A fault diagnosis method for helical gear-rack transmission system based on multi-axis vibration signals

Abstract

This invention discloses a fault diagnosis method for helical gear rack transmission systems based on multi-axis vibration signals. The method includes: obtaining a high-fidelity healthy signal; injecting fault features into the high-fidelity healthy signal to obtain a single-axis simulation signal; constructing a three-axis signal matrix, including radial, tangential, and axial vibration signals, based on the generated single-axis simulation signal through amplitude scaling and random perturbation; performing feature extraction to obtain multi-dimensional feature vectors corresponding to each axis; fusing the multi-dimensional feature vectors to construct a sample set, then using the sample set to train a support vector machine model to obtain a fault diagnosis model; and using the fault diagnosis model to realize gear rack fault diagnosis. This invention not only overcomes the problem of insufficient actual sampling data but also fully utilizes the coupled information of the three-axis signals, combined with the small sample advantage of SVM, to achieve high-precision identification of multiple types of faults in helical gear rack transmission systems, demonstrating good engineering applicability and promotional value.

Description

Technical Field

[0001] This invention belongs to the field of fault diagnosis and intelligent detection technology of mechanical transmission systems, and in particular, it is a fault diagnosis method for helical gear and rack transmission systems based on multi-axis vibration signals. Background Technology

[0002] Helical rack and pinion transmission systems, as a crucial component of precision linear drive devices, are widely used in CNC machine tools, gantry printing and marking equipment, industrial robots, and other high-precision equipment. Due to their advantages such as smooth meshing, high transmission accuracy, and strong load-bearing capacity, these systems have become key fundamental components of modern mechanical equipment. However, during long-term operation, gears and racks are inevitably affected by factors such as alternating loads, deteriorating lubrication conditions, and manufacturing and assembly errors, leading to faults such as tooth breakage, pitting, tooth root fatigue cracks, tooth surface wear, and abnormal axial forces. These faults not only significantly increase system vibration and noise but also cause a decrease in transmission accuracy, and in severe cases, even lead to machine downtime, significantly impacting production efficiency and equipment lifespan.

[0003] Existing research generally employs diagnostic methods based on measured uniaxial vibration signals. This typically involves deploying accelerometers on the equipment to collect signals and then utilizing time-domain, frequency-domain, and time-frequency-domain features for fault identification. However, this approach suffers from several significant problems: First, obtaining real fault data is difficult. Equipment operating in a healthy state for extended periods results in a very limited number of fault samples available for modeling and training. Furthermore, actively creating faults is not only costly but can also cause irreversible damage to the equipment. Second, uniaxial signals cannot fully reflect the three-dimensional dynamic characteristics of gear meshing, especially the axial force effect caused by the helix angle in helical gears, which is often difficult to accurately capture using signals from a single direction. Third, traditional feature extraction and classification methods are highly dependent on operating conditions and susceptible to noise, leading to insufficient diagnostic robustness.

[0004] To address the aforementioned issues, researchers have begun to attempt to generate virtual fault data through dynamic modeling and numerical simulation. However, most existing modeling-based research remains at the level of a single gear pair or a simplified model, lacking a systematic simulation and diagnostic method for helical gear and rack linear transmission systems.

[0005] Furthermore, deep learning methods have received widespread attention in classification in recent years, but their training typically requires a large number of fault samples, making it difficult to meet the needs of small sample conditions in engineering fields. In contrast, Support Vector Machines (SVMs) can maintain strong classification performance and generalization ability even with limited samples, making them particularly suitable for diagnosing multi-class mechanical faults under complex working conditions. Therefore, there is an urgent need for a triaxial vibration signal generation method that can combine physical mechanisms, high-fidelity simulation, and engineering disturbance factors, and to form a complete dataset that can be used for large-scale deep learning training, in order to achieve high-precision and reliable gear and rack fault diagnosis. Summary of the Invention

[0006] The purpose of this invention is to address the problems existing in the prior art by providing a fault diagnosis method for helical gear and rack transmission systems based on multi-axis vibration signals.

[0007] The technical solution to achieve the purpose of this invention is: a fault diagnosis method for a helical gear and rack transmission system based on multi-axis vibration signals, the method comprising the following steps:

[0008] Step 1: Based on the structural parameters, operating conditions and dynamic model of the helical gear rack transmission system, construct a mathematical model of the gear rack system. Based on the mathematical model, simulate the vibration signal of the healthy state, and apply velocity fluctuation, meshing back nonlinearity, friction slip noise, structural modal vibration and motor electromagnetic torque fluctuation to the vibration signal to obtain a high-fidelity healthy signal.

[0009] Step 2: Inject fault features into the high-fidelity health signal to obtain a single-axis simulation signal;

[0010] Step 3: Based on the generated single-axis simulation signal, construct a three-axis signal matrix by amplitude scaling and random perturbation, including radial, tangential and axial vibration signals;

[0011] Step 4: Extract features from the three-axis signals generated in Step 3 to obtain the multi-dimensional feature vectors corresponding to each axis;

[0012] Step 5: Repeat steps 2 to 4 several times to fuse the multidimensional feature vectors to construct a sample set. Then, use the sample set to train a support vector machine model to obtain a fault diagnosis model.

[0013] Step 6: For the helical gear rack transmission system to be diagnosed, acquire triaxial signals, including radial, tangential and axial vibration signals. Then, perform feature extraction as in Step 4, and input the obtained multidimensional feature vector into the fault diagnosis model to realize gear rack fault diagnosis.

[0014] Furthermore, the mathematical model of the gear and rack system constructed in step 1 specifically includes:

[0015] The nonlinear dynamic equations of the transmission system are established based on the meshing principle of helical gear and rack. The equations simultaneously consider the rotational inertia of the gear, the translational mass of the rack, the meshing stiffness, the damping term, and the external excitation force.

[0016] Meshing backlash and tooth profile error are introduced into the nonlinear dynamic equation to characterize the non-ideal nature of the actual meshing process, and the fundamental vibration response signal under healthy conditions is solved by combining the system operating conditions and boundary constraints.

[0017] By applying periodic disturbances generated by electromagnetic torque fluctuations of the motor, high-frequency random noise caused by tooth surface friction slippage, harmonic components caused by nonlinear stiffness changes in gear meshing, and coupled vibration effects caused by structural modal resonance to the fundamental vibration response signal, the fundamental vibration response signal has multi-source excitation and broadband characteristics similar to the signals acquired by actual equipment, providing a high-fidelity healthy vibration signal for subsequent fault injection and feature analysis.

[0018] Furthermore, the high-fidelity health vibration signal is represented as :

[0019]

[0020] In the formula, This is the sensitivity coefficient of stiffness to vibration amplitude. For time-varying meshing stiffness, As a baseline signal for health, This is the hysteresis waveform. For the narrowband response of the structural modes, For motor electromagnetic torque fluctuation, This is broadband noise.

[0021] Furthermore, the fault in step 2 is achieved by introducing parameter disturbances, changes in contact stiffness, changes in damping characteristics, or abnormal excitation forces into the mathematical model.

[0022] Furthermore, step 2 injects fault features, including: tooth breakage, pitting, wear, tooth root fatigue, and abnormal axial force. Tooth breakage is achieved by superimposing an exponentially decaying impact sequence at the time position corresponding to the meshing angle and introducing a periodic stiffness weakening function. Pitting is achieved by applying short-time impact pulses at the periodic phase position of gear meshing and superimposing a frequency doubling modulation component. Wear is achieved by introducing amplitude amplification and phase modulation factors into the fundamental signal to reflect the nonlinear response caused by changes in meshing tooth profile. Tooth root fatigue is achieved by constructing a gradually changing amplitude attenuation window within the rotation period and combining it with a periodic weak impact signal. Abnormal axial force is achieved by superimposing low-frequency and high-frequency harmonic components into the healthy signal and introducing additional random noise to form representative fault vibration signal samples.

[0023] Furthermore, the axial force anomaly specifically refers to the axial force anomaly of the helical gear; based on the high-fidelity healthy signal, a vibration signal model of the helical gear under the abnormal axial force condition is constructed, the specific process of which includes:

[0024] 1) Converting tangential force to axial force:

[0025]

[0026] in, It is an instantaneous tangential force. For instantaneous torque, The pitch circle radius of the gear. For instantaneous axial force, The helix angle of a helical gear;

[0027] 2) Superimpose low-frequency phase modulation terms and high-frequency structural terms related to frequency conversion onto the high-fidelity health signal, and improve the random noise floor:

[0028]

[0029] In the formula, The periodic modulation amplitude characterizing the axial force. The modulation coefficient, Characterizing the high-frequency components of structural coupling, For structural coupling of high frequencies, For axial abnormal noise intensity, This is axial anomalous random noise. For high-fidelity health signals, This represents the vibration signal model of a helical gear under abnormal axial force conditions.

[0030] Furthermore, step 3 involves constructing a three-axis signal matrix, specifically including:

[0031] The reference signal is a single-axis simulation signal in a healthy state or after a fault is injected.

[0032] By linearly mapping the reference signal using a preset amplitude scaling factor, the radial, tangential, and axial signal components are obtained respectively.

[0033] Meanwhile, random disturbances following a Gaussian distribution are introduced into each signal component to simulate the sensitivity differences of sensors at different measuring points to vibration response and measurement noise interference, thereby obtaining a three-dimensional vibration signal matrix, i.e., a triaxial signal matrix, consistent with the actual working conditions.

[0034] Furthermore, the triaxial signal matrix in step 3 is represented as follows:

[0035]

[0036] In the formula, , , These represent the radial, tangential, and axial vibration signals at time t, respectively. , , These are the amplitude scaling factors for radial, tangential, and axial directions, representing the differences in sensitivity and vibration coupling strength in different directions. To follow a Gaussian distribution The random disturbance term is used to simulate measurement noise and local disturbances. This is the reference signal.

[0037] Further, in step 4, feature extraction is performed, specifically: multidimensional features of the triaxial signal are extracted from the perspectives of time domain, frequency domain, envelope demodulation, and energy distribution, and triaxial ratio features are further constructed, specifically including time domain features, envelope analysis features, frequency domain features, piecewise root mean square features, and triaxial vibration ratio features.

[0038] Among them, time-domain features are used to characterize the non-stationarity of impact-type faults, including mean, variance, skewness, and kurtosis; envelope analysis features are used to identify meshing frequencies and harmonic components, including calculating the envelope signal and extracting the mean, standard deviation, and maximum value through Hilbert transform; frequency-domain features are used to highlight periodic modulation effects, including calculating the signal spectrum using fast Fourier transform and extracting the amplitude of the first few frequency points; piecewise root mean square features are used to reflect local energy abrupt changes, including dividing the signal into several segments and calculating the root mean square value (RMS) of each segment; triaxial vibration ratio features are used to introduce correlation parameters between multidimensional signals, making the axial features significant under abnormal axial force conditions of helical gears, specifically the ratio of the RMS of each axis to the total RMS.

[0039] Furthermore, step 5 specifically includes:

[0040] For each set of triaxial signals, the multidimensional feature vectors of the three axes are concatenated to form a triaxial fusion feature vector. Based on this, the triaxial vibration ratio feature is introduced to further characterize the spatial distribution of triaxial vibration energy, resulting in a high-dimensional fusion feature vector of a single sample. Thus, a high-dimensional fusion feature vector sample set is obtained.

[0041] The high-dimensional fused feature vector sample set is normalized to eliminate the differences in dimensions and scale effects between different features;

[0042] Principal component analysis is used to reduce the dimensionality of the feature vectors in the normalized high-dimensional fused feature vector sample set.

[0043] A support vector machine model is trained using a set of high-dimensional fused feature vector samples after dimensionality reduction.

[0044] The support vector machine model maps nonlinear features to a high-dimensional space through a radial basis function kernel, and solves for the optimal hyperplane in this space. The optimization parameters include the regularization coefficient C and the kernel function parameters. By obtaining the optimal parameters through k-fold cross-validation or grid search, multi-category classification of gear and rack faults can be achieved.

[0045] Compared with the prior art, the significant advantages of this invention are:

[0046] (1) This invention adopts a combination of dynamic modeling and numerical simulation, which can directly obtain high-fidelity multi-axis vibration signals without relying on large-scale experimental data collection under actual working conditions for a long time. By injecting multiple typical faults into the model, including tooth breakage, tooth surface pitting, tooth surface wear, and abnormal axial force due to tooth root fatigue, a comprehensive fault condition database can be systematically constructed, thereby overcoming the limitations of existing methods that suffer from insufficient samples due to high experimental costs and long data collection cycles.

[0047] (2) The triaxial vibration signal matrix construction and coupling feature extraction method proposed in this invention can fully reflect the dynamic response characteristics of radial, tangential and axial directions during helical gear meshing, especially highlighting the unique axial force anomaly of helical gears in multi-dimensional channels. By jointly extracting time-domain statistics, frequency-domain energy distribution, envelope demodulation, piecewise root mean square and triaxial ratio features, it not only enhances the sensitivity to non-stationary impact faults, but also significantly improves the ability to characterize the energy distribution of coupled vibrations. Compared with traditional methods that only use single-axis signals or single feature parameters, this invention can capture fault evolution characteristics more comprehensively and accurately.

[0048] (3) In the classification and recognition stage, this invention adopts the Support Vector Machine (SVM) model and achieves robust classification under small sample and multi-feature conditions through standardization, PCA dimensionality reduction and radial basis function kernel mapping. This method has strong generalization ability and can maintain high recognition accuracy and stability under limited training sample conditions, effectively solving the problem that traditional methods are prone to insufficient accuracy due to sample size and feature redundancy in multi-class fault diagnosis.

[0049] (4) This invention not only embodies the advantages of integrating modeling and data-driven approaches in its methodology, but also has broad applicability in engineering applications. This method can be applied to gear and rack transmission mechanisms in marking and printing equipment, and can also be extended to various applications such as CNC machine tools, automated production lines, and heavy-duty linear drive equipment. It does not rely on the parameters of specific transmission components, thus possessing good versatility. Furthermore, the diagnostic results of this invention can be used for predictive maintenance and life management, significantly reducing the risk of sudden equipment downtime and maintenance costs, and improving production efficiency and operational safety.

[0050] The present invention will now be described in further detail with reference to the accompanying drawings. Attached Figure Description

[0051] Figure 1 This is a flowchart of the fault diagnosis method for helical gear and rack transmission system based on multi-axis vibration signals according to the present invention.

[0052] Figure 2 This is a time-domain comparison diagram of gear radial vibration signals under different working conditions simulated in one embodiment.

[0053] Figure 3 This is a time-domain comparison diagram of gear tangential vibration signals under different working conditions simulated in one embodiment.

[0054] Figure 4 This is a time-domain comparison diagram of gear axial vibration signals under different working conditions simulated in one embodiment.

[0055] Figure 5 This is an example of an SVM classification confusion matrix diagram. Detailed Implementation

[0056] To make the objectives, technical solutions, and advantages of this application clearer, the following detailed description is provided in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the scope of this application.

[0057] It should be noted that if the embodiments of the present invention involve directional indicators (such as up, down, left, right, front, back, etc.), the directional indicators are only used to explain the relative positional relationship and movement of the components in a certain specific posture (as shown in the figure). If the specific posture changes, the directional indicators will also change accordingly.

[0058] Furthermore, if the embodiments of this invention involve descriptions such as "first" or "second," these descriptions are for descriptive purposes only and should not be construed as indicating or implying their relative importance or implicitly specifying the number of technical features indicated. Therefore, a feature defined with "first" or "second" may explicitly or implicitly include at least one of those features. Additionally, the technical solutions of the various embodiments can be combined with each other, but this must be based on the ability of those skilled in the art to implement them. If the combination of technical solutions is contradictory or impossible to implement, it should be considered that such a combination of technical solutions does not exist and is not within the scope of protection claimed by this invention.

[0059] In one embodiment, combined Figure 1 This paper provides a fault diagnosis method for helical gear rack transmission systems based on multi-axis vibration signals. The method includes the following steps:

[0060] Step 1: Based on the structural parameters, operating conditions and dynamic model of the helical gear rack transmission system, construct a mathematical model of the gear rack system. Based on the mathematical model, simulate the vibration signal of the healthy state, and apply velocity fluctuation, meshing back nonlinearity, friction slip noise, structural modal vibration and motor electromagnetic torque fluctuation to the vibration signal to obtain a high-fidelity healthy signal.

[0061] Step 2: Inject fault features into the high-fidelity health signal to obtain a single-axis simulation signal;

[0062] Step 3: Based on the generated single-axis simulation signal, construct a three-axis signal matrix by amplitude scaling and random perturbation, including radial, tangential and axial vibration signals;

[0063] Step 4: Extract features from the three-axis signals generated in Step 3 to obtain the multi-dimensional feature vectors corresponding to each axis;

[0064] Step 5: Repeat steps 2 to 4 several times to fuse the multidimensional feature vectors to construct a sample set. Then, use the sample set to train a support vector machine model to obtain a fault diagnosis model.

[0065] Step 6: For the helical gear rack transmission system to be diagnosed, acquire triaxial signals, including radial, tangential and axial vibration signals. Then, perform feature extraction as in Step 4, and input the obtained multidimensional feature vector into the fault diagnosis model to realize gear rack fault diagnosis.

[0066] Furthermore, in one embodiment, step 1, constructing a mathematical model of the gear and rack system, specifically includes:

[0067] The nonlinear dynamic equations of the transmission system are established based on the meshing principle of helical gear and rack. The equations simultaneously consider the rotational inertia of the gear, the translational mass of the rack, the meshing stiffness, the damping term, and the external excitation force.

[0068] Meshing backlash and tooth profile error are introduced into the nonlinear dynamic equation to characterize the non-ideal nature of the actual meshing process, and the fundamental vibration response signal under healthy conditions is solved by combining the system operating conditions and boundary constraints.

[0069] By applying periodic disturbances generated by electromagnetic torque fluctuations of the motor, high-frequency random noise caused by tooth surface friction slippage, harmonic components caused by nonlinear stiffness changes in gear meshing, and coupled vibration effects caused by structural modal resonance to the fundamental vibration response signal, the fundamental vibration response signal has multi-source excitation and broadband characteristics similar to the signals acquired by actual equipment, providing a high-fidelity healthy vibration signal for subsequent fault injection and feature analysis.

[0070] Furthermore, in one embodiment, the high-fidelity health vibration signal is represented as :

[0071]

[0072] In the formula, This is the sensitivity coefficient of stiffness to vibration amplitude. For time-varying meshing stiffness, As a baseline signal for health, This is the hysteresis waveform. For the narrowband response of the structural modes, For motor electromagnetic torque fluctuation, This is broadband noise.

[0073] Specifically, the process of constructing a high-fidelity health vibration signal includes:

[0074] 1.1 Based on the structural parameters, operating conditions, and contact stiffness characteristics of the gear and rack system, the dynamic equations of the system are established. The transmission system is simplified to an equivalent mass-damping-stiffness model, and its differential equations of motion can be expressed as:

[0075]

[0076] in Let M be the system displacement response vector, which varies with time t. M represents the equivalent mass matrix of the system, including the rotational inertia of the gear and the translational mass of the rack. C is the damping coefficient matrix, which reflects the tooth surface friction, lubrication state and structural damping. For time-varying meshing stiffness, the periodic variation and nonlinear characteristics of gear meshing stiffness are taken into account; External excitation forces include electromagnetic torque fluctuations of the motor, structural modal coupling vibrations, friction, and random noise disturbances.

[0077] To achieve rapid synthesis of large batches of training samples and online reproducibility, the meshing relative motion is condensed into a single-degree-of-freedom approximation, preserving the meshing master degrees of freedom and time-varying stiffness (TVMS), significantly reducing the computational load.

[0078]

[0079]

[0080] Among them, meshing relative displacement From the pitch circle radius of the gear Gear angular displacement With rack displacement Jointly determined; equivalent centralized quality ,in For rack mass, The moment of inertia of the gear; This is the equivalent damping coefficient. External incentives.

[0081] Among them, time-varying meshing stiffness The modeling adopts a harmonic expansion form:

[0082]

[0083] In the formula, For average meshing stiffness, This represents the change in the engagement angle over time. and denoted as the amplitude and phase of the h-th harmonic, respectively, where H is the expansion order.

[0084] In engineering implementation, directly solving the above equation to obtain the displacement and acceleration response is costly. Therefore, a semi-analytical synthesis is used to form a healthy baseline signal.

[0085]

[0086] In the formula, The amplitude of the h-th harmonic is... For amplitude modulation (AM) coefficients, For phase modulation (FM) coefficients, The rotational frequency of the gear shaft. It is the h-th order modulation phase.

[0087] 1.2, The meshing hysteresis effect is introduced into the healthy baseline signal through a nonlinear function, and the hysteresis waveform is shown. The nonlinear saturation reflecting the tooth flank clearance is expressed as:

[0088]

[0089] In the formula, This represents the engagement hysteresis amplitude.

[0090] 1.3 Introducing speed fluctuations: To reflect small speed fluctuations during operation, a smoothed perturbation model is adopted for linear velocity.

[0091]

[0092] in, For nominal speed, The maximum speed, The disturbance coefficient is... For low-frequency operating conditions, the frequency varies. Zero-mean, unit-variance white noise This represents the noise intensity.

[0093] This yields the meshing frequency. With frequency conversion :

[0094]

[0095] In the formula, Let z be the end face pitch and z be the number of teeth. From this, we can obtain the meshing phase accumulated over time:

[0096]

[0097] 1.4, Introducing the narrowband response of a specific structural mode. :

[0098]

[0099] 1.5, Introducing electromagnetic torque fluctuations in the motor This indicates first-order, second-order, and other modulations of electromagnetic torque.

[0100]

[0101] 1.6, Introducing broadband noise This indicates frictional slippage and measurement noise:

[0102]

[0103] In the formula, Zero-mean, unit-variance white noise Set parameters for identifiable or empirical purposes.

[0104] 1.7 In summary, the high-fidelity signal formula established in step 1 is as follows. The fundamental signal constructed in this embodiment under healthy conditions not only includes the basic meshing frequency and harmonic components, but also superimposed friction noise, high-frequency disturbances, and low-frequency mode coupling terms, so that the analog signal has broadband characteristics and multi-source excitation characteristics close to the actual measured signal.

[0105]

[0106] in This is the sensitivity coefficient of stiffness to vibration amplitude.

[0107] Furthermore, in one embodiment, the fault in step 2 is achieved by introducing parameter disturbances, changes in contact stiffness, changes in damping characteristics, or abnormal excitation forces into the mathematical model.

[0108] Preferably, the fault features injected in step 2 include: tooth breakage, pitting, wear, tooth root fatigue, and abnormal axial force. Tooth breakage is achieved by superimposing an exponentially decaying impact sequence at the time position corresponding to the meshing angle and introducing a periodic stiffness weakening function. Pitting is achieved by applying short-time impact pulses at the periodic phase position of gear meshing and superimposing a frequency doubling modulation component. Wear is achieved by introducing amplitude amplification and phase modulation factors into the fundamental signal to reflect the nonlinear response caused by changes in meshing tooth profile. Tooth root fatigue is achieved by constructing a gradually changing amplitude attenuation window within the rotation period and combining it with a periodic weak impact signal. Abnormal axial force is achieved by superimposing low-frequency and high-frequency harmonic components into the healthy signal and introducing additional random noise to form a representative fault vibration signal sample.

[0109] Specifically, step 2 includes:

[0110] 2.1 First, to standardize the description of shocks, an exponentially decaying shock sequence is introduced:

[0111]

[0112] In the formula, For the impact amplitude, Let be the exponentially decaying time constant, and let the time offset corresponding to the initial impact trigger phase be . Average rotation period The time point of the kth impact , It is a unit step function, only when time Only then will the exponentially decaying shock term be activated.

[0113] 2.2 Injecting a broken tooth fault: At the time position corresponding to the gear meshing angle, an impact sequence is introduced and superimposed with a stiffness reduction function. Assume one revolution cycle... Each turn An impact is triggered at the phase, and a stiffness attenuation window is introduced into the latter part of that phase:

[0114]

[0115]

[0116] In the formula, For stiffness reduction function, This is the impact phase offset coefficient. This represents the impact amplitude of the broken tooth. It is the attenuation constant. It is a periodic stiffness reduction factor. Modulo operation is performed to calculate the stiffness attenuation window parameters. Used to periodically define the time interval for stiffness reduction, ensuring that the stiffness decay window is effective only in a specific phase segment within each revolution cycle. This represents the vibration signal model under a broken tooth fault.

[0117] 2.3. Pitting corrosion fault: A short-duration impact pulse is applied at the engagement phase position, introducing frequency doubling modulation. Local pitting causes weak impact due to contact micro-disengagement, triggering random phase jitter, and simultaneously generating frequency doubling sidebands and high-frequency roughness modulation.

[0118] Let the reference phase coefficient of pitting impact during the rotation cycle be... Then the trigger time In each revolution Nearby, and add Gaussian dithering. The expression follows a mean of 0 and a variance of . The normal distribution, superimposed with a double frequency conversion modulation term:

[0119]

[0120]

[0121] In the formula, This represents the impact amplitude of pitting corrosion. The pitting impact attenuation constant is... This is twice the frequency conversion modulation amplitude. For time The changing gear rotation frequency, A vibration signal model representing pitting corrosion failure;

[0122] 2.4 Injecting tooth surface wear faults introduces amplitude amplification and phase modulation into the signal. Tooth profile wear leads to changes in the effective contact line, resulting in increased equivalent backlash and meshing error. This causes harmonic amplitude redistribution and increased phase modulation, so amplitude amplification and deeper PM are applied to the fundamental and harmonic frequencies.

[0123]

[0124]

[0125] In the formula, Let h be the amplitude correction factor for the h-th harmonic. For phase modulation coefficients, This is a vibration signal model under tooth surface wear fault.

[0126] 2.5, fatigue cracking at the tooth root introduces a gradual amplitude attenuation within the rotational period. Crack initiation and propagation cause stiffness to decrease in phase segments, resulting in periodic energy dips in specific meshing segments, accompanied by weak impacts. For each revolution at the phase center... A triangular window-type amplitude attenuation is applied, and a weak impact is superimposed:

[0127]

[0128]

[0129] In the formula, This is the amplitude attenuation window width coefficient. This is the crack amplitude attenuation coefficient. It is a set of triangular window attenuation functions. This represents the impact amplitude of the crack. Let be the crack impact decay time constant. This is a vibration signal model for tooth root fatigue crack faults.

[0130] 2.6. Abnormal Axial Force Fault: The axial force of a helical gear is determined by the tangential force and the helix angle. The helical gear generates a steady-state axial force due to the helix angle. When lubrication, assembly, or load is abnormal, the axial force exhibits abnormal modulation with the meshing cycle, significantly manifested in the axial vector channel, while a coupling side band remains along the entire vector.

[0131] First, the tangential force and axial force are converted as follows:

[0132]

[0133] In the formula, It is an instantaneous tangential force. For instantaneous torque, The pitch circle radius of the gear. For instantaneous axial force, The helix angle of a helical gear;

[0134] Next, a low-frequency phase modulation term and a high-frequency structural term related to the frequency conversion are superimposed on the health signal, and the random noise floor is improved:

[0135]

[0136] In the formula, The periodic modulation amplitude characterizing the axial force. The modulation coefficient, Characterizing the high-frequency components of structural coupling, For structural coupling of high frequencies, For axial abnormal noise intensity, This is axial anomalous random noise. For high-fidelity health signals, This represents the vibration signal model of a helical gear under abnormal axial force conditions.

[0137] Furthermore, in one embodiment, step 3, constructing the triaxial signal matrix, specifically includes:

[0138] The reference signal is a single-axis simulation signal in a healthy state or after a fault is injected.

[0139] By linearly mapping the reference signal using a preset amplitude scaling factor, the radial, tangential, and axial signal components are obtained respectively.

[0140] Meanwhile, random disturbances following a Gaussian distribution are introduced into each signal component to simulate the sensitivity differences of sensors at different measuring points to vibration response and measurement noise interference, thereby obtaining a three-dimensional vibration signal matrix, i.e., a triaxial signal matrix, consistent with the actual working conditions.

[0141] Here, considering the sensitivity differences and installation uncertainties of the sensor in the radial (r), tangential (t), and axial (a) directions, the single-axis reference signal s(t) (under healthy or any fault condition) after the fault injection in step 2 is projected onto the radial, tangential, and axial three channels according to the directional sensitivity and installation differences using a reference signal mapping and light random perturbation method, thus constructing a three-axis signal matrix as follows:

[0142]

[0143] In the formula, , , These represent the radial, tangential, and axial vibration signals at time t, respectively. , , These are the amplitude scaling factors for radial, tangential, and axial directions, representing the differences in sensitivity and vibration coupling strength in different directions. To follow a Gaussian distribution The random disturbance term is used to simulate measurement noise and local disturbances. This is the reference signal.

[0144] Further, in step 4, feature extraction is performed, specifically: multidimensional features of the triaxial signal are extracted from the perspectives of time domain, frequency domain, envelope demodulation, and energy distribution, and triaxial ratio features are further constructed, specifically including time domain features, envelope analysis features, frequency domain features, piecewise root mean square features, and triaxial vibration ratio features.

[0145] Among them, time-domain features are used to characterize the non-stationarity of impact-type faults, including mean, variance, skewness, and kurtosis; envelope analysis features are used to identify meshing frequencies and harmonic components, including calculating the envelope signal and extracting the mean, standard deviation, and maximum value through Hilbert transform; frequency-domain features are used to highlight periodic modulation effects, including calculating the signal spectrum using fast Fourier transform and extracting the amplitude of the first few frequency points; piecewise root mean square features are used to reflect local energy abrupt changes, including dividing the signal into several segments and calculating the root mean square value (RMS) of each segment; triaxial vibration ratio features are used to introduce correlation parameters between multidimensional signals, making the axial features significant under abnormal axial force conditions of helical gears, specifically the ratio of the RMS of each axis to the total RMS.

[0146] Specifically, step 4 includes:

[0147] 4.1 Extracting time-domain features of the signal to characterize its impulsiveness and non-stationarity, highlighting the non-Gaussian nature of impulsive fault signals such as gear tooth breakage and pitting, specifically including:

[0148]

[0149] In the formula, This represents the amplitude of the i-th sampling point. This represents the total number of sampling points. The mean, The standard deviation is denoted as .

[0150] 4.2 Extracting signal frequency domain features: The signal spectrum is calculated using Fast Fourier Transform (FFT), and the amplitudes of the first few frequency points are extracted to identify the significance of the meshing frequency and harmonic components. The formula is as follows:

[0151]

[0152] 4.3 Extract the signal envelope demodulation features and obtain the analytic signal through Hilbert transform:

[0153]

[0154] in The Hilbert transform has the following envelope:

[0155]

[0156] Features such as mean, standard deviation, and maximum value are extracted from the envelope signal to highlight the gear meshing frequency and modulation effect.

[0157] 4.4 Extract the segmented root mean square (RMS) features of the signal to reflect local energy abrupt changes. Divide the signal into M segments, and define the RMS of each segment as:

[0158]

[0159] In the formula, L is the length of each segment. Let be the amplitude of the i-th sampling point in the m-th segment.

[0160] 4.5 Extracting triaxial ratio features of the signal to characterize the triaxial energy distribution pattern is suitable for diagnosing axial abnormalities in helical gear transmissions. This elevates the single-dimensional signal features to multi-dimensional coupled features, significantly enhancing the sensitivity to axial abnormalities. The energy proportion feature of the i-th axis... The formula is as follows:

[0161]

[0162] In the formula, express The RMS value of the axis, These represent radial, tangential, and axial directions, respectively.

[0163] Furthermore, in one embodiment, step 5 specifically includes:

[0164] For each set of triaxial signals, the multidimensional feature vectors of the three axes are concatenated to form a triaxial fusion feature vector. Based on this, the triaxial vibration ratio feature is introduced to further characterize the spatial distribution of triaxial vibration energy, resulting in a high-dimensional fusion feature vector of a single sample. Thus, a high-dimensional fusion feature vector sample set is obtained.

[0165] The high-dimensional fused feature vector sample set is normalized to eliminate the differences in dimensions and scale effects between different features;

[0166] Principal component analysis is used to reduce the dimensionality of the feature vectors in the normalized high-dimensional fused feature vector sample set.

[0167] A support vector machine model is trained using a set of high-dimensional fused feature vector samples after dimensionality reduction.

[0168] The support vector machine model maps nonlinear features to a high-dimensional space through a radial basis function kernel, and solves for the optimal hyperplane in this space. The optimization parameters include the regularization coefficient C and the kernel function parameters. By obtaining the optimal parameters through k-fold cross-validation or grid search, multi-category classification of gear and rack faults can be achieved.

[0169] Here, the radial basis function (RBF) is:

[0170]

[0171] SVM obtains the optimal classification hyperplane by solving the following optimization problem:

[0172]

[0173] The constraints are:

[0174]

[0175] in, For sample feature vectors, For sample labels, For kernel parameters, The normal vector of the classification hyperplane, Here, C is the bias term for the classification hyperplane, and C is the penalty factor. As slack variables, This is the feature mapping function.

[0176] In one embodiment, a fault diagnosis system for a helical gear and rack transmission system based on multi-axis vibration signals is provided, the system comprising:

[0177] The first module is used to: construct a mathematical model of the helical gear and rack transmission system based on the structural parameters, operating conditions and dynamic model of the helical gear and rack transmission system; simulate the vibration signal of the healthy state based on the mathematical model; and apply velocity fluctuation, meshing back nonlinearity, friction slip noise, structural modal vibration and motor electromagnetic torque fluctuation to the vibration signal to obtain a high-fidelity healthy signal.

[0178] The second module is used to: inject fault features into a high-fidelity health signal to obtain a single-axis simulation signal;

[0179] The third module is used to: construct a three-axis signal matrix, including radial, tangential and axial vibration signals, based on the generated single-axis simulation signal through amplitude scaling and random perturbation;

[0180] The fourth module is used to perform feature extraction on the three-axis signals generated by the third module to obtain the multi-dimensional feature vectors corresponding to each axis.

[0181] The fifth module is used to: repeatedly execute the second to fourth modules several times to fuse multi-dimensional feature vectors to construct a sample set, and then use the sample set to train a support vector machine model to obtain a fault diagnosis model;

[0182] The sixth module is used to: acquire triaxial signals, including radial, tangential and axial vibration signals, for the helical gear rack transmission system to be diagnosed; then perform feature extraction in the manner of the fourth module; and input the obtained multidimensional feature vector into the fault diagnosis model to realize gear rack fault diagnosis.

[0183] Specific limitations regarding the fault diagnosis system for helical gear and rack transmission systems based on multi-axis vibration signals can be found in the limitations of the fault diagnosis method for helical gear and rack transmission systems based on multi-axis vibration signals mentioned above, and will not be repeated here. Each module in the aforementioned fault diagnosis system for helical gear and rack transmission systems based on multi-axis vibration signals can be implemented entirely or partially through software, hardware, or a combination thereof. These modules can be embedded in hardware or independently of the processor in a computer device, or stored in software in the memory of a computer device, so that the processor can call and execute the corresponding operations of each module.

[0184] As a specific example, in one embodiment, the fault diagnosis method for helical gear and rack transmission system based on multi-axis vibration signals of the present invention will be described in detail.

[0185] In this embodiment, the transmission mechanism example comes from a full-width steel plate marking and printing equipment. The equipment consists of two parallel X-axis with a total length of 24.5 meters, forming a large gantry-type ground rail shaft. Both X-axis are guided by straight guide rails and employ a precision helical gear and rack transmission system. Based on this equipment, the parameters and symbols used in this embodiment are shown in Table 1 below.

[0186] Table 1 Parameters and Symbols

[0187]

[0188] See Figure 1 This embodiment provides a fault diagnosis method for a helical gear and rack transmission system based on multi-axis vibration signals. Taking the aforementioned full-width steel plate scribing and printing equipment as an example, the method is implemented using the following steps:

[0189] Step 1: Based on the structural parameters, operating conditions and dynamic model of the helical gear rack transmission system, construct a mathematical model of the gear rack system. Based on the model, simulate the vibration signal of the healthy state, and apply velocity fluctuation, meshing back nonlinearity, friction slip noise, structural modal vibration and motor electromagnetic torque fluctuation to the signal to obtain a high-fidelity healthy signal.

[0190] Step 2: Inject fault characteristics into the high-fidelity health signal, including: broken teeth, pitting, wear, tooth root fatigue, and abnormal axial force.

[0191] Step 3: Based on the generated single-axis signal, construct a three-axis signal matrix by amplitude scaling and random perturbation, including radial, tangential and axial vibration signals;

[0192] Step 4: Extract features from the generated triaxial signal, including: time-domain statistics, envelope analysis features, frequency-domain features, piecewise root mean square features, and triaxial vibration ratio features.

[0193] Step 5: Use the above feature data to train a support vector machine (SVM) model to achieve multi-category gear and rack fault diagnosis, including healthy, broken teeth, pitting, wear, tooth root fatigue, and abnormal axial force conditions.

[0194] Specifically, step 1, which involves building the model and generating health signals, includes:

[0195] 1.1 Based on the structural parameters, operating conditions, and contact stiffness characteristics of the gear and rack system described above, the dynamic equations of the system are established. The transmission system is simplified to an equivalent mass-damping-stiffness model, and its differential equations of motion can be expressed as:

[0196]

[0197] Where M represents the equivalent mass matrix of the system, including the rotational inertia of the gear and the translational mass of the rack; C is the damping coefficient matrix, which reflects the tooth surface friction, lubrication state and structural damping; For time-varying meshing stiffness, the periodic variation and nonlinear characteristics of gear meshing stiffness are taken into account; External excitation forces include electromagnetic torque fluctuations of the motor, structural modal coupling vibrations, friction, and random noise disturbances.

[0198] To achieve rapid synthesis of large batches of training samples and online reproducibility, the meshing relative motion is condensed into a single-degree-of-freedom approximation, preserving the meshing master degrees of freedom and time-varying stiffness (TVMS), significantly reducing the computational load.

[0199]

[0200] in The time-varying meshing stiffness modeling employs a harmonic expansion form and is discretized.

[0201]

[0202] in, For average meshing stiffness, This represents the change in the engagement angle over time. and Let be the amplitude and phase of the h-th harmonic, respectively. This refers to the expansion order. The example uses... To balance accuracy and speed.

[0203] In engineering implementation, directly solving the above equation to obtain the displacement and acceleration response is costly. Therefore, a semi-analytical synthesis is used to form a healthy baseline signal.

[0204]

[0205] In practical engineering, parameter H is usually taken as 2-4. In this example, H=3 is used, which can cover the main peak and the first two harmonics. Helical gear meshing, under healthy conditions, exhibits multiple harmonic superposition with simultaneous amplitude modulation (AM) and phase modulation (PM). The AM depth can be selected as... PM depth This expression simultaneously generates the main frequency, harmonics, sidebands, and frequency modulation bands, enabling the reproduction of the typical spectral structure of the meshing signal.

[0206] 1.2, The meshing hysteresis effect is introduced through a nonlinear function, and the hysteresis waveform is shown. The nonlinear saturation reflecting the tooth flank clearance is used to simulate the impact vibration during reverse motion. Considering engineering implementation, an approximate discretization is obtained:

[0207]

[0208] in This represents the engagement hysteresis amplitude.

[0209] 1.3 Based on the variation pattern of the equipment's operating speed of 6 m / min and maximum return speed of 30 m / min, considering slow-changing load and periodic disturbances, and limiting the amplitude, speed fluctuations are simulated, and a smooth disturbance model is adopted for the linear velocity:

[0210]

[0211] in For nominal speed, The maximum speed, For low-frequency operating conditions, the frequency varies. The noise is Gaussian white noise. After discretization, the instantaneous velocity sequence is obtained directly through array superposition. From this, the meshing frequency and rotational frequency are derived.

[0212]

[0213] The engagement phase accumulates over time, and then a discrete integral is performed:

[0214]

[0215] 1.4 Based on the inertial characteristics of the total mass and the modal parameters of the gantry structure, a structural resonance term is added to the signal to simulate the structural vibration caused by the motion of a large mass, generating a signal containing structural modal perturbations. :

[0216]

[0217] 1.5, Introducing electromagnetic torque fluctuations in the motor This indicates first-order and second-order modulation of electromagnetic torque. By changing the gear meshing excitation force through the relationship between torque and speed, a signal containing electromagnetic disturbances is obtained.

[0218]

[0219] 1.6, Introducing broadband noise This indicates frictional slippage and measurement noise:

[0220]

[0221] in Zero-mean, unit-variance white noise Set parameters for identifiable or empirical purposes.

[0222] 1.7 In summary, the high-fidelity signal formula established in step 1 is as follows. The fundamental signal constructed in this embodiment under healthy conditions not only includes the basic meshing frequency and harmonic components, but also superimposed friction noise, high-frequency disturbances, and low-frequency mode coupling terms, so that the analog signal has broadband characteristics and multi-source excitation characteristics close to the actual measured signal.

[0223]

[0224] Among them, take This is the sensitivity coefficient of stiffness to vibration amplitude.

[0225] Specifically, step 2, injecting fault features, involves constructing various typical fault signals by perturbing the model parameters and excitation signals based on the healthy signals generated in step 1. The specific process includes:

[0226] 2.1 First, to unify the description of the impact, an exponentially decaying impact sequence is introduced and discretized to obtain:

[0227]

[0228] in is the sampling frequency, is the discrete trigger time of the k-th impact, and is the impact amplitude. Attenuation constant .

[0229] 2.2 Injecting a broken tooth fault: At the time position corresponding to the gear meshing angle, an impact sequence is introduced and superimposed with a stiffness reduction function. Assume one revolution cycle... Each turn An impact is triggered at the phase, and a stiffness attenuation window is introduced into the latter part of that phase:

[0230]

[0231]

[0232] Where the attenuation constant is taken Impact amplitude taken , The periodic stiffness reduction factor is the proportion of the window width. .

[0233] 2.3. Pitting corrosion fault: A short-duration impact pulse is applied at the engagement phase position, introducing frequency doubling modulation. Local pitting causes weak impact due to contact micro-disengagement, triggering random phase jitter, and simultaneously generating frequency doubling sidebands and high-frequency roughness modulation.

[0234] The trigger time is during each revolution. Nearby Gaussian jitter, superimposed with a double frequency shift modulation term:

[0235]

[0236]

[0237] Among them, take .

[0238] 2.4 Injecting tooth surface wear faults introduces amplitude amplification and phase modulation into the signal. Tooth profile wear leads to changes in the effective contact line, resulting in increased equivalent backlash and meshing error. This causes harmonic amplitude redistribution and increased phase modulation, so amplitude amplification and deeper PM are applied to the fundamental and harmonic frequencies.

[0239]

[0240] in, This manifests as harmonic distortion and amplitude amplification. And synthesized using the same TVMS, hysteresis, structure, and noise terms:

[0241]

[0242] 2.5, fatigue cracking at the tooth root introduces a gradual amplitude attenuation within the rotational period. Crack initiation and propagation cause stiffness to decrease in phase segments, resulting in periodic energy dips in specific meshing segments, accompanied by weak impacts. For each revolution at the phase center... A triangular window-type amplitude attenuation is applied, and a weak impact is superimposed:

[0243]

[0244]

[0245] in, This is a triangular window function that takes the center of the window. Width ratio (Window half-width), amplitude attenuation coefficient Weak impact .

[0246] 2.6. Abnormal Axial Force Fault: The axial force of a helical gear is determined by the tangential force and the helix angle. The helical gear generates a steady-state axial force due to the helix angle. When lubrication, assembly, or load is abnormal, the axial force exhibits abnormal modulation with the meshing cycle, significantly manifested in the axial vector channel, while a coupling side band remains along the entire vector.

[0247] First, the tangential force and axial force are converted as follows:

[0248]

[0249] By altering the helix angle to induce an additional axial load, low-frequency and high-frequency harmonics are superimposed on the healthy signal to represent its observed effect, and the random noise floor is enhanced.

[0250]

[0251] The low-frequency modulation amplitude of the axial force High-frequency components of structural coupling The frequency band related to the structure can be selected , Add Gaussian noise .

[0252] Specifically, step 3, the process of constructing the triaxial signal, includes:

[0253] In a gear and rack transmission system, vibration signals are mainly distributed in three directions: radial (r), tangential (t), and axial (a). The signal amplitude and noise level differ in each direction. To obtain a three-dimensional vibration signal matrix that reflects the actual working conditions, a single-axis reference signal (in healthy or faulty condition) is first generated based on steps 1 and 2. Then, a three-axis signal matrix is ​​constructed using amplitude scaling and random perturbation methods.

[0254]

[0255] Among them, take This represents the difference in sensitivity and vibration coupling strength in different directions; To follow a Gaussian distribution The random perturbation term is used to simulate measurement noise and local perturbations. This linear mapping preserves the three-dimensional coupling differences without increasing the model complexity.

[0256] The model generated a triaxial high-fidelity vibration signal through steps 1, 2, and 3, such as... Figure 2 , 3 Figures 4 and 5 show the time-domain comparison of gear axial, tangential, and radial vibration signals under different working conditions.

[0257] Specifically, step 4, which involves feature extraction from the generated triaxial signal, includes:

[0258] Multidimensional features are extracted from the triaxial vibration signal obtained in step 3 from the perspectives of time domain, frequency domain, envelope demodulation, and energy distribution. Furthermore, triaxial ratio features are constructed to enhance the characterization of energy coupling relationships under different working conditions.

[0259] 4.1 Extracting time-domain features of the signal to characterize its impulsiveness and non-stationarity, highlighting the non-Gaussian nature of impulsive fault signals such as gear tooth breakage and pitting, specifically including:

[0260]

[0261] 4.2 Extracting signal frequency domain features: The signal spectrum is calculated using Fast Fourier Transform (FFT), and the amplitudes of the first few frequency points are extracted to identify the significance of the meshing frequency and harmonic components. The formula is as follows:

[0262]

[0263] 4.3 Extract the signal envelope demodulation features and obtain the analytic signal through Hilbert transform:

[0264]

[0265] in The Hilbert transform has the following envelope:

[0266]

[0267] Features such as mean, standard deviation, and maximum value are extracted from the envelope signal to highlight the gear meshing frequency and modulation effect.

[0268] 4.4 Extract the segmented root mean square (RMS) features of the signal to reflect local energy abrupt changes. Divide the signal into M segments, and define the RMS of each segment as:

[0269]

[0270] In the formula, L is the length of each segment. Let be the amplitude of the i-th sampling point in the m-th segment.

[0271] 4.5 Extracting the triaxial ratio features of the signal to characterize the triaxial energy distribution pattern is suitable for diagnosing axial abnormalities in helical gear transmissions. This elevates the single-dimensional signal features to multi-dimensional coupled features, significantly enhancing the sensitivity to axial abnormalities. The RMS ratio feature formula is as follows:

[0272]

[0273] Specifically, step 5, the process of training and testing based on the SVM model, includes:

[0274] 5.1 Constructing training samples:

[0275] ① Generate labeled dataset: Based on the model established above, set the sampling frequency fs=5000Hz and the sampling duration T=1.0s, generate 1000 sample datasets for each of the 6 working conditions, and finally output sample set x(6000, 5000, 3) and label set y(6000, 1).

[0276] ② Data Feature Dimensions: After feature extraction of the original signal in step 4, features are extracted separately for each axis and then fused. A total of 8-dimensional time-domain features, 3-dimensional envelope analysis features, 50-dimensional frequency-domain features, 10-dimensional piecewise root mean square feature values, and 3-dimensional triaxial vibration ratio features are extracted, with a total of 71×3+3=216 features across the three axes. The final output is X_feat(6000,216).

[0277] ③ PCA dimensionality reduction: After obtaining the multidimensional feature vectors, they are normalized, and the PCA method is used to map the high-dimensional features to a low-dimensional space, reducing the dimensionality to 100 dimensions, thus reducing feature redundancy and training time. After the above processing, each signal sample is mapped to a fixed-length feature vector.

[0278] 5.2 Support Vector Machine Classifier: Radial Basis Function (RBF) is selected as the kernel function.

[0279]

[0280] in, The kernel parameters are used. SVM obtains the optimal classification hyperplane by solving the following optimization problem:

[0281]

[0282] The constraints are:

[0283]

[0284] in, For sample feature vectors, For sample labels, For kernel parameters, The normal vector of the classification hyperplane, Here, C is the bias term for the classification hyperplane, and C is the penalty factor. As slack variables, This is the feature mapping function.

[0285] 5.3 Parameter Optimization and Training:

[0286] ① Parameter optimization: Candidate parameters , By using grid search and 5-fold cross-validation (dividing the training set into 5 parts, 4 for training and 1 for validation, iterating 5 times), C and ... are simultaneously optimized. This achieves optimal classification performance.

[0287] ② Dataset partitioning: 80% training set, 20% test set, stratified sampling to ensure consistent proportions for each category.

[0288] ③ Model Training and Experimental Results: The multidimensional feature vectors obtained from the dimensionality reduction step were input into the classification model for training and testing. The overall classification accuracy on the test set reached 97.75%. Figure 5 The output confusion matrix further verifies the diagnostic stability of the model. The final trained classifier can distinguish six states: healthy, broken tooth, pitting, wear, tooth root fatigue, and abnormal axial force. The diagnostic accuracy meets the actual application requirements of industrial sites.

[0289] In summary, the fault diagnosis method for helical gear and rack transmission systems based on multi-axis vibration signals proposed in this invention has the following characteristics: It solves the problem of small sample sizes: through signal generation technology driven by physical mechanisms, high-fidelity fault samples can be generated even when measured data is lacking, eliminating the need for large-scale on-site collection and reducing data acquisition costs; it improves the realism of fault features: superimposed multi-source interference and refined fault injection methods accurately reflect the physical characteristics of the fault; it has wide engineering applicability: the simulation modeling process of this invention can be flexibly adjusted according to the parameters of different devices to generate suitable fault datasets; at the same time, the generated standardized datasets can be directly used for training and verification of algorithms such as deep learning and neural networks, reducing the cost and cycle of subsequent algorithm development, and is suitable for gear and rack fault diagnosis of various high-precision transmission equipment; the model has low computational complexity and can be deployed on embedded devices to achieve real-time diagnosis.

[0290] The foregoing has shown and described the basic principles, main features, and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited to the above embodiments. The embodiments and descriptions in the specification are merely illustrative of the principles of the invention. Any modifications, equivalent substitutions, or improvements made within the spirit and principles of the present invention without departing from its spirit and scope should be included within the protection scope of the present invention.

Claims

1. A method for fault diagnosis of a helical gear-rack transmission system based on multi-axis vibration signals, characterized in that, The method includes the following steps: Step 1: Based on the structural parameters, operating conditions and dynamic model of the helical gear rack transmission system, construct a mathematical model of the gear rack system. Based on the mathematical model, simulate the vibration signal of the healthy state, and apply velocity fluctuation, meshing back nonlinearity, friction slip noise, structural modal vibration and motor electromagnetic torque fluctuation to the vibration signal to obtain a high-fidelity healthy signal. Step 2: Inject fault features into the high-fidelity health signal to obtain a single-axis simulation signal; Step 3: Based on the generated single-axis simulation signal, construct a three-axis signal matrix by amplitude scaling and random perturbation, including radial, tangential and axial vibration signals; Step 4: Extract features from the three-axis signals generated in Step 3 to obtain the multi-dimensional feature vectors corresponding to each axis; Step 5: Repeat steps 2 to 4 several times to fuse the multidimensional feature vectors to construct a sample set. Then, use the sample set to train a support vector machine model to obtain a fault diagnosis model. Step 6: For the helical gear rack transmission system to be diagnosed, acquire triaxial signals, including radial, tangential and axial vibration signals. Then, perform feature extraction as in Step 4, and input the obtained multidimensional feature vector into the fault diagnosis model to realize gear rack fault diagnosis.

2. The fault diagnosis method for helical gear and rack transmission system based on multi-axis vibration signals according to claim 1, characterized in that, Step 1 involves constructing a mathematical model of the gear and rack system, specifically including: The nonlinear dynamic equations of the transmission system are established based on the meshing principle of helical gear and rack. The equations simultaneously consider the rotational inertia of the gear, the translational mass of the rack, the meshing stiffness, the damping term, and the external excitation force. Meshing backlash and tooth profile error are introduced into the nonlinear dynamic equation to characterize the non-ideal nature of the actual meshing process, and the fundamental vibration response signal under healthy conditions is solved by combining the system operating conditions and boundary constraints. By applying periodic disturbances generated by electromagnetic torque fluctuations of the motor, high-frequency random noise caused by tooth surface friction slippage, harmonic components caused by nonlinear stiffness changes in gear meshing, and coupled vibration effects caused by structural modal resonance to the fundamental vibration response signal, the fundamental vibration response signal has multi-source excitation and broadband characteristics similar to the signals acquired by actual equipment, providing a high-fidelity healthy vibration signal for subsequent fault injection and feature analysis.

3. The fault diagnosis method for helical gear and rack transmission system based on multi-axis vibration signals according to claim 2, characterized in that, The high-fidelity health vibration signal is represented as : ； In the formula, This is the sensitivity coefficient of stiffness to vibration amplitude. For time-varying meshing stiffness, As a baseline signal for health, This is the hysteresis waveform. For the narrowband response of the structural modes, For motor electromagnetic torque fluctuation, This is broadband noise.

4. The fault diagnosis method for helical gear and rack transmission system based on multi-axis vibration signals according to claim 1, characterized in that, In step 2, the fault is caused by introducing parameter disturbances, changes in contact stiffness, changes in damping characteristics, or abnormal excitation forces into the mathematical model.

5. The fault diagnosis method for helical gear and rack transmission system based on multi-axis vibration signals according to claim 1, characterized in that, Step 2 injects fault features, including: tooth breakage, pitting, wear, tooth root fatigue, and abnormal axial force. Tooth breakage is achieved by superimposing an exponentially decaying impact sequence at the time position corresponding to the meshing angle and introducing a periodic stiffness weakening function. Pitting is achieved by applying short-time impact pulses at the periodic phase position of gear meshing and superimposing a frequency doubling modulation component. Wear is achieved by introducing amplitude amplification and phase modulation factors into the fundamental signal to reflect the nonlinear response caused by changes in meshing tooth profile. Tooth root fatigue is achieved by constructing a gradually changing amplitude attenuation window within the rotation period and combining it with a periodic weak impact signal. Abnormal axial force is achieved by superimposing low-frequency and high-frequency harmonic components into the healthy signal and introducing additional random noise to form representative fault vibration signal samples.

6. The fault diagnosis method for helical gear and rack transmission system based on multi-axis vibration signals according to claim 5, characterized in that, The axial force anomaly specifically refers to the axial force anomaly of the helical gear. Based on the high-fidelity healthy signal, a vibration signal model of the helical gear under the abnormal axial force condition is constructed. The specific process includes: 1) Converting tangential force to axial force: ； in, It is an instantaneous tangential force. For instantaneous torque, The pitch circle radius of the gear. For instantaneous axial force, The helix angle of a helical gear; 2) Superimpose low-frequency phase modulation terms and high-frequency structural terms related to frequency conversion onto the high-fidelity health signal, and improve the random noise floor: ； In the formula, The periodic modulation amplitude characterizing the axial force. The modulation coefficient, Characterizing the high-frequency components of structural coupling, For structural coupling of high frequencies, For axial abnormal noise intensity, This is axial anomalous random noise. For high-fidelity health signals, This represents the vibration signal model of a helical gear under abnormal axial force conditions.

7. The fault diagnosis method for helical gear and rack transmission system based on multi-axis vibration signals according to claim 1, characterized in that, Step 3 involves constructing a three-axis signal matrix, specifically including: The reference signal is a single-axis simulation signal in a healthy state or after a fault is injected. By linearly mapping the reference signal using a preset amplitude scaling factor, the radial, tangential, and axial signal components are obtained respectively. Meanwhile, random disturbances following a Gaussian distribution are introduced into each signal component to simulate the sensitivity differences of sensors at different measuring points to vibration response and measurement noise interference, thereby obtaining a three-dimensional vibration signal matrix, i.e., a triaxial signal matrix, consistent with the actual working conditions.

8. The fault diagnosis method for helical gear and rack transmission system based on multi-axis vibration signals according to claim 7, characterized in that, The triaxial signal matrix in step 3 is represented as follows: ； In the formula, , , These represent the radial, tangential, and axial vibration signals at time t, respectively. , , These are the amplitude scaling factors for radial, tangential, and axial directions, representing the differences in sensitivity and vibration coupling strength in different directions. To follow a Gaussian distribution The random disturbance term is used to simulate measurement noise and local disturbances. This is the reference signal.

9. The fault diagnosis method for helical gear and rack transmission system based on multi-axis vibration signals according to claim 1, characterized in that, In step 4, feature extraction is performed, specifically: multidimensional features of the triaxial signal are extracted from the perspectives of time domain, frequency domain, envelope demodulation, and energy distribution, and triaxial ratio features are further constructed, including time domain features, envelope analysis features, frequency domain features, piecewise root mean square features, and triaxial vibration ratio features. Among them, time-domain features are used to characterize the non-stationarity of impact-type faults, including mean, variance, skewness, and kurtosis; envelope analysis features are used to identify meshing frequencies and harmonic components, including calculating the envelope signal and extracting the mean, standard deviation, and maximum value through Hilbert transform; frequency-domain features are used to highlight periodic modulation effects, including calculating the signal spectrum using fast Fourier transform and extracting the amplitude of the first few frequency points; piecewise root mean square features are used to reflect local energy abrupt changes, including dividing the signal into several segments and calculating the root mean square value (RMS) of each segment; triaxial vibration ratio features are used to introduce correlation parameters between multidimensional signals, making the axial features significant under abnormal axial force conditions of helical gears, specifically the ratio of the RMS of each axis to the total RMS.

10. The fault diagnosis method for a helical gear and rack transmission system based on multi-axis vibration signals according to claim 9, characterized in that, Step 5 specifically includes: For each set of triaxial signals, the multidimensional feature vectors of the three axes are concatenated to form a triaxial fusion feature vector. Based on this, the triaxial vibration ratio feature is introduced to further characterize the spatial distribution of triaxial vibration energy, resulting in a high-dimensional fusion feature vector of a single sample. Thus, a high-dimensional fusion feature vector sample set is obtained. The high-dimensional fused feature vector sample set is normalized to eliminate the differences in dimensions and scale effects between different features; Principal component analysis is used to reduce the dimensionality of the feature vectors in the normalized high-dimensional fused feature vector sample set. A support vector machine model is trained using a set of high-dimensional fused feature vector samples after dimensionality reduction. The support vector machine model maps nonlinear features to a high-dimensional space through a radial basis function kernel, and solves for the optimal hyperplane in this space. The optimization parameters include the regularization coefficient C and the kernel function parameters. By obtaining the optimal parameters through k-fold cross-validation or grid search, multi-category classification of gear and rack faults can be achieved.

Images