The compound fault interaction analysis of the planet bearing system

Abstract

This paper aims to understand the bearing distributed–localized compound fault interaction in planetary bearing system. To reveal the compound faults coupling mechanism, the planetary gear bearing distributed–localized fault mathematical models were proposed to obtain the dynamic response in the presence of various planet bearing distributed–localized compound fault combinations. The instantaneous angular speed (IAS) measured from the carrier arm has been pre-processed with the consideration of different distributed–localized fault interactions and treated as inputs for the coherence estimation analysis, which has been used as a tool for the individual fault contribution quantification. Six representative bearing compound fault cases have been selected and the results show that the existence of the bearing compound faults interaction phenomenon can weaken the localized fault characteristic frequencies for the bearing fault diagnosis. However, the localized fault still contributes more than the distributed fault on the same bearing component on the condition that the fault magnitude is the same.

Appendices

Appendix A

Planet bearing characteristic frequency for individual distributed fault and localized fault can be summarized here.

The planet bearing localized fault characteristic frequencies are:

The characteristic frequency of the outer race fault can be calculated as,

$$f_{bpfo}^{c} = \frac{{n_{b} }}{2}\left( {1 - \frac{{D_{b} }}{{D_{p} }}\cos \alpha } \right)f_{o}^{c}$$

(A1)

The characteristic frequency of the inner race fault can be calculated as,

$$f_{bpfi}^{c} = \frac{{n_{b} }}{2}\left( {1 + \frac{{D_{b} }}{{D_{p} }}\cos \alpha } \right)f_{o}^{c}$$

(A2)

The characteristic frequency of the rolling element fault can be calculated as,

$$f_{bf}^{c} = 2f_{b}^{c} = \frac{{D_{p} }}{{D_{b} }}\left[ {1 - \left( {\frac{{D_{b} }}{{D_{p} }}\cos \alpha } \right)^{2} } \right]f_{o}^{c}$$

(A3)

The frequency f_c is the carrier arm absolute rotating frequency and the planet gear absolute rotating frequency is,

$$f_{p} = 1 + \frac{{z_{r} }}{{z_{p} }}f_{c}$$

(A4)

The inner race absolute rotating frequency and the outer race absolute rotating frequency are,

$$\begin{gathered} f_{i} = f_{c} \hfill \\ f_{o} = f_{p} \hfill \\ \end{gathered}$$

(A5)

The outer race relative rotating frequency becomes,

$$f_{o}^{c} = f_{p} - f_{c}$$

(A6)

The rolling element cage rotating frequency becomes,

$$f_{bcg}^{c} = \frac{1}{2}\left( {1 + \frac{{D_{b} }}{{D_{p} }}\cos \alpha } \right)f_{o}^{c}$$

(A7)

The rolling element rotating frequency becomes,

$$f_{b}^{c} = \frac{{D_{p} }}{{2D_{b} }}\left[ {1 - \left( {\frac{{D_{b} }}{{D_{p} }}\cos \alpha } \right)^{2} } \right]f_{o}^{c}$$

(A8)

where D_b is the diameter of a rolling element, D_p is the bearing pitch diameter, α is the contact angle.

The planet bearing distributed fault characteristic frequencies are:

For the inner raceway distributed fault, the resultant vibration peaks can be found in the positions of $k{N}_{b}{f}_{cg}^{c} \left(k=\mathrm{1,2},3,\dots \right)$ regardless of the inner raceway fault orders.

For the outer raceway distributed fault, the resultant vibration peaks were found to be centered around the frequency of $k{N}_{b}\left({f}_{oc}-{f}_{cg}^{c} \right) \left(k=\mathrm{1,2},3,\dots \right)$, which was further modulated by the frequency of ${f}_{cg}^{c}$ or ${f}_{o}$.

For the rolling ball distributed fault, the resultant vibration peaks were found to be centered around the frequency of $2k{f}_{bc}^{c} \left(k=\mathrm{1,2},3,\dots \right)$, which was modulated by the frequency of ${f}_{cg}^{c}$.

Appendix B

For the rotary motion of the motor, the motion equation is,

$$I_{{\text{m}}} \ddot{\theta } = T_{in} - T_{s}$$

$$T_{{\text{s}}} = k_{cp} \left( {\theta_{m} - \theta_{s} } \right) + q_{cp} \left( {\dot{\theta }_{m} - \dot{\theta }_{s} } \right)$$

(B1)

For the motion of the sun gear, the differential equation is,

$$\begin{gathered} {\text{m}}_{s} \ddot{x}_{s} + \sum\limits_{i = 1}^{3} {\left[ {F_{spi} \cos \left( {\varphi_{spi} } \right)} \right]} + k_{sx} x_{s} + q_{sx} \dot{x}_{s} { = } {\text{ m}}_{s} x_{s} \Omega^{2} + 2m_{s} \dot{y}_{s} \Omega + m_{s} y_{s} \dot{\Omega } \hfill \\ \end{gathered}$$

$$\begin{gathered} {\text{m}}_{s} \ddot{y}_{s} + \sum\limits_{i = 1}^{3} {\left[ {F_{spi} \sin \left( {\varphi_{spi} } \right)} \right]} + k_{sx} y_{s} + q_{sx} \dot{y}_{s} { = } {\text{ m}}_{s} {\text{y}}_{s} \Omega^{2} + 2m_{s} \dot{x}_{s} \Omega + m_{s} x_{s} \dot{\Omega } \hfill \\ \end{gathered}$$

$$\left( {\frac{{I_{s} }}{{r_{s} }}} \right)\ddot{\theta }_{s} + \sum\limits_{i = 1}^{3} {F_{spi} = \frac{{T_{s} }}{{r_{s} }}}$$

(B2)

where F_spi is the normal contact force between the sun gear and the ith planet,

$$F_{{{\text{spi}}}} = k_{spi} \delta_{spi} + q_{spi} \dot{\delta }_{spi}$$

$$\begin{gathered} \delta_{{{\text{spi}}}} = \left( {x_{s} - x_{pi} } \right)\cos \varphi_{spi} + r_{s} \theta_{s} + r_{p} \theta_{pi} \ { + }\left( {y_{s} - y_{pi} } \right)\sin \varphi_{spi} - r_{c} \theta_{c} \cos \alpha_{sp} \hfill \\ \end{gathered}$$

$$\varphi_{{{\text{spi}}}} = \frac{\pi }{2} - \alpha_{sp} + \varphi_{i}$$

$$\varphi_{{\text{i}}} = \frac{{2\pi \left( {i - 1} \right)}}{3};i = 1,2,3.$$

(B3)

For the motion of the planet gears, the x and y components become,

$$\begin{gathered} {\text{m}}_{pi} \ddot{x}_{pi} - k_{bp} \left( {x_{bp} - x_{p} } \right) - c_{bp} \left( {\dot{x}_{bp} - \dot{x}_{p} } \right){ + }F_{{{\text{spi}}}} \cos \varphi_{spi} + F_{rpi} \cos \varphi_{rpi} \hfill \\ \,\quad\quad\quad = m_{pi} x_{pi} \Omega^{2} { + }2{\text{m}}_{pi} \dot{y}_{pi} \Omega + m_{pi} y_{pi} \dot{\Omega } + m_{pi} r_{c} \Omega^{2} \cos \varphi_{i} \hfill \\ \end{gathered}$$

$$\begin{gathered} {\text{m}}_{pi} \ddot{y}_{pi} - k_{kb} \left( {y_{bp} - y_{p} } \right) - c_{bp} \left( {\dot{y}_{bp} - \dot{y}_{p} } \right) - F_{{{\text{spi}}}} \sin \varphi_{spi} - F_{rpi} \sin \varphi_{rpi} \hfill \\ \,\quad\quad\quad = m_{pi} y_{pi} \Omega^{2} { - }2{\text{m}}_{pi} \dot{x}_{pi} \Omega - m_{pi} x_{pi} \dot{\Omega } + m_{pi} r_{c} \Omega^{2} \sin \varphi_{i} \hfill \\ \end{gathered}$$

$$\left( {\frac{{I_{pi} }}{{r_{p} }}} \right)\ddot{\theta }_{pi} + F_{spi} - F_{rpi} = 0$$

(B4)

where F_rpi is the normal contact force between the ring gear and the ith planet,

$$F_{{{\text{rpi}}}} = k_{rpi} \delta_{rpi} + q_{rpi} \dot{\delta }_{rpi}$$

$$\begin{gathered} \delta_{{{\text{rpi}}}} = \left( {x_{r} - x_{pi} } \right)\cos \varphi_{rpi} + r_{r} \theta_{r} - r_{p} \theta_{pi} { + }\left( {y_{r} - y_{pi} } \right)\sin \varphi_{rpi} - r_{c} \theta_{c} \cos \alpha_{rp} \hfill \\ \end{gathered}$$

$$\varphi_{{{\text{rpi}}}} = \frac{\pi }{2} + \alpha_{rp} + \varphi_{i}$$

(B5)

For the motion of the ith planet gear bearing, the inner race can be modelled as a two degree of freedom system that has translational motion in x and y directions. The equation of motion of the inner race can be written as,

$$\begin{gathered} {\text{m}}_{ibs} \ddot{x}_{ibs} + F_{cpix} - f_{ibx} \hfill \\ =\, m_{ibs} x_{ibs} \Omega^{2} + 2m_{ibs} \dot{y}_{ibs} \Omega + m_{ibs} y_{ibs} \dot{\Omega } + m_{ibs} r_{c} \Omega^{2} \cos \varphi_{i} \hfill \\ \end{gathered}$$

$$\begin{gathered} {\text{m}}_{ibs} \ddot{y}_{ibs} + F_{cpiy} - f_{iby} \hfill \\ = \, m_{ibs} y_{ibs} \Omega^{2} - 2m_{ibs} \dot{x}_{ibs} \Omega + m_{ibs} x_{ibs} \dot{\Omega } + m_{ibs} r_{c} \Omega^{2} \sin \varphi_{i} \hfill \\ \end{gathered}$$

(B6)

The outer race is modelled as a two degree of freedom system that has translational motion in the x and y directions. The equation of motion of the outer race can be written as,

$$\begin{gathered} m_{ibp} \ddot{x}_{ibp} + k_{bp} \left( {x_{bp} - x_{p} } \right) + c_{bp} \left( {\dot{x}_{bp} - \dot{x}_{p} } \right) + f_{ibx} \hfill \\\quad\quad\quad = \, m_{ibp} x_{ibp} \Omega^{2} + 2m_{ibp} \dot{y}_{ibp} \Omega + m_{ibp} y_{ibp} \dot{\Omega } + m_{ibp} r_{c} \Omega^{2} \cos \varphi_{i} \hfill \\ \end{gathered}$$

$$\begin{gathered} m_{ibs} \ddot{y}_{ibp} + k_{bp} (y_{bp} - y_{p} ) + c_{bp} (\dot{y}_{bp} - \dot{y}_{p} ) = \, m_{ibp} y_{ibp} \Omega^{2} - 2m_{ibp} \dot{x}_{ibp} \Omega \,- m_{ibp} x_{ibp} \dot{\Omega } + m_{ibp} r_{c} \Omega^{2} \sin \varphi_{i} \hfill \\ \end{gathered}$$

(B7)

The sprung mass is attached in the y direction. Its equation of motion can be written as,

$$m_{ibr} \ddot{y}_{ibs} - k_{br} \left( {y_{bp} - y_{br} } \right) - c_{br} \left( {\dot{y}_{br} - \dot{y}_{br} } \right) = 0$$

(B8)

where F_cpix and F_cpiy describe the interaction force between the ith planet bearing and the carrier arm in the x and y directions,

$$F_{{{\text{cpix}}}} = k_{ibs} \left( {x_{ibs} - x_{c} } \right) + q_{ibs} \left( {\dot{x}_{ibs} - \dot{x}_{c} } \right)$$

$$F_{{{\text{cpiy}}}} = k_{ibs} \left( {y_{ibs} - y_{c} } \right) + q_{ibs} \left( {\dot{y}_{ibs} - \dot{y}_{c} } \right)$$

(B9)

f_ibx, f_iby describe the summation of the contact forces in the x- and x- directions for a ball bearing with n_b balls and can be calculated as,

$$f_{ibx} = k_{b} \sum\limits_{j = 1}^{{n_{b} }} {\gamma_{ibj} \delta_{ibj}^{1.5} } \cos \phi_{ibj}$$

$$f_{{ib{\text{y}}}} = k_{b} \sum\limits_{j = 1}^{{n_{b} }} {\gamma_{ibj} \delta_{ibj}^{1.5} } \sin \phi_{ibj}$$

(B10)

where k_b is the combined contact stiffness between the rolling element and the races and γ_ibj is the contact state. δ_ibj and Φ_ibj are the overall contact deformation as well as the angular position for the jth rolling element on the ith planet gear bearing, respectively, and they can be described as,

$$\delta_{{{\text{ibj}}}} { = }\left( {x_{ibs} - x_{ibp} } \right)\cos \phi_{ibj} - c + (y_{ibs} - y_{ibp} )\sin \phi_{ibj} {\text{ + w}}^{i}$$

$$j = 1,2,3,...,n_{b}$$

$$\phi_{ibj} = \frac{{2\pi \left( {j - 1} \right)}}{{n_{b} }} + w_{ibc}^{c} dt + \phi_{ibo}$$

(B11)

${\Phi }_{\mathrm{ibo}}$ is the initial cage position and ${\upomega }_{\mathrm{ibc}}^{\mathrm{c}}$ is the cage speed measured in the rotating frame, which can be calculated as,

$$w_{ibc} = w_{ibc}^{c} = \frac{{w_{ii} }}{2}\left( {1 - \frac{{D_{b} }}{{D_{p} }}\cos \alpha } \right) + \frac{{w_{io} }}{2}\left( {1 + \frac{{D_{b} }}{{D_{p} }}\cos \alpha } \right) - w_{c}$$

(B12)

For the motion of the carrier arm, the motion equation is,

$$m_{c} \ddot{x}_{c} + k_{cx} x_{c} + q_{cx} \dot{x}_{c} - \sum\limits_{i = 1}^{3} {F_{cpix} } = m_{c} x_{c} \Omega^{2} + 2m_{c} \dot{y}_{c} \Omega + m_{c} y_{c} \dot{\Omega }$$

$$m_{c} \ddot{y}_{c} + k_{cy} y_{c} + q_{cy} \dot{y}_{c} - \sum\limits_{i = 1}^{3} {F_{cpiy} } = m_{c} y_{c} \Omega^{2} - 2m_{c} \dot{x}_{c} \Omega - m_{c} x_{c} \dot{\Omega }$$

$$\left( {\frac{{I_{c} }}{{r_{c} }}} \right)\ddot{\theta }_{c} + \sum\limits_{i = 1}^{3} {F_{cpix} } \sin \varphi_{i} - \sum\limits_{i = 1}^{3} {F_{cpiy} } \cos \varphi_{i} = \frac{{T_{c} }}{{r_{c} }}$$

(B13)

$$T_{c} = k_{cg} \left( {\theta_{c} - \theta_{out} } \right) + q_{cg} \left( {\dot{\theta }_{c} - \dot{\theta }_{out} } \right)$$

For the motion of the ring gear, the motion equation is,

$$\begin{gathered} m_{r} \ddot{x}_{r} + k_{rx} x_{r} + q_{rx} \dot{x}_{r} + \sum\limits_{i = 1}^{3} {F_{rpi} \cos \varphi_{rpi} } = \, m_{r} x_{r} \Omega^{2} + 2m_{r} \dot{y}_{r} \Omega + m_{r} y_{r} \dot{\Omega } \hfill \\ \end{gathered}$$

$$\begin{gathered} m_{r} \ddot{y}_{r} + k_{ry} y_{r} + q_{ry} \dot{y}_{r} + \sum\limits_{i = 1}^{3} {F_{rpi} \sin \varphi_{rpi} } = \, m_{r} y_{r} \Omega^{2} - 2m_{r} \dot{x}_{r} \Omega - m_{r} x_{r} \dot{\Omega } \hfill \\ \end{gathered}$$

$$\left( {\frac{{I_{r} }}{{r_{r} }}} \right)\ddot{\theta }{}_{r} + \left( {\frac{{q_{rt} }}{{r_{r} }}} \right)\dot{\theta }_{r} + \left( {\frac{{k_{rt} }}{{r_{r} }}} \right)\theta_{r} + \sum\limits_{i = 1}^{3} {F_{rpi} = 0}$$

(B14)

For the rotary motion of the load, the motion equation is,

$$I_{out} \ddot{\theta }_{out} = - T_{out} + T_{c}$$

(B15)