Wind turbine drive train dynamic characterization using vibration and torque signals

Mechanism and Machine Theory 98 (2016) 2–20

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Wind turbine drive train dynamic characterization using vibration and torque signals P. Srikanth, A.S. Sekhar ⁎ Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai 600036, India Department of Mechanical Engineering, IIT-Madras, Chennai 600036, India

a r t i c l e

i n f o

Article history: Received 2 June 2015 Received in revised form 26 September 2015 Accepted 25 November 2015 Available online xxxx Keywords: Wind turbine Drive train Nonlinear non-stationary signal Turbulence Internal resistive torque Dynamic transmission error

a b s t r a c t Dynamic analysis of wind turbine drive train subjected to stochastic aerodynamic loads is carried out in the present study. The longitudinal wind speed at the turbine site normally consists of a mean value superimposed with ramp, gust and turbulence components. In the present study, the aerodynamic torque is obtained by considering wind speed parameters of a typical wind turbine site. The dynamic model accounts for time varying gear mesh stiffness, bearing elasticity and torsional shaft stiffness. The dynamic analysis is done with stochastic aerodynamic loads and the vibration responses are obtained in time and frequency domains. It is observed that the entire spectral content of the vibration signals is confined to low frequency region, whereas higher frequencies are hidden. In order to capture the hidden frequency information from vibration signals, the wavelet decomposition technique is used. The dynamic analysis using torque signals is also discussed. The present study shows that, from the internal resistive torque all the characteristic frequencies can be clearly observed. © 2015 Elsevier Ltd. All rights reserved.

1. Introduction The installation capacity of wind turbines is increasing significantly year by year. A wind turbine drive train is a typical case of rotating equipment operating at slow speeds and subjected to severe stochastic loads. The wind turbine components are always affected by uncertain loads. Such kind of load spectrum always leads to vibration signals which are nonlinear and non-stationary in nature [1]. The dynamic analysis and condition monitoring of most of the rotating machinery focus on rotating components that are subjected to constant, periodic or transient load spectra. These are relatively easier for modeling and analysis, compared to the wind turbine drive trains. Wind turbines are subjected to wind loads which are stochastic in nature. Due to these loads, the chances of failures are higher for critical components, such as gearbox. The downtime and cost associated with the gearbox is more, as handling of this component is difficult. Hence, detailed predictive condition monitoring strategies have to be developed for such components. A huge step-up speed in the range of 40:1 to 135:1 can be achieved from rotor to generator in wind turbine using epicyclic gearbox [2]. Qin et al. [3] carried out wind turbine gearbox dynamic analysis with flexible multi-body modeling technique, taking into account the elastic strain energy due to gears and bearings. Abboudi and Walha [4] carried out dynamic analysis of a two stage external gearbox of wind turbine using an empirical approach based on aerodynamic torque. Guo et al. [5] modeled gravity effect on the vibration response of wind turbine planetary gears and it is concluded that the gravity plays a vital role when compared to gear tooth meshing excitation alone. These results compared well with those of mathematical and experimental models. Rigid multibody modeling with discrete flexibility approach is used by Todorov et al. [6] to assess the torsional dynamic behavior of wind turbine drive train. The dynamic behavior of wind turbine gearbox is evaluated using ⁎ Corresponding author at: Department of Mechanical Engineering, IIT-Madras, Chennai 600036, India. Tel.: +91 44 2257 4709 (O); fax: +91 44 2257 4652. E-mail addresses: [email protected] (P. Srikanth), [email protected] (A.S. Sekhar).

http://dx.doi.org/10.1016/j.mechmachtheory.2015.11.013 0094-114X/© 2015 Elsevier Ltd. All rights reserved.

P. Srikanth, A.S. Sekhar / Mechanism and Machine Theory 98 (2016) 2–20

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three techniques namely pure torsional, rigid multi-body modeling and flexible multi-body modeling [7,8]. Here, finite element approach and test rig experiments are used. By taking variable gear mesh stiffness into account, Shi et al. [9] studied the torsional dynamic behavior of the entire wind turbine drive train subjected to constant aerodynamic torque excitation using Lagrange's approach. Notations ramp and gust amplitudes Ar, Ag C1, C2, C3 low, intermediate and high speed shaft torsional stiffness. Cp(λ, β) power coefficient H1(s), H2(s) spatial and rotational sampling filter I turbulence intensity factor of wind Ji (i = 1 … 8) inertias of rotor, carrier, planets, sun, gear1, gear2, gear3, and generator kri, ksi, kpg time varying mesh stiffness of ring–planet, planet–sun and gear–pinion. Kbj (j = 1 … 6) stiffness values of bearings length scales of the Kaimal and von Karman spectra L1, L2 [M], [C(t)], [K(t)] mass, damping, and stiffness matrices R radius of rotor S(f) auto spectral density function initial and stop times Ti, Ts TR(t), TG(t) rotor and generator torque u(t), U, ur(t), ug(t), ut(t) total, average, ramp, gust and turbulence components of wind speed yi, zi, ys, zs, y1, z1, y2, z2, y3, z3 translational degrees of freedom in y & z directions of planets, sun, gear1, gear2, and gear3. σ wind speed standard deviation γ decay factor λ tip speed ratio ϕ, θ, β incident angle, angle of attack, and pitch angle rotor speed and rotor damping factor ωr, ωd ωM ðpÞ , ωM ðg12Þ , ωM ðg34Þ gear mesh frequency of ring–planet and planet–sun, gear1–2, gear2–3 δri, δsi, δpg dynamic transmission errors of ring–planet, planet–sun, gear–pinion ϕi (i = 1 … 8) Absolute rotational angles of rotor, carrier, planets, sun, gear1, gear2, and gear3, and generator absolute rotational angles. Coupled torsional bending dynamic analysis is carried out for fixed speed wind turbine using Lagrange's approach in [10]. Here, constant gear mesh stiffness, support bearing elasticity and strain energy associated with shafts are used in the formulation. Aerodynamic torque is modeled as a periodic signal based on the empirical relation. Harmonic balance method is used by Ji et al. [11] to obtain the dynamic responses of the wind turbine gearbox. Techniques like finite element analysis and experimental modal analysis are adopted by Haijun et al. [12] for estimating the natural frequencies and mode shapes. Lumped parameter dynamic model of the wind turbine gearbox is developed by Long Quan et al. [13] to estimate the vibration levels. Yongqian et al. [14] elaborated the effect of design parameters on the sensitivity of the natural frequency and dynamic characteristics with respect to gear mesh stiffness in epicyclic wind turbine gearbox. The details of different wind components existing at wind turbine sites are revealed in [15]. Different wind speed models (Weibull, extreme value distribution of type 1 & type 2 and Rayleigh) are proposed by Jang and Lee [16] for the Taiwan area. The random nature of wind is normally represented by the Weibull distribution model, four component composite model, auto regressive and moving average model and power spectral model [17]. Near the wind turbine site, the best approximation to represent wind is the Weibull distribution; this has been concluded based on wind data, over a period of 44 years [18]. There always exist deterministic and stochastic components in wind loads [19]. The main reason for gearbox failures is the wind stochastic load that always creates uneven stochastic aerodynamic torque on the rotor. These loads are distributed unevenly between the bearings and gears [20]. Zhu et al. [21] using the commercial SIMPACK software carried out the dynamic analysis of the wind turbine drive train by considering external excitations due to load spectrum and internal excitations due to the time varying mesh stiffness and transmission errors. These results are compared with the experimentally measured vibration acceleration signals. A simplified method is proposed to estimate the long-term extreme value of the gear transmitted load. This value is estimated based on the cumulative Weibull distribution for one hour mean wind speed [22]. The need to consider the effects of mesh stiffness and impact stress on the dynamic transmission error evaluation of the wind turbine planetary gear system is revealed by Zhao et al. [23]. Wei et al. [24] analyzed the effects of uncertainty in the gearbox system parameters like mesh stiffness, damping and transmission error on the dynamic response of the system. In literature, dynamic analysis of the wind turbine drive train is performed by subjecting the drive train to constant or periodic load components only. Srikanth and Sekhar [25] modeled the wind turbine drive train by including the coupled torsional bending dynamics. Here, the wind turbine is subjected to stochastic loads which are estimated based on the Danish standard DS 472. In addition, time varying mesh stiffness of gear, stiffness of shaft and bearing are considered in the dynamic formulation. Although, the standard model is used, the obtained signals from this method are not close to realistic signals on site. Hence, the authors further improved the wind modeling and carried out the drive train analysis in the present study.

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Unlike in the literature, in the present work, the wind loading acting on the turbine is modeled in detail where the simulated wind or aerodynamic torque signal almost represents the realistic one. Further the dynamic characterization of drive train with torque signals is discussed. To the best of the authors' knowledge, drive train characterization using torque signals is not done in literature. The methodology used in the present study for dynamic characterization of wind turbine drive train is discussed in the following subsection.

1.1. Methodology The wind speed at the hub level of rotor generally consists of average, ramp, gust and stochastic (turbulence) components. In order to capture the wind speed variation over the rotor swept area, the wind speed signal at the hub level has to be passed through aerodynamic filters. The signal at hub level first pass through spatial filter from which high frequency components present in the wind are eliminated; later the output of wind speed from spatial filter has to be passed through a rotational sampling filter. The phenomenon of rotational sampling in the case of wind turbines can be explained as follows. Typically, the wind at the wind turbine site consists of mean and turbulence components in it. The apparent wind speed that the blade is subjected to always depends upon the position of the blade. This phenomenon is called ‘rotational sampling’, because the turbulence spectrum of wind is amplified at multiples of rotor running frequency. The effect of this rotational sampling is very important for the wind turbine drive train dynamic analysis. The aerodynamic torque is obtained from the output wind signal of the rotational sampling filter. The coupled torsional bending model developed by the authors in [25] is considered in the present study. The dynamic analysis of this drive train model subjected to stochastic aerodynamic torque is carried out. The equations of motion are solved for vibration responses in both time and frequency domains. It is observed that the entire frequency spectrum is confined to the low frequency region only and all the frequency components are not seen clearly from the FFT. Hence, in order to extract the hidden high frequencies, the wavelet packets are used on the vibration signal from which all the characteristic frequencies are extracted. The methodology is summarized in the flow chart given in Fig. 1. Later torque signals are also used for drive train dynamic characterization. However, the external torque (output torque) follows a similar trend as the vibration signal, hence one can use wavelet packets to extract all the hidden information. The internal resistive torque signals are also used for the drive train dynamic characterization where all the characteristic frequencies are seen from the FFT itself. The procedure of estimating the gear tooth contact force and internal resistive torque by estimating the dynamic transmission error is discussed.

Turbulence longitudinal wind speed component generation based on PSD Obtain total wind speed by adding average, ramp, gust components to turbulence component Pass the total wind speed through spatial filter Pass the spatial filter output through rotational sampling filter

Obtain aerodynamic torque signal from rotational sampling filter output Perform dynamic analysis

Obtain Vibration Signals

Obtain Torque Signals

External torque

Use Wavelet packets to extract all characteristic frequencies

Internal resistive torque

All characteristic frequencies can be observed clearly from FFT itself.

Fig. 1. Flow chart for the drive train dynamic characterization.

P. Srikanth, A.S. Sekhar / Mechanism and Machine Theory 98 (2016) 2–20

5

0

10

Power spectral density

Von karman model

-1

10

-2

10

-2

-1

10

0

10

10

1

10

Frequency (Hz)

Fig. 2. Power spectral density of the von Karman model.

2. Aerodynamic torque modeling Wind speed variations can be represented as long term, annual and seasonal, synoptic and diurnal, turbulence components in wind resource at the wind turbine site. The long term variation in wind means the amount of variation in wind from one year to another or even over periods of decades. It is difficult to predict such kind of wind variations at the wind turbine site. It is hard to predict the year to year annual variations. However, during the year, wind speed variations can be represented well with a probability distribution. The hourly mean wind speed over the year follows the Weibull distribution at the wind turbine site. Representation of hourly mean wind speed by a probability distribution clearly indicates the random variation in it. Wind speed variations are more random and less predictable on a shorter time scale compared to seasonal variations. These variations have frequency content peaks at around 4 days. Weather fronts associated with high and low pressure areas are the cause for such variations. On the other hand, diurnal variations are associated with a frequency spectrum that has peak at a frequency of 24 h. This is due to the heating and cooling of air in a day. It is the best way to represent wind as a time series with a mean value superimposed by the turbulence fluctuations averaged over a 10 min period. The mean component is accounted for through seasonal, synoptic and diurnal effects [26]. The longitudinal wind acting at the wind turbine site always consists of basically four components, as given in the following equation, uðt Þ ¼ U þ ur ðt Þ þ ug ðt Þ þ ut ðt Þ;

ð1Þ

where u(t) = total longitudinal wind speed; U = mean wind speed; ur(t) = ramp component; ug(t) = gust component; and ut(t) = turbulence component. The ramp component is expressed in terms of initial time (Ti), stop time (Ts) and ramp amplitude (Ar) as given in [27]. The ramp component does not have much physical meaning except that it always tries to increase the wind speed mean value. 0

0 B t−T i B ur ðt Þ ¼ @ Ar T s −T i 0

t b Ti

1

C Ti b t b Ts C A:

ð2Þ

t NT s

Table 1 Wind speed parameters and blade data. Parameter

Value

Mean speed of wind (U) Turbulence intensity (I) Length Scale (L2) Ramp amplitude (Ar) Gust amplitude (Ag) Initial time (Ti) Stop time (Ts) Rotor damping factor (ωd) Rotor Radius (R) Pitch Angle (β)

6 m/s 0.10 64.5 m 4 m/s 7 m/s 2.5 s 7.5 s 0.8765 rad/s 28 m 2°

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10

Single-Sided Amplitude Spectrum

4

power spectral density

(a) 10

10

2

X: 0.02686 Y: 10.24

0

X: 0.05737 Y: 6.561

X: 0.1184 Y: 5.946 X: 0.2551 Y: 0.57

10

-2

X: 0.8997 Y: 0.03985

X: 1.798 Y: 0.02129

-4

10 -3 10

10

-2

10

-1

10

0

10

1

Frequency (Hz) Single-Sided Amplitude Spectrum

4

10

Power spectral density

Gain of nearly one at all freqencies

(b)

2

10

X: 0.02686 Y: 10.21

X: 0.05737 Y: 6.504

Amplification around rotor characteristic frequency and harmonics

X: 0.1184 Y: 5.619 X: 0.2551 Y: 0.5498

0

10

X: 0.8997 Y: 0.06585

X: 1.798 Y: 0.0254

-2

10

-4

10 -3 10

-2

-1

10

10 Frequency (Hz)

0

10

1

10

Fig. 3. Power spectral density (PSD) of wind signal (a) before filtering and (b) after filtering.

The gust component of the wind is a function of initial, stop time and the gust amplitude which is expressed in terms of a deterministic function as given in [27], 1 t b Ti  0  C B Ag t−T i ug ðt Þ ¼ B Ti b t b Ts C A: @ 2 1− cos 2π T −T s i 0 t N Ts 0

ð3Þ

However, the gust component is in general, stochastic in nature at the wind turbine site under severe loading conditions. In general, the turbulence component of wind is expressed in terms of power spectral density. The turbulence component of wind follows either the Kaimal or von Karman spectrum at the wind turbine site. The power spectral density associated with the longitudinal wind speed component which is represented in the Kaimal model in terms of frequency, length scale and average wind speed is given as [27], Sð f Þ 4L1 =U ¼ ; σ2 ð1 þ 6 f L1 =U Þ5=3

ð4Þ

Fig. 4. Cross-section of wind turbine blade.

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where S(f), L1, U, f and σ represent the auto spectral density function, length scale, average wind speed, frequency and standard deviation respectively. The length scale depends upon surface roughness and height above the ground. The longitudinal wind speed component standard deviation (σ) is constant with respect to height above the ground. However, this standard deviation depends upon the turbulence intensity (I) and average wind speed (U). The turbulence intensity, I is given by I¼

σ : U

ð5Þ

The wind turbulence can also be represented by the von Karman spectrum especially for complex terrain, whose power spectral density can be represented in the following form as given in [27], Sð f Þ 4L2 =U ¼ 5=6 : σ2 1 þ 70:8ð f L2 =U Þ2

ð6Þ

4

velocity (m/s)

Ramp component of wind 3

(a)

2

1

0 0

1

2

3

4

5 Time (S)

6

7

8

9

10

9

10

9

10

9

10

7

velocity (m/s)

6

Gust component of wind

5

(b)

4 3 2 1 0 0

1

2

3

4

5 Time (S)

6

7

8

Wind speed (m/s)

20

(c) 15

10

5

0 0

Aerodynamic Torque ( N-m)

2

x 10

1

2

3

4

5 Time (S)

6

7

8

6

(d)

1.5

1

0.5

0 0

1

2

3

4

5 Time (S)

6

7

8

Fig. 5. Components of wind and obtained aerodynamic torque (a) ramp component, (b) gust component, (c) total wind speed (after filtering) and (d) aerodynamic torque.

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Fig. 6. Schematic diagram of wind turbine drive train.

Table 2 Drive train data. J1 — inertia of the rotor (kg m−2) J2 — inertia of the carrier (kg m−2) mp — mass of the planet (kg) J3 — inertia of the planet (kg m−2) J4 — inertia of the sun (kg m−2) J5 — inertia of gear1 (kg m−2) J6 — inertia of the gear2 (kg m−2) J7 — inertia of the gear3 (kg m−2) J8 — generator inertia (kg m−2) C1 — low speed shaft stiffness C2 — intermediate shaft stiffness C3 — stiffness of the high speed shaft Kri — ring–planet gear mean mesh stiffness Ksi — sun–planet gear mean mesh stiffness Kg12 — gear1–2 mean mesh stiffness Kg23 — gear2–3 mean mesh stiffness K1 — stiffness of bearing1 K2 — stiffness of bearing2 K3 — stiffness of bearing3 K4 — stiffness of bearing4 K5 — stiffness of bearing5 K6 — stiffness of bearing6 Crp — contact ratio of ring–planet gear pair Csp — contact ratio of sun–planet gear pair Cg12 — gear1–2 contact ratio Cg23 — gear 2–3 contact ratio rc — radius of carrier (mm) r3 — radius of planet (mm) r4 — radius of sun (mm) r5 — radius of gear1 (mm) r61 — 2nd stage gear2 radius (mm) r62 — 3rd stage gear2 radius (mm) r7 — gear3 radius (mm) α — pressure angle Gear mesh frequency of ring–planet and planet–sun gears (Hz) Gear1–2 gear mesh frequency (Hz) Gear2–3 gear mesh frequency (Hz) Gear ratio Cr — root chord (m); Ct — tip chord (m) Ω = speed of rotor (rpm)

4.18 × 106 57.72 57.79 1.12 0.86 14.32 1.62 0.2 93.22 7.19 × 107 1.4 × 107 0.15 × 107 0.73 × 108 0.73 × 108 2.02 × 109 0.11 × 108 8.04 × 109 4.08 × 109 4.08 × 109 2.16 × 109 2.16 × 109 2.16 × 109 1.9342 1.6242 1.6616 1.5984 270 160 110 290 95 185 80 20 25.2 83.8 326.5 34.654 3.3; 0.9 18 (0.3 Hz)

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Table 3 Characteristic frequencies of wind turbine drive train. Characteristic frequency component

Frequency (Hz)

Gust frequency First torsional natural frequency Rotor characteristic frequency (three times the rotor rotational speed) Second natural frequency Gear mesh frequency of ring–planet and planet–sun gear Gear mesh frequency of gear1–2 Gear mesh frequency of gear 2–3.

0.2 0.4 0.9 4.7 25.2 83.8 326.5

With the Kaimal model, three dimensional spatial wind components can be well represented and matched with experimental data. However, the von Karman model fits better than the Kaimal model in the sense of power levels. Hence, the von Karman model is used in the present study. Fig. 2 represents the power spectral density of the von Karman model used for the generation of stochastic process.

100

Response (rad/s2)

(a) 50

0

-50

-100 0

1

2

3

4

5 Time (S)

6

7

8

9

10

300

(b)

Response (rad/s2)

200 100 0 -100 -200 -300 0

1

2

3

4

5 Time (S)

6

7

8

9

10

800

(C)

Response (rad/s2)

600 400 200 0 -200 -400 -600 0

1

2

3

4

5 Time (S)

6

7

8

9

10

800

(d)

Response (rad/s2)

600 400 200 0 -200 -400 -600 0

1

2

3

4

5 Time (S)

6

7

8

9

10

Fig. 7. Time domain representation of vibration acceleration response: (a) gear1, (b) gear2, (c) gear3 and (d) generator.

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The standard method of generating a stochastic process, the Shinozuka method [28] is used to obtain the wind stochastic time series from the turbulence spectrum of the power spectral density. In this method, at each point in the time series, the wind speed can be obtained by using weighed cosine series and a random phase angle, as given by ut ðt Þ ¼

XN qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f Sð f n ÞΔf cosð2π f n t þ ϕn þ ΔϕÞ; n¼1

ð7Þ

where ϕn is the uniformly distributed random phase angle, Δf is the incremental frequency, and fn is the frequency in an array which is formed by dividing the frequency range of power spectral density model into Nf number of incremental frequencies. The above approach takes into account the turbulence component of wind speed only; however, the ramp and gust components of wind are also taken into account in the present approach by adding average, gust and ramp components (see Single-Sided Amplitude Spectrum of Gear1 20

(a) Amplitude

15

10

5

0 0

50

100

150

200

250 300 350 Frequency (Hz) Single-Sided Amplitude Spectrum of Gear2

400

450

500

50

(b)

Amplitude

40 30 20 10 0 0

50

100

150

200

250 300 350 Frequency (Hz) Single-Sided Amplitude Spectrum of Gear3

400

450

500

140 120

X: 5.127 Y: 135

(c)

Amplitude

100 80 60 40 20 0 0

50

100

150

200

250 300 Frequency (Hz)

350

400

450

500

Single-Sided Amplitude Spectrum of Generator 140

(d)

120

Amplitude

100 80 60 40 20 0 0

50

100

150

200

250 300 Frequency (Hz)

350

400

450

500

Fig. 8. Frequency domain representation of vibration acceleration response: (a) gear1, (b) gear2, (c) gear3 and (d) generator.

P. Srikanth, A.S. Sekhar / Mechanism and Machine Theory 98 (2016) 2–20

11

Single-Sided Amplitude Spectrum of Gear3 140 120

Amplitude

100 80 60 40 20 0 0

1

2

3

4

5 Frequency (Hz)

6

7

8

9

10

Fig. 9. Vibration signal of gear3 at a frequency range of 0–10 Hz.

Eqs. (2) & (3)) to the wind signal obtained in Eq. (7). Hence, the longitudinal wind speed component (Eq. (1)) is the algebraic sum of the average, ramp, gust and turbulence components. The obtained wind signal based on the above approach represents closely, the realistic wind speed variation with time at the wind turbine site. However, the above wind signal represents wind speed time series at the hub level only. The longitudinal wind speed is not constant along the blade length and over the rotor swept area. Hence, the influence of wind speed variation over the swept area has to be accounted for in evaluating the mechanical aerodynamic torque. The procedure of accounting this is summarized below. The wind parameters selected for the aerodynamic torque simulation are given in Table 1. To account for the effect of wind speed variation over the rotor swept area, aerodynamic filters are used in the present work based on literature [27]. The obtained wind speed signal at the hub level is the input to these aerodynamic filters and the output signal is the wind signal that accounts for the effect of rotor swept area. The wind speed signal obtained at the hub level is passed first through the spatial filter by which the high frequency components present in the wind are dampened. Then the signal has to be passed through rotational sampling filter by which the effect of wind variation over the rotor swept area can be captured. The spatial filter is represented by the Laplace transfer function as given by, pffiffiffi 2 þ bs H1 ðsÞ ¼ pffiffiffi pffiffiffi pffiffiffi  ; 2 þ bs a 1 þ b= as

ð8Þ

where a = 0.55, b = γR/U, R is the rotor radius, U is the average speed of wind at the hub height and γ is the disc decay factor (in general γ = 1.3). The rotational sampling filter can be represented with the Laplace transfer function as given by, 2

H2 ðsÞ ¼

2

2

ωr þ ωd ðs þ ωr Þ ; ðs þ ωd þ jωr Þðs þ ωd −jωr Þ ωr 2

ð9Þ

where ωr = 2πnN/60 and ωd is the rotor damping factor at a rotor speed of ωr, N is number of blades and n is the speed of the rotor. The procedure for selecting the damping factor for rotational sampling filter as given in [29] is summarized here. Petru [29] used shaft torque obtained from an advanced and simplified approach through aerodynamic filters to estimate the damping factor in the rotational sampling filter of a typical wind turbine. He observed that in a frequency range of 0.5 to 2.5 Hz, the wind speed amplification by the filter is higher. The variation of damping factor with mean wind speed has been plotted. In the present study,

Fig. 10. Schematic representation of wavelet decomposition of vibration signal up to the 3rd level.

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Single-Sided Amplitude Spectrum

-3

2

x 10

X: 326.7 Y: 0.001696

Gear mesh frequency of gear 2-3

Amplitude

1.5

1

0.5

0 0

50

100

150

200 Frequency (Hz)

250

300

350

400

Fig. 11. FFT of the decomposed signal at the 3rd level in frequency band 312–375 Hz.

an approximate damping factor is assumed from the aforesaid plot for simulation and is indexed in Table 1. The significance of the rotational sampling frequency is explained in the following paragraph. Fig. 3 represents the wind speed signal power spectral density (PSD) obtained at the fixed point (preferably hub level) and PSD of the wind speed signal after passing through aerodynamic filters. It can be clearly observed that the filter gain is nearly equal to one at the lower frequencies and around the high frequency region, frequency content is clearly dampened. The amplification in PSD value is observed at around rotor characteristic frequency and its higher harmonics. In literature, the more detailed way of modeling rotational wind speed turbulence for large wind turbines is discussed and some of which is summarized here. Rotational wind speed turbulence through shaping filters and correlation techniques is proposed for large scale wind turbines in [30]. From the numerical results, it is observed that good approximation is found between the rotational sampling spectrum by correlation technique and synthesized shaping filter. The effect of location along the blade on the rotational sampling effect is also captured. It is also illustrated that for the calculation of aerodynamic torque the use of PSD at four fifths of the rotor radius is recommended. The rotor aerodynamic power Pa is given by [27], Pa ¼

1 2 3 πρR u C p ðλ; βÞ: 2

ð10Þ

The power coefficient (Cp(λ, β)) is specific to a given turbine that can be defined as the ability of the turbine to convert its kinetic energy into the mechanical energy. The power coefficient is a function of pitch angle β and the tip speed ratio λ. Fig. 4 shows a typical cross-section of the wind turbine blade, where ϕ, θ, and β represent incident angle, angle of attack and pitch angle respectively. The maximum achievable value of Cp for wind turbine is 0.59. Generally in practice, the maximum achievable Cp is 0.5 for two bladed high speed turbines and 0.2–0.4 for low speed multi-bladed turbines [31]. The coefficient of performance Cp can be estimated using the following empirical relation as given in [31].  1 2 −0:17λ : λ−0:022β −5:6 e 2

ð11Þ

Single-Sided Amplitude Spectrum

0.14

Gear mesh freqeuncy ring-planet, planet-sun (25.2 Hz)

0.12 0.1

Amplitude

C p ðλ; βÞ ¼

0.08 0.06 0.04 0.02 0 0

5

10

15 20 Frequency (Hz)

25

30

Fig. 12. FFT of decomposed signal at the 5th level at a frequency band of 15.625–31.25 Hz.

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Single-Sided Amplitude Spectrum

-3

1.5

13

x 10

Amplitude

Gear mesh freqeuncy of gear1-2 (83.8 Hz) 1

0.5

0 0

10

20

30

40

50 60 Frequency (Hz)

70

80

90

100

Fig. 13. FFT of decomposed signal at the 6th level at a frequency band of 85.94–93.75 Hz.

The tip speed ratio (λ) can be defined as,

λ¼

ωR R ; u

ð12Þ

where R is the rotor radius, ωR is the rotational speed of the rotor and u is the total wind speed. The rotor aerodynamic torque is the ratio of rotor aerodynamic power Pa to the shaft angular velocity. Hence by using Pa in Eq. (10), the torque is given by,

TR ¼

1 2 3 πρR u C p ðλ; βÞ: 2ωR

ð13Þ

The total wind speed is obtained using parameters in Table 1. The power coefficient is estimated using the blade data given in Table 1. For the estimation of aerodynamic torque, the total wind speed and power coefficient estimated from Eq. (11) are used. The aerodynamic torque obtained from the above approach is used for the simulation of wind turbine dynamics. Fig. 5 represents the ramp, gust, total wind speed and the aerodynamic torque signal in time domain obtained from the above approach. This obtained wind speed signal follows the same trend as the measured longitudinal wind speed signal at the site as given in [32,33]. 3. Modeling of dynamics of drive train Fig. 6 shows a schematic diagram of a wind turbine drive train. The main components of the drive train are rotor, gearbox and generator. Generally, the wind turbine is equipped with an epicyclic gearbox which contains two external and one epicyclic spur gear stages. The coupled torsional bending dynamic model of wind turbine gearbox is developed based on the Lagrange's formulation and the dynamic equations of motion are formulated in [25]. The same modeling is used in the present study, hence, the details of the dynamic model are summarized briefly as follows. The time varying gear mesh stiffness, torsional shaft stiffness and bearings elasticity are accounted for in the dynamic model. Rigid multibody modeling with discrete flexibility approach is used for the development of equations of motion. In this model, there exist eight rotational and ten transverse degrees of freedom. Single-Sided Amplitude Spectrum 140 120

Amplitude

100 80

Gust freqeuncy

Peak at first torsional natural freqeuncy

60 40 Rotor Characteristic freqeuncy ( 1X and 2X components)

20 0 0

0.2

0.4

0.6

0.8

1 1.2 Frequency (Hz)

1.4

1.6

1.8

2

Fig. 14. Frequency domain representation of gust, peak at first natural frequency and rotor characteristic load frequency and its harmonics (8th level decomposition).

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P. Srikanth, A.S. Sekhar / Mechanism and Machine Theory 98 (2016) 2–20

6

x 10

4

Torque (N-m)

4 2 0 -2 -4 -6 0

1

2

3

4

5 Time (S)

6

7

8

9

10

Fig. 15. Time domain representation of external output torque.

The generalized coordinates to represent the degrees of freedom are as follows, T

fqg ¼ fϕ1 ϕ2 ϕ3 ϕ4 ϕ5 ϕ6 ϕ7 ϕ8 yi zi ys zs y1 z1 y2 z2 y3 z3 g :

ð14Þ

where ϕi (i = 1, 2, … ,8) are the rotor, carrier, planets, sun, gear1, gear2, gear3 and generator rotational degrees of freedom respectively and yi, zi, ys, zs, y1, z1, y2, z2, y3, and z3 are the planets, sun, gear1, gear2 and gear3 transverse degrees of freedom in the y and z directions respectively. Lagrange's technique is used for developing the equation of motion where kinetic and potential energies of rotating and orbiting gears are considered. The kinetic energy of orbiting and rotating gears can be represented as, K:E: ¼

 1 _2 1 1 1 1 1 1 1 3 3 1 2 2 2 2 2 2 2 2 2 2 2 J 1 ϕ1 þ J 2 þ 3mi r c1 ϕ_ 2 þ ð3 J 3 Þϕ_ 3 þ J 4 ϕ_ 4 þ J 5 ϕ_ 5 þ J 6 ϕ_ 6 þ J 7 ϕ_ 7 þ J 8 ϕ_ 8 þ mi y_ i þ mi z_ i þ ms y_ s 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 2 2 2 2 2 2 2 þ ms z_ s þ m1 y_ 1 þ m1 z_ 1 þ m2 y_ 2 þ m2 z_ 2 þ m3 y_ 3 þ m3 z_ 3 : ð15Þ 2 2 2 2 2 2 2

Mainly the gear mesh stiffness, bearings and shaft stiffness contribute to the elastic potential energy. The potential energy due to the time varying gear mesh stiffness is represented as, P:E:1 ¼

n n 1X 1X 1 pg 2 2 2 kri δri þ k δ þ kpg δ pg ; 2 i¼1 2 i¼1 si si 2

ð16Þ

where kri, ksi, and kpg are the ring–planet, planet–sun and external gear pinion mesh stiffness and δri, δsi, and δpg are the fixed ring–planet, sun–planet and external gear–pinion dynamic transmission errors respectively. The time varying gear mesh stiffness is represented in terms of Fourier series whose details along with dynamic transmission error are discussed in [25]. Bearings associated with elastic potential energy, whose contribution to the total elastic potential energy is represented as, 6   1X 2 2 k y þ zb j ; 2 j¼1 b j b j

ð17Þ

Single-Sided Amplitude Spectrum of torque 14000 12000

Amplitude (N-m)

P:E:2 ¼

10000 8000 6000 4000 2000 0 0

50

100

150

200

250 Frequency (Hz)

300

350

Fig. 16. Frequency domain representation of external output torque.

400

450

500

P. Srikanth, A.S. Sekhar / Mechanism and Machine Theory 98 (2016) 2–20

15

4

Carrier internal torque (N-m)

5

x 10

4

(a)

3 2 1 0 -1 -2 -3 0

x 10

Sun Internal torque (N-m)

4

1

Gear 1 internal torque (N-m)

4

5 Time (S)

6

7

8

9

10

(b)

2 0 -2 -4

x 10

6

1

2

3

4

5 Time (S)

6

7

8

9

10

4

(c) 4 2 0 -2 -4 0

x 10

1

Gear2 internal torque (N-m)

3

4

-6 0

1

2

3

4

5 Time (S)

6

7

8

9

10

4

(d) 0.5

0

-0.5

-1 0

Gear3 internal torque ( N-m)

2

1

2

3

4

5 Time (S)

6

7

8

9

10

4000

(e)

2000 0 -2000 -4000 -6000 -8000

0

1

2

3

4

5 Time (S)

6

7

8

9

10

Fig. 17. Internal resistive torque due to time varying gear mesh stiffness (a) carrier, (b) sun, (c) gear1, (d) gear2 and (e) gear3.

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P. Srikanth, A.S. Sekhar / Mechanism and Machine Theory 98 (2016) 2–20

where kbj represents bearing stiffness, and ybj and zbj represent transverse displacements of bearings in the y and z directions respectively. The elastic potential energy due to the torsional shaft stiffness is represented as P:E:3 ¼

1 1 1 2 2 2 C ðϕ −ϕ1 Þ þ C 2 ðϕ5 −ϕ4 Þ þ C 3 ðϕ8 −ϕ7 Þ ; 2 1 2 2 2

ð18Þ

where C1, C2, and C3 represent the low speed, intermediate and high speed shaft stiffness respectively. The details of evaluating strain energy associated with bearings and shafts are discussed in [25]. Lagrange's formulation is used to extract the coupled torsional bending dynamic equations of motion for the gearbox. The equations of motion associated with rotational and translational coordinates are indexed in Appendix A. The generator torque is represented in the form of, T G ðt Þ ¼

T R ðt Þ ; G

ð19Þ

where G is the epicyclic gearbox gear ratio. The damping is included in the dynamic system using the Rayleigh's method of proportional damping. The damping ratio of 5% is considered in this study, the justification of this value is discussed in detail in [34]. The generalized equations of motion are represented in the matrix form as, €g þ ½C ðt Þfq_ g þ ½K ðt Þfqg ¼ fT 0 ðt Þg; ½Mfq

ð20Þ

where [M], [C(t)] and [K(t)] represent the mass, time varying damping and stiffness matrices respectively. Newmark time integration algorithm is used to solve the equation of motion and the vibration responses are obtained in time and frequency domains. 4. Results and discussion The wind turbine drive train parameters as considered in [25] are used in the present study and are tabulated in Table 2. The generation of the aerodynamic torque time series is discussed in Section 2. The vibration responses in the time and frequency domains are obtained from these signals. In the present study, vibration, external torque and internal resistive torques are used for the dynamic characterization of drive train. Table 3 represents the characteristic frequencies that can be expected from the vibration signal of a typical wind turbine drive train, for the data considered in Table 2. 4.1. Vibration signal Typically, wind turbine is an example of slow speed rotating machinery, rotating at a constant speed and subjected to stochastic loads at the power production stage (in the case of both fixed and variable speed wind turbines). Over a finite period of time, the obtained vibration signal due to stochastic loads can be assumed to be continuous and aperiodic, hence, in the present study, FFT is applied. Figs. 7 and 8 represent the time and frequency domain (FFT) representation respectively of the vibration acceleration signals of gear1, gear2, gear3 and the generator. From the time domain signals, it is clearly evident that the vibration signals of wind turbine drive train are nonlinear and non-stationary in nature due to the type of impressed load. From the frequency domain representation, a clear peak is observed at a frequency close to the second natural frequency (see Table 3) and all the energy content of the vibration signal is confined to the low frequency region only. Further, the vibration signal is zoomed at low frequencies to extract the characteristic frequencies of the drive train. Fig. 9 represents the zoomed vibration signal from 0–10 Hz of gear3, where no clear information is available except the peaks close to the second natural frequency. There exist a number of hidden frequencies such as, gear mesh frequency, gust amplitude frequency, peak at the first torsional natural frequency and characteristic load frequencies in the frequency spectrum, all of which are to be identified. From the previous work of authors, it is found that gear3 (see Fig. 1) is subjected to more vibration acceleration of all gears in the wind turbine gearbox [25]. Hence, in the present work, wavelet decomposition is applied to the vibration signal of gear3 to identify all the expected characteristic frequencies. A sym8 wavelet is used with a sampling frequency of 1000 Hz for decomposition. The signal is decomposed up to the 8th level with the use of commercial software package MATLAB 7.10.0. Fig. 10 shows a typical wavelet decomposition up to the 3rd level. From the FFT of gear3 vibration signal, no clear information is found about the characteristic frequencies of the gearbox. However, FFTs are applied to the decomposed signals from which all the characteristic peaks are seen. The gear mesh frequency of gear2–3 (326.5 Hz) is seen clearly in the 3rd level decomposition at a frequency band of 312–375 Hz (see Fig. 11). This gear mesh frequency is clearly seen from wavelet decomposition from the fact that no frequencies exist in the gearbox except that at a range of 250–500 Hz. Fig. 18. Frequency domain representation of internal resistive torque (a) carrier, (b) sun, (c) gear1, (d) gear2 and (e) gear3.

P. Srikanth, A.S. Sekhar / Mechanism and Machine Theory 98 (2016) 2–20

17

Single-Sided Amplitude Spectrum 5000 Peak near second natural frequency

Amplitude (N-m)

4000

(a)

Ring-planet, planet-sun gear mesh freqeuncy and higher harmonics (25.2 Hz)

3000 2000 1000 0 0

50

100

150

Amplitude (N-m)

250 300 Frequency (Hz)

350

400

450

x 10

(b)

1.5

1

500

Single-Sided Amplitude Spectrum

4

2

200

peak near second natural freqeuency Sun-planet gear mesh frequency

0.5

0 0

50

100

150

200

250 300 Frequency (Hz)

350

400

450

500

Single-Sided Amplitude Spectrum 15000

Amplitude (N-m)

(c) 10000

Gear1-2 gear mesh frequency (83.8 Hz)

5000

0 0

50

100

150

200

250 300 Frequency (Hz)

350

400

450

500

Single-Sided Amplitude Spectrum 1200

(d)

Amplitude (N-m)

1000 Gear 2-3 gear mesh freqeuncy (326.5 Hz)

800 Gear 1-2 gear mesh frequency (83.8 Hz)

600 400 200 0 0

50

100

150

200

250 300 Frequency (Hz)

350

400

450

500

Single-Sided Amplitude Spectrum 1600 1400

(e)

Amplitude (N-m)

1200 1000 800

Gear 2-3 gear mesh freqeuncy (326.5 Hz)

600 400 200 0 0

50

100

150

200

250 300 Frequency (Hz)

350

400

450

500

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P. Srikanth, A.S. Sekhar / Mechanism and Machine Theory 98 (2016) 2–20 Single-Sided Amplitude Spectrum 500

Amplitude (N-m)

Gust amplitude frqeuncy

400

Peak near first torsional natural freqeuncy.

Peak at charecterisitc load freqeuncy ( 1X, 2X, 3X and 4X)

300 200 100 0

0.5

1

1.5

2 Frequency (Hz)

2.5

3

3.5

4

Fig. 19. Enlarged view of frequency spectrum of gear3 internal torque.

Figs. 12 & 13 bring out gear mesh frequencies of ring–planet, planet–sun and gear1–2 (25.2 and 83.5 Hz) through wavelet decomposition. These gear mesh frequencies are observed at 5th and 6th levels of decomposition. The wavelet packet at the 8th level is used to capture the characteristic load frequency of the rotor, the first natural frequency and gust frequency (see Fig. 14). Due to the rotational sampling effect (as discussed earlier with respect to Fig. 3), 1×, and 2× peaks are observed near the rotor characteristic load frequency within the specified frequency limits. Hence, by wavelet decomposition it is possible to extract the hidden frequency information which exists in the gearbox. 4.2. Torque signals The possibility of understanding the dynamics of drive train through torque signals is discussed in this section. Generally, there exists internal resistive torque in addition to external torque in the gearbox. It is clearly evident from equation of motion (see Eq. (A1(h)) in Appendix A) that the output external torque can be obtained from the algebraic sum of generator inertia torque and internal resistive torque due to high speed shaft stiffness. Hence, this signal depends upon the vibration acceleration of generator, vibration displacements of gear3 and generator. Figs. 15 & 16 show the time and frequency representation respectively of external output torque signal obtained from vibration. It can be noticed that the external torque signal also behaves similar to the vibration signal and if one needs a detailed study of it, wavelet decomposition is the best strategy for dynamic characterization of such signals. The possibility of characterizing the drive train with internal resistive torque signals is also discussed as in the following. The internal resistive torques exist due to gear mesh stiffness, damping present in the system and inertia effects. These types of torques are represented in equations of motion (Eqs. (A1(a–h)) in Appendix A). For example, from the equation of motion (Eq. (A1(h))), the internal resistive torque due to stiffness is presented in the form of modulation of time varying gear mesh stiffness, dynamic transmission error and radius of gear3. Similar kind of forms can also be observed for internal resistive torques of other gears also from Eqs. (A1(a–g)). Figs. 17 and 18 represent the internal resistive torque in time and frequency domains respectively that exist between gears, due to time varying gear mesh stiffness. In Fig. 17, it can be noticed that in gear tooth contact force time domain representation, gear mesh frequencies are overlapped over the dynamic transmission error. From frequency domain representation, it is evident that the gear mesh frequencies are equally dominating with low frequency region peaks. It is also observed that frequency spectrum corresponding to gear2 internal resistive torque contains gear mesh frequencies of both gear1–2 and gear2–3. From the frequency spectra all the gear mesh frequencies and their higher harmonics are also clearly observed. Fig. 19 represents the enlarged view of internal resistive torque of gear3 in the low frequency region. From this it is evident that clear peaks are observed at gust frequency, first torsional natural frequency and multiples of rotor characteristic frequency (1×, 2×, 3× and 4×) from FFT itself besides the peaks at the second natural frequency and gear mesh frequency. Hence, one can notice that FFT of internal resistive torque signal itself reveals all the characteristic frequencies of the drive train. It may be observed from the above discussion that, it is difficult to measure the internal resistive torques directly in the wind turbine drive train. Hence, a strategy of measuring such signals is discussed in the present study. From the equation of motion (Eq. (A1) in Appendix A) it can be understood that the internal resistive torque due to stiffness is a function of time varying gear mesh stiffness and dynamic transmission error. Hence, internal resistive torque can be measured indirectly by measuring the above two parameters. It is easy to measure the dynamic transmission error which exists at any gear stage. The detailed procedure of dynamic transmission measurement is discussed in [35]. Similarly, the gear mesh stiffness depends upon parameters like load and rotational position of gears. It is difficult to measure the gear mesh stiffness directly, however one can measure this indirectly by measuring the dynamic transmission error over the angular position by subjecting the gearbox to a constant torque. Theoretically it is possible to obtain vibration responses individually for each and every gear by rigid multi body modeling with discrete flexibility approach. However in practice, the torsional vibration exists in the coupled mode and the dynamic transmission error itself is the torsional vibration that exists between the two gears [36]. Hence, dynamic transmission error is the indication of torsional vibration in the case of gears. Thus, in practice it is possible to measure time varying gear mesh stiffness of two gear pairs.

P. Srikanth, A.S. Sekhar / Mechanism and Machine Theory 98 (2016) 2–20

19

Since internal resistive torque due to stiffness is a function of measurable parameters such as, dynamic transmission error and time varying mesh stiffness, it is recommended to use these two parameters to extract the frequency information. 5. Conclusions The dynamic analysis of wind turbine drive train is carried out. The dynamic model as developed by the authors using Lagrange's formulation is used in the present study. The aerodynamic torque signal is obtained by taking into account the average, ramp, gust and turbulence components of the wind. This signal represents the almost typical wind signal at the site. The vibration responses and external toque responses are obtained to get the dynamic characterization of the wind turbine gearbox. It is revealed that these signals are very less sensitive to loads and require a detailed signal processing to get the full information. Hence, wavelet packets are used in the present approach from which all the characteristic frequencies are clearly seen in the decomposed signal at different levels. The study shows that dynamic characterization can be done through internal resistive torque signals also. It is noted that, there exists a number of hidden characteristic frequencies of the drive train in vibration signal, which needs detailed signal processing. However, if internal resistive torque is used all such frequencies can be clearly observed in FFT itself. Further, the possibility of indirect measurement of internal resistive torque signals by dynamic transmission error and time varying gear mesh stiffness measurement is elaborated. The study envisages the suitability of internal resistive torque measurement for wind turbine drive train condition monitoring. Appendix A The equation of motion in rotational coordinates are given by € þ C ðϕ −ϕ Þ ¼ T ðt Þ J1 ϕ 1 1 1  2 R 2 € 2 2 J 2 þ 3m3 r c1 ϕ 2 þ C 1 ðϕ2 −ϕ1 Þ þ 3K ri ðt Þr c1 cosðα Þϕ2 þ 3K si ðt Þr c1 cosðα Þϕ2 −3K si ðt Þr 3 r c1 cosðα Þ þ 3K ri ðt Þr 3 r c1 cosðα Þ−3K si ðt Þr c1 r 4 cosðα Þϕ4 ¼ 0 € þ 3K ðt Þr 2 ϕ þ 3K ðt Þr 2 ϕ þ 3r r K ðt Þϕ 3J 3 ϕ 3 ri 3 3 si 3 3 s i si 4 þ3r c1 r i cosðα ÞK si ðt Þϕ2 þ 3r c1 ri cosðα ÞK ri ðt Þϕ2 ¼ 0  € þ K ðt Þr r cosðα Þϕ þ K ðt Þr r ϕ þ C þ K ðt Þr 2 ϕ −C ϕ ¼ 0 J4 ϕ 4 si c1 4 2 si 3 4 3 2 si 4 4 2 5

ðaÞ ðbÞ ðcÞ ðdÞ

€ þ C ðϕ −ϕ Þ þ K ðt Þr 2 ϕ þ K ðt Þr r ϕ þ K ðt Þr sinðα Þy J5 ϕ 5 2 5 4 g12 5 5 g12 5 61 6 g12 5 1 ðeÞ þK g12 ðt Þr 5 cosðα Þz1 −K g12 ðt Þr 5 sinðα Þy2 −K g12 ðt Þr 5 cosðα Þz2 ¼ 0 € J 6 ϕ6 þ K g23 ððy2 −y3 Þ sinðα Þ þ ðz2 −z3 Þ cosðα Þ þ r 62 ϕ6 þ r7 ϕ7 Þr 62 ðfÞ þK g12 ððy1 −y2 Þ sinðα Þ þ ðz1 −z2 Þ cosðα Þ þ r5 ϕ5 þ r 61 ϕ6 Þr 61 ¼ 0 € þ C ðϕ −ϕ Þ þ K ðt Þððy −y Þ sinðα Þ þ ðz −z Þ cosðα Þ þ r ϕ þ r ϕ Þr ¼ 0 ðgÞ J7 ϕ 7 3 7 8 g23 2 3 2 3 62 6 7 7 7 € þ C ðϕ −ϕ Þ ¼ −T ðt Þ J8 ϕ ðhÞ 8 3 8 7 G

ðA1Þ

where TR(t), and TG(t) are the rotor and generator torques respectively. The equations of motion associated with translational coordinates are given by €i þ  3my k1 yi −1:5K si ðt Þys ¼ 0 ðaÞ  si ðbÞ 3m€zi þ k1 −1:5K ri ðt Þ−1:5K si ðt Þ zi þ 1:5K si ðt Þzs ¼ 0     k þ 9k3 k −3k3 €s −ð1:5K si ðt ÞÞyi þ 2 −1:5K ri ðt Þ ys þ 2 y ¼0 ðcÞ ms y 4 4  1    k þ 3k3 k −3k3 þ 1:5K si ðt Þ zs þ 2 z1 ¼ 0 ðdÞ ms €zs −ð1:5K si ðt ÞÞzi þ 2 4    4  k −3k3 k þ k3 €1 þ 2 y þ 2 y þ K g12 ðt Þððy1 −y2 Þ sinðα Þ þ ðz1 −z2 Þ cosðα Þ þ r 5 ϕ5 þ r61 ϕ6 Þ sinðα Þ ¼ 0 ðeÞ m1 y 4  s  4  1  : k −3k3 k þ k3 zs þ 2 z1 þ K g12 ðt Þððy1 −y2 Þ sinðα Þ þ ðz1 −z2 Þ cosðα Þ þ r 5 ϕ5 þ r 61 ϕ6 Þ cosðα Þ ¼ 0 ð f Þ m1 €z1 þ 2 4 4   €2 þ ðk4 þ k5 Þy2 −K g12 ðt Þððy1 −y2 Þ sinðα Þ þ ðz1 −z2 Þ cosðα Þ þ r 5 ϕ5 þ r 61 ϕ6 Þ sin α g ðgÞ m2 y þK g23 ðt Þððy2 −y3 Þ sinðα Þ þ ðz2 −z3 Þ cosðα Þ þ r 62 ϕ6 þ r 7 ϕ7 Þ sinðα Þ ¼ 0 m2 €z2 þ ðk4 þ k5 Þz2 −K g12 ðt Þððy1 −y2 Þ sinðα Þ þ ðz1 −z2 Þ cosðα Þ þ r 5 ϕ5 þ r 61 ϕ6 Þ cosðα Þ þK g23 ðt Þððy2 −y3 Þ sinðα Þ þ ðz2 −z3 Þ cosðα Þ þ r 62 ϕ6 þ r 7 ϕ7 Þ cosðα Þ ¼ 0ðhÞ €3 þ k6 y3 −K g23 ðt Þððy2 −y3 Þ sinðα Þ þ ðz2 −z3 Þ cosðα Þ þ r 62 ϕ6 þ r 7 ϕ7 Þ sinðα Þ ¼ 0 m3 y ðiÞ m3 €z3 þ k6 z3 −K g23 ðt Þððy2 −y3 Þ sinðα Þ þ ðz2 −z3 Þ cosðα Þ þ r 62 ϕ6 þ r 7 ϕ7 Þ cosðα Þ ¼ 0 ð jÞ

ðA2Þ

References [1] Y. Li, Discussion on the principles of wind turbine condition monitoring system, International Conference on Materials for Renewable Energy and Environment ICMREE-2011, 20–22 May 2011, Shanghai, China, 2011. [2] U. Giger, K. Arnaudov, High efficiency high torque gearbox for multi megaWatt wind turbines, Mach. Technol. Mater. (1) (2010) 19–24.

20

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[3] D. Qin, J. Wang, T.C. Lim, Flexible multibody dynamic modeling of a horizontal wind turbine drive train system, J. Mech. Des. ASME 131 (2009) 114501(1)–114501(6). [4] K. Abboudi, L. Walha, Dynamic behavior of a two stage gear train used in a fixed speed wind turbine, Mech. Mach. Theory 46 (2011) 1888–1900. [5] Y. Guo, J. Keller, R.G. Parker, Nonlinear dynamics and stability of wind turbine planetary gear sets under gravity effects, Eur. J. Mech. A Solids 47 (2014) 45–57. [6] M. Todorov, Dobrev, F. Massouh, Analysis of torsional oscillation of the drive train in horizontal-axis wind turbine, Proceedings of 2009 ELECTROMOTION EPE Chapter Electric Drives Joint Symposium, 1–3 July 2009, Lille, France, 2009. [7] J. Helsen, F. Vonhollebeke, Multibody modeling of varying complexity for modal behavior analysis of wind turbine gearboxes, Renew. Energy 36 (2011) 3098–3113. [8] J. Helsen, F. Vonhollebeke, Insights in wind turbine drive train gathered by validating advanced models on a newly developed 13.2 MW dynamically controlled test rig, Mechatronics 21 (2011) 737–752. [9] W. Shi, C.W. Kim, C.W. Chung, H.C. Park, Dynamic modeling and analysis of wind turbine drive train using the torsional dynamic model, Int. J. Precis. Eng. Manuf. 14 (2013) 153–159. [10] P. Srikanth, A.S. Sekhar, Dynamic analysis of fixed speed wind turbine drive train, Proceedings of the 2013 International Congress on Sound and Vibration, 7–11 July 2013, Bangkok, Thailand, 2013. [11] D.D. Ji, Y.M. Song, J. Zhan, Dynamics of gear train set in wind turbine, Mater. Sci. Forum 698 (2012) 701–705. [12] W. Haijun, D.U. Xuesong, X.U. Xiangyang, Dynamic analysis and experimental study of MW wind turbine gearbox, Appl. Mech. Mater. 86 (2011) 739–742. [13] L. Quan, L. Yongqian, Y. Yongping, Vibration Response Analysis of Gear Driven System of Wind Turbine, 3, IEEE, 2010 380–383. [14] L. Yongqian, L. Quan, Y. Yongping, Sensitivity analysis of natural frequency of gear driven system of wind turbine, Proceedings of the CMCE 2010 Conference, 24– 26 Aug 2010, IEEE 2010, pp. 559–562. [15] H. Sipeng, Z. Yangpei, X. Li, Y. Yue, Equivalent Wind Speed Model in Wind Farm Dynamic Analysis, IEEE, 2011 1751–1755. [16] J.J. Jang, J.R. Lee, Analysis of design wind speed distribution of Taiwan area, J. Mar. Sci. Technol. 5 (1997) 55–63. [17] Z. Xuesong, L. Ji, M. Youjie, Review on wind speed model research in wind power systems dynamic analysis, International Conference on Sustainable Power Generation and Supply SUPERGEN '09, April 2009, Nanjing, China, 2009. [18] W. Dong, Y. Zing, T. Moan, Z. Gao, Time domain based gear contact fatigue analysis of a wind turbine drive train under dynamic conditions, Int. J. Fatigue 48 (2013) 133–146. [19] S. Ahmed, Wind Energy Theory and Practice, PHI Learning Private Limited, New Delhi, India, 2010. [20] A.M. Ragheb, M. Ragheb, Wind Turbine Gearbox Technologies, 1st ed. Intech Europe, Croatia, 2011 189–206. [21] C. Zhu, S. Chen, H. Liu, H. Haung, G. Li, F. Ma, Dynamic analysis of the drive train of a wind turbine based upon the measured load spectrum, J. Mech. Sci. Technol. 28 (6) (2014) 2033–2040. [22] A.R. Nejad, Z. Gao, T. Moan, Long term analysis of gear loads in fixed offshore wind turbines considering ultimate operating conditions, Energy Procedia 35 (2013) 187–197. [23] Y. Zhao, M. Ahmat, K. Bari, Nonlinear dynamics modeling and analysis of transmission error of wind turbine planetary gear system, J. Multi-body Dyn. 228 (2014) 438–448. [24] S. Wei, J. Zhao, Q. Han, F. Chu, Dynamic response analysis on torsional vibrations of wind turbine geared transmission with uncertainty, Renew. Energy 78 (2015) 60–67. [25] P. Srikanth, A.S. Sekhar, Dynamic analysis of wind turbine drive train subjected to nonstationary wind load excitation, Proc. Inst. Mech. Eng. C J. Mech. Eng. Sci. 229 (3) (2015) 429–446. [26] T. Burton, D. Sharpe, M. Jenkins, E. Bossanyi, Wind Energy Handbook, Wiley, Chichester, UK, 2001. [27] A.P. Martins, P.C. Costa, A.S. Carvalho, Coherence and wakes in wind models for electro mechanical and power systems standard simulations, European Wind Energy Conference and Exhibition EWEC 2006, 27 February–2 March 2006, Athens, 2006. [28] M. Shinozuka, C.M. Jan, Digital simulation of random process, J. Sound Vib. 25 (1) (1972) 111–128. [29] T. Petru, Modeling of Wind Turbine System for Power System Studies(Phd thesis) , 2003, ISBN 91-7291-306-1. [30] A. Burlibasa, E. Cenga, Rotationally sampled spectrum approach for simulation of wind speed turbulence in large wind turbines, Appl. Energy 111 (2013) 624–635. [31] P.O. Ochieng, A.W. Manyonge, A.O. Oduor, Mathematical analysis of tip speed ratio of a wind turbine and its effect on power coefficient, 4 (1) (2014) 61–66. [32] T. Barszcz, M. Bielexka, A. Bielecki, M. Wojcik, Wind speed modeling using Weierstrass function fitted by a genetic algorithm, J. Wind Eng. Ind. Aerodyn. 109 (2012) 68–78. [33] A. Bielecki, T. Barszcz, M. Wojcik, Modeling of a chaotic load of wind turbine drive train, Mech. Syst. Signal Process. 55 (2015) 491–505. [34] S. Chauhan, M.H. Hansen, D. Tcherniak, Application of operational modal analysis and blind source separation/independent component analysis techniques to wind turbines, Conference nd Exposition on Structural Dynamics IMAC-XXVII, 9–12 February 2009, Orlando, Florida, USA, 2009. [35] D.R. Houter, G.W. Blankenship, Methods for measuring gear transmission error under load and at operating speeds, SAE 8918691989. [36] Z. Hameed, Y.S. Hong, Y.M. Cho, S.H. Ahn, C.K. Song, Condition monitoring and fault detection of wind turbines and related algorithms: a review, Renew. Sust. Energ. Rev. 13 (2009) 1–39.