flutter control using vibration test data: theory, rig design and...

Flutter control using vibration test data: theory, rig design and preliminary results

E. Papatheou1, X. Wei1, S. Jiffri1, M. Prandina1, M. Ghandchi Tehrani2, S. Bode1, K. Vikram Singh3, J. E. Mottershead1, J. E. Cooper4 1University of Liverpool, Centre for Engineering Dynamics, Liverpool L69 3GH, United Kingdom. e-mail: [email protected] 2Southampton University, Institute for Sound and Vibration Research, United Kingdom. 3Miami University, Department of Mechanical and Manufacturing Engineering, Oxford, OH. 4University of Bristol, Department of Aerospace Engineering, United Kingdom.

Abstract The problem of flutter suppression in aeroelasticity may be treated using eigenvalue assignment. The conventional approach to active vibration control requires not only the structural mass, damping and stiffness terms, but also aeroelastic damping and stiffness contributions. This requirement becomes unnecessary when using the receptance method which depends only upon vibration measurements. In the cases considered in this paper it requires wind-tunnel ‘wind on’ measurement of frequency response functions. The set-up includes a two degree-of-freedom pitch-plunge aerofoil, which allows for adjustment of the open-loop eigenvalues and extrapolation of the closed-loop eigenvalues with wind speed enabling the prediction of flutter velocities. One of the advantages of using the receptance method is that in principle the controller can be continuously corrected using in-flight measurements with consequent improvements to aircraft manoeuvrability and possibly survivability in the event of damage to the aircraft.

The paper describes the first experimental study of flutter suppression carried out in the Liverpool low-speed wind-tunnel including physical tests and simulated application of the receptance method using test data. Actuation of a flap is achieved by two piezo-stacks in a ‘V’ configuration and the vibration response is measured using ICP accelerometers mounted externally to the wind-tunnel. The purpose of the research is to demonstrate the delayed onset of flutter by increasing the flight envelope of stable air speeds. Preliminary experimental results are presented.

1 Introduction

Flutter of an aircraft, or its components, is a dynamic instability associated with aeroelastic systems. It involves interaction and coupling of modes (typically wing bending/torsion, wing torsion/control surface or wing/engine) due to the surrounding aerodynamic forces such that energy is extracted from the airstream leading to negatively damped modes and unstable oscillations. For a given Mach number, at some critical speed (flutter speed) the system eigenvalues exhibit instability leading to sustained oscillations and eventually fatigue or failure. In flutter analysis the eigenvalues (natural frequencies and damping) of an aeroelastic system are computed for varying speeds, altitudes and Mach number. In aeroelastic control, the problem is to suppress flutter or extend the flutter boundary by assigning stable poles using feedback control forces, preferably supplied by available control surfaces; this problem may

3047

be treated using eigenvalue assignment. In addition to the usual structural mass, damping and stiffness terms the conventional vibration control approach based on the state-space formulation requires the matrices of aeroelastic damping and stiffness, obtainable by double-lattice subsonic lifting surface theory (DLM) available in commercial codes such as MSC-NASTRAN or ASTROS.

The problem of flutter control has been the subject of research investigation for decades. In the 1980s and 1990s mainly analytical and computational procedures were developed. Ueda and Dowel [1] and Lee and Tron [2] studied the effect of aerodynamic and structural nonlinearities respectively. Lee [3] developed an iterative procedure for flutter analysis in large-scale systems; Tang and Dowell [4] considered flutter instability and forced response of a non-rotating helicopter blade; and Lee, Jiang and Wong [5] analysed the flutter dynamics of an aerofoil with a cubic restoring force. Towards the end of the 1990s the first experimental studies of binary flutter in aerofoils appeared and studies of this kind have continued up to the present day. Block and Strganac [6] used full-state feedback with an optimal observer to stabilise linear and nonlinear systems beyond the open-loop flutter speed; Platanitis and Strganac [7] applied feedback linearisation to a nonlinear wing section using control surfaces at both the leading and trailing edges and Trickey et al. [8] made an experimental study of limit cycle oscillations in a 3 degree of freedom aerofoil.

The receptance method of eigenvalue assignment was developed by Ram and Mottershead [9]. One of the advantages of the approach is that the controller is formulated entirely on measurements rather than on the conventional matrix theory, typically in state space. In practice, the methodology requires the in-flight measurement of frequency response functions (FRFs) so that there is no requirement to know or to evaluate the structural mass, damping and stiffness matrices (denoted A, D, E in this paper) or the aerodynamic matrices (denoted B, C). Also there is no requirement for model reduction or for the use of an observer to estimate the unmeasured state variables. In principle the receptance-based controller can be continuously corrected using in-flight measurements with consequent improvements to aircraft manoeuvrability and possibly survivability in the case of damage to the aircraft. The receptance method has been developed for (i) partial pole placement (so that selected poles are assigned while other chosen poles remain unchanged) [10] and (ii) the assignment is carried out robustly with respect to noise on the measured receptances [11]. It was applied for vibration control of an AgustaWestland W30 helicopeter airframe using electro-hydraulic actuators originally designed to work as active vibration isolators separating the airframe and the raft-mounted engine and gearbox [12].

This paper describes the first stages of an experimental study of flutter suppression carried out in the Liverpool low-speed wind-tunnel. The test set-up allows for the adjustment of open-loop eigenvalues at different wind speeds and the extrapolation of closed-loop eigenvalues with wind speed enables the prediction of flutter velocities. The practical application of the receptance method is described and simulated results, based on test data, are presented.

2 Theory

The governing equation of an aeroelastic system may be cast in matrix form [13],

( ) ( ) pfqECqDBqA +=++++ 2VV ρρ &&& (1) where, A, B, C, D, E are the structural inertia, aerodynamic damping, aerodynamic stiffness, structural damping and structural stiffness matrices respectively, ( )tq is the vector of generalised coordinates, ( )tf is the control force and ( )tp represents an external disturbance such as measurement noise or turbulence. In aeroelasticity, both circulatory and noncirculatory forces generated by the wake, for a chosen Mach number and reduced frequency, are expressed as additional contributions to the system matrices. In equation (1) these terms appear as matrices B and C which, in general, are frequency dependent. Often these wake-induced forces are combined together in the form of the aeroelastic influence coefficient (AIC) matrix at a set of discrete frequencies. Finite element codes such as MSC-NASTRAN or ASTROS provide the necessary modelling routines.

3048 PROCEEDINGS OF ISMA2012-USD2012

For the open-loop homogenous system and by separation of variables, ( ) ( )tt jn

jjj λα exp

2

1∑==

φq , the

eigenvalue equation of the jth mode may be expressed as,

( ) ( )( ) 0φECDBA =++++ jjj VV 22 ρλρλ (2)

where jα is the jth modal coordinate. The complex eigenvalues, in equation (2), may be written in terms of

the j th damping and natural frequency, the real and imaginary parts of the following expression. When the matrices, A, B, C, D, E are strictly real the eigenvalues (and eigenvectors) appear in complex conjugate pairs

2

, 1i jjjjjnj ζωωζλ −±−=+ , .,,2,1 nj K= (3)

The real part of the eigenvalues defines the stability of the system and hence, when the real part of the eigenvalues jλ in equation (3) is positive, the system is said to be unstable (resulting in flutter). For all other values the aeroelastic system is either stable or marginally stable.

The receptance FRF matrix may be expressed in theory as the inverse of the aeroelastic dynamic stiffness matrix,

( ) ( ) ( )( )122 ii

−++++−= ECDBAH VV ρρωωω (4)

In practice, when applying the receptance method, ( )ωiH is estimated from measured vibration data using well established techniques. Curve fitting to the estimated ( )ωiH , for example by the LMS PolyMAX routine [13], allows the determination of ( )sH by substituting s for iω in the curve-fitted approximation [10]. In this paper we assume that the matrix of receptances can be determined from ‘wind on’ measurements carried out in the low-speed wind tunnel. It will be demonstrated that any input and output signals may be used in aeroelastic eigenvalue assignment, in which case the dynamics of the actuators and sensors (including the effects of time delay) may be included in the measurement, rendering unnecessary the need for mathematical models to approximate the behaviour of actuators and sensors.

The receptance method depends upon an elegant result from the linear algebra, namely the Sherman-

Morrison formula, which produces a modified inverse matrix, ( ) 11 −− += TuvZZ , when a known rank-1 modification, Tuv , and original inverse matrix, 1−Z , are available such that,

( ) ( ) ( ) ( ) ( )( ) ( ) ( )sss

ssssss

T

T

uZv

ZvuZZZ

1

1111

1)( −

−−−−

+−= (5)

In single-input control, the control force is typically given by,

( ) ( )tut bf = (6) and

( ) qgqf &&&TTtu += (7)

for velocity and acceleration feedback.

It is easily demonstrated that ( )Tss gfb 2+ represents a rank-1 modification to the open-loop dynamic stiffness matrix of the aeroelastic system. Then by combining equations (5) and (7) it is found that,

( ) ( ) ( ) ( )( ) ( )bHgfHgfbH

HHsss

ssssss

T

T

2

2

1)(

++

+−= (8)

SELF-EXCITED VIBRATIONS 3049

where ( ) ( )ss 1−= ZH and ( ) ( ) ( )ssss TT vugfb =+ 2 The poles of the closed-loop system are given by the zeros of the denominator of equation (11). Thus for

the assignment of the complex-conjugate pair of poles *1 jj µµ =+ the control gains f, g must be chosen to satisfy the equations,

( ) ( )( ) ( ) nkkjjTjj

jT

jj ,,2,1,121

1

12

11

2

K=−=

=++

=++

+++ 0bHgf

0bHgf

µµµ

µµµ (9)

Re-arranging and combining equations (9) into a single matrix expression leads to,

−

−−

=

1

1

1

Mf

gG (10)

with

=

Tnn

Tnn

TT

TT

22222

22222

11121

rr

rr

rr

G

µµ

µµµµ

MM (11)

where,

( ) ( )( ) ( ) nkkjjjj

jjj,,2,1,12

111K=−=

==

+++ bHr

bHr

µµµµ

(12)

which allows the determination of f and g by inversion of the matrix G. Ram and Mottershead [9] showed that (a) G is invertible when the system is controllable and the poles ( )n221 µµµ L are distinct, and (b) f and g are real when G is invertible and the set ( )n221 µµµ L are closed under conjugation.

When G is a square matrix, there is a unique solution for ( )TTT fg and when the system (11) is under-determined (fewer poles to be assigned than the number of gain terms - f, g) a minimum norm solution is available for the minimization of control effort. Alternatively, in the latter case, the gains may be chosen that assign the chosen eigenvalues, while at the same time minimizing the sensitivity of the assigned poles to inaccuracy and noise in the measured receptances. A robust pole-placement approach to noise on the measured receptances was described by Tehrani et al. [11].

3 The Experimental Rig

The wind tunnel experiment consists of a working section containing a NACA0018 aerofoil (chord = 0.35 m, span = 1.2m), supported by adjustable vertical and torsional leaf springs. The aerofoil can be modelled as a 2D system with pitch and plunge degrees of freedom as illustrated in Figure 1.

The design allows the adjustment of stiffnesses kh and kθ (the vertical and torsional springs) as well as the possibility of adding external masses me to vary the location of the centre of mass C, the mass of the wing mw and its moment of inertia IC. The maximum air speed for the wind tunnel used is around 20 m/s. The aim of the design is to explore regions close to the flutter speed of the system.

The vertical spring arrangement is shown in Figure 2. By varying the clamp location in the direction shown in the figure, it is possible to vary the stiffness of the vertical springs, one on each side of the wind tunnel, which support the wing (attached to the shaft on the left of the figure). The variation of the vertical


stiffness versus the beams length was determined analytically and is shown in Figure 3 and may be varied from about 200 to 23000 N/m.

The adjustable torsional spring is shown in Figure 4. By moving the device in the direction indicated by the arrows it is possible to increase or reduce the torsional stiffness in the range shown in Figure 5, approximately from 10 to 320 Nm/rad.

Figure 1: Pitch and plunge 2D model

Figure 2: Adjustable vertical spring

Using these ranges of stiffness, it is possible to vary the flutter speed of the aeroelastic system approximately between 10 and 70 m/s. The open working section (with sides removed and separated from the wind tunnel) is shown in Figure 6. A torsional bar is used in order to maintain the same vertical displacement on the two sides of the test section. The external mass of the system was calculated to be around 6.5 Kg.

Plunge motion

Adjustable clamp location


Figure 3: Vertical stiffness range

Figure 4: Adjustable torsional spring

Active vibration control described in [15, 16] is achieved by means of a ‘V’-stack piezoelectric actuator shown in Figure 7 acting on the control surface of the wing, allowing a flap deflection of about ±7°. The actuator consists of two piezo-stacks (Noliac SCMAP09-H80-A01) in a ‘V’ formation. The flap is actuated when one arm of the ‘V’ is made to extend while the other retracts by an equal amount – caused by applying equal voltages to the two piezo stacks but with 180° phase difference. Khron Hite wideband power amplifiers, model 7500, were used. The ‘V’ stack actuator [15] is known to behave as a pure gain provided that its natural frequency is well above the frequencies of the assigned poles. The receptance method was applied with voltage input to the stacks and acceleration output, measured using four Kistler ICP accelerometers mounted externally to the wind-tunnel on small rigid beams (two accelerometers on

Pitch motion


each side). Thus both the pitch and plunge degrees of freedom were measureable. The term receptance is used unconventionally in this case as a generalised input-output FRF.

Figure 5: Torsional stiffness range

Figure 6: Open Working Section

4 Preliminary Results

A modal test of the experimental rig was carried out using hammer excitation. Figure 8 shows a measured FRF in the frequency band 0-102.4 Hz with resolution 0.1 Hz. The first two modes are the two designed pitch and plunge modes of the system, at the currently used setup - pitch at 3.9 Hz and plunge at 6.7 Hz.

Torsion Bar

Torsional Stiffness

Aerofoil

Vertical Stiffness

Flap


The first bending and torsion modes were found at around 40 Hz (well away from the modes to be controlled) and other modes, between 20 and 40 Hz were found to be modes of the supporting structure. Work is presently going on to stiffen these latter modes to take them further away from the controlled pitch and plunge modes. All the experimental results were obtained using an LMS SCADAS III data acquisition system running on a DELL desktop PC.

(a) (b)

Figure 7: (a) ‘V’ Stack Actuator; (b) Actuator In-Situ

Figure 8: Preliminary FRF of the whole experimental rig under impact hammer excitation

In order to realise active vibration control, as described in [15, 16], FRFs were recorded with the ‘V’ stack actuator acting as an input to the system while the acceleration was recorded through the externally mounted ICP accelerometers at different wind speeds. Figures 9-12 show the FRFs obtained from stepped sine tests in the frequency band 0.5 – 12.5 Hz with resolution 0.05 Hz using just one of the four accelerometers. The wind speeds used were 6, 7, 9 and 9.5 m/s. The effects of A/D and D/A conversion and of the filters implemented using dSPACE were included in the measured FRFs. A low-pass Butterworth filter with a cut-off frequency at 10 Hz was included.

The FRFs were found to be noisier than those obtained by regular vibration test procedures, probably because of the nature of the excitation, aerodynamically by the flap moving in the airflow, and possibly due to the piezo stacks which often produced audible noise. As a first step in the eigenvalue assignment, a single input – single output (SISO) approach was used with the response from one of the four accelerometers. The open loop FRFs recorded are shown in Figures 9-12 along with curve-fitted


approximations obtained using Structural Dynamics toolbox [17] in MATLAB with a pole/residue model. It should be noted that although the FRFs were all of the accelerance type, the actual sensitivity of the transducers was not used, so Figures 9-12 simply display voltage/voltage transfer functions.

The curve fitted FRFs were then used in order to calculate the feedback gains f and g. Velocity feedback was applied by integration of the acceleration signal in dSPACE. The pole of the second, plunge, mode at 6.7 Hz was assigned a damping increase from 1 % to 2 %. Therefore the assigned closed-loop pole became -0.9 ± 41.63i from the open-loop pole which was -0.48 ± 41.63i. The actuator gains were calculated from the curve fitted FRF measured with wind speed at 7 m/s and were found to be: g = 77.103 and f = -1756.7.

The same gains were subsequently applied for all the remaining cases of 6, 9 and 9.5 m/s. The simulated closed-loop FRFs were calculated by using the Sherman-Morrison formula in equation (8) with curve fitted data from the experimental measurements. Figures 13-16 show the open- and simulated closed-loop FRFs when the damping of second mode of the system was increased for all the wind speed cases. The simulated response of the closed-loop system is shown using a linear scale (Figures 13-16) in order to clarify the effect of the damping increase in the plunge mode of the system. It was found that although the measured FRFs were noisy, the method was adequately robust and worked successfully for all cases. The damping of the plunge mode was always increased, but the effect was not exactly the same for all wind speeds, something which was expected since the actuator gains were calculated at a specific speed of 7 m/s and subsequently implemented at the other speeds.

Figure 9: Measured and curve fitted FRF with wind speed at 6 m/s, stepped sine test

5 Discussion and Conclusions

This paper describes the first results from a flutter-suppression study carried out in the Liverpool low-speed wind-tunnel using the receptance method. The theory, along with the rig design is described and experimental and simulated results are presented. Acceleration and velocity feedback was used together with test measurements to obtain the first simulated closed-loop results. Acceleration was measured though transducers and velocity was derived by integration. FRFs were measured at different wind speeds and pole assignment by an increase in the damping of the plunge mode was performed on the curve fitted data originating from a test with a wind speed at 7 m/s. The actuator gains g and f were then used for other


wind speeds, higher and lower, and it was found that the simulated closed-loop system behaved precisely as expected.

Overall, this paper has shown that FRFs can be measured with the ‘V’ stack actuator acting as an input to the system and although there was some noise in the measured data, the receptance method worked well in simulation providing confidence for full experimental closed-loop implementation in real time for flutter suppression and the extension of flutter boundaries.




Figure 12: Measured and curve fitted FRF with wind speed at 9.5 m/s, stepped sine test

Figure 13: Pole placement (damping increase) of the second mode, closed-loop is simulated, at wind

speed of 7 m/s


Figure 14: Pole placement (damping increase) of the second mode at wind speed of 6 m/s when g,f gains

were calculated from the curve fitted FRF with wind speed at 7 m/s

Figure 15: Pole placement (damping increase) of the second mode at wind speed of 9 m/s when g,f gains

were calculated from the curve fitted FRF with wind speed at 7 m/s


Figure 16: Pole placement (damping increase) of the second mode at wind speed of 9.5 m/s when g,f

gains were calculated from the curve fitted FRF with wind speed at 7 m/s

Acknowledgement

The authors gratefully acknowledge the support of the US Air Force, EOARD Grant FA8655-10-1-3054 and EPSRC Grant EP/J004987/1.

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