Jian-xing ZHOU, Dong-chang DAI, Jian-jie ZHANG

(School of Mechanical Engineering, Xinjiang University, Urumqi 830047, China)

Abstract: In the present paper, an analytic model of the planetary gear transmission for the 1.5MW wind turbine generator is established by using lump mass method, in which the linear spring is adopted to simulate the gear tooth meshing under the effect of external load. After that the differential equation of motion and the calculation method of dynamic load coefficient for each gear pair of planetary gear are derived. Based on the above conditions, the paper analyses the influences of various external load on the system. The following conclusion are drawn that the meshing frequency is to be enlargement on the increase of stable wind load, meanwhile, the energy of meshing frequency component is highest; As a result, the primary influence factor of whole system changes from the stiffness excitation to the load excitation; In addition the dynamic load component becomes various, with the low frequency band zone and side band frequency arising. Regarding extreme gust with changeable intensity, the external load undergoes instantaneous change, which leading to a surge of the dynamic load and its coefficient; Regarding extreme gust with changing direction, the meshing force between X-direction and Y-direction appears phase angle, besides, external meshing force along three planetary gears exists phase difference, resulting in that the growth ratio of dynamic load of system is to be once more than gust load and load carrying between gear teeth becomes larger.

Key words: Wind turbine, Planetary gear transmission, Meshing frequency, Dynamic load

As a new type of regenerative clean energy, wind energy has been extensively utilized. However, since it has been used in abominable operational environment with changeable load from time to time, characterized by aerial erection and difficult maintenance, which necessitates strict requirements for dynamic performances of gear transmission system. Therefore, the research of dynamic characteristic of gear in operating condition featuring of random variation has become one of the important tasks in design of wind driven generator.

Many researchers have made an in-depth research into this respect. Bartelmus conducted testing and analysis of planetary gear vibration in operation conditions of changeable loads, then real time data obtained and dynamic response of system analyzed[1]; Zhou Zhigang set up random wind load by(WLS-SVM)vector machine, and the analysis of dynamic reliability and fatigue life prediction of gear transmission system are carried out based on the WLS[2]; A. Heege built multi-body dynamic model by using the FEM, then successively completed calculation of pneumatic load of wind turbine generator and the variation of gear box in emergency braking[3]; Qin Datong studied the external excitation under the condition of changeable wind velocity, according to this, the dynamic differential equations of the system are solved by using mode superposition method, the vibration response of the system are obtained[4].

The paper takes external load of wind turbine as excitation to establish dynamic analytical model for planetary gear transmission system. Based on the model, dynamic characteristics of system under the condition of stable wind and two types of extreme gusts respectively are researched, and the changeable pattern of the system dynamic load in various operating conditions are concluded as well.

1 Building of analytical model

1.1 Analysis of planetary gear transmission model and load bearing

When wind turbine generator is operating, the external wind load is driving fan blades for movement, and then the load is transmitted to hub to drive planetary gear transmission system rotating through principal axis, finally the power is outputted to the generator. The paper presents planetary gear transmission system for 1.5 MW wind turbine generator as an object for analysis, as shown in Fig.1(a), in which, sun gear is shown asS, planetary gear is indicated asP, planetary carrier is represented byC, ring gear is denoted byR,Vis velocity of wind, planetary carrier is the input end; ring gear is fixed; sun gear is output end.

Fig.1(b) shows three-dimensional assembly diagram of the planetary gear system, its basic parameters are as shown in table1, in which moment of inertia and mass are obtained from UG NX based on solid modeling.

Fig.1 Force analysis of wind turbine planetary gear transmission and three-dimensional model

1.2 Dynamic model for planetary gear system

The dynamic model of the planetary gear transmission system is as shown in Fig.2. The origin of coordinatesois center of planetary carrier, the horizontal direction is defined asX-direction and vertical direction is defined asY-direction. The support stiffness and torsional stiffness of the sun gear are indicated byksandksθ; the support stiffness of planetary gears are shown bykpi(i=1,2,3); the meshing stiffness between the sun gear and the planetary gear as well as planetary gear and annular gear are represented bykspiandkrpirespectively; the ring gear constraint on four symmetric positions, and the support stiffness of ring gear is definedkr.

Table 1 Parameters of planetary gear transmission system

ParameterSun gearPlanetary gearAnnulargearNumber of teeth343196Modulus/mm666Pressure angle/(°)282828Face width/mm424242Mass/kg2.72.55.3 Moment of inertia/(kg·m2)0.0080.0070.178

Fig.2 The translational-rotational dynamic model of planetary gears train

The system contains transverse and torsional micro-displacement of all components, in generalized coordinate. Then transverse micro displacement is expressed byxandy, torsional micro displacement is indicated byθ. Suppose that gear has degree of freedom along transverse and torsional directions, the generalized displacement vector of the system:

{X}={XsYsθs,Xp1Yp1θp1, …，XrYrθr}T

The dynamic model for the sun gear and the planetary gear is as shown in Fig.3, it is advisable to adopt linear spring to simulate meshing relation of the gear pair. The micro displacement of every gear convert to meshing spring, and then define that meshing spring is positive for compression and negative for tension, so:

δspi=xscosφspi+yssinφspi+us-xpisinα-

ypicosα-upi-espi(t)

φspi=φc-α+φpi+π/2

us=θsrs;upi=θpirpi;φc=ωct

(1)

Whereαis meshing angle,φis phase angle(φpi=2π(i-1)/3);rsis base circle of sun gear,rpiis base circle of planetary gears;espi(t) is meshing error.

Fig.3 Dynamic model of gear pair meshing force

Now, the meshing force of gear is expressed as:

(2)

According to the stress equilibrium relation among components and Newtonian mechanics, the differential equation for gear movement can be obtained as follows:

(3)

Eq.3 is written as matrix(4), with wind load introduced to the model in the form of external excitation.

(4)

WhereMis mass matrix;Cis damping matrix;Kis rigidity matrix;Xis displacement vector;Pis exciting load column vector.

The load of meshing line of gear is composed of two parts, namely static load and dynamic load, fourth order Runge- Kutta method is adopted to solve above mentioned differential equation.

2 Response analysis of system in stable wind velocity

2.1 Influences of wind load on dynamic load of system

The excitation which acts on gear transmission system includes external excitation and internal excitation in a broad meaning. On the premise of only taking the variation of internal meshing rigidity into consideration, when external wind velocity is lower, its acting force on hub load of wind turbine shall be smaller, the rigidity excitation has greater influence on system, as shown in Fig.4.

Fig.4 Dynamic response of gears system under light load

The meshing force is shown in Fig.4. When the load is smaller, the deformation of gear tooth along meshing direction is imperceptible, the comprehensive stiffness of tooth mesh is to play a leading role in system. Since gear meshing status is changed between single tooth and double teeth, the significant step-wise abrupt variation is appeared in the gear mesh time-varying stiffness curve. In order to describe it more clearly in the figure, the variation in dynamic load for the first 10 s is taken. As the figure shown, the fluctuation in mesh cycle of 1.36 s is indicated in the whole mesh cycle, which contains double teeth meshing zone and single tooth meshing zone. Under this conditions, the main frequency component of system is simple, in which the mesh frequency playing a leading role. When stable wind velocity gets higher, its driving action on wind turbine hub is on the increase.

With external load gets higher, dynamic load almost doubles which make gear system under greater stress. As is shown from the frequency spectrum diagram 5(c), system response includes, in addition to mesh frequency, obvious increase in amplitude of SHG and low frequency band. Besides, the sideband frequency shows at mesh frequency and frequency multiplication. External load has a significant exciting action.

Fig.5 Dynamic response of gears system under heavy load

2.2 Influences of continuously changeable wind velocity on system

The wind velocity in external environment is changeable randomly at any moment. When wind velocity is continuous increasing, its acting force on wind turbine hub is also on the increase. When the velocity is higher than rated wind velocity (11.5 m/s), the system is to automatically adjust blade direction to prevent wind turbine structure from being damaged. At this moment, the variation in hub load is getting stable, as shown in Fig.6.

Fig.6 Hub load fluctuation under different wind speeds

The main exciting components of planetary gear transmission system are the mesh frequency and frequency multiplication components. The intensity of system vibration is under the immediate influences of all exciting components. In order to research what type of exciting component has influences on system vibration, the paper presents calculation of the dynamic responses of planetary gear transmission system due to external excitation generated from various wind velocities.

The spectral cascades of the meshing force between the sun and planetary wheels is shown in Fig.7. The mesh frequency isfm(fm=(ns-nc)zs/60). As the wind velocity continually grows, mesh frequency is also increase linearly. When external steady wind is up to 11.5 m/s as rated wind velocity that can be borne by system, mesh frequency is to maintain unchanged, being 36.06 Hz. In case that external load is smaller, the main frequency components is simple and composed of nothing but mesh frequency. With the increase of load excitation, energy at doubling and tripling of mesh frequency are to be on the increase. Meanwhile the components on the low frequency band zone is appeared, the sideband frequency components at mesh frequency and frequency multiplication are also arisen, but the mesh frequency is still the primary component of system. It can be seen that on the premise of increase in external load, the system response is getting complex, but the role of rigidity excitation is decreasing, meanwhile frequency component in system is diversifying.

Fig.7 Spectral cascades of the meshing force between the sun and planetary wheels

3 Responses of system in extreme condition

3.1 Extreme gust model (variation in wind intensity)

Gust is the fluky wind on velocity and intensity in fleeting time, the gust is result of air disturbance. The extreme operating gust is to rise and fall twice intermittently, finally end to initial value (as shown in Fig.8). The amplitude and duration of gust are to change with the change of repetition period.

Fig.8 Variation diagram of wind speed

As Fig.9 shown Time history of wheel hub load wind velocity is 11.5 m/s.

It is observed variation in wind intensity is happened on windward side of wind turbine generator during 15-30 s, and greater fluctuation in hub load is appeared in the interval. As is shown from section h in the figure, when the wind intensity decreases suddenly, the hub load decreases to the lowest in a twinkling; Then hub load increase by 39.1% in a second exerts greater influence on planetary gear transmission system due to the wind intensity is on the increase by a flash.

Fig.9 Time history of wheel hub load

Since each planetary gears in the system are endure both the internal meshing force between the planetary gear and the ring gear and the external meshing force between the sun gear and the planetary gear. Fig.10(a) and (b) show external meshing force of planetary gear along the directionxandy, it is observed that abrupt variation in dynamic load has greater influences on vibration along directionx; Fig.10(c) shows circumferential external meshing force of planetary gear. It is also observed from the figure that variation in dynamic coefficient (the figure shows four steps, namely a, b, c and d steps), as shown in following figure.11, with curve smoothed to show that dynamic coefficient is to increase with the increase of dynamic load.

3.2 Model for continuous gust with changeable direction

Extreme wind direction variation means that wind direction is subject to cosine curve shaped variation in succession, with its amplitude and duration dependent on variation in repetition period. As is shown from Fig.12, at 18-30 s time, the included angle between wind direction and windward side of wind turbine is up to 40° in a second, and the time at the highest is 24 s; in addition, the included angle is 10° at the other times.

Fig.10 Dynamic load curve excited by load

Fig.11 Curve of dynamic coefficient

The load on hub is also to change with the change thereof due to influence of variation in wind direction, with abrupt increase by 11%. When external wind condition gets stable, its impact on wind turbine generator is on the decrease to stable state.

The mesh force curve along directionxandyis shown in Fig.14. It is observed that external vibration alongX-direction andY-direction are significant, with amplitude fluctuating from -2 000 to 2 000 N in a periodic. The difference is that there is phase angle between the two. It is observed from that the changeable pattern in external meshing force along directionxand directionyare similar.

Fig.12 Variation diagram of wind direction

Fig.13 Time domain process of wheel hub load

It is observed that dynamic load of system underwent stepped abrupt change in case of abrupt change in external load (as shown section o in the Fig.14),which results in greater impact in system and larger fluctuation in dynamic load of system in other periods with stable external load (section m and n shown in Fig.14). It is observed from Fig.15 that dynamic coefficient doubles compared with external load, indicating that in case that external wind load gets involved, gear system of wind turbine generator is to undergo intense vibration, with greater impact. The results give damage to system.

Fig.14 Dynamic load curve excited by load

Fig.15 Curve of hub load and dynamic coefficient

4 Conclusions

The paper presents dynamic simulation model for meshing force of planetary transmission system for 1.5MW wind turbine. According to analysis of the influence of steady wind and extreme wind on planetary gear transmission, some conclusions are obtained.

(1) In case of steady wind, the primary excitation for gear system is from rigidity excitation to load excitation. All harmonic components of dynamic load are shown in radial distribution, meanwhile, harmonic components bear greater energy at mesh frequency, with low frequency band and sideband frequency appearing.

(2) Regarding extreme gust with changeable intensity, as hub load increases by 39.1%, its corresponding dynamic load is to increase by almost 60%, and the related dynamic load coefficient maximizes. All gears are in contact with each other all along, and the load bearing in operating process is changing.

(3) Regarding extreme gust with changing direction, dynamic load of system undergoes significant change in amplitude along X-direction and Y-direction, but with phase difference up to 90°; three pairs’ external meshing force are identical, and phase difference is 60°; under the action of external load, dynamic load doubles with the intense impact.

Acknowledgements

This paper is supported by National Natural Science Foundation of China(No.51665054).