A gearbox fault diagnosis method based on physical information neural network

Abstract

The application discloses a gearbox fault diagnosis method based on a physical information neural network. First, a vibration signal acquisition system is arranged in a gearbox, vibration and angular velocity signals are acquired, a labeled measured data set is established, and a labeled simulation data set is obtained by establishing a gear dynamics model. Then, the physical information neural network composed of the long short-term memory network (LSTM) is trained by using the measured and simulation data sets, vibration and angular velocity signals are taken as inputs, and a time-varying meshing stiffness curve is output. Next, the meshing stiffness curve data are extracted for features, a fault classification training set is formed, and a support vector machine (SVM) model is trained by using the data set. Finally, the actually acquired vibration data are input into the physical information neural network and the SVM model, and accurate diagnosis of the gearbox fault is realized.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of gear box fault diagnosis, and particularly relates to a gear box fault diagnosis method based on a physical information neural network. BACKGROUND

[0002] As a core component in mechanical transmission systems, the running state of a gear box is directly related to the reliability and production efficiency of the entire device. However, due to the long-term operation of the gear box in a complex working environment, the gear box is prone to failure under the influence of various factors such as load fluctuation, insufficient lubrication and component wear. Common faults include gear wear, fatigue fracture, bearing damage, etc. Especially in large industrial equipment, the failure of the gear box not only causes production stagnation, but also may cause serious safety hazards. Therefore, how to diagnose the gear box failure in a timely and accurate manner is crucial for equipment maintenance and production safety.

[0003] Existing gear box fault diagnosis methods mainly rely on vibration signals, temperature changes, sound signals and other sensor data to identify potential gear box failures by analyzing the abnormal characteristics of these physical quantities. For example, vibration analysis can effectively monitor the vibration pattern of the gear mesh inside the gear box. When the gear is damaged, fatigued or failed, its vibration signal will show obvious abnormalities. However, due to the fact that vibration signals are easily disturbed by external environmental noise and different types of faults may exhibit similar characteristics under complex working conditions, the accuracy of fault diagnosis is greatly limited.

[0004] In recent years, with the rapid development of deep learning technology, researchers have gradually applied neural networks to gear box fault diagnosis, automatically extracting features from vibration signals through the construction of end-to-end models, and performing classification and identification, thereby greatly improving the efficiency and accuracy of fault diagnosis. Common neural network architectures include convolutional neural networks (CNN), recurrent neural networks (RNN), long short-term memory networks (LSTM), etc. However, these data-driven deep learning methods do not fully consider the physical characteristics of the gear box system, and the model lacks physical constraints during training, resulting in insufficient generalization ability, especially in complex working conditions and unobserved fault patterns.

[0005] Therefore, a gear box fault diagnosis method based on a physical information neural network is proposed, which embeds the dynamics equation of the gear box as a physical constraint into the training process of the neural network, so that the model not only relies on data learning, but also follows the physical laws of the gear box system. This method can improve the generalization ability of the model, reduce the influence of noise interference on the diagnosis result, and achieve accurate diagnosis of gear box faults under complex working conditions. SUMMARY

[0006] In light of this, this paper proposes a gearbox fault diagnosis method based on a physical information neural network, enabling efficient and accurate diagnosis of gearbox faults while improving the level of diagnostic automation. By combining physical constraints with neural networks and effectively utilizing real-world and simulated data for modeling, this method can accurately diagnose a variety of gearbox faults, and the diagnostic results exhibit good robustness and generalizability.

[0007] The present invention is achieved through the following technical solutions:

[0008] A gearbox fault diagnosis method based on physical information neural network includes the following steps:

[0009] S1: Obtain the design parameters of the gears in the gearbox, arrange vibration signal sensors around the gearbox bearings, and arrange gear speed sensors on the top side of the gears;

[0010] S2: Select a batch of healthy gears and faulty gears as samples. After the gearbox is driven by a motor to run smoothly, the vibration signal and angular velocity signal of each gear are obtained through the vibration signal sensor and gear speed sensor arranged in step S1.

[0011] Classify and label the obtained gear vibration signals and angular velocity signals to obtain a labeled measured data set;

[0012] S3: Establish a gear dynamics model and solve it using a numerical solution method to obtain a labeled simulation data set;

[0013] S4: Using the labeled measured data set and the labeled simulation data set to train the physical information neural network model, to obtain a trained physical information neural network model;

[0014] S5: Import the labeled measured data set obtained in S2 and the labeled simulation data set obtained in S3 into the trained physical information neural network model to obtain a time-varying mesh stiffness change curve data set, classify and label the obtained time-varying mesh stiffness change curve data set, and obtain a labeled time-varying mesh stiffness curve data set;

[0015] S6: Calculate various features of the labeled time-varying mesh stiffness curve data set obtained in S5 as a fault classification training set;

[0016] S7: Use the fault classification training set obtained in S6 to train a multi-classification algorithm model to obtain a trained multi-classification algorithm model;

[0017] S8: Collect the gear vibration signal of the gearbox under actual working condition to obtain the gear vibration data set to be tested;

[0018] S9: Input the actual collected vibration data of the gear to be tested into the physical information neural network model trained in S4, output the time-varying meshing stiffness change curve, and then calculate the various features in step S6 based on the output time-varying meshing stiffness change curve. Finally, input the features into the multi-classification algorithm model trained in S7 to obtain the classification result, which is the gearbox fault result.

[0019] Further preferably, a gearbox fault diagnosis method based on physical information neural network comprises the following steps:

[0020] Step S1: Vibration signal sensors are placed around the gearbox bearings, and a gear speed sensor is placed on the top side of the gear to obtain the gearbox's vibration signal and angular velocity signal. Specifically, vibration sensors are placed below the bearing seats of the driving and driven pulleys of the gearbox to collect vertical and horizontal vibration signals, and the gear speed sensor is used to obtain the gear's angular velocity signal.

[0021] Step S2: Select a batch of healthy and faulty gears as samples. Fault types for the faulty gears include root fracture and tooth flaking, and obtain a labeled real-world training dataset. Specifically, run both the faulty and healthy gearboxes for 10 seconds, collecting the corresponding gear vibration and angular velocity signals to form a labeled real-world dataset.

[0022] Step S3: Establish a healthy gear dynamics model and various faulty gear dynamics models, calculate the theoretical gearbox vibration signal and angular acceleration signal through a numerical solution method, form a labeled simulation data set, and use the labeled simulation data set and the labeled measured data set to train a physical information neural network model with dynamic equations as physical constraints. The model uses a long short-term memory network (LSTM) as the neural network structure of the physical information neural network. The physical information neural network has 6 inputs (vibration signals and angular accelerations of the driving wheel and the driven wheel in the horizontal and vertical directions) and 1 output (time-varying meshing stiffness), and a trained physical information neural network model is obtained.

[0023] Step S4: Input the labeled measured data set and the labeled simulation data set into the trained physical information neural network model to obtain a time-varying meshing stiffness change curve data set.

[0024] Step S5: label the time-varying mesh stiffness curve data set with fault classification to form a labeled time-varying mesh stiffness curve data set.

[0025] Step S6: extracting time domain features, frequency domain features and time-frequency mixed features from the labeled time-varying mesh stiffness curve data set obtained in step S5 as feature vectors for fault classification, and constructing a fault classification training set.

[0026] Step S7: Use the above fault classification training set to train a multi-classification algorithm model for gear fault classification.

[0027] Step S8: collecting the gear vibration signal of the gearbox under actual working state to obtain a gear vibration data set to be tested;

[0028] Step S9: Input the actual collected vibration data of the gear to be tested into the physical information neural network trained in S3, output the time-varying meshing stiffness change curve, and then calculate the various features in step S6 based on the output time-varying meshing stiffness change curve. Finally, input it into the multi-classification algorithm model trained in S7 to obtain the classification result, which is the gearbox fault result.

[0029] Improved features:

[0030] In step S1, the signals include: vibration signals in the horizontal and vertical directions below the gear driving wheel and driven wheel shaft, and gear angular velocity signals.

[0031] A high-frequency dynamic acquisition system was installed, and charge-type vibration sensors were used to collect horizontal and vertical vibration signals under the shafts of the gearbox driving and driven wheels. A signal amplifier was used to amplify the signals and convert them to a 10kHz acquisition device for data acquisition, which was finally saved as a healthy data set in TDM format.

[0032] In step S2, the faults include: tooth root fracture and tooth surface peeling.

[0033] Fault types include tooth root fracture and tooth surface spalling. After the fault is injected, the machine is kept rotating at a constant speed for 10 seconds, and vibration signals are collected to form a labeled measured data set.

[0034] The method for obtaining a gear with tooth root fracture fault is to partially cut the root of the gear using a grinding wheel.

[0035] The method for obtaining gears with tooth surface spalling faults is to use acidic solution for corrosion.

[0036] In step S3, when training the physical information neural network, two types of loss functions are used: a pure data-driven loss function and a physical-driven loss function based on the gear meshing dynamics equation to ensure the effectiveness of physical constraints.

[0037] During the training and optimization of the physical information neural network in step S3, the loss function used is:

[0038] loss total =loss1+loss2

[0039] Among them, loss1 is a purely data-driven loss function, and loss2 is a physical-driven loss function built based on the gear meshing dynamics equation to better capture timing dependencies.

[0040] The specific formulas of loss1 and loss2 loss functions are:

[0041]

[0042] loss2=r1+r2+r3+r4+r5+r6

[0043]

[0044]

[0045]

[0046]

[0047]

[0048]

[0049] Among them, y, y pre are the actual results and the predicted results of the neural network, m1 and m2 are the masses of the driving wheel and the driven wheel, is the horizontal acceleration of the driving wheel and the driven wheel, is the vertical acceleration of the driving wheel and the driven wheel, c x1 , c x2 , c y1 , c y2 is the support damping of the driving wheel and the driven wheel in the x and y directions, is the horizontal speed of the driving wheel and the driven wheel, is the vertical speed of the driving wheel and the driven wheel, k x1 , k x2 , k y1 , k y2 is the support stiffness of the driving wheel and the driven wheel in the x and y directions, x1, x2 are the horizontal displacements of the driving wheel and the driven wheel, y1, y2 are the vertical displacements of the driving wheel and the driven wheel, is the angular acceleration of the driving wheel and the driven wheel, R1, R2 is the base circle radius of the driving wheel and the driven wheel, J1, J2 is the mass inertia moment of the driving wheel and the driven wheel, T1, T2 is the torque of the driving wheel and the driven wheel, F m is the gear meshing force, and α is the gear meshing angle.

[0050] The features extracted from the time-varying meshing stiffness variation curve in step S6 include: time domain features, frequency domain features, and time-frequency mixed features.

[0051] Compared with the prior art, the present invention has the following advantages:

[0052] The present invention achieves high-precision classification of gearbox faults and can effectively identify common gear fault types by combining physical information neural networks with support vector machines. In particular, in the process of fault diagnosis, the combination of simulation data and measured data makes the diagnostic model have good robustness and accuracy. By introducing physical information neural networks, the diagnostic model not only relies on data, but also combines the gear dynamics model, further improving the reliability and intelligence level of fault diagnosis. In addition, the neural network structure of the physical information neural network in the present invention adopts a long short-term memory neural network to replace the fully connected neural network structure in the traditional physical information neural network, which can better capture the temporal dependency. BRIEF DESCRIPTION OF THE DRAWINGS

[0053] Figure 1 Schematic diagram of the process of this method.

[0054] Figure 2 is the gear meshing dynamics model used in this method.

[0055] Figure 3 The neuron structure diagram of LSTM used in this method.

[0056] Figure 4 For training and verification loss curves.

[0057] Figure 5 The diagram shows the time-varying mesh stiffness of gears under different predicted fault levels. DETAILED DESCRIPTION

[0058] In order to enable those skilled in the art to better understand the technical solutions in this specification, the technical solutions in the embodiments of this specification will be clearly and completely described below in conjunction with the drawings in the embodiments of this specification. Obviously, the described embodiments are only part of the embodiments of this application, not all of the embodiments.

[0059] In the description of the embodiments of this application, words such as "for example" or "for instance" are used to indicate examples, illustrations, or explanations. Any embodiment or design described as "for example" or "for instance" in the embodiments of this application should not be construed as being preferred or advantageous over other embodiments or designs. Rather, the use of words such as "for example" or "for instance" is intended to present the relevant concepts in a concrete manner.

[0060] In the description of the embodiments of the present application, the term "multiple" means two or more. For example, multiple systems refer to two or more systems, and multiple screen terminals refer to two or more screen terminals. In addition, the terms "first" and "second" are used for descriptive purposes only and are not to be understood as indicating or implying relative importance or implicitly indicating the indicated technical features. Thus, the features defined as "first" and "second" may explicitly or implicitly include one or more of the features. The terms "including", "comprising", "having" and their variations all mean "including but not limited to", unless otherwise specifically emphasized.

[0061] like Figure 1 As shown, the present invention discloses a gearbox fault diagnosis method based on physical information neural network, which specifically includes the following steps:

[0062] S1: Arrange a dynamic vibration signal acquisition system to obtain the vibration signal and angular acceleration signal processed in the gearbox:

[0063] S11: Install a charge-type vibration sensor below the fixed position of the driving wheel and driven wheel gear shaft of the gearbox.

[0064] S12: Install the gear speed sensor 2 mm above the driving and driven pulleys of the gearbox.

[0065] S13: Use a high-frequency dynamic acquisition system to respectively acquire the acceleration signals in the horizontal and vertical directions below the gearbox driving wheel and driven wheel shaft, as well as the angular velocity signals of the gears. The sampling frequency Fs used by the high-frequency dynamic acquisition system is 10 kHz.

[0066] S2: Gear vibration signals and angular velocity signals are collected during the actual operation of the gearbox to form a measured data set:

[0067] S21: Select a batch of faulty gears and a batch of healthy gears and classify them.

[0068] S22: Use the servo motor to drive the faulty gear and the healthy gear in S21 to rotate at a constant speed of 1200 r / min. After 10 seconds of stable operation, collect 5 seconds of gear vibration and angular velocity signals.

[0069] S23: The acquired data is timestamped according to the acquisition time and stored in TDM format to form a measured data set including the health and fault status of the gear.

[0070] S3: Solve the kinetic equations to obtain the theoretical data training set:

[0071] S31: Establish Figure 2The six-degree-of-freedom gear dynamic model shown here uses the following differential equations to solve:

[0072]

[0073] Among them, m1 and m2 are the masses of the driving wheel and the driven wheel, is the horizontal acceleration of the driving wheel and the driven wheel, is the vertical acceleration of the driving wheel and the driven wheel, c x1 , c x2 , c y1 , c y2 is the support damping of the driving wheel and the driven wheel in the x and y directions, is the horizontal speed of the driving wheel and the driven wheel, is the vertical speed of the driving wheel and the driven wheel, k x1 , k x2 , k y1 , k y2 is the support stiffness of the driving wheel and the driven wheel in the x and y directions, x1, x2 are the horizontal displacements of the driving wheel and the driven wheel, y1, y2 are the vertical displacements of the driving wheel and the driven wheel, is the angular acceleration of the driving wheel and the driven wheel, θ1, θ2 are the rotation angles of the driving wheel and the driven wheel, R1, R2 are the base circle radii of the driving wheel and the driven wheel, J1, J2 are the mass inertia moments of the driving wheel and the driven wheel, T1, T2 are the torques of the driving wheel and the driven wheel, F m is the gear meshing force, and α is the gear meshing angle.

[0074] Some design parameters of the gear system are:

[0075] parameter driving wheel driven wheel Module / mm 27 73 Tooth width / mm 2 2 Pressure angle / (°) 20 20 <![CDATA[质量惯性 / (kg·m 2 )]]> 0.00017 0.0072 <![CDATA[支撑刚度 / (N·m- 1 )]]> 1 x 10 7 ]] 1 x 10 7 ]] <![CDATA[支撑阻尼 / (N·m- 1 )]]> <![CDATA[1×10 3 ]]> <![CDATA[1×10 3 ]]>

[0076] S32: The dynamic model in S21 is numerically solved by the Runge-Kutta method to obtain the theoretical dynamic response.

[0077] S33: The dynamic response obtained by theoretical calculation and the time-varying meshing stiffness curve used for calculation are used as simulation data sets.

[0078] S4: Training the physical information neural network model:

[0079] S41: Construct a physical information neural network model, which includes a long short-term memory network structure. The loss function of the physical information neural network includes the gear meshing dynamics model as a physical constraint. The specific loss function is as follows:

[0080] loss total =loss1+loss2

[0081] wherein loss1 is a pure data-driven loss function, and loss2 is a physics-driven loss function based on the gear meshing dynamics equation to better capture the time-dependent relationship.

[0082] The specific formula of the loss1 and loss2 loss functions is:

[0083]

[0084] loss2 = r1 + r2 * r3 + r4 * r5 + r6

[0085]

[0086]

[0087]

[0088]

[0089]

[0090]

[0091] wherein y, y pre are actual results and neural network prediction results, m1, m2 are the masses of the driving wheel and the driven wheel, are the accelerations of the driving wheel and the driven wheel in the horizontal direction, are the accelerations of the driving wheel and the driven wheel in the vertical direction, c x1 , c x2 , c y1 , c y2 are the support dampings of the driving wheel and the driven wheel in the x and y directions, are the velocities of the driving wheel and the driven wheel in the horizontal direction, are the velocities of the driving wheel and the driven wheel in the vertical direction, k x1 , k x2 , k y1 , k y2 are the support stiffnesses of the driving wheel and the driven wheel in the x and y directions, x1, x2 are the displacements of the driving wheel and the driven wheel in the horizontal direction, y1, y2 are the displacements of the driving wheel and the driven wheel in the vertical direction, are the angular accelerations of the driving wheel and the driven wheel, R1, R2 are the base circle radii of the driving wheel and the driven wheel, J1, J2 are the mass moments of inertia of the driving wheel and the driven wheel, T1, T2 are the torques of the driving wheel and the driven wheel, F m is the gear meshing force, and a is the gear meshing angle.

[0092] In this example, a single-layer long short-term memory network (LSTM) is used as the structure of the physical information neural network. The LSTM structure includes two layers of LSTM, where the first layer has 64 neurons and the second layer has 32 neurons. The LSTM neurons are as follows: Figure 3 As shown, the next layer is a fully connected layer (Dense Layer) with 16 neurons, which finally outputs the time series signal of 1 neuron, and the input is 6-channel time series data with a length of 300.

[0093] S42: The design parameters and vibration response are used as the input of the model (300 length, 6 channels), and the time-varying mesh stiffness is used as the output of the model (300 length, 1 channel).

[0094] S43: Using the measured data set obtained in S23 and the simulation data set obtained in S23, the data set is divided into a training set, a test set, and a validation set according to the ratio of 80%, 10%, and 10%, respectively. The model is trained using the training set, and the performance of the model is evaluated and optimized using the test set and validation set. Finally, a trained physical information neural network model is obtained. The change of the loss function during the training process is shown in the figure below. Figure 4 shown.

[0095] S5: Constructing a time-varying mesh stiffness curve dataset:

[0096] S51: Input the measured data set obtained in S15 and the simulation data set obtained in S23 into the trained physical information neural network model obtained in S33 to obtain the corresponding time-varying meshing stiffness curve, thereby forming a time-varying meshing stiffness change curve data set.

[0097] S52: Classify and label the time-varying meshing stiffness change curve data set obtained in S51 to obtain a labeled time-varying meshing stiffness curve data set.

[0098] S6: Construct a multi-classification support vector machine fault classification dataset:

[0099] S61: Feature extraction is performed on the labeled time-varying mesh stiffness curve data set obtained in S52. A total of 32 features are selected. The extracted features are shown in the following table:

[0100]

[0101] In the table, N is the total number of signal points, and x(n) is the value corresponding to the nth sampling point of the signal.

[0102] S62: Classify and label the extracted data set to obtain a fault classification training set.

[0103] S7: Use the fault classification training set obtained in step S62 as training samples to train a multi-classification support vector machine to build a classification model from time-varying meshing stiffness characteristics to fault types.

[0104] S8: Using the vibration signal dynamic acquisition system arranged in step S1 to collect the gear vibration signal of the gearbox under the actual working state, to obtain a vibration data set of the gear to be tested;

[0105] S9: Input the vibration data of the gear to be tested actually collected in step S8 into the physical information neural network model trained in step S4, and obtain the following Figure 5 The predicted time-varying mesh stiffness curves of gears under different fault degrees are shown. Then, the 32 features in step S6 are calculated based on the output time-varying mesh stiffness change curves. These features are then input into the support vector machine model trained in S7 to finally obtain the classification result, which is the gearbox fault result.

[0106] It will be apparent to those skilled in the art that the present invention is not limited to the details of the exemplary embodiments described above and that the invention can be embodied in other specific forms without departing from the spirit or essential characteristics of the invention. Therefore, the embodiments should be considered in all respects as illustrative and non-restrictive, and the scope of the invention is defined by the appended claims, not the foregoing description, and all variations within the meaning and range of equivalents of the claims are intended to be included therein. Any reference sign in a claim should not be construed as limiting the claim to which it relates.

[0107] In addition, it should be understood that although this specification is described in terms of implementation methods, not every implementation method contains only one independent technical solution. This narrative method of the specification is only for the sake of clarity. Those skilled in the art should regard the specification as a whole. The technical solutions in each embodiment can also be appropriately combined to form other implementation methods that can be understood by those skilled in the art.

Claims

1. A gearbox fault diagnosis method based on physical information neural network, characterized in that: The following steps are involved: S1: Obtain the design parameters of the gears in the gearbox, arrange vibration signal sensors around the gearbox bearings, and arrange gear speed sensors on the top side of the gears; S2: Select a batch of healthy gears and faulty gears as samples. After the gearbox is driven by a motor to run smoothly, the vibration signal and angular velocity signal of each gear are obtained through the vibration signal sensor and gear speed sensor arranged in step S1. Classify and label the obtained gear vibration signals and angular velocity signals to obtain a labeled measured data set; S3: Establish a gear dynamics model and solve it using a numerical solution method to obtain a labeled simulation data set; S4: Using the labeled measured data set and the labeled simulation data set to train the physical information neural network model, to obtain a trained physical information neural network model; S5: Import the labeled measured data set obtained in S2 and the labeled simulation data set obtained in S3 into the trained physical information neural network model to obtain a time-varying mesh stiffness change curve data set, classify and label the obtained time-varying mesh stiffness change curve data set, and obtain a labeled time-varying mesh stiffness curve data set; S6: Calculate various features of the labeled time-varying mesh stiffness curve data set obtained in S5 as a fault classification training set; S7: Use the fault classification training set obtained in S6 to train a multi-classification algorithm model to obtain a trained multi-classification algorithm model; S8: Collect the gear vibration signal of the gearbox under actual working condition to obtain the gear vibration data set to be tested; S9: Input the actual collected vibration data of the gear to be tested into the physical information neural network model trained in S4, output the time-varying meshing stiffness change curve, and then calculate the various features in step S6 based on the output time-varying meshing stiffness change curve. Finally, input the features into the multi-classification algorithm model trained in S7 to obtain the classification result, which is the gearbox fault result.

2. The gearbox fault diagnosis method based on physical information neural network according to claim 1 is characterized in that: In step S2, the faults include: tooth root fracture and tooth surface peeling.

3. The gearbox fault diagnosis method based on physical information neural network according to claim 1 is characterized in that: In step S2, the gear vibration signal includes: horizontal and vertical vibration signals of the gear driving wheel and driven wheel shaft.

4. The gearbox fault diagnosis method based on physical information neural network according to claim 1 is characterized in that: In step S4, the physical information neural network model adopts a long short-term memory network as the network structure and a dynamic equation as the physical constraint, takes the gear vibration signal and angular velocity signal in the gearbox as input, and outputs the time-varying meshing stiffness.

5. The gearbox fault diagnosis method based on physical information neural network according to claim 1 is characterized in that: In step S4, the physical information neural network model is trained, and the total loss function used is: loss total =loss1+loss2 Among them, loss1 is a purely data-driven loss function, and loss2 is a physical-driven loss function based on the gear meshing dynamics equation; The specific formulas of loss1 and loss2 loss functions are: loss2=r1+r2+r3+r4+r5+r6 Among them, y, y pre are the actual results and the predicted results of the neural network, m1 and m2 are the masses of the driving wheel and the driven wheel, is the horizontal acceleration of the driving wheel and the driven wheel, is the vertical acceleration of the driving wheel and the driven wheel, c x1 , c x2 , c y1 , c y2 is the support damping of the driving wheel and the driven wheel in the x and y directions, is the horizontal speed of the driving wheel and the driven wheel, is the vertical speed of the driving wheel and the driven wheel, k x1 , k x2 , k y1 , k y2 is the support stiffness of the driving wheel and the driven wheel in the x and y directions, x1, x2 are the horizontal displacements of the driving wheel and the driven wheel, y1, y2 are the vertical displacements of the driving wheel and the driven wheel, is the angular acceleration of the driving wheel and the driven wheel, R1, R2 is the base circle radius of the driving wheel and the driven wheel, J1, J2 is the mass inertia moment of the driving wheel and the driven wheel, T1 and T2 are the torques of the driving wheel and the driven wheel, and F m is the gear meshing force, and α is the gear meshing angle.

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