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1.State Key Laboratory of Mechanics and Control of Mechanical Structures，Nanjing University of Aeronautics and Astronautics，Nanjing 210016，P.R.China；2.China Aerodynamics Research and Development Center，Mianyang 621000，P.R.China

Abstract: In wind tunnel tests，long cantilever stings are usually used to support aerodynamic models. However，this kind of sting support system is prone to vibration problems due to its low damping，which limits the test envelope and affects the data quality. It is shown in many studies that the sting vibration can be effectively reduced by using active sting dampers based on piezoelectric actuators. This paper attempts to review the research progress of piezoelectric vibration control in wind tunnel tests，covering the design of active sting dampers，control methods and wind tunnel applications. First of all，different design schemes of active sting dampers are briefly introduced，along with the vibration damping principle. Then，a comprehensive review of the control methods for active sting dampers is presented，ranging from classic control methods，like PID control algorithm，to various intelligent control methods.Furthermore，the applications of active sting dampers and controllers in different wind tunnels are summarized to evaluate their vibration damping effect. Finally，the remaining problems that need to be solved in the future development of piezoelectric vibration control in wind tunnel tests are discussed.

Key words：wind tunnel；sting vibration；active damper；piezoelectric actuator；active vibration control

0 Introduction

It is well known that many mechanical struc⁃tures，especially in the field of aerospace，are often subjected to vibrations introduced by operational or environmental effects，which are usually detrimen⁃tal，resulting in structural fatigue and the decrease of safety［1-10］. Among various studies on the attenua⁃tion of vibration in aviation structures，this paper mainly discusses the recent research on vibration suppression of the cantilever sting support system used in wind tunnels.

Wind tunnel tests，which are usually operated to simulate flight environments and evaluate aerody⁃namic characteristics，are of vital importance in air⁃craft design process. To avoid support interference on test section flow，long cantilever stings are wide⁃ly used in wind tunnels to mount test models.Gener⁃ally，such cantilever sting support system consists of a test model，a wind tunnel balance and a tapered hollow sting connected to a rigid model attitude sup⁃port system［11-13］. The length of the cantilever sting varies from three to five times the length of the mod⁃el to avoid aerodynamic interference of the support system［14-15］. As the sting length increases， the damping and stiffness of the entire structure will be lower. As a consequence，undesirable large-ampli⁃tude and low-frequency vibrations occur easily on the sting support system when the test model sweeps to a large angle-of-attack or model flow sep⁃aration appears，leading to test envelope limitation and data quality degradation. Moreover，this kind of harmful oscillations also result in wind tunnel bal⁃ance overloads and threaten the safety of wind tun⁃nel tests. In wind tunnel test history，many perfor⁃mance tests of wind tunnel models have been severe⁃ly affected by the sting vibrations. Therefore，in or⁃der to ensure planed test envelope and obtain highquality data，vibration suppression in wind tunnel tests has always been an active research topic.

Over the past decades，a variety of passive，ac⁃tive and hybrid control techniques to tackle structur⁃al vibration problems have been developed［16-22］.However，the current trend towards lightweight flexible structures leads to new challenges on vibra⁃tion control technology. Conventional vibration damping materials and control methods are limited in a lot of applications involving flexible structures due to the inability of vibration control at low fre⁃quencies and the need for more space，energy and weight. The cantilever sting used in wind tunnel is one such flexible structure. The dynamic response of the sting support system exposed to wideband force excitation is dominated by low frequency modes. In addition，the test environments of wind tunnels also have strict restrictions，including space and flow field requirements. It is obvious that con⁃ventional control technology is not suitable for the vibration suppression in wind tunnels. In the early wind tunnel tests，passive control methods were mainly used to attenuate sting vibrations，such as us⁃ing tuned mass dampers and viscoelastic materi⁃als［23-25］. The limitations of this method are that the low frequency vibration suppression effect is small，and each passive damper is only suitable for a specif⁃ic test component，indicating that the passive damp⁃er may be less effective when the test component changes. As a result，considerable efforts to find new solutions for vibration control of the cantilever stings used in wind tunnels are constantly spared by many scholars and institutes.

In this context，the advent of smart materials has opened up new routes for vibration control of flexible structures. Among various smart materials，piezoelectric materials have emerged as the most widely used materials for vibration control，thanks to their excellent properties，such as lightweight，small volume，high electro-mechanical coupling co⁃efficient，large frequency bandwidth of operation，and easy integration with flexible structures［26-30］.The principle of piezoelectricity can be divided into the direct piezoelectric effect and the converse piezo⁃electric effect. Piezoelectric materials will generate electric charges when subjected to mechanical loads（direct piezoelectric effect），and conversely，they will generate mechanical stress or strain when sub⁃jected to external electrical fields（converse piezo⁃electric effect）. Based on converse piezoelectric ef⁃fect，different types of piezoelectric actuators have been designed for active vibration control，which is the main application form of piezoelectric materials in wind tunnel tests［31-35］.

The applications of piezoelectric materials in ac⁃tive vibration control provide wind tunnel research⁃ers with a new idea for vibration damping. Up to now，many studies have been conducted on the use of piezo actuators to suppress sting vibrations. The Europe transonic wind-tunnel（ETW）first conduct⁃ed the relevant research on active vibration control of cantilever stings using piezoelectric elements. A piezoelectric-based active damping device，called the anti-vibration-system（AVS）was developed to counteract vibrations in pitch plane of the sting sup⁃port system［36-37］. Nevertheless，the vibration con⁃trol effect of this system was not satisfactory for strong vibrations due to the drawbacks of structural design［38］. Based on the similar design philosophy，Balakrishna et al.［39-41］developed various active sting dampers using piezoelectric stack actuators accord⁃ing to different test models，and a series of evalua⁃tion tests were conducted in different wind tunnels，such as YiGYAN’s low speed wind tunnel，NASA Langley Research Center National Transonic Facili⁃ty（NTF）and Ames Research Center 11×11 Foot Transonic Wind Tunnel（11’TWT）. Experimental results showed that the damping ratio of the support system and angle-of-attack testing range were great⁃ly improved by using active piezoelectric sting damp⁃ers. However，most studies of ETW and NTF only focused on the design and validation tests of active sting dampers，and there were few studies on con⁃trol algorithms. It is undoubtedly that control algo⁃rithms are vital to active vibration damping systems.Recently，many scholars have also investigated the control algorithms for vibration suppression of the cantilever stings used in wind tunnels. The early re⁃search used the proportional integral derivative（PID）algorithm to control the piezoelectric actua⁃tors in active sting dampers［42］. Since it always takes a long time to obtain satisfying PID control parame⁃ters， novel intelligent control algorithms have aroused the interest of many researchers. Based on the PID algorithm，artificial neural network PID（NNPID）algorithm and linear quadratic regulator（LQR）optimal control algorithm，Shen et al.［43］de⁃veloped three different controllers for active sting dampers，and the results indicated the superiority of intelligent control algorithms. Liu et al.［44］proposed a self-adaptive fuzzy PD control algorithm to control active sting dampers，which realized the self-tuning of PD control parameters.

A larger number of valuable results have been obtained in wind tunnel vibration control using piezo⁃electric actuators. However，exiguous review arti⁃cles can be found in summarizing the relevant re⁃search results and the existing problems. In order to provide the necessary research background and the latest developments for the researchers interested in wind tunnel vibration control，this paper presents a review of piezoelectric vibration control in wind tun⁃nel tests. Different design schemes of active sting dampers，as well as the vibration damping principle are reviewed according to the installing positions of piezo actuators. Various investigations on vibration control methods for active sting dampers are then discussed. Additionally，an assessment is made about the applications of active sting dampers and controllers in different wind tunnels. The final sec⁃tion discusses the shortcomings to be improved in the future developments.

1 Active Sting Dampers Using Piezoelectric Actuators

1.1 Structural design

As shown in Fig.1，the cantilever sting support system used in the wind tunnel can be regarded as a Bernoulli-Euler beam. The continuous sting system can be converted into a mass-stiffness-damping sys⁃tem with finite degrees of freedom by discretization，which can be represented by

Fig.1 Schematic of a typical cantilever sting support system

whereM，C，Kare the mass，the damping and the stiffness matrixes；F(t) is the external force vector of the sting support system andx(t) the vibration displacement vector.

When no control force is acted on the sting sup⁃port system，the external force vector can be ex⁃pressed asF(t)=Faero(t)，whereFaero(t) is the aerodynamic force vector. During wind tunnel tests，the aerodynamic force acting on the aircraft model can be divided into the static lift force，and the dy⁃namic force excited by flow turbulence with multifrequency components. The static force carries the aerodynamic information of the test model，while the dynamic force is the major cause of sting vibra⁃tions. Moreover，the sting support system tends to respond dynamically in eigen modes when exposed to aerodynamic forces. Generally，most wind tunnel sting support systems show six dominant eigen modes，including sting modes in pitch and yaw planes，coupled pitch-yaw sting modes manifesting as roll mode，modal-balance modes in pitch and yaw planes，and an axial translational modal-bal⁃ance mode.

Since the slenderness of the cantilever sting is supposed to be as small as possible，the structural damping of the sting system is poor，resulting in large vibration amplitude and long oscillation dura⁃tion. By appending extra damping to the sting，the vibration will be substantially reduced. Therefore，a lot of active sting dampers，which can improve the damping of the sting system，have been designed to suppress sting vibrations.

Hefer［36］from ETW proposed a piezoelectricbased active vibration damping device called the counter vibration generator in a patent for the sting support system used in wind tunnels.It is the first ef⁃fective design scheme that applies piezoelectric ele⁃ments to vibration control in wind tunnel tests. As shown in Fig.2（a），it was composed of six piezo⁃electric elements evenly distributed around the axis of the sting. The component 1 represents the coun⁃ter vibration generator and the component 2 repre⁃sents the piezoelectric element. Furthermore，it could be arranged not only between the sting and the wind tunnel balance，but also in the middle of the sting as shown in Fig.2（b）.

Fig.2 Schematic of the counter vibration generator designed by Hefer[36]

According to Hefer’s study， the design scheme was improved and an active AVS for full span model testing in the ETW was developed by Fehren et al［37］. As illustrated in Fig.3，the active damping structure which consisted of 14 piezoelec⁃tric actuators was installed between the 100 mm di⁃ameter flange of the six-component strain gauge bal⁃ance and the sting. A carbon fiber sleeve was intro⁃duced to withstand the loads in terms of pressure，tension and shear，which enhanced the structural safety. Additionally，the vibrations in all degrees of freedom except the roll direction could be attenuated by using AVS.

Fig.3 Schematic of ETW’s AVS[37]

Similarly，the researchers from the NASA Langley Research Center NTF began to study vibra⁃tion suppression of wind tunnel model support sys⁃tems in the 1990s. Through the research on the dy⁃namics of several standard models with violent vi⁃bration in wind tunnels，Young et al.［45］proposed the idea of applying smart materials to active vibra⁃tion suppression. As a result，an active sting damp⁃er design concept［39］was presented in Balakrishna’s study，which considered mounting a cluster of four piezoelectric devices in the sting，as shown in Fig.4.In this symmetric configuration，the piezo actuators were embedded with structural integrity in the sting in a cruciform pattern. Based on the balance signal feedback，the actuators could generate restoring mo⁃ment in the pitch and yaw planes to counteract vibra⁃tions at the eigen frequencies of the sting system.

Fig.4 Design concept of an active sting damper using piezo⁃electric actuators[39]

The optimization of piezo-actuator cluster con⁃figuration was also investigated，as well as the mounting locations of the piezo-actuator cluster in the sting［40］. Two piezo cluster configurations and two types of active sting dampers were proposed as shown in Fig.5. The sting-tip damper was more suit⁃able for high lift models，while the sting-root damp⁃er was more suitable for low lift models. Moreover，the piezo cluster configuration 1 could improve the structural rigidity of the sting system，compared with the second configuration. On this basis，three different types of active dampers were designed for the NTF Pathfinder-I check standard model，the crew launch vehicle（CLV）model and the NASA common research model （CRM） ， respective⁃ly［39-41，46-48］.

Fig.5 Configurations of the active sting damper[40]

In addition to the design of an active damper connected to the sting，the piezoelectric actuators can also be embedded in the sting directly. Accord⁃ing to Liu’s study，four piezoelectric stack actuators were embedded in the mounting grooves of the sting［44］. In order to ensure that the active damping device can output both pressure and tension，a pretightening mechanism was also designed in the mounting grooves. The schematic diagram of the ac⁃tive damping structure is shown in Fig.6.

Fig.6 Schematic of the sting-root embedded active damping device[44]

It is well known that the output displacement of the piezoelectric stack actuator is in the micron or⁃der under electrode voltage，which limits the con⁃trol performance of the active sting damper. There⁃fore，corresponding micro-displacement amplifiers have also been designed to improve the output per⁃formance of piezoelectric stack actuators. An active sting damper with a flexure hinge was proposed by Dai et al.［49］and four bolts were used to pre-tighten the piezoelectric actuators. Besides，only two highvoltage piezoelectric stack were used to generate the control force. Fig.7 shows the specific components of the active sting damper.

Fig.7 Components of the active sting damper with a flexure hinge[49]

1.2 Vibration damping principle

It can be concluded that all active sting damp⁃ers currently used in wind tunnel tests work accord⁃ing to the closed-loop feedback control theory. The principle of wind tunnel vibration reduction based on active sting dampers is illustrated in Fig.8.The aero⁃dynamic load，composed of the static lift force and the wideband stochastic dynamic force，acts at the pressure center of the test model with zero mo⁃ments，namely

Fig.8 Principle of vibration suppression using the active sting damper

whereFs(t) andFd(t) represent the static and dy⁃namic forces，respectively.

Furthermore，the aerodynamic force will cause a bending moment that progressively increases along the axial direction of the cantilever sting.Then，the bending moment at the cross section of the active sting damper can be expressed as

whereLrepresents the distance between the section of the active sting damper to the center of pressure.

To counteract the bending moment that causes sting vibrations，the piezoelectric stack actuators are introduced in the active sting damper to generate an equal and opposite restoring moment，which can be expressed as

wherenrepresents the number of piezoelectric stack actuators，fpieis the dynamic force generated by one piezoelectric stack，anddthe distance from the equivalent action point of the piezoelectric stack to the neutral axis.

Hence，the net moment at the cross section of the active sting damper becomes zero，thereby sup⁃pressing the corresponding vibrations. However，it is noticeable that only the dynamic bending moment needs to be eliminated for vibration suppression. As a consequence，the time derivative of the bending moment is used as the feedback signal in the closedloop control system. Such rate feedback enssures that the active sting damper will not work against the static moment generated in a wind tunnel test and only the dynamic moment is cancelled.

2 Control Methods for Wind Tun⁃nel Vibration Suppression

The use of piezoelectric actuators for vibration suppression in wind tunnel tests has been proved to be effective. However，in addition to ensuring a good structural design of the active sting damper，it is also important to adopt appropriate control meth⁃ods to improve the vibration damping performance.Based on the control algorithm，the research on vari⁃ous control methods used for wind tunnel vibration suppression can be divided into two categories：One covering the work in classic vibration control meth⁃ods，and the other covering the studies with adap⁃tive and intelligent methods.

2.1 Classic vibration control methods

2.1.1 PID controller

The first controller in question is the propor⁃tional，integral，and derivative（PID） controller.As the most widely used controller，the PID con⁃troller maintains a dominant position in engineering due to its robustness and universality. Similarly，it has been used by several researchers to suppress sting vibrations in wind tunnel tests［42，50］. A classical PID controller in a feedback loop is illustrated in Fig.9. The control output consists of PID compo⁃nents based on error signals. The mathematical ex⁃pression in time domain is given as follows

Fig.9 Diagram of a classical PID controller

whereKp，Ki，Kdare the proportional coefficient，the integral coefficient，and the derivative coeffi⁃cient，respectively.

Considering that a digital implementation of a PID controller requires a discretization of the error signals，two discrete PID controllers are further de⁃veloped by backward finite differences，namely the positional and incremental PID controllers.

Since the sting system will always turn back to the original equilibrium point after attenuation，there is no need to consider the steady-state error.As a result，the PID controller is mainly used in the actual damping applications. However，it is still a difficult problem for wind tunnel test engineers to adjust the control parameters to the optimal values of the desired control response，even though there are only three parameters to be determined. Be⁃sides，the values of control parameters usually need to be changed when the test model change for differ⁃ent wind tunnel tests，leading to great inconve⁃nience for engineers.

2.1.2 LQR controller

Among the active vibration damping systems used in wind tunnels，the balance is located at the tip of the sting，while the piezoelectric stack actua⁃tors are arranged at the root of the sting，thereby forming a non-collocated configuration with the drawbacks of poor stability and robustness. There⁃fore，the LQR controller has been used to achieve the control of sting vibrations in wind tunnel tests［51］. In order to design the effective LQR con⁃troller，the state-space model of the sting system is essential and different system identification methods have been studied up to now，such as the observer/Kalman filter identification method and the statespace model identification method. In LQR design，the performance criteria for each mode of the sting system can be determined independently and the overall system cost can be calculated as the sum of all modal costs. The total quadratic performance in⁃dex can be defined as

whereJiis the performance cost of a single mode，and subscriptiindicates the mode.nis the number of modes to be suppressed. The performance cost of a single mode can be assumed as

whereQiis the positive error weighted matrix andria positive control weighted factor.

2.2 Adaptive and intelligent control methods

As mentioned above，most classical vibration control methods cannot adapt to the changes in con⁃trolled objects or environments，which limits the vi⁃bration damping performance. Especially for the PID controller，constant control parameters are not suitable for all tests models，and it usually takes a long time for wind tunnel test engineers to obtain the optimal values of control parameters again. With development of intelligent control theory，active vi⁃bration control of a variety of structures has been un⁃dertaken with many adaptive and intelligent control⁃lers and the cantilever sting system used in wind tun⁃nels is no exception.

2.2.1 Fuzzy PID controller

It is well known that the fundamental difficulty with the PID control is the tuning of control parame⁃ters. As a consequence，a self-adaptive fuzzy PID controller was proposed by Liu et al.［44］to realize control parameters adjustment automatically for vari⁃ous test conditions. According to the angle-of-at⁃tack，the airflow speed and the performance of the wind tunnel，the fuzzy rules were set based on the expert experience. As shown in Fig.10，the propor⁃tional and derivative parameters are adjusted accord⁃ing to the velocity error and the velocity error rate.

Fig.10 Schematic of self-adaptive fuzzy PID controller[44]

2.2.2 Neural network PID controller

In addition to the fuzzy-based intelligent con⁃trollers，neural networks are also widely used in the field of intelligent control due to their powerful selflearning capabilities. For the vibration control in wind tunnel tests，the research on neural networks mainly focuses on the design of neural network PID controllers，such as developing a self-learning PID controller based on a back propagation neural net⁃work（BPNN）with three layers［49-50］. Fig.11 shows a BPNN-PID controller in a feedback loop. The in⁃put layer has two neurons，which take the expected value and the actual value of vibration as inputs，while the hidden layer consists of three neurons func⁃tioning as the PID control parameters. The output layer obtains the control output by calculating the sum of the control parameters in the hidden layer.

Fig.11 Schematic of a BPNN-PID controller with three layers[50]

Furthermore，the most extensive gradient de⁃scent（delta learning）rule is adopted to update the weight vectors. The formula for a BPNN-PID con⁃troller is

whereep(k)，ei(k)，ed(k) are the errors of the pro⁃portional coefficient，the integral coefficient，and the derivative coefficient，respectively.

To save the resource of the self-tuning pro⁃cess，the learning rateβis usually set the same val⁃ue for three control parameters. Then according to the delta learning rule，the chain rule，and the equiv⁃alent conversion，the final self-learning formula of the BPNN-PID can be described as

3 Wind Tunnel Applications

The large-amplitude and low-frequency vibra⁃tions of the cantilever sting used in wind tunnels have influenced the normal operation of many wind tunnel tests，which also restricts the development of a new generation of aircraft to a certain extent. To suppress such sting vibrations，different piezoelec⁃tric-based active damping devices and control meth⁃ods have been developed and applied to the various wind tunnels. Here，we list the applications of these active sting dampers in wind tunnels，as well as their control methods.

Several typical wind tunnel applications of the active sting dampers are shown in Fig.12. It can be concluded that most of the active sting dampers are used in low-speed or transonic wind tunnels around the world，while there are few attempts in superson⁃ic wind tunnels. Additionally，there are mainly two design schemes of the active sting damper，namely，the sting-tip damper and the sting-root damper. The sting-tip damper is designed to be mounted between the balance and the sting，which has poor structural rigidity. The sting-root damper is considered as an alternative to improve the structural rigidity. How⁃ever，it also has higher requirement for the perfor⁃mance of piezoelectric actuators. Moreover，the long distance between actuators and sensors might create spatial and temporal wave propagation is⁃sues，which needs to be considered in vibration con⁃trol. In addition to active sting dampers，it is essen⁃tial to use an effective control method for vibration suppression. However，most relevant foreign litera⁃tures did not present their control methods in detail.By contrast，the domestic scholars have conducted many research on wind tunnel vibration control methods. The control effects of the active damper with their control methods are shown in Fig.13.Since different models need to be evaluated in wind tunnel tests，most classic control methods will be less effective when the test model changes. As a re⁃sult，increasingly more novel intelligent and adap⁃tive control methods are investigated to suppress sting vibrations in wind tunnel tests. Various wind tunnel applications of the active sting dampers are concluded in Table 1.

Table 1 Wind tunnel applications of piezoelectric⁃based active sting dampers

Fig.13 Control effects of different active sting dampers with their control methods

4 Conclusions

Over the past three decades，piezoelectric actu⁃ators have been widely used in active vibration con⁃trol of various structures. Particularly in wind tunnel tests，extensive studies on the design of piezoelec⁃tric-based active sting dampers and control methods have been conducted to suppress the vibrations of the cantilever sting support system. This paper pro⁃vides an overview of the recent developments of piezoelectric vibration control for wind tunnel tests.Two different types of active sting dampers are high⁃lighted，according to the mounting locations of piezoelectric actuators. Both classic control methods and novel adaptive and intelligent control methods have been developed to improve the vibration damp⁃ing effect. Furthermore，the extensive and impres⁃sive wind tunnel applications were investigated to prove the feasibility of using piezoelectric actuators for active vibration control of the cantilever stings.

Despite the significant progress，several issues that remains in the field of wind tunnel vibration con⁃trol are worth probing，as detailed in the following.

（1）By means of piezoelectric stacks as damp⁃ing devices，the sting vibrations in low-speed and transonic wind tunnels can be attenuated，which has been experimentally proved. However，the existing active sting dampers have not been widely applied to supersonic wind tunnels and there are few perfor⁃mance evaluation studies in supersonic wind tun⁃nels. Future research should focus on the impact of different test conditions on the damp performance of active sting dampers and extend the application to supersonic wind tunnels.

（2）To increase the Reynolds number of wind tunnel tests，low-temperature wind tunnel has re⁃ceived more and more attention recently. Neverthe⁃less，the driving performance of piezoelectric actua⁃tors will drop significantly at low temperatures. Al⁃though the wind tunnel researchers have noticed this problem，there is few relevant research on improv⁃ing the control performance of active sting dampers at low temperature. Development of a heat preserva⁃tion device with flexible temperature control ability is essential to solve this problem.

（3）Currently，the control target of most stud⁃ies only focuses on the first pitch mode of sting vi⁃brations while the higher order modes are not con⁃sidered.In addition，when the test model is a bilater⁃al symmetrical structure like a rocket or missile，it is necessary to suppress the vibration both in the pitch and yaw directions at the same time. For fu⁃ture developments of piezoelectric vibration control in wind tunnel tests，novel adaptive and intelligent control methods for multi-modal and multi-dimen⁃sional vibration should be designed to improve the effect of vibration reduction.