large as 6◦ . We assume that the curve that delineates the two types of behavior for 24.8 kPa < pd < 27.9 kPa represents the separatrix between the stable equilibrium state and the flutter
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Fig. 2. Experimental results obtained with a sinusoidal forcing of the leading edge control surface. In the depicted flap amplitude vs. dynamical pressure plane, crosses represent conditions that lead to self-sustained oscillations, and circles represent conditions that produce stable equilibrium behavior. The nominal flutter transition pressure is pd ∼ = 27.9 kPa.
∼ 24.8 kPa represents the position of the saddle-node bifurcation in state, and that the threshold at pd = which the unstable LCO involved in the subcritical transition are stabilized. The results of the above experiments have subsequently been reanalyzed in order to construct a proper bifurcation diagram. The accelerometer signal was numerically integrated twice to obtain the velocity and deflection signals for the wing model, and phase plots were prepared for different initial conditions and various dynamical pressures. This has allowed us to determine the variation of the amplitudes for both the stable and the unstable limit cycle as a function of the dynamic pressure. As depicted in Fig. 3, the bifurcation diagram clearly illustrates the subcritical nature of the flutter transition. The crosshatched area represents the significant uncertainty in locating the separatrix (unstable limit cycle). Fig. 4 displays the results of a particular wind tunnel experiment. Here, the dynamic pressure has been initiated at a value beyond the nominal flutter pressure, the flutter control has been switched off, and we follow the changes in the accelerometer signal as the dynamic pressure is gradually reduced. At time t ∼ = 11.5 s the dynamic pressure reaches the saddle-node bifurcation, stable self-sustained oscillations cease to occur, and the accelerometer signal drops from an amplitude of about 6 V to less than 1 V. This latter value represents the stable equilibrium state excited by the random fluctuations in the air flow. Fig. 5a shows the power spectrum of the signal recorded from the accelerometer during the initial period of self-excited oscillations in Figs. 4 and 5b displays the spectrum for the noise excited equilibrium state attained for t > 12 s. The spectrum in Fig. 5a clearly reveals not only the frequency of the flutter mode (f ∼ = 22 Hz), but also its higher harmonics up to the order of seven or more. Once the transition to the noise excited equilibrium state has occurred, one no longer observes the narrow peaks in the spectral distribution. The power spectrum displayed in Fig. 6 applies to a state of noise excited equilibrium dynamics for a dynamical pressure pd ∼ = 27.0 kPa, i.e. below but relatively close to the nominal flutter pressure. This spectrum exhibits two narrow and relatively close peaks in the frequency range of 20–25 Hz characteristic
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Fig. 3. Experimentally obtained bifurcation diagram illustrating the subcritical nature of the flutter transition. The diagram was constructed from phase plots obtained with different initial conditions and six different values of the dynamic pressure. Note that the self-sustained oscillations in the 1 m long wing structure reach amplitudes of the order of 12 mm.
Fig. 4. Variation of the accelerometer signal as the dynamic pressure is gradually reduced. At t ∼ = 11.5 s, the amplitude abruptly drops from about 6 V to less than 1 V as the aeroelastic system passes the saddle-node bifurcation in Fig. 3 and stable self-sustained oscillations cease to exist.
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Fig. 5. Power spectrum for the flutter oscillation observed in Fig. 4 for t < 10 s (a). Here, one can observe the build up of lines in the spectrum corresponding to the flutter frequency and its harmonics. Spectrum observed in the noise excited equilibrium state attained in Fig. 4 for t > 12 s (b).
of flutter oscillations in the considered wing model. This result, which seems to be reproduced fairly consistently, indicates that two different modes may be involved in the flutter transition. The beating phenomenon, i.e. the periodic modulation of the flutter amplitude observed in Fig. 4 points to the same conclusion. We consider this spectrum as an example of the emergence of noisy precursors, i.e. of relatively strong
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Fig. 6. Power spectrum for a noise excited equilibrium state immediately below the point of nominal flutter transition. Note the appearance of two nearby lines. Such noisy precursors may be used in the development of active flutter control.
components of the main flutter frequency and its harmonics even before the Hopf bifurcation has occurred [18]. Such signals may play an important role in the development of methods for active flutter control [22]. A particular test with active flutter control has produced a very different behavior [16]. In this experiment the dynamical pressure was held constant at a value pd ∼ = 28.3 kPa, i.e. slightly above the nominal flutter ∼ pressure (pd = 27.9 kPa). When the flutter control was switched off, large amplitude flutter oscillations were observed to build up with an incubation time of the order of 5 s. To stop the oscillations the flutter control was activated and then again released. This time a low amplitude beating oscillation appeared to finally again give way to large amplitude flutter oscillations. After a new activation and release of the flutter control, the low amplitude beating oscillation reappeared. However, this time it seemed to be stable. Moreover, upon reduction of the dynamic pressure the oscillation amplitude decreased smoothly as in the presence of a supercritical (or soft) bifurcation. The above results indicate the possibility that two different flutter modes may occur for the same dynamical pressure, and that one of the modes arises via a supercritical Hopf bifurcation [16]. Unfortunately, in spite of much work we have never been able to reproduce these results or to find anything resembling them. However, the results again seem to indicate that two different aerodynamic modes can become critical almost simultaneously, and that cooperative effects of mode-to-mode interactions and noise excitation might influence the dynamics near the bifurcation point.
3. Finite state aeroelastic model The purpose of this section is to sketch the development of a finite state aeroelastic model for our experimental wing system, and to present the main results of a linear stability analyses for the onset of
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flutter [13,14]. In Section 4, we shall consider a reduced nonlinear model in order to examine some of the phenomena that can arise via the interaction of noise with a subcritical Hopf bifurcation. The structural dynamics of the wing are represented by its six basic modes: four bending modes and two torsion modes, and each of these modes is described by its generalized coordinate q and velocity q. ˙ In the linearized analysis, the phase shift associated with the unsteady airflow is represented by a set of six delay variables [24], and in this way we arrive at an equation of motion of the following standard form: x˙ = Ax with x = [q, q, ˙ z]T ∈ R18 where x denotes the 18-dimensional state vector. The system matrix A = {aij } ∈ R18×18 takes the form 0 I −1 A = −(M − A0 ) (K − A2 ) −(M − A0 )−1 (C − A1 ) Ba 0
(3.1)
0 −(M − A0 )−1 Λ
(3.2)
where M, C, and K are the mass, damping coefficient and stiffness matrices for the considered structural modes, and I is the unit matrix. The matrices Ai , i = 0, 1, and 2 and the matrix Ba are coefficient matrices in the finite state expression of the aerodynamic forces, and Λ is a diagonal matrix of reciprocal delay times. Methods to obtain these matrices have been discussed by several authors. In the present analyses we have adopted the method described by Roger [24]. To determine the coefficients of the various matrices in the structural model a number of vibration tests were performed. Using both sinusoidal and random forcing, the wing model was vibrated at selected points, and the deflections at 39 points over the wing surface were measured by means of a laser deflection meter. The results were treated with the help of a modal analysis software (FEM analysis) to determine the natural frequencies, damping coefficients and shapes (nodal lines) for the six basic modes. The mass distribution across the wing and the variation of rigidity were adjusted to obtain the best possible fit between the linear model and the vibration results. Table 1 lists the corresponding natural frequencies. For all six modes, the analytically calculated frequencies agree with the experimentally determined frequencies to within an error of 3%, and the nodal lines also show good agreement. Table 1 Natural frequencies in Hz for the basic modes of the wing modela Mode
First bending Second bending First torsion Third bending Second torsion Fourth bending a
Excitation
Model
Random
Sinusoidal
12.5 37.0 44.0 86.5 134.0 173.0
12.22 37.23 45.11 87.20 134.64 177.33
12.52 36.23 44.01 90.10 138.06 172.45
The frequencies of all six modes agree within an error of 3%. The nodal lines also show good agreement between experiments and the linear model [13].
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Fig. 7. Root locus analysis for the eigenvalues of the six basic modes as functions of the air speed. Only the lowest mode undergoes a Hopf bifurcation. The corresponding frequency is 22 Hz.
The aerodynamic forces depend on the air speed and are distributed across the wing surface. In order to perform a linear, finite-mode analysis these forces must be approximated by lumped parameters for the individual modes, and the generalized forces calculated in this way must be expanded with respect to the position, velocity and phase shift variables. Fig. 7 presents the results of an eigenvalue analysis for the coefficient matrix A in terms of a root locus plot with the air speed as a parameter. Inspection of the figure shows how a pair of complex conjugated eigenvalues for the lowest mode (the first bending mode) start very close to the imaginary axis and with increasing air speed first move in the direction of increasing damping to subsequently move to the right and cross the imaginary axis for air speeds of the order of 280 m/s. The point of crossing corresponds to a frequency of approximately 26 Hz. Several of the other modes also have fairly low damping coefficients at low air speeds. However, as the air speed increases the damping becomes stronger, and the eigenvalues move away from the imaginary axis. We must conclude that the analytical results do not provide support to the idea that two modes of nearly the same frequency become unstable at approximately the same wind speed. A similar conclusion may be drawn from a reduced two-mode analysis of the problem [13]. When comparing the experimental results with the analytic calculations it is also observed that the root locus analysis predicts a flutter dynamic pressure that is only about one-third of the nominal flutter pressure determined in the experiments. To account for this disagreement, a factor of 0.368 has been applied to the generalized aerodynamic forces. We suppose that a better agreement could be achieved if the description of the dynamics of the unsteady air flow was improved, e.g. by replacing the six first order differential equations for the delay variables by six equations of motion of second order.
4. Noise response The above discussion has left us with a number of questions. In particular, the linear analyses shows that only a single structural mode is under bifurcation, whereas some of the experimental results (beating
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Fig. 8. Bifurcation diagram for the extended normal-form model (4.1). The diagram illustrates the subcritical Hopf bifurcation and the saddle-node bifurcation in which the unstable limit cycle oscillations are stabilized. Here, α = −1 and β = 0.2.
of the flutter amplitude and double-peaked spectra) indicate that two critical modes of nearly the same frequency might be involved. Another problem is the observation (in a single experiment) of a flutter transition that could appear to be supercritical. We cannot resolve these problems at present. However, in view of the relatively strong noise level in the air flow (presumably caused by a weak separation on the wing surface associated with shock waves that exist even at these relatively low wind speeds) it appears worthwhile to try to understand some of the phenomena that can arise when a subcritical Hopf bifurcation is excited by a random external signal. The wind tunnel experiments (see, e.g. Fig. 4) show an amplitude for the noise response of the equilibrium state of nearly 20% of the fully developed flutter amplitude. As a first approach to this problem we have considered a simple model representing the normal-form of a subcritical Hopf bifurcation, but extended with higher order terms to account for the stabilization of the unstable LCO in a saddle-node bifurcation. Neglecting some of the possible cross terms and assuming a symmetrical form for the nonlinear terms, our simplified model may be expressed as x˙ = µx − ωy + αx(x 2 + y 2 ) + βx(x 4 + y 4 ) and y˙ = ωx + µy + αy(x 2 + y 2 ) + βy(x 4 + y 4 )
(4.1)
Here, ω represents the angular frequency of the oscillations at the bifurcation point µ = 0 where the damping vanishes. The Hopf bifurcation is subcritical for α < 0 and supercritical for α > 0. A typical bifurcation diagram is illustrated in Fig. 8. Here, α = −1 and β = 0.2. Fig. 9a illustrates the dynamics of the extended normal-form model when the stable equilibrium point is perturbed by a noise signal of relatively low amplitude, and Fig. 9b shows the corresponding power spectrum. Here, µ = −1. As expected, the trajectory fluctuates around the stable equilibrium point, and the spectrum shows a peak near the natural frequency of the system (22.4 Hz). With higher amplitude it becomes possible for the noise to excite the system forth and back across the unstable limit cycle, i.e. between the noise perturbed equilibrium state and fully developed flutter oscillations. This is illustrated in Fig. 10. A similar behavior cannot be observed in the wind tunnel experiments because the wing model is not allowed to operate in the flutter state for extended periods of
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Fig. 9. Temporal behavior of the extended normal-form model subjected to a low amplitude noise signal (a). Corresponding power spectrum showing a peak near the resonance frequency of the model (b). Here, µ = −1.
time. However, the phenomenon is characteristic of coherence resonance in a bistable system and will be discussed in more detail in Section 5. When applying a low level noise signal to the extended normal-form model below but relatively close to the Hopf bifurcation point, a relatively strong response is elicited at the resonant flutter frequency. In many ways this response may appear like a low amplitude limit cycle. As the distance to the bifurcation point
Fig. 10. In the presence of a noise signal of larger amplitude the system may jump forth and back between a state of noise-perturbed equilibrium dynamics and fully developed flutter oscillations µ = −1.
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Fig. 11. A gradual shift of the operation point away from the Hopf bifurcation causes the noise response of the equilibrium state to decrease. The phenomenon may explain the observation in a particular experiment of an apparent supercritical flutter transition.
is increased by changing the bifurcation parameter, the damping of the resonant mode also increases, and one can observe something that looks like a smooth transition. This is illustrated in Fig. 11. It is possible that this phenomenon provides an explanation to the result obtained in a single wind tunnel experiment and interpreted as a supercritical Hopf bifurcation. A particular problem concerns the influence of noise on the presence of higher harmonics of the flutter frequency in the power spectra. Such harmonics are quite conspicuous in some of the experimental results, but nearly absent in others. Fig. 12 shows a sequence of power spectra calculated for the extended normal-form model with increasing noise amplitude. Here, µ = 0.2, and we are operating just above the Hopf bifurcation point. Fig. 12a was calculated for the noise free system, and we clearly observe the higher harmonics of the limit cycle oscillations. As the noise amplitude is increased, the higher harmonics are gradually suppressed (Fig. 12b) to finally disappear almost completely (Fig. 12c). After this introductory investigation, we have considered the effects of noise on a nonlinear 2 DOF structural model representing the dynamics of the first bending and the first torsion mode, i.e. x˙ = A(v)x + b(x, v) with x = (q1 , q2 , q˙1 , q˙2 , z1 , z2 )T ∈ R 6
(4.2)
Here, v denotes the air speed. The coefficients of the matrix A(v) were determined as described in Section 3 for the full 6 DOF version of the model. The nonlinear contribution b(x, v) contains second and fourth order terms in the bending mode coordinate q1 and is added to the acceleration term for this mode, i.e.
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Fig. 12. Power spectra obtained for the extended normal-form model in the presence of noise signals of increasing amplitude. Here, µ = +0.2, and the model is operating immediately above the Hopf bifurcation point.
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Fig. 12. (Continued).
0 0 a (v)(αq 2 + βq 4 ) 33 1 1 b(x, v) = 0 0 0
(4.3)
There is no detailed physical argument for this expression, and other authors have used a formulation in which the nonlinear terms depend on the velocity q˙1 rather than on the coordinate q1 itself [25]. With a proper choice of the parameters, the two approaches appear to give similar results. The aerodynamic forces were adjusted such that the model reproduced the experimentally observed flutter transition point, and the nonlinear parameters α and β were chosen to produce a proper position of the saddle-node bifurcation point and a reasonable amplitude of the flutter oscillations. With α ∼ = −1 and β∼ = 0.3, the nonlinear 2 DOF model can reproduce the experimentally observed bifurcation diagram in Fig. 3. A root locus analysis for the (linearized) 2 DOF model shows that for this model it is the second mode (i.e. the first torsion mode) that becomes unstable. At the Hopf bifurcation point the frequency of this mode is approximately 22 Hz. An exchange of the active modes must, of course, be considered with a certain caution. However, none of the modes are pure, and a mixing of the two modes has occurred well before the Hopf bifurcation takes place.
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Fig. 13. A series of power spectra obtained for the nonlinear 2 DOF structural model immediately above the Hopf bifurcation point. As in Fig. 12 we find that increasing noise intensity suppresses the higher harmonics of the flutter oscillations.
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Fig. 13. (Continued).
Application of noise to the nonlinear 2 DOF structural model reproduces all the same results as we have obtained for the extended normal-form model. In particular, when operating in the subcritical regime (i.e. at dynamic pressures below the Hopf bifurcation point), with the proper initial conditions the trajectory stays close to the stable equilibrium point at low noise levels to jump into the state of fully developed flutter oscillations at larger noise amplitudes. At even larger noise amplitudes, the system starts to jump forth and back between the two states in a manner similar to the behavior depicted in Fig. 10. The 2 DOF model can also reproduce the smooth reduction of the noise response as the point of operation is shifted away from the bifurcation point. This is similar to the results displayed in Fig. 11. As a further example of the similarity between the results obtained with the two models, Fig. 13 shows a sequence of power spectra calculated for the supercritical regime immediately above the Hopf bifurcation point. Again we see how increasing the noise intensity gradually suppresses the higher harmonics in the spectrum while leaving the peak at the flutter frequency relatively unaffected.
5. Conclusion We presented the results of a series of transonic flutter experiments performed at the NAL in Japan and outlined the basic aeroelastic theory used to analyze the results. In order to understand better some of the observed phenomena, particularly the significance of random fluctuations in the airflow in the presence of a subcritical flutter transition, we considered two nonlinear models. First an extended normal-form
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model and secondly a nonlinear version of a 2 DOF structural model. For both models the findings were the same. • When operating in the subcritical regime (i.e. at dynamic pressures below the nominal flutter pressure), one can observe a gradual increase in the noise excited fluctuations of the wing as the bifurcation point is approached. Particularly, the resonant components of the spectrum tend to grow. Under proper conditions, this phenomenon may appear like a supercritical transition. • At higher noise amplitudes, the system may jump from the noise perturbed equilibrium state into the state of fully developed flutter (i.e. across the separatrix provided by the unstable limit cycle oscillations). This causes a reduction of the effective flutter transition point relative to the deterministically calculated bifurcation point. • At even higher noise amplitudes the system may start to jump forth and back between the noise perturbed equilibrium state and the flutter state. This produces the phenomenon of coherence resonance [17,18]. • When operating in the supercritical regime immediately above the Hopf bifurcation, the presence of noise is found to suppress the higher harmonics of the flutter frequency. Coherence resonance is a phenomenon by which the application of noise to a nonlinear system can elicit a coherent component in the response. This phenomenon is related to some extent to the more well-known phenomenon of stochastic resonance [26] by which the application of noise to a nonlinear system in the presence of a (weak) periodic signal can enhance the periodic component. Stochastic resonance has been applied, for instance, to explain the pseudo-regular transitions of the earth’s climatic system between ice ages and more normal conditions [27] or the ability of crayfish to detect their preys [28]. Experimental observation of coherence resonance has recently been reported for cascaded excitable systems [29]. In order to characterize the switching behavior between the noise perturbed equilibrium state and the flutter state in the subcritical regime (as displayed, for instance, in Fig. 10), one can consider the degree of regularity in the jumps for a very long time series. At the lower end of the interval of noise amplitudes that can elicit this phenomenon, jumps from one of the states will be quite rare, and a spectral analysis of the temporal behavior will reveal a very broadband spectrum. In the other end of the interval, i.e. for very large noise amplitudes, jumps will occur quite frequently and will again be random in nature. However, there may be an optimal choice of the noise amplitude (typically when the jump intervals are nearly the same for the two directions) when the spectral analyses reveals a clear peak. When this is the case, we observe a noise induced time scale in the switching behavior, and a time scale which is not related to noisy precursors of the deterministic dynamics or to any other inherent deterministic periodicity. A more quantitative description of the degree of regularity may be obtained by calculating the entropy S(s) for the spectral distribution of the switching dynamics relative to the entropy S(n) of a completely random switching process or, alternatively, the so-called regularity parameter γ =1−
S(s) S(n)
(5.1)
As applied here, the spectral entropy is defined by N S(s) = − si ln si i=1
(5.2)
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Fig. 14. Variation of the regularity parameter γ with the noise level and the air speed for the nonlinear 2 DOF structural model. When scanning the model vertically between the two bifurcations, one can observe a maximum in the regularity of the switching process. The scale to the right defines the size of the regularity parameter. This parameter is found to vary between 0.02 and 0.20.
Fig. 15. Variation of the lifetime in the noise-perturbed equilibrium state as a function of the noise level and the air speed. For a noise level of 1 m/s, transition to flutter oscillations occurs nearly 5% below the nominal flutter speed. The scale to the right defines the lifetime in the noise-perturbed equilibrium state in seconds.
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where si represents the relative contribution to the power spectrum from the frequency interval #Ωi = Ω/N, N being the total number of spectral bins and Ω the frequency range of the considered spectrum. For a random process (with a flat spectrum), S(s) = S(n) = ln N, and the regularity parameter is γ = 0. For a completely monochromatic signal, on the other hand, S(s) = 0 and γ = 1. Fig. 14 shows the variation of the regularity parameter with the noise level and the air speed for the nonlinear 2 DOF structural model discussed in Section 4. The subcritical Hopf bifurcation occurs at v ∼ = 266 m/s, and ∼ the saddle-node bifurcation at v = 248 m/s. In the interval between these two bifurcations, the unstable limit cycle represents a separatrix between the equilibrium state and the flutter state. When scanning the diagram vertically in this interval one can clearly observe that there is a maximum in the regularity of the switching dynamics. Fig. 15 shows the lifetime for the system in the noise perturbed equilibrium state as a function of the noise level and the air speed, now at somewhat lower noise levels. Inspection of the figure shows how the effective threshold for excitation of flutter oscillations moves to lower and lower air speeds as the noise level is increased. Moreover, this shift appears to be nearly linear. For a noise level of 1 m/s, transition to flutter oscillations occurs within 20 s at an air speed of v ∼ = 252 m/s, or nearly 5% below the nominal flutter speed. References [1] A.M. Cunningham Jr., Practical problem: air-planes, in: D. Nixon (Ed.), Unsteady Transonic Aerodynamics, Progress in Astronautics and Aeronautics, Vol. 120, AIAA, 1989, p. 75. [2] E.H. Dowell, Nonlinear Aeroelasticity, Flight-Vehicle Materials, Structures and Dynamics, Vol. 5, ASME, 1993, p. 213. [3] O.O. Bendiksen, G.-Y. Hwang, Nonlinear flutter calculations for transonic wings, in: Proceedings of the CEAS International Forum on Aeroelasticity and Structural Dynamics, Rome, Italy, 1997, p. 105. [4] S. Schulze, Transonic aeroelastic simulation of a flexible wing section, in: Proceedings of the AGARD Structures and Materials Panel Workshop on Numerical Simulation of Unsteady Aerodynamics, AGARD R-822, Aalborg, Denmark, 1997, pp. 10–11. [5] H. Kheirandish, G. Beppu, J. Nakamichi, Flutter simulation by Navier–Stokes equations, in: Proceedings of the 34th Aircraft Symposium, FSASS, 1996, p. 545. [6] J.J. Meijer, A.M. Cunningham Jr., Applications of a semi-empirical method for predicting transonic limit cycle oscillation characteristics of fighter aircraft, in: Proceedings of the International Forum on Aeroelasticity and Structural Dynamics, 1995, p. 75.1. [7] H. Timmermann, L. Tichy, Application of the Hilbert transform to flight vibration testing, in: Proceedings of the International Forum on Aeroelasticity and Structural Dynamics, 1995, p. 78.1. [8] G. Schewe, H. Deyhle, Experiments on transonic flutter of a two-dimensional supercritical wing with emphasis on the nonlinear effects, in: Proceedings of the Royal Aeronautical Society Conference on Unsteady Aerodynamics, 1996. [9] A. Knipfer, G. Schewe, Investigations of an oscillating supercritical two-dimensional wing section in a transonic flow, in: Proceedings of the 37th Aerospace Sciences Meeting and Exhibition, AIAA 99-0653, Reno, USA, 1999. [10] D.R. Dreim, S.B. Jacobson, R.T. Britt, Simulation of nonlinear transonic aeroelastic behavior on the B-2, in: Proceedings of the CEAS/AIAA/ICASE/NASA Longley International Forum on Aeroelasticity and Structural Dynamics, 1999, p. 511. [11] N.S. Namachchivaya, H.J. Van Roessel, Unfolding of degenerate Hopf bifurcation for supersonic flow past a pitching wedge, J. Guidance Control Dynamics 9 (1986) 413. [12] N.S. Namachchivaya, H.J. Van Roessel, Unfolding of double-zero eigenvalue bifurcations for supersonic flow past a pitching wedge, J. Guidance Control Dynamics 13 (1990) 343. [13] H. Matsushita, M. Hashidate, K. Saitoh, Y. Ando, K. Fujii, K. Suzuki, D.H. Baldelli, Transonic flutter control of a high aspect ratio wing: mathematical modeling, control law design and wind tunnel tests, in: Proceedings of the 19th ICAS Congress, ICAS-94-5.6.2, Anaheim, CA, 1994.
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