Gear fault diagnosis using gear meshing stiffness identified by gearbox housing vibration signals

doi:10.1007/s11465-022-0713-3

Front. Mech. Eng.

2022, Vol. 17

Issue (4) : 57 https://doi.org/10.1007/s11465-022-0713-3

RESEARCH ARTICLE

Gear fault diagnosis using gear meshing stiffness identified by gearbox housing vibration signals

Xiaoluo YU¹, Yifan HUANGFU¹, Yang YANG², Minggang DU², Qingbo HE¹(

), Zhike PENG¹

¹. State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University, Shanghai 200240, China
². Science and Technology on Vehicle Transmission Laboratory, China North Vehicle Research Institute, Beijing 100072, China

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Abstract

Gearbox fault diagnosis based on vibration sensing has drawn much attention for a long time. For highly integrated complicated mechanical systems, the intercoupling of structure transfer paths results in a great reduction or even change of signal characteristics during the process of original vibration transmission. Therefore, using gearbox housing vibration signal to identify gear meshing excitation signal is of great significance to eliminate the influence of structure transfer paths, but accompanied by huge scientific challenges. This paper establishes an analytical mathematical description of the whole transfer process from gear meshing excitation to housing vibration. The gear meshing stiffness (GMS) identification approach is proposed by using housing vibration signals for two stages of inversion based on the mathematical description. Specifically, the linear system equations of transfer path analysis are first inverted to identify the bearing dynamic forces. Then the dynamic differential equations are inverted to identify the GMS. Numerical simulation and experimental results demonstrate the proposed method can realize gear fault diagnosis better than the original housing vibration signal and has the potential to be generalized to other speeds and loads. Some interesting properties are discovered in the identified GMS spectra, and the results also validate the rationality of using meshing stiffness to describe the actual gear meshing process. The identified GMS has a clear physical meaning and is thus very useful for fault diagnosis of the complicated equipment.

Keywords gearbox fault diagnosis meshing stiffness identification transfer path signal processing

Corresponding Author(s): Qingbo HE

Just Accepted Date: 15 June 2022 Issue Date: 23 November 2022

Cite this article:

Xiaoluo YU,Yifan HUANGFU,Yang YANG, et al. Gear fault diagnosis using gear meshing stiffness identified by gearbox housing vibration signals[J]. Front. Mech. Eng., 2022, 17(4): 57.

URL:

https://academic.hep.com.cn/fme/EN/10.1007/s11465-022-0713-3
https://academic.hep.com.cn/fme/EN/Y2022/V17/I4/57

Fig.1 Gearbox vibration modeling by combining dynamics and transfer path analysis.

Fig.2 Assembly method of the gear?shaft?bearing system matrix.

Fig.3 Flowchart of gear meshing stiffness identification with its application.

Tab.1 Parameters of gears and shafts

Fig.4 Simulated gear meshing stiffness (a) waveforms and (b) spectra of healthy and two typical gear faults.

Fig.5 Simulated housing vibration response at two measuring points.

Fig.6 Identified bearing dynamic forces compared with theoretical ones.

Fig.7 Identified gear meshing stiffness: (a) waveforms of healthy, pitting, and wear gear; and (b) spectra of healthy, pitting, and wear gear.

Tab.2 Similarity index results of the simulation example under 600 r/min

Tab.3 Gear parameters of the single-stage gearbox test rig

Fig.8 Gearbox test rig and different fault gears.

Fig.9 Decoupling frequency response functions (FRFs) test (take excitation at 1x as an example).

Fig.10 Spectra of the measured vibration signals for different fault types.

Fig.11 Spectra of the identified bearing dynamic forces in the experiment.

Fig.12 Dynamic modeling of the gear?shaft?bearing system of the test rig.

Fig.13 Identified gear meshing stiffness in the experiment: (a) waveforms of healthy, pitting, wear gear, (b) spectra of healthy, pitting, wear gear, and (c) fault characteristic frequency magnitude of healthy, pitting, wear gear.

Fig.14 Comparison study of the proposed gear meshing stiffness identification and traditional fault diagnosis approach: (a) waveform and time?frequency distribution of the intrinsic mode function and (b) normalization RMS of different measuring points and the identified gear meshing stiffness (GMS).

Fig.15 Identified gear meshing stiffness in the experiment: fault characteristic frequency magnitude of healthy, wear, and pitting gear using (a) measuring points 2#?4# and (b) measuring points 3#?5#.

Fig.16 Identified gear meshing stiffness in the experiment: fault characteristic frequency magnitude of healthy, wear, and pitting gear under another speed.

Tab.4 Similarity index results under different speed using the housing vibration signals or the identified GMS under 1800 r/min and 9 N?m conditions as the basis

Fig.17 Identified gear meshing stiffness in the experiment: fault characteristic frequency magnitude of healthy and pitting gear under another load.

Tab.5 Similarity index results under different load using the housing vibration signals or the identified GMS under 1800 r/min and 9 N?m conditions as the basis

Fig.18 Spectrum noise level of the identified gear meshing stiffness indicates gear fault severity. Cond1: measuring points 1#?3#, 1800 r/min, and 9 N?m; Cond2: measuring points 2#?4#, 1800 r/min, and 9 N?m; Cond3: measuring points 3#?5#, 1800 r/min, and 9 N?m; Cond4: measuring points 1#?3#, 2400 r/min, and 9 N?m; Cond5: measuring points 1#?3#, 1800 r/min, and 3 N?m.

Abbreviations
DOF	Degree of freedom
FEM	Finite element method
FT	Fourier Transform
FRF	Frequency response function
GMS	Gear meshing stiffness
GMSI	Gear meshing stiffness identification
IMF	Intrinsic mode function
IFT	Inverse Fourier Transform
Variables
$D$	System damping matrix
$f i x R,$ , $f i x I$	Real and imaginary parts of the x component of bearing dynamic force at bearing i, respectively
$f m$	Meshing frequency
$f d$	Intermediate variable of the decoupled bearing dynamic force excitation spectra vector
$F$	External excitation
$F 12$	External load
$F b$	Bearing dynamic force excitation spectra vector
$F B$	Bearing force
$F d$	Decoupled bearing dynamic force excitation spectra vector
$F e$	Load caused by no-load transfer error of the gear pair
$H i x R j x R, H i x I j x I$	Frequency response functions
$H$	Transfer function matrix
$H d$	Decoupled transfer function matrix
$H T$	Transition matrix
I₁, I₂	Transverse moments of inertia of gears 1 and 2, respectively
J₁, J₂	Polar moments of inertia of gears 1 and 2, respectively
$k 12$	Meshing stiffness of the gear pair
$k m i$	Meshing stiffness of gear pair i
$k x x$ , $k y y$ , $k zz$	Bearing stiffnesses of translational DOF in x-, y-, and z-direction, respectively
$k θ x θ x$ , $k θ y θ y$	Bearing stiffnesses of rotational DOF in x and y directions, respectively
$k b$	Bearing flexibility vector
$k B$	Bearing stiffness vector
$k ~ m$	Temporary variable containing meshing stiffness of all gear pairs
$K$	System stiffness matrix
$K 12$	Meshing stiffness matrix of the gear pair
$K i s$	Unit stiffness matrix of the system
$K ~ m$	Estimated whole meshing stiffness matrix
$K m i$	Meshing stiffness matrix of gear pair i
$m$	Number of bearing dynamic force excitation channels
$m g$	Mass of gear
$m 1,$ $m 2$	Masses of gears 1 and 2, respectively
$M$	System mass matrix
M_g	Mass matrix of a lumped mass model
$M 12$	Mass matrix of the gear pair
$n$	Number of response channels
$n i$	Node number
$r b 1, r b 2$	Radii of base circle
S₂	2-norm similarity index
S_e	Energy ratio similarity index
T₁, T₂	Torques applied on the input shaft and output shaft, respectively
$u$	Nodes displacement vector
$u e$	DOF of the beam model
$u f$	Vibration response of all nodes in the frequency domain
$w i$	Element of the weighted matrix
W	Weighted matrix
$x$	Translational DOF of beam in x direction
$x 1, x 2$	Translational DOFs of gears 1 and 2 in x direction, respectively
$x b$	Translational DOF of bearing in x direction
$x j x R, x j x I$	Real and imaginary parts of the x component of housing vibration response at location j, respectively
$x$	Response channel spectra vector
$x ~ b$	Vibration response of all bearing nodes
$x B$	Displacement vector of bearing
$x d$	Decoupled response channel spectra vector
$X 12$	Generalized coordinate of the gear pair
$y$	Translational DOF of beam in y direction
$y 1,$ $y 2$	Translational DOFs of gears 1 and 2 in y direction, respectively
$y b$	Translational DOF of bearing in y direction
$z$	Translational DOF of beam in z direction
$z 1,$ $z 2$	Translational DOFs of gears 1 and 2 in z direction, respectively
$z b$	Translational DOF of bearing in z direction
$α$	Weighted factor
$α 12$	Projection vector of the gear pair
$ψ 12$	Angle between the positive y axis and the meshing surface
Ψ	Projection matrix of all gear nodes
$δ$	Numerical error
$ω$	Angular frequency
$θ x$ , $θ y$ , $θ z$	Rotational DOFs of beam in x, y, and z directions, respectively
$θ x 1,$ $θ x 2$	Rotational DOFs of gears 1 and 2 in x direction, respectively
$θ y 1,$ $θ y 2$	Rotational DOFs of gears 1 and 2 in y direction, respectively
$θ z 1,$ $θ z 2$	Rotational DOFs of gears 1 and 2 in z direction
$θ x b$ , $θ y b$ , $θ z b$	Rotational DOFs of bearing in x, y, and z directions, respectively
$Θ i$	Projection matrix

Abbreviations
DOF	Degree of freedom
FEM	Finite element method
FT	Fourier Transform
FRF	Frequency response function
GMS	Gear meshing stiffness
GMSI	Gear meshing stiffness identification
IMF	Intrinsic mode function
IFT	Inverse Fourier Transform
Variables
$D$	System damping matrix
$f i x R,$ , $f i x I$	Real and imaginary parts of the x component of bearing dynamic force at bearing i, respectively
$f m$	Meshing frequency
$f d$	Intermediate variable of the decoupled bearing dynamic force excitation spectra vector
$F$	External excitation
$F 12$	External load
$F b$	Bearing dynamic force excitation spectra vector
$F B$	Bearing force
$F d$	Decoupled bearing dynamic force excitation spectra vector
$F e$	Load caused by no-load transfer error of the gear pair
$H i x R j x R, H i x I j x I$	Frequency response functions
$H$	Transfer function matrix
$H d$	Decoupled transfer function matrix
$H T$	Transition matrix
I₁, I₂	Transverse moments of inertia of gears 1 and 2, respectively
J₁, J₂	Polar moments of inertia of gears 1 and 2, respectively
$k 12$	Meshing stiffness of the gear pair
$k m i$	Meshing stiffness of gear pair i
$k x x$ , $k y y$ , $k zz$	Bearing stiffnesses of translational DOF in x-, y-, and z-direction, respectively
$k θ x θ x$ , $k θ y θ y$	Bearing stiffnesses of rotational DOF in x and y directions, respectively
$k b$	Bearing flexibility vector
$k B$	Bearing stiffness vector
$k ~ m$	Temporary variable containing meshing stiffness of all gear pairs
$K$	System stiffness matrix
$K 12$	Meshing stiffness matrix of the gear pair
$K i s$	Unit stiffness matrix of the system
$K ~ m$	Estimated whole meshing stiffness matrix
$K m i$	Meshing stiffness matrix of gear pair i
$m$	Number of bearing dynamic force excitation channels
$m g$	Mass of gear
$m 1,$ $m 2$	Masses of gears 1 and 2, respectively
$M$	System mass matrix
M_g	Mass matrix of a lumped mass model
$M 12$	Mass matrix of the gear pair
$n$	Number of response channels
$n i$	Node number
$r b 1, r b 2$	Radii of base circle
S₂	2-norm similarity index
S_e	Energy ratio similarity index
T₁, T₂	Torques applied on the input shaft and output shaft, respectively
$u$	Nodes displacement vector
$u e$	DOF of the beam model
$u f$	Vibration response of all nodes in the frequency domain
$w i$	Element of the weighted matrix
W	Weighted matrix
$x$	Translational DOF of beam in x direction
$x 1, x 2$	Translational DOFs of gears 1 and 2 in x direction, respectively
$x b$	Translational DOF of bearing in x direction
$x j x R, x j x I$	Real and imaginary parts of the x component of housing vibration response at location j, respectively
$x$	Response channel spectra vector
$x ~ b$	Vibration response of all bearing nodes
$x B$	Displacement vector of bearing
$x d$	Decoupled response channel spectra vector
$X 12$	Generalized coordinate of the gear pair
$y$	Translational DOF of beam in y direction
$y 1,$ $y 2$	Translational DOFs of gears 1 and 2 in y direction, respectively
$y b$	Translational DOF of bearing in y direction
$z$	Translational DOF of beam in z direction
$z 1,$ $z 2$	Translational DOFs of gears 1 and 2 in z direction, respectively
$z b$	Translational DOF of bearing in z direction
$α$	Weighted factor
$α 12$	Projection vector of the gear pair
$ψ 12$	Angle between the positive y axis and the meshing surface
Ψ	Projection matrix of all gear nodes
$δ$	Numerical error
$ω$	Angular frequency
$θ x$ , $θ y$ , $θ z$	Rotational DOFs of beam in x, y, and z directions, respectively
$θ x 1,$ $θ x 2$	Rotational DOFs of gears 1 and 2 in x direction, respectively
$θ y 1,$ $θ y 2$	Rotational DOFs of gears 1 and 2 in y direction, respectively
$θ z 1,$ $θ z 2$	Rotational DOFs of gears 1 and 2 in z direction
$θ x b$ , $θ y b$ , $θ z b$	Rotational DOFs of bearing in x, y, and z directions, respectively
$Θ i$	Projection matrix

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