Dynamic coupled vibration analysis of a large wind turbine gearbox transmission system

2.1. Lumped-parameter analytical model of helical gear

Considering the influence of input/output and support bearing, et al., the static model of helical gear-shaft-bearing system is shown in Fig. 3. The dynamic model of the transmission system, as shown in Fig. 4, is established. $m_1$ and $m_2$ are the equivalent mass of helical gears respectively. $J_1$ and $J_2$ represent moment of inertia; $m_{bi}$ ($i=$ 1-4) is equivalent mass at bearing position. $J_d$ and $J_l$ indicate moment of inertia in input/output terminal. ${\rho }_1$ and ${\rho }_2$ represent the eccentricity. $F_{xi}$, $F_{yi}$ and $F_{zi}$ ($i=$ 1-4) are nonlinear bearing forces in $x$, $y$ and $z$ directions.

Fig. 3Static model of helical gear-rotor-bearing system

Fig. 4Dynamic mode of helical gear-rotor-bearing system

In Fig. 3, the fixed coordinate system $A_i$-$x_i{y}_i{z}_i$ ($i=$ 1, 2) is set up at $A_i$, which is the ideal center of the helical gears. $B_i$ ($i=$ 1-4) represents the ideal center of bearing. $O_1(x_1,y_1,z_1)$ and $O_2(x_2,y_2,z_2)$ are the geometric rotational centers of the driving and driven gears; $G_1(x_{g1},y_{g1},z_{g1})$ and $G_2(x_{g2},y_{g2},z_{g2})$ represent centroid respectively. To deduce kinetic equations of the helical gear-shaft-bearing system, the stress of the cylindrical gear with helical teeth should be analyzed primarily. Therefore, the dynamic model of gear meshing is shown in Fig. 4. ${\mathrm{\Phi }}_1$ and ${\theta }_1$ ($j=$ 1, 2, $d$, $l$) are the rotation angle and angular vibration displacement of driving gear, driven gear, input terminal and output terminal. $r_{b1}$ and $r_{b2}$ are the base radius of driving and driven gears respectively; $F_t$, $F_r$ and $F_a$ represent the tangential force; radial force and axial force. ${\alpha }_t$ and ${\alpha }_n$ are the pressure angles of the pitch circle in the end face and normal direction. $\beta$ and ${\beta }_t$ are helix angles of pitch circle and base circle. ${\alpha }_1$ indicates the angle between center line and vertical direction.

According to the previous geometrical relationship, the angle displacements of the input/output, driving and driven gears can be expressed by the following equations:

The relationships between $G_1\left(x_{g1},y_{g1},z_{g1}\right)\text{,}$$G_2(x_{g2},y_{g2},z_{g2})$ and the $O_1\left(x_1,y_1,z_1\right)\text{,}$$O_2(x_2,y_2,z_2)$ are expressed as follows:

In order to ensure the contact of teeth surface on meshing performance, it is assumed that the relative deformation of gear pair is completely changed into elastic deformation on teeth surface along with the meshing direction. The meshing gear pair is connected through the spring and damper [20]. Therefore, the comprehensive deformation of two helical gears in the meshing direction is expressed as:

Based on the viscoelasticity theory, the meshing force of gear can be described as:

where $k_m$ and $c_m$ represent the average meshing stiffness and damping; $r_{b1}$ and $r_{b2}$ indicate the base radius of helical gears; $e\left(t\right)$ is the meshing error.

It is assumed that the dynamic meshing force which is from the driving gear to the driven gear is positive. So the meshing forces in $x$, $y$ and $z$ directions can be described respectively as:

2.2. Lumped-parameter analytical model of planetary gear

The planet carrier is regarded as the input terminal, and the sun gear is the output terminal. The gear mesh is modeled as linear spring acting along the pressure line direction. The other supports/bearings are modeled as nonlinear springs. The friction forces due to gear teeth contact and other dissipative effects are captured using modal damping. Therefore the translational-rotational model of the planetary gear system is shown in Fig. 5.

Fig. 5The dynamic model of planetary transmission system

In Fig. 5, the $OXY$ is the fixed-coordinate system. $Oxy$ represents the motional-coordinate system of the planet carrier. The $O{x}_i$ points to the theory center of the planet gear, and the ideal angular velocity of the carrier is ${\omega }_c$. $s$, $r$, $c$, $p_i$ are the subscripts of the sun, ring, carrier and planet respectively, and the subscripts from 1$-i$ ($i=$ 1, 2, 3) designate the planets. $x_j$, $y_j$ ($j=s$, $r$, $c$) represent the lateral displacements (the projections in the fixed coordinate system OXY). $x_{pi}$, $y_{pi}$($i=$ 1, 2, 3) indicate the vibration displacements projections of the $i$th planet gear in $o_{pi}{x}_{pi}{y}_{pi}$ coordinate system. $r_{bj}$ ($j=s$, $r$, $c$, $p_i$) is the base radius of gears or radius of the carries. ${\theta }_j$ ($j=s$, $r$, $c$, $p_i$) represents the rotational displacement of the sun, ring, carrier and planet respectively. Besides, it is assumed that the counterclockwise is the positive direction. ${\phi }_i$ is the circumferential position of the $i$th planet around the sun gear and ${\phi }_i=2\pi (i-1)/n$, $\alpha$ is the sun-planet and ring-planet operating pressure angles. $k_{sj}$, $c_{sj}$ ($j=s$, $t$, $b$) are the bending, torsional of sun gear and the bearing stiffness and damping in $x$-direction and $y$-direction respectively, $k_{rj}$, $c_{rj}$ ($j=s$, $t)$ represent the bending of ring and the bearing stiffness and damping in $x$-direction and $y$-direction, $k_{cj}$, $c_{cj}$ ($j=s$, $t$, $b$) indicate the bending, torsional of the carrier and the bearing stiffness and damping in $x$-direction and $y$-direction respectively, $k_{bpi}$, $c_{bpi}$ are the $i$th bearing stiffness and damping of planet gear in lateral-direction, $k_{pis}$, $c_{pis}$, $k_{pir}$$c_{pir}$ are the mesh stiffness and damping of sun-planet and ring-planet in the pressure line direction.

2.3. Vibration differential equations of the wind turbine gearbox transmission system

Based on the dynamic models of the helical gear and the planetary gear transmission, a generalized displacement vector of the whole gearbox transmission system can be defined with respect to the global co-ordinate system by:

where ${\theta }_i$ ($i=c$, $r$, $p_i$, $s$, 1, 2, 3, 4) represent the angular vibration displacement respectively, $x_i$, $y_i$, $z_i$ ($i=c$, $r$, $p_i$, $s$, 1, 2, 3, 4) are the linear vibration displacements in $x$, $y$, $z$ directions.

According to the dynamic analysis of the wind turbine gearbox, taking into the mesh stiffness, input/output torque, the kinetic energy $T$, the potential energy $U$ and the dissipation function $R$, are established of the gearbox system. Utilizing the Lagrange equation, the differential equations of the train system can be expressed as follows:

The vibration differential equation of the carrier:

The vibration differential equation of the ring gear:

The vibration differential equation of the $i$th planet gear:

The vibration differential equation of the sun gear:

The vibration differential equation of the intermediate-speed driving gear:

The vibration differential equation of the intermediate-speed driven gear:

The vibration differential equation of the high-speed driving gear:

The vibration differential equation of the high-speed driven gear:

3.1. External excitation

According to the aerodynamic theory [21-22], taking the AR wind speed model as an external excitation, the excitation can be expressed as follows:

where $P_{in}$ is the input power of the wind turbine, $\rho$ represents the air density, $R$ indicates the radius of the wind turbine, $v\left(t\right)$ is the stochastic wind speed, $C_p$ is power coefficient.

The input torque and output torque of the wind turbine can be expressed by the following equations:

where $\omega$ is the angular velocity of the wind turbine, $i$ represents the transmission ratio of the gearbox.

According to the foregoing formulas, the simulation results of the wind speed and the wind turbine input torque are shown in Fig. 6.

Fig. 6Random wind speed and driving torque

a) Waveform of wind speed

b) Waveform of input torque

3.2. Internal excitation

3.2.1. Dynamic model of the angular contact rolling bearing

Angular contact ball bearing model is shown in Fig. 7. Bearing outer ring is fixed in bearing chock, and inner ring is fixed in shaft. The balls are uniformly distributed between outer ring and inner ring. The velocities $v_i$ and $v_0$ at the contact point between the rolling ball and outer/inner ring was analyzed, which can be expressed as:

20

$v_i={\omega }_{i}r,\mathrm{}\mathrm{}\mathrm{}\mathrm{}{v}_o={\omega }_{o}R,$

where $R$ and $r$ represent the radius of the bearing outer and inner rings; ${\omega }_i$ and ${\omega }_0$ are the angular velocities.

Assuming that it is the pure rolling between ball and outer/inner ring, and the velocity of cage can be expressed:

21

$v_b=\frac{\left(v_o+v_i\right)}{2}=\frac{\left({\omega }_{o}R+{\omega }_{i}r\right)}{2}.$

Generally, the bearing inner ring rotates together with shaft, and the bearing outer ring is fixed. Utilizing the relationships ${\omega }_0=$0, ${\omega }_i=\omega$. So the angular velocity of the cage is as follows:

22

${\omega }_b=\frac{2v_b}{(R+r)}=\frac{{\omega }_{i}r}{\left(R+r\right)}.$

The rotation angle ${\phi }_i$ of the $i$th rolling ball at $t$ moment can be expressed:

23

${\phi }_i={\omega }_{b}t+\frac{2\pi \left(i-1\right)}{N_b},\left(i=1,2,3,\dots ,N_b\right),$

where $N_b$ represents the number of rolling ball.

In Fig. 7, $d_b$ is the diameter of rolling ball, $d$ indicates the diameter of shaft, $d_i$ and $d_o$ represent the diameters of the bearing outer and inner orbits, $d_m$ is the diameter of pitch circle, and $d_m=(d_i+d_o)/2$.

Fig. 7Schematic diagram of angular contact rolling bearing

When the angular contact ball bearing is at a low speed, the centrifugal force and gyroscopic moment can be neglected. $F_r$ and $F_a$ represent the radial force and axial force of bearing respectively. The ball bearing and geometric deformation of different position are shown in Fig. 8. ${\alpha }_0$ is the initial contacting angular of before being loaded, ${\alpha }_0^{\text{'}}$ indicates the angular after being loaded; $P_0$ represents the center of the bearing outer ring before being loaded; $P_0^{\text{'}}$ is the one after being loaded. Because the outer ring is fixed, the $P_0$ and $P_0^{\text{'}}$ are the same position; $P_i$ is the center of inner ring before being loaded; $P_i^{\text{'}}$ represents the after being loaded; ${\delta }_{ai}$, ${\delta }_{ri}$ and ${\theta }_i$ are the axial deformation, radial deformation and angular deformation respectively; $R_i$ is the curvature radius of inner orbit; ${\phi }_i$ is the position angular; ${\delta }_i$, ${\delta }_0$ and ${\delta }_{bi}$ represent the contacting deformation and total deformation between rolling ball and outer and inner orbits.

Fig. 8Ball bearing and deformation of different positions

a) Before being loaded

b) After being loaded

c) Geometric deformation relation

Because the centrifugal force and gyroscopic moment are ignored, the contacting angular between rolling ball and bearing orbit and the contacting force are the same. In Fig. 8(b), the deformation of rolling ball at position angular ${\phi }_i$ can be expressed as:

24

${\delta }_{bi}={\delta }_i+{\delta }_o=S-A.$

According to geometric relationship in Fig. 8(c), the distance $S$ between the curvature center of inner orbit and outer orbit can be expressed by the following equation:

25

$S={\left[{\left(A\mathrm{s}\mathrm{i}\mathrm{n}{\alpha }_o+z+R_i{\theta }_i\mathrm{c}\mathrm{o}\mathrm{s}{\phi }_i\right)}^2+{\left(A\mathrm{c}\mathrm{o}\mathrm{s}{\alpha }_o+x\mathrm{c}\mathrm{o}\mathrm{s}{\phi }_i+y\mathrm{s}\mathrm{i}\mathrm{n}{\phi }_i\right)}^2\right]\mathrm{}}^{1/2}.$

The normal contacting deformation between the $i$th rolling ball and the bearing orbit is as follow:

26

${\delta }_{bi}=S-A={\left[{\left(A\mathrm{s}\mathrm{i}\mathrm{n}{\alpha }_o+{\delta }_a+R_i{\theta }_i\mathrm{c}\mathrm{o}\mathrm{s}{\phi }_i\right)}^2+{\left(A\mathrm{c}\mathrm{o}\mathrm{s}{\alpha }_o+{\delta }_r\right)}^2\right]}^{1/2}-A.$

Utilizing the relationships ${\delta }_a=z$, ${\delta }_r=x\mathrm{c}\mathrm{o}\mathrm{s}{\phi }_i+y\mathrm{s}\mathrm{i}\mathrm{n}{\phi }_i$, the Eq. (26) can be expressed by:

27

${\delta }_{bi}=S-A=[{\left(A\mathrm{s}\mathrm{i}\mathrm{n}{\alpha }_o+z+R_i{\theta }_i\mathrm{c}\mathrm{o}\mathrm{s}{\phi }_i\right)}^2+{\left(A\mathrm{c}\mathrm{o}\mathrm{s}{\alpha }_o+x\mathrm{c}\mathrm{o}\mathrm{s}{\phi }_i+y\mathrm{s}\mathrm{i}\mathrm{n}{\phi }_i\right)}^2{]}^{1/2}-A,$

where $x\text{,}$$y$ and $z$ represent the linear vibration displacement respectively; $R_i=d_m/2+\left(r_i-\left(d_b/2\right)\right)\mathrm{c}\mathrm{o}\mathrm{s}{\alpha }_0$; $A$ is the initial distance between curvature center of inner orbit and outer orbit, and $A=r_i+r_0+{\gamma }_0-d_b$.

At the positon angular ${\phi }_i$, the actually contacting angular ${\alpha }_0^{\text{'}}$ can be expressed as:

28

$\mathrm{t}\mathrm{a}\mathrm{n}{\alpha }_o^{\mathrm{\text{'}}}=\frac{A\mathrm{s}\mathrm{i}\mathrm{n}{\alpha }_o+z+R_i{\theta }_i\mathrm{c}\mathrm{o}\mathrm{s}{\phi }_i}{A\mathrm{c}\mathrm{o}\mathrm{s}{\alpha }_o+x\mathrm{c}\mathrm{o}\mathrm{s}{\phi }_i+y\mathrm{s}\mathrm{i}\mathrm{n}{\phi }_i}.$

According to nonlinear Hertz contact theory, the contact force between the $i$th rolling ball and the bearing orbits is $f_{bi}$, at the same time, the normal stress can be only generated between the rolling ball and the bearing orbits when the ${\delta }_{bi}$ is more than zero. So the force can be expressed by:

29

$f_{bi}=K_c({\delta }_{bi}{)}^{3/2}\cdot H\left({\delta }_{bi}\right)$
$\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}=K_c{\left\{{\left[\right(A\mathrm{s}\mathrm{i}\mathrm{n}{\alpha }_o+z+R_i{\theta }_i\mathrm{c}\mathrm{o}\mathrm{s}{\phi }_i{)}^2+\left(A\mathrm{c}\mathrm{o}\mathrm{s}{\alpha }_o+x\mathrm{c}\mathrm{o}\mathrm{s}{\phi }_i+y\mathrm{s}\mathrm{i}\mathrm{n}{\phi }_i{)}^2\right]}^{1/2}-A\right\}}^{3/2}$
$\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\cdot H\left\{\left[\right(A\mathrm{s}\mathrm{i}\mathrm{n}{\alpha }_o+z+R_i{\theta }_i\mathrm{c}\mathrm{o}\mathrm{s}{\phi }_i{)}^2+{\left(A\mathrm{c}\mathrm{o}\mathrm{s}{\alpha }_o+x\mathrm{c}\mathrm{o}\mathrm{s}{\phi }_i+y\mathrm{s}\mathrm{i}\mathrm{n}{\phi }_i\right)}^2{]}^{1/2}-A\right\}.$

The normal load can be known by the Eq. (29), and the axial and radial forces can be expressed as below:

30

$f_{ri}=f_{bi}\mathrm{c}\mathrm{o}\mathrm{s}{\alpha }_0^{\mathrm{\text{'}}}=K_c({\delta }_{bi}{)}^{3/2}\mathrm{c}\mathrm{o}\mathrm{s}{\alpha }_0^{\mathrm{\text{'}}},f_{ai}=f_{bi}\mathrm{s}\mathrm{i}\mathrm{n}{\alpha }_0^{\mathrm{\text{'}}}=K_c({\delta }_{bi}{)}^{3/2}\mathrm{s}\mathrm{i}\mathrm{n}{\alpha }_0^{\mathrm{\text{'}}}.$

So the bearing forces ($F_{bx}$, $F_{by}$ and $F_{bz}$) in $x$, $y$ and $z$ directions can be expressed by:

31

$F_{bx}=\sum _{i=1}^{N_b}{f}_{ri}\mathrm{c}\mathrm{o}\mathrm{s}{\phi }_i=\sum _{i=1}^{N_b}{K}_c({\delta }_{bi}{)}^{3/2}\mathrm{c}\mathrm{o}\mathrm{s}{\alpha }_0^{\mathrm{\text{'}}}\mathrm{}H\left({\delta }_{bi}\right)\mathrm{c}\mathrm{o}\mathrm{s}{\phi }_i,$
$F_{by}=\sum _{i=1}^{N_b}{f}_{ri}\mathrm{s}\mathrm{i}\mathrm{n}{\phi }_i=\sum _{i=1}^{N_b}{K}_c({\delta }_{bi}{)}^{3/2}\mathrm{c}\mathrm{o}\mathrm{s}{\alpha }_0^{\mathrm{\text{'}}}H\left({\delta }_{bi}\right)\mathrm{s}\mathrm{i}\mathrm{n}{\phi }_i,$
$F_{bz}=\sum _{i=1}^{N_b}{f}_{ai}=\sum _{i=1}^{N_b}{K}_c({\delta }_{bi}{)}^{3/2}\mathrm{s}\mathrm{i}\mathrm{n}{\alpha }_0^{\mathrm{\text{'}}}H\left({\delta }_{bi}\right).$

4.1. Dynamic response of the wind turbine gearbox

By applying the coupled vibration model of a wind turbine gearbox transmission system, the vibration analysis and calculations are carried out. In this paper, the dynamic response of the MW wind turbine gearbox with the external load excitation, meshing stiffness, input and output torques, meshing error, nonlinear characteristics of the support bearing and gravity effects are investigated using the Runge-Kutta numerical simulation method that includes simultaneous internal and external excitations. This method using the elastic mechanics theory is applied to a lumped-parameter wind turbine gearbox model. The coupled dynamic mechanical analysis of the system is performed so as to include all the components of influence factors.

Fig. 9Vibration waveform of the planet and sun

a1)

b1)

a2)

b2)

Fig. 9-Fig. 11 and Fig. 12-Fig. 14 show the vibration response of the partial component in $x$, $y$, $\theta$ directions, in which the carrier driver, having a rated speed of 17 r.p.m., drives the wind turbine gearbox to rotate. The response curves (vibration waveform and spectrum) of components are made appropriate interception in order to analyze the dynamic characteristics of the wind turbine gearbox train system more accurately. The lateral vibration displacement response of planet gear and sun gear in the low-speed stage is shown in Fig. 9. The negative value in figure indicates the opposite direction with respect to the predetermined positive direction in the model. In this paper, it can be found from comparison with Fig. 9-Fig. 11 that the planet gear has the minimum vibration amplitude, the vibration amplitude of medium-speed helical gear stage takes second place, and the vibration amplitude of high-speed helical gear stage has the maximum. In addition, the vibration waveform of each component in the train system presents a different changing trend due to the influences of the internal and external excitations.

Fig. 10Vibration waveform of the gear in medium-speed stage and high-speed stage

a1)

b1)

c1)

a2)

b2)

c2)

a3)

b3)

c3)

a4)

b4)

c4)

It can be seen from Fig. 9 that the dynamic characteristics of the planet gear and sun gear in low-speed stage become more complex due to the time-varying load and nonlinear characteristics of the support bearing. The vibration responses of the planet and sun are significant higher than others in low-speed planetary gear stage. (Because of the limited space, this paper only gives the vibration response of planet and sun). Moreover, high-harmonic components are significant shown in time-domain response. It can be found from comparison Fig. 9 with Fig. 6 that the vibration responses of the planet and sun are similar change with the external excitation.

Fig. 10 shows the lateral and axial vibration displacement responses of the medium-speed helical gear stage and high-speed helical gear stage. It can be seen from Fig. 10 that the vibration responses of driving gear in $x$, $y$, $z$ directions are slight higher than one of the driven gear. Comparing Fig. 10 with Fig. 9, it can be seen that the vibration response of the medium-speed helical gear stage become more complex due to the effects on the low-speed planetary gear stage and high-speed helical gear. In addition, it can be also found the low-speed stage is mainly affected by the external load, and the vibration of the high-speed stage mainly causes by load. The vibration amplitudes are also obviously increasing and the magnitude of amplitude reach the 0.1 mm due to the effects of the internal and external excitation. The support bearings in the low-speed stage and high-speed stage inevitability stand heavy dynamic load due to the effect of the gear eccentricity, mesh error and gravity. Hence, the bearing appears earlier failure. The dynamic response analysis for the wind turbine gearbox should fully be considered the influences of the bending vibration, torsional vibration, axial vibration and coupled vibration.

Fig. 11 shows angular vibration displacement response of the partial component. This comparison proves that the torsional vibration displacement of each component is obviously higher than those of lateral and axial vibration displacement response. So the torsional vibration is the main vibration and torsional amplitude obviously appears different vibration features due to the effects of the external excitation. In addition, the torsional vibration responses of the components appear complex vibration characteristics, which are caused by the nonlinear features of the support bearing and other excitations. In fact, bearing is one of main factor controlling vibration in a power transmission system because they play a key role in determining the system’s natural frequencies, and they can also be responsible for vibration amplification and transmission to other parts.

Fig. 11Torsional vibration of the partial component in gearbox

a)

b)

c)

d)

4.2. Frequency response of the gearbox

In order to accurately analyze the dynamic of the wind turbine gearbox transmission system, Figs. 12-14 show the frequency domain response of the partial component in lateral-direction, axial-direction and torsional-direction. It can be seen that the frequency domain response appear a lot of frequency multiplication and frequency combination components. The meshing frequency component of the low-speed stage is $f_m$ ($f_m=f_c{z}_r=$ 29.45 Hz), medium-meshing frequency component is $f_{m1}$ ($f_{m1}=z_1{n}_1/60=$163.6Hz), and the meshing frequency component of the high-speed stage $f_{m2}$ ($f_{m2}=z_3{n}_3/60=$707 Hz) in the wind turbine gearbox transmission system. Fig. 12 show the lateral frequency domain response of planet gear and sun gear in the low-speed stage. Analysis of the Fig. 12(a), (b) shows that there is a certain degree of continuous spectrum at low-frequency stage due to the effects of the support bearing and the meshing between the gears, which mainly includes frequency combination $\text{(2}{f}_m+f_{bc}\text{,}$$\text{4}{f}_m+f_{bp}\text{,}$$\text{5}{f}_m+\text{3}{f}_{bp}+f_{bc}$,…) and frequency multiplication. Besides, it also appears a high-harmonic component 536.5 Hz ($\text{18}{f}_m+\text{3}{f}_s+f_p$). For the sun gear spectrum Fig. 12(c), (d), the frequency components are relatively simple, but it also appears complex frequency combination ($\text{2}{f}_m+f_s$, $\text{4}{f}_m+\text{10}{f}_p$, $\text{5}{f}_m+\text{3}{f}_{bs}-f_s$,…) and frequency multiplication. In addition to the high-harmonic component 536.5 Hz, there is also another obvious high-harmonic component 571.5 Hz ($\text{19}{f}_m+f_{bs}$). The result indicate that the vibration frequency low-speed stage has a very obvious probability distribution, which is mainly concentrated in the frequency range below 200 Hz. This phenomenon is caused by the nonlinear characteristics of the support bearing and external excitation.

Fig. 12Lateral spectrogram of the planet and sun in gearbox

a)

b)

c)

d)

Fig. 13 shows the lateral and axial frequency domain response of the medium-speed helical gear stage and high-speed helical gear stage. It can be seen from Fig. 13 that the own rotating frequency and the rotating frequency with meshing gear are the main frequencies in $x$, $y$, $z$ directions. For example, the driving rotating frequency $f_1$ ($f_1=n_1/60=$ 1.57 Hz) and the driven rotating frequency $f_2$ ($f_2=n_2/60=$7.13 Hz) in medium-speed helical gear stage. The driving rotating frequency $f_3$${\text{(}f}_3=n_3/60=\text{7.13 Hz})$ and the driven rotating frequency $f_4$${\text{(}f}_4=n_4/60=$28.71 Hz) in high-speed helical gear stage. The meshing frequencies $f_{m1}$ ($f_{m1}=z_1{n}_1/60=$ 163.6 Hz) and ($f_{m2}=z_3{n}_3/60=$ 707 Hz) are not immediately obvious, which aren’t mark in the spectrogram.