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1.School of Aeronautics，Northwestern Polytechnical University，Xi’an 710072，P.R.China；2.The First Aircraft Institute of AVIC，Xi’an 710089，P.R.China

Abstract:The frequent occurrence of control surface vibration has become one of the key problems affecting aircraft safety.The source of the freeplay of the control surface is studied，and a measurement device is developed.A nonlinear flutter analysis method under trimmed flight condition is proposed based on the discrete state-space method.Consequently，the effects of center-type freeplay and the freeplay with preload on flutter characteristics are analyzed，and the effects of preload on nonlinear flutter are verified by wind tunnel tests of a single wing model.

Key words：freeplay；nonlinearity；flutter；preload；wind tunnel test

0 Introduction

The vibration response tends to diverge as the aircraft approaches its critical flutter speed［1］.The real physical phenomena are not entirely linear and the limit-cycle oscillation（LCO）usually occurs at speeds below the linear flutter speed when the aero?elastic system contains the freeplay nonlinearity.This issue is commonly seen in aircraft design.Usu?ally，the freeplay nonlinearity results in a steady vi?bration response with finite amplitude.This will lead to structural damage when the vibration ampli?tude exceeds the capacity of the structure，but more typically leads to degradation of flight performance and fatigue of the airframe structure［2］.In addition，the freeplay nonlinearity will affect the closed-loop aeroservoelastic stability and active flutter suppres?sion system［3］.Some LCO problems are solved by eliminating the control system freeplay，while oth?ers are tackled by conservative measures such as limiting flight speed.The specifications for the free?play of control surface are clearly stated in GJB 67.7A—2008［4］，while the Federal Aviation Admin?istration（FAA）considers the current specifications to be too stringent to be met during manufacturing and requires verification through analysis and flight tests［5］.Therefore，in order to ensure the flight safe?ty，it is essential to carry out nonlinear flutter analy?sis of freeplay in the design phase.

Nonlinear aeroelasticity research mainly in?cludes theoretical analysis，wind tunnel tests and flight tests.In recent years，the analytical methods based on the nonlinear dynamics theory have been gradually applied to the nonlinear aeroelastic analy?sis with effective results［4-6］.The exploration of the basic theories and methods of nonlinear dynamics，on the one hand，is helpful to deeply understand and reveal the rules and mechanisms of nonlinear aero?elasticity of various structures，and provides theoret?ical basis for aeroelastic design of aircraft in the fu?ture.On the other hand，it also provides necessary means for preventing and eliminating aeroelastic in?stability.It is of great theoretical and engineering significance.The mathematical model of nonlinear aeroelastic system is a nonlinear differential equa?tion，and it is difficult to obtain accurate analytical solutions as there is no general and effective method to solve the equation.Based on the discrete statespace method，a nonlinear flutter analysis method is proposed under trimmed flight condition in this pa?per.

1 Design Requirements

1.1 Sources of freeplay

The freeplay mainly comes from some structur?al links，such as the pivot of the all-moving surface，the rotation of the control surface，nacelle pylon，and the wing folding mechanism.Besides the free?play incurred in design，manufacturing and assem?bly，it will also be enlarged due to wear during inservice.

The elevator of a certain aircraft is a point-topoint control.The control joint adopts the structure of two ears，while the bushing is pressed into the ear hole.There is the interference fit between the bushing and the bearing，while the freeplay fit be?tween the bushing and the bolt and between the bolt and the bearing.The structure is shown in Fig.1，where the freeplay value between the bushing and the bolt is（0.001 27±0.001 27）cm，and the value between the bolt and the bearing is（0.002 54±0.001 27）cm.Therefore，the maximum freeplay that may exist after assembling is 0.006 35 cm.Con?sidering the potential wear of the bearing，an addi?tional freeplay of 0.002 54 cm will be generated.The most severe case（freeplay with 0.008 89 cm）will be used for calculation and analysis in this paper.

Fig.1 Schematic diagram of freeplay of joint

1.2 Design specifications

The freeplay design specifications are derived from Joint Service Specification Guides（JSSG），which is based on a series of wind tunnel flutter model tests carried out by the Wright-Patterson Air Force Base in the mid-to-late 1950s［7］.The test re?sults show that if these freeplay requirements are ap?plied in use，there will be no significant reduction in the flutter speed margin.These freeplay values can also be found in MIL-A-8870（ASG）［8］.However，as these requirements are too severe，even F-22 cannot meet the specifications［9］.For most of the control surfaces the values of freeplay of the life cy?cle exceeded the specifications in the JSSG.In 2000，the Federal Aviation Administration（FAA）stated that these requirements were considered too conservative and too small to be practically con?trolled in service life［5］.In such cases，the manufac?turers have provided analyses and/or flight tests to confirm the adequacy of the freeplay.In 2014，AC25.629-1B added the adequate requirements for wear of components such as control surface actua?tors，hinge bearings，and engine mounts in order to maintain aeroelastic stability margins［10］.Freeplay requirements can also be found in the British Air Force and Navy Aircraft Design Requirement AP 970（Aeroelastic Part），with a simpler version（a.Normal control surface 0.1°.b.Full control sur?face 0.05°）［11］.

Domestic aircraft design requirements of free?play are based on the requirements of U.S.military specifications，which are quite identical to the re?quirements of MIL-A-8870（ASG）［8］，see Section 3.2.1.8.4.in GJB 67.7A—2008［4］.However，due to the limitation of technology development，there is a lack of freeplay nonlinear flutter analysis and free?play measurement methods.The freeplay require?ments have rarely been considered in previous air?craft design，or a rough evaluation method has been adopted.At present，with frequent occurrence of freeplay problems，more and more companies begin to pay attention to the issues with focus not only on the freeplay control of the structure in the develop?ment process，but also on a series of freeplay tests and evaluations during in-service life.

1.3 Measurement requirements

In order to ensure the freeplay of control sur?face meets the design requirements，it is necessary to obtain the freeplay through reliable measurement.Some simple measurement methods were used in the early stage，such as the marking，splint-holding shaking and so on［12］.Today，sensors，microme?ters，image measurement and more accurate and ad?vanced methods are gradually adopted［13-16］.In this paper，a direct freeplay detection device driven by servo-motor is designed，as shown in Fig.2，which is fixed on the stabilizer and the control surface through the clamping device respectively.The force and deflection angle curves are recorded in real time by the sensor and the measuring instrument，and the angle of the freeplay will be read directly by the intersection point of the measured curves and the co?ordinate axis with an accuracy of 0.002°.In addition to the direct measurement method described above，some indirect measurement methods have been de?veloped，such as frequency response measurement method［17］，preload measurement method［18］，phase lag measurement method and so on［19］.The frequen?cy response measurement method is based on the principle that the rotation frequency of control sur?face is directly related to its freeplay size.The phase lag measurement method is based on the description function method of nonlinear system as the funda?mental component of the output signal is constant in any frequency range.The phase-frequency character?istic curve is a flat line，and the lag angleφcan be obtained as［19］

Fig.2 High-precision measurement

whereAandcare amplitude of the input signal and the freeplay of control surface，respectively.

The analysis shows that the input amplitude cannot be too small or too large，otherwise the mea?surement error of freeplay is large.The recommend?ed value ofAis 2 times that ofc，and the lag angle is about 6°.

As the freeplay measurement is a highly accu?rate ground test，it is necessary to consider the influ?encing factors.The loading force is used in most of the freeplay measurement methods.Therefore，it is important to eliminate the influence of elastic defor?mation in measurement，or to select a suitable load?ing force，which can measure the freeplay value but will not cause large elastic deformation.In the case of an aircraft with T-tail，the loading force will cause large elastic deformation of the stabilizer if it is too high when used to measure the freeplay of the elevator.This paper calculates and compares the dif?ferent displacement of the trailing edge of the eleva?tor under different loading forces，and shows that the deformation of the trailing edge of the whole air?craft is larger than that of the single elevator.As shown in Figs.3，4，if the loading force is too large and the displacement of the trailing edge isL1andL2，the freeplay value calculated according to the slope of the curve isB，which will be smaller than the actual freeplay value.Therefore，the influence of loading force and other factors should be taken in?to account in the measurement of the freeplay of con?trol surface.

Fig.3 Deformation morphologies of elevator

Fig.4 Displacement curves of elevator

2 Nonlinear System Formulations

At present，the main quantitative analyses of the freeplay nonlinear flutter are semi-analytical and numerical integration methods［20］.The descriptive function method is a commonly used semi-analytical method.When the system satisfies certain hypotheti?cal conditions，the output of the nonlinearity in the system under the action of sinusoidal signal is ap?proximated by the first harmonic component.Thus，the approximate linear characteristic of the nonlinear characteristic is obtained.The descriptive function method is therefore an equivalent linearization meth?od.Since the flutter system has good filtering char?acteristics，the deflection angleα(t) can be obtained by

whereωfis the flutter frequency，andαfthe ampli?tude of the deflection angle.

As shown in Fig.5，assuming a sinusoidal in?putx=Asinωt，whenωt=a，the excitation ampli?tude is equal to the freeplay size.In the case of sinu?soidal input，the restoring forceyis written as

Fig.5 Input-output relationship with freeplay

whereKis the normal stiffness，andKbecomes the constantKαwhenα>E.ais the freeplay andEthe input corresponding freeplay.

For periodic output signals，ycan be expanded into a Fourier series

The descriptive function method generally con?siders first-order harmonics，and only gives a rough approximation of flutter，whose accuracy decreases with the increase of nonlinear stiffness，LCO ampli?tude，and sometimes even fails.The numerical inte?gration method will solve the problem that the semianalytical method cannot solve.The current re?search focuses on the time-domain simulation based on CFD/ CSD coupling［21］，which improves the cal?culation accuracy，but is time-consuming and ineffi?cient.

The discrete state-space method is different from the general time-domain simulation meth?od［22］.As shown in Fig.6，the nonlinear system is di?vided into subsystems by means of nonlinear param?eter configuration.These subsystems can form a set of piecewise discrete time-domain state-space equa?tions.However，the discrete gust or the control in?put can be designated as the external disturbance of the nonlinear system，and the stability of the system can be judged by the response analysis.The basic as?sumption of the discrete state-space method is that the nonlinear characteristics of an aeroelastic system can be represented by a set of system parameters.The equations of nonlinear aeroelastic systems can be expressed as function with various discrete val?ues［22］

Fig.6 Calculation flow of discrete state-space method

whereare the nonlinear parameters；Mllij，BllijandKllijthe generalized mass，damping and stiffness matrices，respectively；Plljjthe unsteady aerodynam?ic force；P0ijthe gravitation and the trim force，and｛ε｝the generalized coordinate.

Eq.（6）is similar to the equation of motion of a linear aeroelastic system，except that the system matrix is a function of nonlinear parameters.When the nonlinear parameters are timed，these system matrices can be obtained at various discrete values.At each value，it is assumed that the aeroelastic sys?tem is locally linear.In this way，the time-domain state-space equation of the open-loop aeroelastic sys?tem is obtained

wherecontains the structural and aerodynamic states；uaecontains the deflection，rate and accelera?tion vectors of the control surface；andandare the state space matrices.

The research on nonlinear flutter mainly fo?cused on the verification of freeplay，but the actual flight will be affected by the aerodynamic loads that cause the equilibrium position of center-type free?play had shifted to freeplay with preload，as shown in Fig.7.Ep-Eis the freeplay with preload.

Fig.7 Freeplay type

Before the nonlinear flutter analysis of the free?play with preload is carried out，an aircraft trim cal?culation is required.The equation of trim system is written as

whereare the stiffness，mass and aerodynamic influence coefficient matrices.Dis the rigid body model matrix，the acceleration of rigid body，andxthe elastic displacement.{a} contains the trim parameters of the angle of attack，sideslip angle，roll rate，pitch rate，yaw rate，and deflection angle of control surface.is the derivative of aero?dynamic force corresponding to the rigid body with respect to trimmed parameters，andq∞the dynamic pressure.

After the parameters are obtained，the freeplay nonlinear flutter analysis can be carried out according to the discrete state-space method described above.

3 Numerical Results

3.1 Linear flutter analysis

In order to ensure the flight safety，the nonlin?ear flutter analysis of the freeplay of control surface is required.The vibration modes of airfoil and con?trol surface are shown in Fig.8.The accuracy of non?linear modeling is first verified by the results of fre?quency-domain flutter analysis.Thev-g-fcurves are shown in Fig.9.The critical flutter speed is 228.0 m/s and the flutter frequency is 53.1 Hz.Then the structural response without freeplay is cal?culated by the nonlinear discrete state-space meth?od.1-cos discrete gust is employed as the external excitation and the response curves are shown in Figs.10，11.The horizontal tail diverges with a fre?quency of 52.0 Hz when the flight speed is 228.0 m/s，which is consistent with the results in frequency-domain flutter analysis.

Fig.8 Vibration modes of airfoil and control surface

Fig.9 v-g-f curves of flutter in frequency domain

Fig.10 Response curves at 227 m/s without freeplay

Fig.11 Response curves at 228 m/s without freeplay

3.2 Nonlinear flutter analysis of center?type freeplay

Considering the freeplay of 0.2°（corresponding to the height of the rocker arm of 2.5 cm，the larg?est freeplay of 0.008 89 cm，see Section 1.1），the structural responses of the center-type freeplay at different speeds are calculated by the discrete statespace method and results are shown in Figs.12—15.The curves show that：The tip response of horizon?tal tail is attenuation motion when the flow speed is 60 m/s；the response is LCO when the flow speed is 70 m/s，with a frequency of 15.6 Hz；the re?sponse is also LCO when the flow speed is 240 m/s，with a frequency of 17.2 Hz；and the re?sponse is divergent motion when the flow speed is 243 m/s，with a frequency of 52.1 Hz.Different from the linear flutter case，the center-type freeplay causes an earlier occurrence of LCO before the fre?quency-domain flutter speed but the divergence speed is higher than that calculated in linear flutter analysis.

Fig.12 Response curves at 60 m/s in center-type freeplay

Fig.13 Response curves at 70 m/s in center-type freeplay

Fig.14 Response curves at 240 m/s in center-type freeplay

Fig.15 Response curves at 243 m/s in center-type freeplay

3.3 Nonlinear flutter analysis of freeplay with preload

Different from the center-type freeplay，an air?craft trim is required to calculate the deflection an?gle at different speeds before implementing the freeplay with preload flutter analysis.Taking the longitudinal trim as an example，this paper investi?gates the effects of preload on the freeplay of the elevator by adjusting the pitch angle and the deflec?tion angle of the elevator.The results are shown in Table 1.

Table 1 Trim angle and deflection angle of elevator

The nonlinear model is established according to the deflection angle and freeplay of the elevator，and the structural responses are calculated by the discrete state-space method.The results are shown in Figs.16，17.The response curves show that the tip response of the horizontal tail is attenuation mo?tion when the flight speed is 220 m/s，and the tip re?sponse is divergent motion when the flight speed is 225 m/s with the frequency of 52.1 Hz.Unlike the central-type freeplay，there is no LCO in the free?play with preload，and the divergence speed is close to the linear flutter speed.

Fig.16 Response curves at 220 m/s in freeplay with preload

Fig.17 Response curves at 225 m/s in freeplay with preload

4 Test Results

In order to verify the effect of preload on the flutter of nonlinear freeplay，a single wing model with freeplay of control surface is selected for the wind tunnel flutter test.A 1/6-scale test model is designed as wind tunnel flutter tests are normally performed at speeds no more than 40 m/s.The free?play is achieved through a control mechanism，as shown in Fig.18.The force and deflection angle curve obtained by the high-precision measurement are shown in Fig.19，and the measured value of free?play is twice of 0.7°.The model is connected verti?cally to the wind tunnel floor to eliminate gravity ef?fects，and the angle of attack can be adjusted as shown in Fig.20.In wind tunnel tests，the signal-tonoise ratio is poor and presents a certain degree of nonlinearity as the model is influenced by loads，damping and noise.The original acceleration signal is converted by FFT after the DC component and the trend are removed，and the frequency domain speed signal is obtained by integral method.Then，the time-domain speed signal is obtained by IFFT.Next，the displacement signal is obtained by fre?quency domain speed integral，and lastly the timedomain displacement signal is obtained by IFFT.

Fig.18 Freeplay control structure

Fig.19 Curves of values of freeplay

Fig.20 Wind tunnel test of single wing model

Figs.21—23 are the results of wind tunnel tests.The results show that：The model starts an approximation of LCO with constant amplitude from the speed of 30 m/s，which continues as the flow speed increases.Meanwhile，it can be seen that the oscillation is accompanied by the low fre?quency movement，which is caused by the constant change of the equilibrium position of the freeplay un?der static aerodynamic loads.When the flow speed reaches 40 m/s the oscillation disappears，and the flutter divergence appears.Although the vertically mounted model is used to eliminate the effects of preload，the static load on the wing of asymmetric airfoil increases with the increase of flow speed，and the nonlinear effect caused by the freeplay is elimi?nated under the preload.

Fig.21 Results of wind tunnel test with flow speed of 30 m/s

Fig.22 Results of wind tunnel test with flow speed of 37 m/s

Fig.23 Results of wind tunnel test with flow speed of 40 m/s

According to the curves of the main vibration frequencies（nz-01-nz-05）with different flow speeds in Fig.24，the main frequency of the wing vi?bration signal is about 6 Hz with the speed of 30 m/s，and the frequency fluctuates within a small range as the flow speed increases.The main frequency will produce a step when the flow speed reaches 40 m/s，roughly reaching 8.0 Hz.The divergence speed of the model is close to the result of critical speed of linear flutter，which further proves that the effects of nonlinear freeplay on flutter can be elimi?nated with preload.

Fig.24 Vibration frequencies with different flow speeds

5 Conclusions

Based on numerical results of the discrete statespace method and results of the wind tunnel tests，this paper investigates the influence of the freeplay on nonlinear flutter as well as the effects of the pre?load on the nonlinear flutter of freeplay by the wind tunnel tests of a single wing model.The main con?clusions are as follows：

（1）LCO occurs prior to the linear flutter speed when the center-type freeplay is considered.When the freeplay with preload is considered，the calcula?tion results differ significantly from the center-type freeplay and the divergence speed is close to that of linear flutter.

（2）In the wind tunnel tests，there is LCO in the wing with center-type freeplay when the flow speed is small.When the flow speed is large，it leads to the freeplay with preload，and the experi?mental divergence speed is close to the linear flutter speed.

（3）The aircraft is always subjected to loads in the course of flight，therefore，the deflection of the control surface only passing through the freeplay sec?tion causes the transient oscillation and then leads to the bearing wear and other fatigue problems.