
==== Front
Heliyon
Heliyon
Heliyon
2405-8440
Elsevier

S2405-8440(24)13070-8
10.1016/j.heliyon.2024.e37039
e37039
Research Article
Experimental study on the influence of centroid position of wind turbine section on flutter
Zheng Qi abc
Gao Zhiying hawkwarm@imut.edu.cn
abc⁎
Zhao Baozhong abc
Bai Yefei abd
Su Rina abc
Han Xiaoliang e
Zhao Feng abc
Dong Xueqing abc
Wang Jianwen abc
a Key Laboratory of Wind Energy and Solar Energy Utilization Technology of Ministry of Education, Inner Mongolia University of Technology, Hohhot, 010051, China
b Inner Mongolia University Renewable Energy Engineering Research Center, Inner Mongolia University of Technology, Hohhot, 010051, China
c College of Energy and Power Engineering, Inner Mongolia University of Technology, Hohhot, 010051, China
d School of Civil Engineering, Inner Mongolia University of Technology, Hohhot, 010051, China
e Inner Mongolia Electric Power Survey and Design Institute, Hohhot, 010020, China
⁎ Corresponding author. Key Laboratory of Wind Energy and Solar Energy Utilization Technology of Ministry of Education, Inner Mongolia University of Technology, Hohhot, 010051, China. hawkwarm@imut.edu.cn
29 8 2024
15 9 2024
29 8 2024
10 17 e3703917 5 2024
5 8 2024
27 8 2024
© 2024 The Authors
2024
https://creativecommons.org/licenses/by-nc/4.0/ This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).
Aiming at the large aspect ratio characteristics of large wind turbines, it is easy to cause irreversible damage due to flutter during operation. A two-degree-of-freedom (plunge and pitch) flutter test bench was built using the time domain and frequency domain analysis methods of dynamic signals. The influence of different centroid positions on the flutter boundary was studied. The test shows that the closer the centroid position is to the leading edge of the wing segment, the better the aeroelastic stability of the wing segment is. Under the linear condition, the forward movement of the centroid position has a more significant influence on the flutter. In addition, the main reason for the wing flutter is related to the decrease of net damping and the coupling of the aeroelastic natural frequency of the pitching and plunging motions. The pitch motion is dominant in the two-degree-of-freedom motion. The farther the centre of mass is from the torsion axis, the greater the pitch and plunge motion displacement. The pitch motion has a more significant impact on the system than the plunge motion. Therefore, the study of flutter suppression should focus on pitch motion.

Keywords

Wing segment
Centroid position
Flutter
Aeroelastic stability
==== Body
pmc1 Introduction

In the context of international agreements such as the ' Kyoto Protocol ' (UU.NN., 1998), the ' Paris Agreement ' (2015) (UU.NN., 2015), the European 2030 Climate and Energy Framework (European Commission, 2014) [1] and the domestic ' double carbon ' goal, the wind power generation technology has been further developed, and the size of the blade is getting larger and larger. In 2023, the diameter of the wind wheel can reach 260m, and the problems of blade stability and safety caused by this are becoming more and more prominent. Flow separation under large deformation will lead to dynamic stall and aeroelastic issues, and the alternating load will increase under dynamic stall [2], leading to flutter problems. The earliest flutter problems appeared on long-span bridges [3] and high-speed aircraft wings. Flutter is a non-decaying self-excited vibration caused by the coupling of aerodynamic force, elastic force, and inertial force. Traditional linear flutter modeling and analysis methods are relatively mature [4,5]. For nonlinear systems such as wings and wind turbine blades, the flutter is usually described as ' limit cycle oscillation ' [6].

The research on dynamic stalls of wind turbine airfoils mainly includes simulation calculation [[7], [8], [9]], empirical model [[10], [11], [12]], and wind tunnel test [13,14]. In the current foreign-related research: Literature [15] explored the parameters affecting the stall flutter of NACA 0012 airfoil and analyzed that the change of limit cycle vibration from symmetric to asymmetric in the stall flutter phenomenon is not controlled by the change of dynamic stall mechanism. Literature [16] established a two-degree-of-freedom vibration model and measured the model under different subcritical and supercritical flow rates in the wind tunnel. The wind tunnel experiment and two-dimensional CFD numerical simulation were combined to calculate the maximum static pressure at the minimum pitch angle, based on DIC technology. Literature [17] studied the fluid-solid coupling problem of the wing in a certain Reynolds number range under different centroid positions of two degrees of freedom. Literature [18] explored the flow field characteristics of wing self-excited vibration by changing the spring stiffness, free flow velocity, and initial angle of attack. Literature [19], based on the modified Theodorsen theory, the effects of blade mass center deviation and wind speed on flutter limit were studied according to the proposed blade aeroelastic model. Results: The flutter limit is sensitive to the change of the center of mass shift. Literature [20] used a system with two degrees of freedom (plunge-pitch) with similar natural frequencies to study the transformation of classical flutter and stall flutter of the system. Literature [21] developed a solver in the Open Foam framework to calculate the large amplitude motion of a two-dimensional rigid structure, which proves that the existing calculation method can capture the self-excited vibration of the stall flutter blade with the accuracy allowed by the project. Literature [22] used a new semi-experimental method to measure the unsteady aerodynamic force and aeroelastic characteristics of a three-dimensional wing. Literature [23] studied the stall flutter boundary of the NACA0012 airfoil by measuring the pressure and flow field when the airfoil is forced to vibrate. Literature [24,25] used a two-degree-of-freedom NACA64-418 airfoil vibration system with similar stiffness (pitch-plunge) to conduct wind tunnel experiments. The limit cycle vibration of the airfoil was observed and the aerodynamic characteristics of the airfoil during the limit cycle vibration were analyzed. Literature [26] studied the dynamic response of a two-degree-of-freedom plate under classical coupled mode flutter. Literature [27] studied the high-order frequencies in the limit cycle vibration and found that the fifth harmonic is related to the vortex shedding frequency of the dynamic stall. Literature [28] studied the limit cycle vibration of the NACA0012 vibration system with different degrees of freedom and found that for stall flutter, the plunge degree of freedom has no significant effect on the stall flutter of the system, which is only controlled by the pitch degree of freedom. Literature [29] studied the flutter phenomenon of the airfoil in a certain Mach number range and found that the curve showed an obvious double-loop structure when the airfoil fluttered through the pitch-plunging curve. Literature [30] used a two-degree-of-freedom airfoil flutter model to verify the flutter margin (PFM) method. It was found that there was a good consistency between the conventional flutter test and the PFM method. Literature [31] found through the flutter test of the wing section in a certain Mach number range that the relative size of the pitch and the natural frequency of the plunge have a significant effect on the flutter of the wing section. Literature [32] studied the flutter limit of IEA 15 MW wind turbine blades and realized the bending-torsion-torsion coupling by changing the carbon fiber angle of the skin and the wing beam. The results show that changing the carbon fiber angle of the wing beam will increase the flutter speed. Literature [33] studied the flutter simulation of bionic blades with V-shaped stripe webs. The results show that the blade has an anti-trembling performance compared with the prototype blade. In addition, literature [34] constructed a bionic airfoil for flutter suppression research based on dragonfly wing ripples. The results show that the structure improves the aerodynamic characteristics and structural stiffness, and plays a flutter suppression effect. Literature [35] studied the influence of parameter changes (stiffness, centroid offset, and radius of gyration) in different regions of NREL 5 MW wind turbine blades on flutter characteristics. The results show that the parameter changes in the tip region of the blade have the greatest influence on the flutter. Literature [36] studied the torsional vibration characteristics and flow field development mechanism of airfoils at different wind speeds and angles of attack. The results show that the formation and shedding of the leading edge vortex will significantly affect the stall flutter characteristics. Literature [37] re-hypothesized Theodorsen's theory, which is widely used to predict the flutter of wind turbine blades, and studied its influence on the flutter of onshore and offshore wind turbines. The results show that it has the greatest influence on the flutter speed of the propeller mode of the onshore wind turbine, and has no obvious influence on the pitch flutter of the offshore wind turbine tower. Literature [38] carried out flutter coupling experiments using scaled models. The research shows that under the critical wind speed of the flutter, the second shimmy mode and the first torsion will have modal coupling, which will induce flutter. In domestic-related research, literature [39] studied the dynamic response characteristics and aerodynamic characteristics of the vibrating wing section by placing a wing section that can be regarded as a rigid body vertically in the incoming flow through wind tunnel experiments. Based on the v-g method, literature [40] derived that whether flutter occurs is closely related to the structure from the ups and downs and pitch motion of the two-dimensional wing segment. Literature [41] combined the mode superposition method with the eigenvalue method to simulate the NACA 4415 airfoil. It is found that the smaller the distance between the center of mass and the torsion center, the larger the bending stiffness in the beating direction, and the smaller the bending stiffness in the waving direction can reduce the possibility of flutter. Literature [42] calculated the flutter boundary of the NREL 5 MW wind turbine when it stopped and operated by FAST software. When the flutter occurred, the response amplitude of the blade in all directions and the load amplitude of the blade root increased, and the load of the tower did not change significantly. Literature [43] solved the flutter velocity and limit cycle response by numerical simulation and discussed the influence of structural parameters on the flutter boundary. It is found that the larger the mass block is, the closer the position is to the end, and the more likely Hopf bifurcation occurs. Literature [44] studied the coupled modal flutter of wind turbine blades from structural parameters and aerodynamic parameters. The results show that the centroid offset has the least influence on the structural parameters, but has a great influence on the flutter speed of the blade at the tip of the blade. Literature [45] carried out CFD numerical simulation on NACA0012 airfoil with an aspect ratio of 16 and studied flutter boundary by changing material density and stiffness. With the increase of the stiffness of the material, the flutter speed and flutter frequency are increased. Literature [46] took the blade with a span of 80.74 m as the research object and analyzed the influence of air density, blade chordwise position, and torsional flapping frequency parameters on flutter speed. Literature [47] studied the aeroelastic stability of IEA 15 MW large wind turbine blades. The results show that the torsional stiffness is the dominant factor affecting the flutter critical speed, and the flutter margin can also be improved by reducing the blade mass and the center of gravity of the forward section.

Most of the existing flutter research is based on aircraft wings, rotors, and bridges, and there are relatively few studies on wind turbine blades as flutter models. For large wind turbines, the blade length is up to hundreds of meters, which is the most prone to flutter. The experimental study of the whole machine has a high reference value. Because the flutter can cause irreversible damage to the structure in a very short time, it isn't easy to study the flutter of the whole machine. For the complete blade, the blade tip is the most obvious part of the flutter. In addition, in a large number of flutter research and solution methods, the two-dimensional wing segment model is a simple but very classic aeroelastic system, which is feasible to simulate the structural dynamic behavior of the actual wing in the wind tunnel experiment [48]. Flutter suppression is mainly studied from the aspects of blade profile design, structural design, vibration characteristics of composite blades, and aeroelastic cutting technology [49]. According to the three-center design principle, the change of the center of mass has a great influence on the stability of the blade, and the influence of the center of mass position on the flutter of the wing segment needs to be further studied. Therefore, based on the previous research results on the flutter boundary and flutter characteristics of the wing segment, this paper takes the NACA64-618 airfoil at the tip of the large wind turbine blade with good aerodynamic characteristics and obvious aeroelastic effect as the research object and explores the influence of the blade structure characteristics-centroid position on the flutter, To provide a reference for the design of preventing and inhibiting flutter of leaves.

2 Testing system

2.1 Test tunnel

This experiment was conducted in the closed test section of the B1/K2 low-speed wind tunnel of the Key Laboratory of Wind and Solar Energy Utilization Technology of the Ministry of Education of Inner Mongolia University of Technology. The overall appearance of the wind tunnel is shown in Fig. 1. The flutter test of this paper is completed in the closed test section of the wind tunnel. The size of the closed test section is 2m (length) × 0.9m (width) × 0.9m (height), as shown in Fig. 2 The wind speed range is 0–60 m/s, the wind speed is continuously adjustable, and the turbulence intensity in the closed test section 0.5 %.Fig. 1 Experimental wind tunnel.

Fig. 1

Fig. 2 Wind tunnel closed test section.

Fig. 2

In this experiment, a two-degree-of-freedom support mechanism (as shown in Fig. 3) is designed to provide two degrees of freedom motion for the wing segment model in the closed test section of the wind tunnel. The support mechanism and the wing segment are installed in the closed section of the wind tunnel, as shown in Fig. 4 The maximum allowable amplitude of the pitching motion is ±25°, and the maximum allowable amplitude of the plunging motion is ±80 mm. The maximum wind tunnel blocking ratio of the wing segment model in the test is 4.96 %.Fig. 3 Schematic diagram of test bench structure.

Fig. 3

Fig. 4 Experiment bench physical map.

Fig. 4

2.2 Wing segment model

The wing segment model (as shown in Fig. 5) is made of resin materials with light weight, high strength, and good toughness. The left and right sides of the wing segment are provided with mass blocks to install bolt holes.Fig. 5 Wing segment physical map.

Fig. 5

Model parameters are shown in Table 1.Table 1 Wing segment model parameters.

Table 1Parameter item	Parameter value	
chord length/c	160 mm	
Length/L	560 mm	
String ratio	3.5	
Materials	Resin (3D)	
Torsion shaft position/mm	0.4212c	

2.3 Acquisition equipments

In this experiment, the magnetostrictive displacement sensor is used to measure the displacement of the plunging motion, the angle sensor is used to measure the angular displacement of the pitching motion, and the dynamic signal acquisition and analysis system is used for data acquisition. The sampling rate of each channel can reach 256 kHz. Sensor parameters are shown in Table 2. The main parameters of the model under the three centroid positions of the wing segment are shown in Table 3.Table 2 Sensor parameters.

Table 2Sensors	Precision	Resolution	Response time	Range	
Magnetostrictive displacement sensor	0.02%FS	0.02%FS	0.5 ms	±180 mm	
Angle sensor	0.3%FS	0.022°	0.6 ms	±50°	

Table 3 The main parameters of three centroids of wing segment.

Table 3Centroid position	Mass block configuration	XEG/c	Mass block material	Model quality/g	
Front centroid	Eccentric + Eccentric	≈0.029	Pure lead	3251.3 g	
Centered centroid	Center + Center	≈0	Pure lead	3251.3 g	
Postposition centroid	Central + Eccentric	≈−0.062	Pure lead	3251.3 g	

3 Determination of three centroid positions

Three-bolt holes are designed inside the wing section to fix the mass block. The copper screw and nut are used to fix the positions of the two mass blocks on both sides of the end plate of the wing section to change the centroid position of the wing section. The position of the center of mass and the torsion axis of the wing section designed in the test is shown in Fig. 6. The torsion axis is the origin of the coordinate, and the trailing edge direction is the positive direction of the X axis. The three centroid positions are shown in Fig. 7. The distance between the three centroids (G) of the wing segment and the center of the torsion axis (E) are set as, xEGΙ, xEGΙΙ, and xEGΙΙΙ, respectively. The distance xEGΙ between the front centroid and the torsion axis is −4.71 mm, the distance xEGΙΙ between the center centroid and the torsion axis is −0.68 mm, and the distance xEGΙΙΙ between the rear centroid and the torsion axis is 9.89 mm.Fig. 6 Schematic diagram of the centroid position and torsion shaft of wing segment.

Fig. 6

Fig. 7 Schematic diagram of the three centroid positions of the wing segment.

Fig. 7

4 Experimental investigation

In the experiment of exploring the influence of centroid position on wing flutter, the experimental process is shown in Fig. 8. The initial angle of attack is 16°, and the initial angle of attack refers to the angle between the chord of the wing segment and the direction of the incoming flow. The external excitation is applied to the leading edge of the wing segment through the exciter to generate a pitching pulse.Fig. 8 Flow chart of experiment.

Fig. 8

4.1 Research on front centroid flutter boundary

The physical image of the front centroid wing segment is shown in Fig. 9.Fig. 9 Front centroid physical map.

Fig. 9

According to the flow chart of the experimental scheme, the flutter boundary of the front centroid is studied. As shown in Fig. 10, the vibration time-domain diagram of the wing section when the pre-centroid is located. It is observed that at the thirteenth wind speed (corresponding to the wind speed of 18 m/s) after the pulse excitation is applied, the limit cycle vibration state appears in the wing section. Therefore, the flutter critical wind speed of the lower wing section of the front centroid is 18 m/s. At the same time, because the wing section is a cantilever beam structure and is placed horizontally, the wind speed continues to increase, and the lift force on the wing section continues to increase, resulting in an initial position upward shift as shown in Fig. 10(b).Fig. 10 Vibration time domain diagram of front centroid wing segment.

Fig. 10

Subsequently, to more clearly analyze the vibration state of the wing segment at each wind speed, the time domain diagram of the pitch motion of the front centroid wing segment is amplified. As shown in Fig. 11, it can be found that with the increase in wind speed, the speed of vibration attenuation after applying pulse excitation to the wing segment gradually slows down, which indicates that the aerodynamic damping acting on the wing segment gradually decreases with the increase of wind speed. As shown in Fig. 11(m), the wind speed is increased from 17.5 m/s to 18 m/s.After the wind speed is stable, the pulse excitation is applied to the wing segment. The vibration of the wing segment does not attenuate but maintains a constant amplitude. At this time, the vibration of the wing segment reaches the limit cycle vibration state, and the net damping of the wing segment is attenuated to zero. From the perspective of linearization theory, flutter will cause the amplitude to increase exponentially until the system completely collapses. However, due to the existence of nonlinear sources of the system (structural and aerodynamic nonlinearity), the amplitude of the wing vibration is stable to a finite value. Under the limit cycle vibration, the maximum amplitude of the instantaneous pitch angle of the wing segment is 9.07°, and the maximum vibration of the instantaneous plunging displacement is 2.16 mm.Fig. 11 Local enlarged time domain diagram of wing segment pitch motion when front centroid.

Fig. 11

4.2 Research on centered centroid flutter boundary

The physical image of the centered centroid wing segment is shown in Fig. 12.Fig. 12 Centered centroid physical map.

Fig. 12

As shown in Fig. 13, the time domain diagram of the wing segment vibration when the center of mass is centered. When the wind speed of the wind tunnel increases to 17 m/s, the limit cycle vibration appears after the pulse excitation is applied. Therefore, the flutter critical wind speed of the lower wing section of the centered centroid is 17 m/s The maximum pitch amplitude of the wing segment in this state is 6.97°, and the maximum plunge amplitude is 1.19 mm. After maintaining the limit cycle vibration for a while, the wind tunnel is shut down.Fig. 13 Vibration time domain diagram of centered centroid wing segment.

Fig. 13

Fig. 14 shows the local vibration time-domain diagram of the wing segment under different wind speeds. When the wind speed is 16.5 m/s, the vibration attenuation time of the wing segment is longer than that of the front centroid of mass, and the trend of limit cycle vibration is more prominent.Fig. 14 Local enlarged time domain diagram of wing segment pitch motion when centered centroid.

Fig. 14

4.3 Research on postposition centroid flutter boundary

The physical image of the postposition centroid wing segment is shown in Fig. 15.Fig. 15 Postposition centroid physical map.

Fig. 15

As shown in Fig. 16, the time domain diagram of the wing segment vibration when the rear center of mass is set. When the wind speed of the wind tunnel increases to 15 m/s, the limit cycle vibration occurs after the pulse excitation is applied at this wind speed. Therefore, the flutter critical wind speed of the lower wing section of the postposition centroid is 15 m/s.In this state, the maximum pitch amplitude of the wing segment is 10.55°, and the maximum plunge amplitude is 0.9 mm.Fig. 16 Vibration time domain diagram of postposition centroid wing segment.

Fig. 16

It can be found from Fig. 17 that the vibration attenuation characteristics are very similar to those of the first two centroids, but what is special is that after the third pulse excitation at 14.5m/s, the wing segment has sustained vibration for nearly 20 s, which may be related to the magnitude of the applied pulse excitation. This phenomenon is similar to the effect of initial pulse excitation on the limit cycle vibration of the wing segment found by N.A.Razak [20]. However, the difference is that the vibration does not persist but decays to a stationary state.Fig. 17 Local enlarged time domain diagram of wing segment pitch motion when postposition centroid.

Fig. 17

According to the wing vibration time-domain diagram of the three centroid positions, the change of the centroid position with the flutter critical wind speed is drawn as shown in Fig. 18. It can be found that the smaller the XEG, the closer the centroid position of the corresponding wing segment to the leading edge, the greater the flutter critical wind speed, and the less likely the wing segment to flutter. The larger the XEG is, the closer the centroid position of the corresponding wing section is to the trailing edge, the smaller the flutter critical wind speed is, and the more likely the wing section is to flutter. The centroid correlation coefficient is defined as the ratio of the flutter critical wind speed variation to the centroid position coordinate variation. The centroid correlation coefficient is defined as the ratio of the flutter critical wind speed variation to the centroid position coordinate variation. Under the linear condition, the correlation coefficient of the center of mass moving forward is 24.8 %, and the center of mass moving backward is 18.9 %, indicating that the influence of the center of mass moving forward on the flutter critical wind speed is more significant. In addition, through the analysis of Fig. 11, Fig. 14, Fig. 17, it is found that the degree of vibration attenuation is slowing down at a wind speed moment before flutter. From this point of view, the center of mass moves to the trailing edge, and the aeroelastic stability of the wing segment is worse.Fig. 18 Relationship between centroid position change and flutter critical wind speed.

Fig. 18

4.4 Natural frequency measurement

Under the condition of no wind, by changing the position of the mass block, the wing segment model is subjected to impulse excitation in the pitch and plunge directions, respectively. Then, the dynamic response signals at different centroid positions are measured. The time domain data collected by the sensor are transformed into frequency domain data by fast Fourier transform (FFT), and the natural frequencies of the wing segment are analyzed. Pay attention to the difference between the natural frequency and the aeroelastic natural frequency [50] the aeroelastic natural frequency is the system vibration frequency produced by the system under the action of aerodynamic force. The pitching and plunging natural frequencies of the wing segment under the three centroid positions are shown in Table 4. fpitch is used to represent the pitch natural frequency, and fplunge is used to represent the plunge natural frequency. The change of the centroid position does not change the natural frequency of the system itself.Table 4 Natural frequencies of the system at three centroid positions.

Table 4Centroid position	fplunge (Hz)	fpitch (Hz)	
Front centroid	3.75	8.00	
Centered centroid	3.75	8.00	
Postposition centroid	3.75	8.00	

4.5 Research on aeroelastic natural frequency

Through the FFT analysis of the time domain diagram of the wing vibration under different wind speeds, the change of the aeroelastic natural frequency of the system under different wind speeds is obtained, as shown in Fig. 19. It is found in Fig. 19(a) that with the increase of wind speed, the natural frequencies of the system are gradually close to each other, and flutter occurs at 18 m/s.At this time, the two frequencies completely coincide at 7.31 Hz. This phenomenon of aeroelastic natural frequency coupling is caused by the interaction between the airflow and the vibrating wing segment, which leads to the unsteady aerodynamic force in the flow field.Fig. 19 The schematic diagram of the change of the natural frequency of the system’s pitching and plunging aeroelasticity with wind speed under three different centroid positions.

Fig. 19

Similarly, the analysis of Fig. 19(b) shows that: At the wind speed of 16.5 m/s, the aeroelastic natural frequency of the wing section's plunge and pitch motion has been coupled at 7.70 Hz, but there is no flutter phenomenon at this time, because the energy provided by the airflow to the wing segment is not enough to offset the structural damping of the system. When the wind speed of the wind tunnel is further increased, the pulse excitation is applied to the wing segment again. At this time, the wing segment begins to flutter, and the coupled aeroelastic natural frequency increases to 7.73 Hz.

It is found from Fig. 19(c) that the natural frequencies of pitch and plunge aeroelasticity of the system are 7.36 Hz at 14.5 m/s wind speed, which is the same as the reason why the vibration attenuation of the lower wing segment of the center mass center is slow before flutter, which is because the net damping does not attenuate to zero at this wind speed. The limit cycle vibration of the wing segment was successfully observed at the following wind speed, and the aeroelastic natural frequency of the coupled system became 7.38 Hz.

The analysis of Fig. 19 shows that the aeroelastic natural frequency of pitch motion tends to be stable with the increase of wind speed, while the aeroelastic natural frequency of plunge motion increases, and finally coincides near fpitch = 8 Hz. In addition, the natural frequency of the plunging aeroelastic is an apparent mutation in the process of approaching the natural frequency of the pitching aeroelastic. This is due to the sudden drop of the lift and the instantaneous destruction of the system stability when the flutter occurs, which is the bifurcation behavior studied by relevant scholars. In addition, when the aeroelastic natural frequencies of pitch and plunge are coupled, but the net damping is not completely reduced to zero, the wing segment will not vibrate continuously. Therefore, it is not rigorous to judge whether flutter occurs only by two-frequency coupling.

Then, the effective power spectral density-frequency analysis of the limit cycle oscillation time-domain diagram at three centroid positions is shown in Fig. 20. Under the three centroid positions, the energy value of the pitch motion is much larger than the energy value of the plunge motion. Therefore, when the flutter phenomenon occurs, the pitch motion is always more intense than the plunge motion. In the two-degree-of-freedom flutter phenomenon, the pitch motion is dominant. In addition, when the centroid position is closer to the torsional shaft position, the coupling frequency during flutter is closer to the natural frequency of the system pitch. It is due to the fixed position of the aerodynamic center and the torsion center. The closer the center of mass is to the torsion shaft, the smaller the effect of the inertial moment is, which is dominated by the aerodynamic force and the elastic force.Fig. 20 Flutter coupling frequency diagram of the system under three centroid positions.

Fig. 20

Firstly, it is stipulated that the peak-to-peak value Y* of pitching motion is used to characterize the severity of pitching motion at the flutter moment. The magnitude of the positive displacement Ymax of the pitching motion is used to characterize the intensity of the bow direction of the pitching motion at the flutter moment; the magnitude of the negative displacement Ymin value of the pitching motion is used to characterize the intensity of the pitching motion head-up direction at the flutter moment. The absolute difference Ymax-min value between the positive displacement Ymax value and the negative displacement Ymin value of the pitch motion is used to characterize the intensity of the pitch-downward motion relative to the head-upward motion at the flutter moment. Similarly, the peak-to-peak value H* of the plunging motion is used to characterize the intensity of the plunging motion at the flutter moment. As shown in Fig. 21, Y* is 16.175°, H* is 1.26 mm, Ymax is 11.975°, Ymin is −4.200°, and Ymax-min is 7.775° under the action of the front centroid. Under the action of the centered centroid, Y* is 13.06°, H* is 0.75 mm, Ymax is 8.946°, Ymin is −4.114°, Ymax-min is 4.832°. Under the action of postposition centroid, Y* is 20.776°, H* is 1.48 mm, Ymax is 10.432°, Ymin is −10.334°, and Ymax-min is 0.098°.Fig. 21 Pitching and plunging motion parameters diagrams at different centroid positions.

Fig. 21

According to the data of Ymax-min value, the relationship curve between the change of centroid position and Ymax-min value is drawn as shown in Fig. 22. It is found that the centroid position moves forward, and the Ymax-min value increases from 0.098° to 7.775°, which indicates that when the centroid position moves to the leading edge, the flutter moment moves more violently in the direction of pitching lower head than in the direction of rising head. The reason is that the position of the centroid position moves forward relative to the torsion center, the inertial moment increases, the bow moment around the torsion center increases, and the amplitude of the pitch bow motion increases.Fig. 22 The relationship curve between centroid position change and Ymax-min value.

Fig. 22

The relationship between the change of centroid position and Y* value, H* value, and ΔY-H value are shown in Fig. 23. It is found that the farther the centroid position is from the torsion axis, the Y* value increases from 13.060° to 20.776°, which indicates that the farther the centroid position is from the torsion shaft, the more severe the pitch motion is. The farther the centroid position is from the torsion axis, the H* value increases from 0.75 mm to 1.48 mm, which indicates that the farther the centroid position is from the torsion shaft, the more intense the plunge motion is. At the same time, the difference between the Y* value and H* value ΔY-H value is specified to characterize the intensity of pitch relative plunge motion. It can be obtained that the farther the centroid position is from the torsion shaft, the higher the ΔY-H value is from 12.33 to 19.29, which indicates that the farther the centroid position is from the torsion shaft, the more intense the pitch relative plunge motion is, and the greater the influence of pitch motion on the flutter phenomenon is. Because the torsion center (E) and the aerodynamic center (A) are fixed, the elastic force and the aerodynamic force acting point remain unchanged. Changing the position of the center of mass will change the inertial force acting point (G). The farther away from the torsion axis, the greater the inertial moment, the greater the moment around the torsion center, and the stronger the pitch motion. The displacement in the plunging direction is composed of two parts: one is the displacement with the plunging motion of the torsion center, and the other is the displacement in the plunging direction caused by the rotation around the torsion center. Therefore, the increase in the displacement of the pitching motion will lead to an increase in the displacement of the plunging motion. The mechanism is shown in Fig. 24. In the two-degree-of-freedom motion, the pitch motion is dominant. Hence, the farther the centroid position is from the torsion axis, the greater the influence of the pitch relative to the plunge motion on the system.Fig. 23 The relationship between the change of centroid position and Y* value, H* value and ΔY-H value.

Fig. 23

Fig. 24 Schematic diagram of flutter mechanism.

Fig. 24

As shown in Fig. 25, the correlation between ΔY-H value and centroid position is analyzed, and the correlation coefficient is defined as the ratio of ΔY-H value change to centroid position coordinate change. It is found that the correlation coefficient of the ΔY-H value of the centroid position moving to the leading edge is 64.2 %, and the correlation coefficient of the ΔY-H value moving to the trailing edge is 65.8 %. The correlation is roughly the same, indicating that the proportion of the influence of the center of mass moving back and forth and the pitch relative plunge motion on the system is unchanged.Fig. 25 Correlation calculation of ΔY-H value.

Fig. 25

5 Conclusion

In this paper, a two-degree-of-freedom test bench installed in the closed section of the wind tunnel is designed and built based on the flutter characteristics of the wing section. The wing segment made of NACA64-618 airfoil is taken as the research object. By adjusting the change of the centroid position of the mass block at different positions on both sides of the wing section, and using the time domain and frequency domain analysis methods of dynamic signals, the dynamic response characteristics of the wing section at different centroid positions are studied respectively, to analyze the flutter boundary of the wing segment. The conclusions are as follows:(1) The wing segment with three different centroid positions is the research object. It is found that the closer the centroid position of the wing segment is to the leading edge, the larger the flutter critical wind speed is, the less likely it is to induce flutter. Under the linear condition, the forward movement of the centroid position has a more significant influence on the occurrence of flutter. Therefore, when designing the blade, the centroid position should be as close as possible to the leading edge.

(2) Two-degree-of-freedom frequency coupling is a sign of flutter occurrence, and the coupling frequency is kept near the pitch natural frequency. In the mutual coupling of aeroelastic natural frequencies, there is a sudden change in the natural frequency of the plunge aeroelastic. The closer the centroid position is to the torsion axis, the closer the coupling frequency is to the pitch natural frequency. Therefore, the coupling frequency can be used as a warning value in the study of anti-tremor.

(3) The limit cycle vibration is mainly pitch motion. When the centroid position moves to the leading edge, the motion in the pitching direction is more intense than that in the rising direction at the flutter moment. The farther the centroid position is from the torsion shaft, the more intense the pitching and plunging motions are, and the influence of pitching motion on flutter is greater than that of plunging motion. Therefore, the study of flutter suppression should focus on pitch motion.

Funding

This work was supported by No.2024MS05063 is a 10.13039/501100004763 natural science fund project of Inner Mongolia Autonomous Region , which is from Inner Mongolia Autonomous Region of China. No.51866012 and 52966014 are 10.13039/501100001809 National Natural Science Foundation of China ( NSFC ) projects from China. Number : RH2300003313 is a horizontal project of enterprise cooperation, sourced from China.

Data availability

Data included in article/supp. Material/referenced in article.

Ethic statement

Not applicable.

CRediT authorship contribution statement

Qi Zheng: Writing – original draft, Methodology, Data curation. Zhiying Gao: Writing – review & editing, Investigation, Data curation. Baozhong Zhao: Writing – review & editing, Data curation. Yefei Bai: Funding acquisition. Rina Su: Supervision, Formal analysis. Xiaoliang Han: Supervision. Feng Zhao: Formal analysis. Xueqing Dong: Conceptualization. Jianwen Wang: Formal analysis, Conceptualization.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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