Modal testing was used to show that the roundabout swing was a natural vibration mode of the wind wheel of a horizontalaxis wind turbine (HAWT). During the vibration, the blade root was simultaneously subjected to bending and rotary shear stresses. A method for indirect testing and determination of the dynamic frequencies of the typical vibrations of the wind wheel was developed, based on the frequencyholding characteristic of each subsignal during the transmission of the multiple mixedvibration signals. The developed method enabled simple and accurate acquisition of the dynamic frequencies without destruction of the flow and structural fields. The dynamic vibration stress of the roundabout swing was found to be significantly stronger than those of the first and secondorder flexural vibrations of the blades. By a combination of numerical simulations and tests, it was determined that the pneumatic circumferential force was the primary determinant of the roundabout swing vibration frequencies, the relationship being quadratic. The roundabout swing vibration potentially offers new explanations and analytical pathways regarding the behavior of horizontalaxis wind turbines, which have been found to be frequently involved in fatiguedamage accidents within periods shorter than their design lives.
The wind wheel is a key power component and the primary excitation source of a wind turbine. Its natural vibration characteristics are thus considered to be the basic considerations in the safe structural design of a wind turbine [
Researchers have tended to focus on the first and secondorder vibration modes of the wind wheel and the accurate determination of the vibration parameters [
Traditional idea of inertia obtained from previous studies: researchers have held the idea that the bending vibrations caused by the aerodynamic force were the most important aspect of the vibration characteristics of a wind wheel [
Technical challenges of testing: wind wheels are rotating machineries, and those used in HAWTs particularly often have higher design speeds for higher power coefficients. This makes it difficult to install sensors directly on the blade surface [
Numerical simulation challenges: the numerical determination of the dynamic vibration characteristics of a wind wheel is a typical fluidstructure interaction problem, and studies along this line have been significantly restricted by the hysteretic development of fluidstructure interaction theories and inadequate computational methods [
In view of the foregoing, the present study employed modal testing of the lowfrequency vibration characteristics of 1.4 m diameter wind wheel to obtain valuable information. The aim was to develop an indirect testing method for determining the dynamic frequencies of a wind wheel through a combination of spectral analysis and modal testing, with the purpose of determining the vibration stresses in various vibration modes using the classical Newtonian mechanics formula. This is expected to enable analysis of the different vibration stresses in the static and dynamic states, as well as their significance. We also used a combination of numerical calculations and tests to analyze the response characteristics of the lowfrequency vibration of the wind wheel with respect to the aerodynamic and centrifugal forces.
Three different types of blades were considered for the investigated 1.4 m diameter wind wheel. All the blades were made from wood and their roots were connected by the flange method using double plywood. Figure
Three considered blade types.
Blade number 1 had an innovative airfoil recently developed by the authors. Blade number 2 had the same airfoil but a different connection mode, which was used to examine whether the roundabout swing was due to the connection mode. Blade number 3 had NACA4415 airfoil and was larger than blade number 1. It was used to examine whether the roundabout swing was due to the airfoil.
The test system was PULSE 19.1 system developed by the Danish B&K Company. The test was conducted on a dedicated test bench installed in front of the opening of B1/K2 lowspeed wind tunnel. The test method involved transient excitation, onepoint driving, and multipoint response. The test system and setup are shown in Figure
Modal testing.
PULSE 19.1 system
Flange connection
Double plywood
The test data was processed by the ME′ scope 5.1 software. The distribution of the measurement points on the wind wheel is shown in Figure
Distribution of measurement points on wind wheel.
As an example, the acquired natural frequencies of blade number 1 are shown in Figure
Vibration frequencies of blade number 1.
The typical secondorder following vibration modes and frequencies of the three wind wheel blades are presented in Table
Typical secondorder following vibration frequencies.
Blade number 







1  9.95  17.10  23.40  24.90  72.70  77.20 
2  9.73  17.90  19.80  23.30  87.70  90.10 
3  8.00  15.00  21.60  24.50  81.80  87.90 
The following can be deduced from the test results in Figure
The roundabout swing is a natural vibration mode of a wind wheel. Considering the structural distinctions of the three blades, the roundabout swing is not caused by the blade connection method or the airfoil structure.
The quality of the wind wheel was stable during the test. Based on the classical formula
Vibration mode of roundabout swing.
Considering blade number 1 as an example, the roundabout swing vibration mode is shown in Figure
The vibration process of the roundabout swing is illustrated by the nine contiguous dynamic screenshots in Figure
The vibration frequency of the roundabout swing is lower than that of the firstorder bending vibration. This is because the overall effectiveness of the wind turbine is significantly higher than that of a single blade. The vibration causes the roots of the blades to bear the stress, which comprises curved shear stress and rotary shear stress. This is because a single blade is similar to a cantilever beam with significant first and secondorder bending vibration performances. The entire wind wheel is thus characterized by whirling vibration owing to the three blades forming a circular frame. Considering that the roundabout swing is easily triggered over time, it hastens fatigue damage of the blades, thereby shortening their service life.
During the rotation of a wind wheel, forced vibration is induced by aerodynamic, centrifugal, and gravitational forces. This external excitation increases the deformation and stiffness of the blade, resulting in the typical vibration frequencies of the wind wheel exceeding the natural vibration frequencies. Although the external excitation is different, it may affect the vibration frequency, amplitude, and phase change, but not the vibration mode. The natural modes and frequencies of the wind wheel can thus be initially determined by modal testing, based on which the dynamic vibration frequency can be obtained by spectral analysis.
The test object and system were the same as those of the modal test. Considering that the installation of the acceleration sensors on the surfaces of the blades would significantly alter their shapes and distort the aerodynamic force and mass distribution, an indirect test and analysis method for determining the typical dynamic vibration frequencies of the wind wheel was developed. The developed method is based on the frequencyholding characteristic of each subsignal during the transmission of multiple mixedvibration signals. Specifically, the acceleration signal from the accelerometer installed at the front end of the generator is received, and its spectral signature is then obtained by FFT. Finally, the dynamic frequencies are distinguished based on their mode shapes and spectral signatures.
Figures
Dynamic frequency test system.
Test setup.
Test bench
Data acquisition
Arrangement of sensors
The vibration signals were received by four threedimensional acceleration sensors. As shown in Figure
Using blade number 1 as an example, we analyzed the axial vibration signals collected by the four sensors at a wind velocity of 10 m/s and tip speed ratio of 6. The vibration spectrums are shown in Figure
Dynamic frequency spectrums.
Sensor number 1
Sensor number 2
Sensor number 3
Sensor number 4
An examination of Figure
In the case of sensor number 3, which was farther from the wind wheel, the peak vibration spectrum gradually blurred with the reduction progressively increasing. This was due to the significant effect of the vibration of the generator excitation coil. The signals of sensor number 4 were even more significantly affected by the vibrations of the generator excitation coil and the tower.
The foregoing indicates that the dynamic frequency of the typical vibration mode of the wind wheel can be more ideally determined using the dynamic frequency curve of sensor number 1. The determination of the dynamic frequency of the wind wheel is illustrated in Figure
Determination of dynamic frequency of wind wheel.
Modal test spectrum
Dynamic test spectrum
From Figure
The roundabout swing is primarily of concern under the operation conditions because its vibration stress is much higher than those of the other vibrations. The dynamic vibration frequencies of the roundabout swing vary with the operation conditions, especially the aerodynamic and centrifugal forces, and require intensive study because they are primary considerations in the structural safety design of a wind turbine.
The following analysis was performed by numerical simulation considering its convenience, requiring only the loading of the aerodynamic and centrifugal forces.
Steady computing using the shear stress transport (SST) equation was employed, taking into consideration the effect of the turbulent shear stress. The import boundary was based on the inlet velocity, while the outlet boundary utilized pressure export, with a relative pressure of 0. The energy, momentum, continuity, and SST equations were used to obtain the coupling solution. The calculation was performed using the unidirectional fluidstructure interaction computation method. The precalculated flow field data for different operation conditions were imputed, and the centrifugal force was then added to solve the structural dynamics equations and obtain the dynamic vibration parameters of the wind wheel, such as the vibration modes and vibration frequencies.
Using blade number 1 as an example, the overall computational domain was generated to include the B1/K2 experimental wind tunnel, as shown in Figures
Wind turbine.
Computational domain.
The computation domain was divided into two parts, namely, the rotation field and the static domain. The rotation field enclosed the blades, the rotation of which was simulated by rotating the field. The rotation region of the wind wheel was generated by a sliding mesh, with a noslip condition at the stationary wall. The rotation domain was generated by a hexahedral mesh, with the static domain comprising a tetrahedral mesh and a sliding grid adopted between the two domains. The data transfer was done by interface technology.
To better capture the flow field information, a bodyfitted grid was applied in the vicinity of the blade surface. The grid expansion technique was implemented in the static domain to partition the mesh layer. The full application of different available grid partitioning technologies generally enables efficient utilization of computing resources.
To properly assess the effectiveness of the mesh generation technique, the grid scale in Figure
Meshing.
Using blade number 1 as an example, the secondorder following vibration modes are shown in Figure
Dynamic frequencies of typical vibrations.
Category 







Experimental value (Hz)  10  17.5  38  40.5  109  115.5 
Calculated value (Hz)  9.8  17.3  36.7  42.2  107.4  119.7 
Fractional error (%)  −2.0  −1.1  −3.4  4.2  −1.5  3.6 
Determined vibration modes: (a) axial movement, (b) roundabout swing, (c) firstorder antisymmetric vibration, (d) firstorder symmetric vibration, (e) secondorder antisymmetric vibration, and (f) secondorder symmetric vibration.
As shown in Figure
To analyze the effects of the aerodynamic and centrifugal forces on the vibrating frequencies of the roundabout swing, the two following example numerical calculations were performed:
The tip speed of the wind wheel was 40 m/s; the wind speed was varied within 5–10 m/s. Two cases involving the application of only the aerodynamic force and the simultaneous application of the aerodynamic and centrifugal forces were, respectively, considered. The results are shown in Figure
Variations of vibration frequencies of roundabout swing with working conditions defined by given tip speed and varying wind velocity.
The wind velocity was 8 m/s, and the tip speed ratio of the wind wheel was varied within 4–8. The two cases of example 1 were also considered here. The results are shown in Figure
Variations of vibration frequencies of roundabout swing with working conditions defined by given wind velocity and varying tip speed ratio.
The vibration frequencies of the roundabout swing for the different wind speeds were expressed as
The vibration frequencies of the roundabout swing for the different tip speed ratios were expressed as
It can be determined from Figures
For an indepth analysis of the effect mechanism of the aerodynamic force on the vibration frequencies of the roundabout swing, an aerodynamic force that acted vertically on the surfaces of the blades was decomposed into an axial force in the axial direction of the wind wheel and a circumferential force in the circumferential direction of the wheel. The variations of the axial and circumferential forces with the working conditions are shown in Figures
Variations of axial and circumferential forces with working conditions.
Variations of axial and circumferential forces with working conditions.
An increase or decrease in the vibration frequencies of a wind wheel is due to a stressinduced increase or decrease in the blade stiffness. Based on this theory, the variations of the pneumatic axial forces in Figures
To verify the above conclusion, the calculation data for the synchronous application of the aerodynamic and centrifugal forces was tested by comparison with experimental results. The relative error was determined to be less than 6%.
The roundabout swing is a natural vibration form of a wind wheel, with its vibration frequency less than the firstorder bending vibration frequency and the vibration stress often significantly higher than the other present vibration stresses. This causes the blade roots to bear the compound stress, which comprises bending shear stress and rotary shear stress.
The proposed indirect test method provides a solution to the technical difficulties associated with monitoring and distinguishing the dynamic frequencies of the typical vibrations of a wind wheel under operation conditions. The various vibration stresses in the static and dynamic tests of the present study were found to be characterized by
The pneumatic circumferential force is the primary determinant of the vibration frequencies of the roundabout swing, the relationship being quadratic. The effect of the centrifugal force on the vibration frequencies is relatively weak.
The authors declare that they have no competing interests.
This study was supported by the National Natural Science Foundation of China (Grant no. 51466012), the Inner Mongolia Natural Science Foundation of China (Grant no. 2016MS0509), and the Open Foundation of Key Laboratory of Wind Energy and Solar Energy Technology of the Ministry of Education of China (Grant no. 201408).