Dynamics analysis of the pitch control reducer for MW wind turbine

2.1. Lumped-parameter analytical model

The lumped-parameter analytic model for single stage is established as shown in Fig. 1. The planet gears are equally spaced. All planets at the same stage are assumed to have identical mass, rotational inertia, support stiffness and time-invariant gear mesh stiffness. It is worthwhile mentioning that the gear backlash, radial bearing clearance, frictional force arising from tooth sliding motion, gear tooth spacing error and misalignment of the gears are not considered in this study.

Fig. 1Lumped-parameter analytic model

In the model, the gear mesh is treated as a linear time-invariant spring and a damping acting along the mesh line [10]. All other supporting bearings are modeled as linear springs. $K_{bsi}$ and $K_{bri}$ present the supporting stiffness of the sun and ring at the $i$ stage, $K_{bpij}$ is the supporting stiffness of the $j$ planet at the $i$ stage. $K_{bci}$ represents the supporting stiffness of the carrier at the $i$ stage. $K_{us}$, $K_{ur}$ and $K_{uc}$ are the rotational stiffness of the sun, ring and carrier respectively, and $K_{spij}$ and $K_{prij}$ are the mesh stiffness between the $j$ planet and the sun or the ring, and $C_{spij}$ and $C_{prij}$ are the mesh damping.

The configuration of the wind turbine pitch control reducer which consists of three stages of planetary system is shown in Fig. 2. The power is transformed from the sun at the first stage to the carrier at the last stage, which is connected to the output shaft of the pitch control reducer. The carrier connects the sun at the next stage by involutes spline. The spline connection is treated as a torsion spring between the carrier and sun, providing torsion support stiffness $K_{ei}$ for the sun and carrier in the dynamics model. $K_{bz}$ and $K_{ez}$ are the supporting stiffness and torsion support stiffness of the output shaft respectively in the dynamics model.

Fig. 2Structure of the wind turbine pitch control reducer

2.2. Dynamics equations of the system

There are both three planets at the first and second stage, and four planets at the third stage, including the sun, ring and the carrier at all three stages, so the component numbers are 6, 6, 7 from the first stage to the last stage orderly, and then the output shaft connected to the third carrier also be considered, there are totally 20 components in the wind turbine pitch control reducer. Three degrees of freedom (DOF) have been considered for each component including one rotational DOF and two translational DOF, so 60 DOF for the wind turbine pitch control reducer must be researched.

The differential equations of motion for all the components in three-stage planet gear train are:

where $\mathbf{M}$ is the inertia matrix, $\mathbf{C}$ is the damping matrix, $\mathbf{K}$ is the stiffness matrix, $\mathbf{F}$ is the force vector of externally applied torque. $\mathbf{x}$ is the vector of 60 degrees of freedom:

where $x$ is the lateral displacement in the fixed $XOY$ coordinate, $y$ is the vertical displacement and the rotational degree $u$ is replaced by the line displacement along the line of mesh, $u=r\theta$, where $\theta$ is the rotation angle of component and $r$ is the base circle radius for the sun, ring and planets and center radius for the carrier. $\phi$ and $\tau$ represent the radial and tangential displacement respectively in the movable $\phi o\tau$ coordinate on the planet.

The degree of freedom (DOF)1-3 are $x_{s1}$, $y_{s1}$, $u_{s1}$ for the sun in the first stage, DOF 4-6 are $x_{r1}$, $y_{r1}$, $u_{r1}$ for the ring, DOF 7-15 are ${\phi }_{p1i}$, ${\tau }_{p1i}$, $u_{p1i}$ for the planets, DOF 16-18 are $x_{c1}$, $y_{c1}$, $u_{c1}$ for the carrier in the first stage. DOF 19-21 are $x_{s2}$, $y_{s2}$, $u_{s2}$ for the sun in the second stage, DOF 22-24 are $x_{r2}$, $y_{r2}$, $u_{r2}$ for the ring, DOF 25-33 are ${\phi }_{p2i}$, ${\tau }_{p2i}$, $u_{p2i}$ for the planets, DOF 34-36 are $x_{c2}$, $y_{c2}$, $u_{c2}$ for the carrier in the second stage, DOF 37-39 are $x_{s3}$, $y_{s3}$, $u_{s3}$ for the sun in the third stage, DOF 40-42 are $x_{r3}$, $y_{r3}$, $u_{r3}$ for the ring, DOF 43-54 are ${\phi }_{p3i}$, ${\tau }_{p3i}$, $u_{p3i}for$the planets, DOF 55-57 are $x_{c3}$, $y_{c3}$, $u_{c3}$ for the carrier in the third stage, DOF 58-60 are $x_z$, $y_z$, $u_z$ for the output shaft.

The inertia matrix $\mathbf{M}$ in the differential motion equation is given as:

where $I_{si}$, $I_{ri}$, $I_{pij}$ and $I_{ci}$ ($i=$ 1, 2, 3; $j=$ 1, …, $n$) present the rotational inertia for the sun, ring, planet and the carrier, $m_{si}$, $m_{ri}$, $m_{pij}$ and $m_c$ present the mass for the sun, ring, planet and the carrier.

The stiffness factor method is suitable to model stiffness matrix $\mathbf{K}$ for the problem containing plenty of complex coupled elements among these components. Each stiffness element in the stiffness matrix associates with the translational force and rotational moment in corresponding DOF. The stiffness is defined to be the force inducing one unit deformation. Mesh conditions of the sun and planet at all stages are treated identical. All the force, mesh stiffness and support stiffness for the sun and a planet are shown in Fig. 3.

Fig. 3The mesh between the sun and planet

The mesh stiffness $K_{spij}$ along the line of action between the sun and every planet can be transformed to $x$, $y$ and $u$ directions in the fixed coordinate. Assuming that the sun deforms one unit in the $x$ direction, the support force $F_{xsi}$ and the mesh force of all planets will be applied on it, the decomposed components for all planets in the $x$ direction and the support stiffness $K_{bsi}$ are superimposed to compose the first element in the stiffness sub-matrix ${\mathbf{K}}_{si}$ of the sun. There is no coupled element in $y$ and $z$ directions for the deformation of the sun in the $x$ direction, so the corresponding elements are 0 in the sub-matrix. The other stiffness elements can be obtained by the same method. The three order stiffness sub-matrix ${\mathbf{K}}_{si}$ of the sun at single stage is:

where $K_{spij}$ is the mesh stiffness between the sun and the $j$ planet at the $i$ stage, ${\phi }_{ij}$ is phase angle of the $j$ planet, ${\alpha }_s$ is external gearing angle. $K_{ei}$ is torsion spring between the carrier and sun, $K_{e1}$ is zero because there is no spline connected the first sun.

The force and mesh status between the planet and ring is shown in Fig.4 (a), the support spring, mesh spring, and the force of the planet are represented in the movable $\phi o\tau$ coordinate, while that of the ring are represented in the $XOY$ fixed coordinate. The force status of the carrier and planet is shown in Fig. 4(b).

Fig. 4The applied force status of mesh between components

a) The ring and planet

b) The carrier and planet

The stiffness submatrix ${\mathbf{K}}_{ri}$ for the ring at the $i$-stage is obtained by the stiffness coefficient method:

where $K_{rpij}$ is the mesh stiffness of the ring and the $j$ planet at the $i$ stage, ${\alpha }_r$ is the internal mesh angle, ${\alpha }_r={\alpha }_s$.

The stiffness sub-matrix for the carrier at the $i$ stage ${\mathbf{K}}_{ci}$ can be obtained as follows:

Based on the applied force of the sun, ring and carrier on the planet in three directions in the movable $\phi o\tau$ coordinate, the sub-matrix ${\mathbf{K}}_{pij}$ for the $j$ planet is:

The other coupled stiffness elements between the sun and planet, the ring and planet, the carrier and planet in the whole matrix are also modeled, the stiffness sub-matrix ${\mathbf{K}}_{sipj}$ coupled between the $j$ planet and sun at the $i$ stage is shown as follows:

The stiffness sub-matrix ${\mathbf{K}}_{ripj}$ coupled between the $j$ planet and ring at the $i$ stage is:

The stiffness sub-matrix ${\mathbf{K}}_{cipj}$ coupled between the $j$ planet and carrier at the $i$ stage is:

All the stiffness sub-matrices at the same stage are concentrated to form the whole stiffness matrix for each stage, which is 18×18 orders for the first and second stage and 21×21 orders for the third stage. The three subsystems of stiffness matrix are integrated as follows, where the uncoupled parts are replaced by sub-matrix 0:

According to the force status of the output shaft, the stiffness sub-matrix ${\mathbf{K}}_z$ is:

The overall system stiffness matrix concentrating the three subsystems and the output shaft for the wind turbine pitch control reducer is:

Rayleigh damping $C$ is used in the dynamics equation which is proportion to the stiffness and inertia [6]:

Rayleigh damping coefficients $\alpha$ and $\beta$ are defined by the method [23] shown in Eq. (3) according to the damping ratio ${\xi }_i$:

There is a damping ratio ${\xi }_i$ corresponding to every order natural frequency ${\omega }_i$. The first and second order damping ratios are treated as the same, which are 0.007 for the steel material here [22]. So, based on Eq. (6) and the first and second order natural frequencies, $\alpha$ and $\beta$ can be obtained.

The applied force consisting of the input and output parts is referred to Eq. (4):

The external torque applies rotational force, so the force matrix for the three stages is referred as:

2.3. Structure parameters of the pitch control reducer

The structure parameters of the pitch control reducer are listed in Table 1. The module is 2 mm for the first and second stages, 4 mm for the third stage. The external and internal mesh angle is 23.7°, 22.8° and 20° from the first to the third stage.

The mesh stiffness for all contact gears is calculated by the Ishikawa method [24] according to the structure parameters in Table 1, and the rotation stiffness and support stiffness for each stage are calculated by static finite element method. The rotation stiffness and support stiffness calculated with the applied force and displacement in the FEM models are displayed in Table 2.

4.1. Decoupling of system vibration model

There are a large number of coupled elements in the stiffness matrix, so it is necessary to decouple the vibration formulation when computing the dynamic response. The linear coordinate transformation method [12] transforms displacement vector $\mathbf{x}$ for all DOFs from physical coordinate to modal coordinate, the transformation process will uncouple the matrix. The uncoupled equation is as follows:

where $\mathbf{\Phi }$ is main vibration mode matrix, $\mathbf{\eta }$ is modal coordinate array.

Substituting Eq. (6) to dynamics Eq. (1), the vibration formulation of the undamped dynamic response is transformed as:

Assuming ${\mathbf{\Phi }}^{\mathbf{T}}\mathbf{M}\mathbf{\Phi }={\mathbf{M}}_{\mathbf{d}}$, ${\mathbf{\Phi }}^{\mathbf{T}}\mathbf{K}\mathbf{\Phi }={\mathit{K}}_d$. The system suffers external harmonic force, Eq. (7) will be transformed as:

where ${\mathbf{F}}_m$ is basic force of the harmonic excitation, ${\mathbf{F}}_0$ is the amplitude of the harmonic force, $\omega$ is harmonic angular frequency.

The vibration displacement in the modal coordinate of the three-stage planet can be calculated as:

where ${\eta }_i\left(t\right)$ is vibration displacement in the $i$ modal coordinates.

Analytical solution of the vibration differential formulation in physical coordinate can be obtained by transformation with Modal Summation technique [25].

4.2. Undamped dynamic response

The input parameters of the pitch control reducer are rotational speed 1600 r/min and torque 38.2 Nm for the first stage, and the speed and torque at the other stages can be computed by the transmission ratio. Based on the above mentioned parameters of the pitch control reducer and the uncoupled process, the undamped dynamic response of the planet system can be solved. Partial results are given in Fig. 7(a)-(f) which present vibration displacement for some DOFs. The minimum vibration amplitude 0.113 mm belongs to the sun at the first stage in the rotation direction as in Fig. 7(a), the maximum displacement amplitude of the system is 0.589 mm which is related to the output shaft in the rotational direction.

Fig. 7Undamped displacement for part of response

a) Rotational displacement $u_{s1}$

b) Radial displacement ${\phi }_{p11}$

c) Lateral displacement $x_{r2}$

d) Rotational displacement $u_{r2}$

e) Lateral displacement $x_z$

f) Rotational displacement $u_z$

4.3. Damping dynamic response

The Rayleigh damping coefficients $\alpha$ and $\beta$ are 4.906 and 9.96×10^-6 respectively [6]. Based on the calculated frequencies, Eq. (3) and the linear modal coordinate transformation in Eq. (6), substituting the proportion damping $C$ in Eq. (2) to dynamics Eq. (1), the damping dynamic response of the pitch control reducer under the nominal load condition is solved with the above Modal Summation Technique, part of the results is illustrated in Fig. 8(a)-(f). Compared to the undamped vibration system, dynamic response of damping vibration obviously decreases. The displacement amplitudes of the ring at the second stage illustrated in Fig. 8(c) and (d) are 0.092 mm in the $x$ direction and 0.283 mm in rotational direction. The peak-peak values are 0.193 mm and 0.621 mm in the $x$ and rotational directions respectively. As illustrated in Fig. 8(e) and (f), the displacement amplitudes of the output shaft are 0.304 mm in the lateral direction and 0.197 mm in rotational direction. The peak-peak value is 0.575 mm and 0.401 mm in the lateral and rotational direction respectively.

5.1. Testing rig

A testing rig is set up to measure the vibration performance of the pitch control reducer under the input speed 1600 r/min and input power 6.4 KW. A tested gearbox of wind turbine pitch control is tested with another accompanied gearbox on the rig as shown in Fig. 9(a). The power is supplied by a direct current motor equipped with electronic speed control, and then transmitted from the torque speed sensor, the tested gearbox, idler and the accompanied gearbox to the direct current generator. The rotational speed decreases through the tested gearbox and then increases through the accompanied gearbox. The vibration sensors, pressure sensors and temperature sensors are all powered and fastened on the tested gearbox. The PLC and a data acquisition card are equipped on the control cabinet orderly. The data collected by the sensors is tackled with the LabView software, and then displayed on the screen as shown in Fig. 9(b).

On the tested gearbox, two vibration sensors were located at the ring at the second stage and output shaft as shown in Fig. 10, three-dimension vibration acceleration in the radial, tangential and axis direction can be tested. The type of vibration sensor is CA-YD-141 in the two positions with 1-6000 Hz frequency response. The tangential displacement of the pitch control reducer in test rig is the rotational displacement in analytic model.

Fig. 8Damping displacement for part of response

a) Rotational displacement $u_{s1}$

b) Radial displacement ${\phi }_{p11}$

c) Lateral displacement $x_{r2}$

d) Rotational displacement $u_{r2}$

e) Lateral displacement $x_z$

f) Rotational displacement $u_z$

Fig. 9Test rig of the pitch control reducer

a) Testing part

b) Operating and display part

Fig. 10Tested positions of the pitch reducer

5.2. Experiment data analysis

The vibration signal collected from the sensors is integrated to obtain the vibration displacement. The radial displacement $r_{r2}$ and tangential displacement $u_{r2}$ for the ring at the second stage are shown in Fig. 11(a) and (b) respectively. The peak displacement $r_{r2}$ and peak-peak value for the ring is 0.104 mm and 0.210 mm respectively, the vibration amplitude of tangential displacement $u_{r2}$ is 0.311 mm, while the peak-peak value is 0.632 mm. The radial displacement $r_z$ and tangential displacement $u_z$ for the output shaft are shown in Fig. 11(c) and (d). The radial peak vibration $r_z$ for the output shaft is 0.293 mm, while the peak-peak value is 0.597 mm, the peak tangential vibration and the peak-peak value of $u_z$ for the output shaft are 0.223 mm and 0.413 mm respectively.

Fig. 11Tested vibration displacement

a) Vertical vibration $x_{r2}$ of the ring at second stage

b) Rotational vibration $u_{r2}$ of the ring at second stage

c) Vertical vibration $x_z$ of the output shaft

d) Rotational vibration $u_z$ of the output shaft

5.3. Validation of the theoretical model

The calculated damping displacement and tested vibration of the ring and the output shaft are illustrated in Table 4. The result shows that the amplitudes for the computation and test are generally similar. The tested tangential displacement amplitude and peak-peak value are close to the analytic rotational amplitude and peak-peak value.

The small deviation of the amplitude for the ring exists between the computation and test because the rigid ring in the analytic model is flexible component in the tested gearbox and that the damping of stirring lubrication oil is not considered in the model. The vibration of the output shaft is influenced by other connected components, the vibration of the test rig and the operating of generator also affect the test vibration, so the tested vibration of the output shaft is little larger than the analytic result.

The experimental result generally agrees well with the theoretical computation and validates the effectiveness of the theoretical model. The lumped-parameter dynamics model for the MW wind turbine gearbox pitch control is fairly precise, and can provide theoretical basis for the research of the dynamics of the planet gearbox.