A fault prediction method for spur gear pairs based on dynamic and gradual coupled torsional vibration model of spur gear pairs

Abstract

The present invention provides a fault prediction method for a spur gear pair based on a dynamic and gradually coupled torsional vibration model of a spur gear pair, and relates to the technical field of fault monitoring of spur gear pairs. The specific steps include: using parameters to construct a static transmission error function and a nonlinear tooth side clearance function; then, generating time-varying meshing stiffness of a fault-free and single broken tooth by data processing, and combining to generate a comprehensive time-varying meshing stiffness; then, constructing a single-degree-of-freedom nonlinear dynamic equation for torsion of a spur gear pair, generating dynamic loads, time-varying bending stress at the tooth root, and time-varying contact stress on the tooth surface; finally, calculating the fatigue damage amount of the tooth root and the tooth surface, constructing a fault coefficient, and comparing it with a preset threshold to predict the fault condition of the spur gear pair. The present invention avoids the hysteresis of fault identification of traditional methods, prevents sudden damage to equipment, solves a general dynamic model of a spur gear pair with any degree of damage under any working condition, and has high numerical simulation accuracy.

Description

Technical Field

[0001] The invention relates to the technical field of spur gear pair fault monitoring, and in particular to a spur gear pair fault prediction method based on a spur gear pair dynamic and gradual coupling torsional vibration model. Background Art

[0002] Gear transmission is the most common transmission mode in mechanical transmission. Due to its advantages of large bearing capacity, high transmission efficiency, high transmission accuracy and compact installation, it is the most widely used component of electromechanical equipment system in national production industry. Among them, the stress concentration, cycle and various impacts caused by the vibration of the gear pair will inevitably cause damage and failure of the gear structure. The structural damage will directly reduce the strength and stiffness of the gear, which will further affect the vibration response of the gear pair. This cycle is repeated, causing damage expansion and even failure. Therefore, the gear transmission process obviously presents dynamic and gradual coupling behavior.

[0003] In mechanical transmission systems, spur gear pairs are one of the core components, and their performance and reliability directly affect the stability and efficiency of the entire transmission system. However, during long-term operation, spur gear pairs are often affected by various complex factors, resulting in wear, broken teeth and other faults. Traditional gear fault prediction methods are mostly based on static analysis or simple dynamic models. Although the dynamic characteristics of gear pairs are taken into account, the processing of factors such as meshing stiffness, torsional displacement and tooth side clearance is simplified, and the dynamic behavior of gear pairs under complex working conditions cannot be fully captured. Due to the lack of effective analysis of gradual fault states, traditional methods have lags in fault identification and cannot provide timely warnings at the early stage of faults, resulting in sudden failure of gear pairs. In addition, traditional nonlinear dynamic models of gear systems have limitations in adaptability, quantitative characterization, coupling analysis and simulation accuracy.

[0004] The above information disclosed in this Background section is only for enhancement of understanding of the background of the present disclosure and therefore it may contain information that does not constitute the prior art that is already known to one of ordinary skill in the art. Summary of the invention

[0005] The object of the present invention is to provide a fault prediction method for a spur gear pair based on a dynamic and gradually coupled torsional vibration model of a spur gear pair, so as to solve the problems raised in the above background technology.

[0006] To achieve the above object, the present invention provides the following technical solutions:

[0007] A fault prediction method for a spur gear pair based on a dynamic and gradually coupled torsional vibration model of a spur gear pair, the specific steps comprising:

[0008] S1. Collecting physical parameters and operating parameters of the spur gear pair, the physical parameters include the number of teeth of the driving gear and the passive gear, the base circle radius of the driving gear and the passive gear, the moment of inertia of the driving gear and the passive gear, the input shaft torsional damping coefficient, the output shaft torsional damping coefficient, the input shaft torsional stiffness coefficient and the output shaft torsional stiffness coefficient, the operating parameters include the torsion angle of the driving gear, the torsion angle of the passive gear, the input torque of the driving gear, the load torque of the passive gear, the motor speed frequency of the driving gear and the relative displacement of the meshing point of the driving gear and the passive gear;

[0009] S2. Obtain the average meshing stiffness according to the collected input shaft torsional stiffness coefficient and output shaft torsional stiffness coefficient, process the average meshing stiffness, input shaft torsional damping coefficient, output shaft torsional damping coefficient, base circle radius of the driving gear, base circle radius of the driven gear, moment of inertia of the driving gear and moment of inertia of the driven gear to obtain meshing viscoelastic damping, construct a static transmission error function according to the motor rotation frequency of the driving gear, and construct a nonlinear tooth side clearance function according to the relative displacement of the meshing point between the driving gear and the driven gear;

[0010] S3. Process the average mesh stiffness and the amplitude of the fault-free stiffness change to generate the time-varying mesh stiffness of the fault-free spur gear pair, process the amplitude of the single-break tooth stiffness change and the number of teeth of the driving gear to generate the time-varying mesh stiffness of the single-break tooth spur gear pair, process the time-varying mesh stiffness of the fault-free spur gear pair and the time-varying mesh stiffness of the single-break tooth spur gear pair to generate the comprehensive time-varying mesh stiffness of the spur gear pair;

[0011] S4. Process the torsion angle of the driving gear, the torsion angle of the driven gear, the base circle radius of the driving gear, the base circle radius of the driven gear and the static transmission error function to generate the torsion displacement of the gear pair. According to the comprehensive time-varying meshing stiffness of the spur gear pair, the torsion displacement of the gear pair, the meshing viscoelastic damping and the nonlinear tooth side clearance function, construct the single-degree-of-freedom nonlinear dynamic equation of the torsion of the spur gear pair. According to the interval range of the torsion displacement of the gear pair, generate the dynamic load when the spur gear teeth are meshed. According to the dynamic load when the spur gear teeth are meshed, generate the time-varying bending stress of the tooth root and the time-varying contact stress of the tooth surface.

[0012] S5.According to the time-varying bending stress at the tooth root and the time-varying contact stress at the tooth surface, the time-varying bending fatigue damage amount at the tooth root and the time-varying contact fatigue damage amount at the tooth surface are obtained. According to the time-varying bending fatigue damage amount at the tooth root and the time-varying contact fatigue damage amount at the tooth surface, a fault coefficient is constructed. The comprehensive time-varying meshing stiffness of the spur gear pair, the single-degree-of-freedom nonlinear dynamic equation of the torsion of the spur gear pair and the fault coefficient are coupled to construct a dynamic and gradual coupling torsional vibration model of the spur gear pair. The fault coefficient is compared with the preset fault coefficient factor threshold to predict the fault condition of the spur gear pair.

[0013] According to the collected input shaft torsional stiffness coefficient and output shaft torsional stiffness coefficient, the average meshing stiffness is obtained according to the following formula:

[0014]

[0015] in, is the average meshing stiffness, k1 is the input shaft torsional stiffness coefficient, k2 is the output shaft torsional stiffness coefficient;

[0016] The meshing viscoelastic damping is obtained by processing the average meshing stiffness, input shaft torsional damping coefficient, output shaft torsional damping coefficient, base circle radius of the driving gear, base circle radius of the driven gear, moment of inertia of the driving gear and moment of inertia of the driven gear. The formula is as follows:

[0017]

[0018] Where c is the meshing viscoelastic damping, c1 is the input shaft torsional damping coefficient, c2 is the output shaft torsional damping coefficient, r b1 is the base circle radius of the driving gear, r b2 is the base circle radius of the passive gear, I1 is the moment of inertia of the active gear, and I2 is the moment of inertia of the passive gear.

[0019] Furthermore, according to the motor rotation frequency of the driving gear, a static transmission error function is constructed based on the following formula:

[0020] e(t)=e a sin(2πft)

[0021] Among them, e(t) is the transmission error value at time t, e a is the error amplitude, f is the motor speed frequency of the driving gear, and t is the time variable;

[0022] According to the relative displacement of the meshing point between the active gear and the passive gear, a nonlinear tooth backlash function is constructed based on the following formula:

[0023]

[0024] Among them, g(x) is a function of the nonlinear tooth backlash changing with the relative displacement of the meshing point between the driving gear and the driven gear, which represents the tooth backlash value at the displacement x, x is the relative displacement of the meshing point between the driving gear and the driven gear, and b is the tooth backlash threshold.

[0025] Furthermore, the Fourier series is used to process the average mesh stiffness and the amplitude of the fault-free stiffness change to generate the time-varying mesh stiffness of the fault-free spur gear pair, based on the following formula:

[0026]

[0027] Among them, K m0 (t) is the time-varying mesh stiffness of the fault-free spur gear pair, k a0 is the amplitude of the fault-free stiffness change, n is the term index of the Fourier series, indicating the component number of the sine term, n = 1, 3, 5, ..., ∞, n is an odd number;

[0028] The Fourier series is used to process the amplitude of the change in the single-break tooth stiffness and the number of teeth of the driving gear to generate the time-varying meshing stiffness of the single-break tooth spur gear pair, based on the following formula:

[0029]

[0030] Among them, K m1 (t) is the time-varying mesh stiffness of the single-broken-tooth spur gear pair, k a1 is the amplitude of the change in stiffness of a single broken tooth, z1 is the number of teeth of the driving gear, and n is the index of the number of terms in the Fourier series;

[0031] The time-varying mesh stiffness of the fault-free spur gear pair and the time-varying mesh stiffness of the single-broken-tooth spur gear pair are processed to generate the comprehensive time-varying mesh stiffness of the spur gear pair, based on the following formula:

[0032] K(t)=(1-ε)K m0 (t)+εK m1 (t)

[0033] Among them, K(t) is the comprehensive time-varying mesh stiffness of the spur gear pair, and ε is the fault factor.

[0034] Furthermore, the torsion angle of the driving gear, the torsion angle of the driven gear, the base circle radius of the driving gear, the base circle radius of the driven gear and the static transmission error function are processed to generate the torsional displacement of the gear pair according to the following formula:

[0035] q(t)=r b1 θ1-r b2 θ2-e(t)

[0036] Among them, q(t) is the torsional displacement of the gear pair, θ1 is the torsional angle of the driving gear, and θ2 is the torsional angle of the driven gear.

[0037] Furthermore, based on the comprehensive time-varying meshing stiffness of the spur gear pair, the torsional displacement of the gear pair, the meshing viscoelastic damping and the nonlinear tooth side clearance function, the single-degree-of-freedom nonlinear dynamic equation of the torsion of the spur gear pair is constructed, and the formula is as follows:

[0038]

[0039] in, is the second-order derivative of the torsional displacement, is the first-order derivative of the torsional displacement, g(q(t)) is the nonlinear tooth backlash, m e is the equivalent mass, F m is the total normal force, F m =J1 / r b1 =J2 / r b2 , J1 is the input torque of the driving gear, J2 is the load torque of the passive gear, is the second-order derivative of the auxiliary variable transmission error.

[0040] Furthermore, according to the range of the torsional displacement of the gear pair, the dynamic load when the spur gear is meshing is generated according to the following formula:

[0041]

[0042] Among them, W d is the dynamic load when the spur gear is meshing, k3 is the nonlinear stiffness coefficient, E2 is the elastic modulus of the second material, h2 is the width of the passive gear tooth, χ is the displacement coefficient, λ is the loading position coefficient, λ=(r n2 -r h2 ) / τ,r n2 is the pitch radius of the passive gear at the meshing point, r h2 is the radius of the top circle of the passive gear, τ is the gear module of the passive gear, A2, B2, C2 are all nonlinear adjustment coefficients;

[0043] According to the dynamic load when the spur gear is meshing, the time-varying bending stress at the tooth root and the time-varying contact stress on the tooth surface are generated according to the following formula:

[0044]

[0045]

[0046] Among them, σ F is the time-varying bending stress at the tooth root, σ H is the time-varying bending stress on the tooth surface, K is the load distribution coefficient K = K a K α K β , K a is the amplitude of stiffness change, K α is the inter-tooth load distribution coefficient, K β is the tooth load distribution coefficient, Y Fa is the tooth form factor, Y Sa is the stress correction factor, α n is the pressure angle of the gear pair, h max is the large gear tooth width, is the gear module, Z H is the contact stress coefficient, Z E is the elastic influence coefficient, U is the gear ratio, that is, the ratio of the number of teeth on the driven gear to the number of teeth on the driving gear, d1 is the pitch diameter of the driving gear, E1 is the elastic modulus of the first material, μ1 and μ2 are the first Poisson's ratio and the second Poisson's ratio respectively, and z2 is the number of teeth on the passive gear.

[0047] Furthermore, according to the time-varying bending stress at the tooth root and the time-varying contact stress at the tooth surface, the time-varying bending fatigue damage amount at the tooth root and the time-varying contact fatigue damage amount at the tooth surface are obtained, and the formulas are as follows:

[0048]

[0049] Among them, D F (t) is the time-varying bending fatigue damage of the tooth root, D H (t) is the time-varying bending fatigue damage of the tooth surface, ρ is the fatigue life parameter of the material, P is the cumulative strength limit of the material, s is the stress amplitude, u F is the mean value of the time-varying bending stress at the tooth root, u H is the mean value of the time-varying bending stress on the tooth surface, is the average value of the time-varying bending stress at the tooth root, is the average value of the time-varying contact stress on the tooth surface.

[0050] Furthermore, the failure coefficient is constructed based on the time-varying bending fatigue damage of the tooth root and the time-varying contact fatigue damage of the tooth surface, and the formula is as follows:

[0051]

[0052] Where gx is the failure coefficient, ξ fH is the damage weight coefficient;

[0053] Damage weight coefficient ξ fH ∈[0,1], when the whole tooth root is bending fatigue, ξ fH =1, when the whole tooth surface is in contact fatigue, ξ fH =0, when there is both tooth root bending fatigue and tooth surface contact fatigue, 0<ξ fH <1;

[0054] The comprehensive time-varying meshing stiffness of the spur gear pair, the single-degree-of-freedom nonlinear dynamic equation of the torsion of the spur gear pair, and the fault coefficient are coupled to construct the dynamic and gradual coupling torsional vibration model of the spur gear pair. The model is as follows:

[0055]

[0056] in, is the angular velocity of the driving gear, is the angular velocity of the driven gear, is the angular acceleration of the driving gear, is the angular acceleration of the driven gear;

[0057] Compare the fault coefficient gx with the preset fault coefficient threshold yz to predict the fault condition of the spur gear pair:

[0058] When gx < yz, the spur gear pair is in normal working condition;

[0059] When gx ≥ yz, the spur gear pair has a fault.

[0060] Compared with the prior art, the beneficial effects of the present invention are:

[0061] By collecting the physical parameters and operating parameters of the driving gear and the driven gear, the present invention constructs a non-linear flank clearance function and a single-degree-of-freedom non-linear dynamic equation, which can more accurately simulate the dynamic response of the gear pair during actual operation, improve the accuracy of fault prediction. By generating the comprehensive time-varying meshing stiffness of the spur gear pair, it can reflect the dynamic characteristics of the gear pair under different fault states in real time, provide the basic data of dynamic changes for fault diagnosis, make the fault prediction closer to the actual working conditions, construct the fault coefficient and conduct coupling analysis, and by comparing with the preset fault coefficient threshold, it can timely detect and warn the potential faults of the gear pair, avoid the lag of fault identification in traditional methods, prevent the sudden damage of equipment. On the basis of the traditional non-linear dynamic model of the gear system, it is proposed to establish a quantitative characterization of the dynamic load response to fatigue damage, solve the general dynamic model of the spur gear pair under any working conditions and any damage degree. This model intuitively explains the continuous coupling effect between the structural vibration and fatigue damage of the spur gear pair, and has high numerical simulation accuracy. BRIEF DESCRIPTION OF THE DRAWINGS

[0062] Figure 1 is the schematic diagram of the overall method flow of the present invention;

[0063] Figure 2 is the waveform diagram of the meshing stiffness curve of the spur gear pair without fault of the present invention;

[0064] Figure 3 is the waveform diagram of the meshing stiffness curve of the spur gear pair with a single tooth break of the present invention;

[0065] Figure 4 is the schematic diagram of the driving gear and the driven gear of the present invention. DETAILED DESCRIPTION OF THE INVENTION

[0066] To make the objectives, technical solutions and advantages of the present invention clearer and more understandable, the present invention will be further described in detail below with reference to specific embodiments.

[0067] It should be noted that, unless otherwise defined, the technical terms or scientific terms used in the present invention should be understood by people with ordinary skills in the field to which the present invention belongs. The words "first", "second" and similar words used in the present invention do not indicate any order, quantity or importance, but are only used to distinguish different components. "Include" or "comprise" and similar words mean that the elements or objects appearing before the word include the elements or objects listed after the word and their equivalents, without excluding other elements or objects. "Connect" or "connected" and similar words are not limited to physical or mechanical connections, but may include electrical connections, whether direct or indirect. "Up", "down", "left", "right" and the like are only used to indicate relative positional relationships. When the absolute position of the described object changes, the relative positional relationship may also change accordingly.

[0068] Example:

[0069] See also Figures 1 to 4 , the present invention provides a technical solution:

[0070] A fault prediction method for a spur gear pair based on a dynamic and gradually coupled torsional vibration model of a spur gear pair, the specific steps comprising:

[0071] S1. Collecting physical parameters and operating parameters of the spur gear pair, the physical parameters include the number of teeth of the driving gear and the passive gear, the base circle radius of the driving gear and the passive gear, the moment of inertia of the driving gear and the passive gear, the input shaft torsional damping coefficient, the output shaft torsional damping coefficient, the input shaft torsional stiffness coefficient and the output shaft torsional stiffness coefficient, the operating parameters include the torsion angle of the driving gear, the torsion angle of the passive gear, the input torque of the driving gear, the load torque of the passive gear, the motor speed frequency of the driving gear and the relative displacement of the meshing point of the driving gear and the passive gear;

[0072] S2. Obtain the average meshing stiffness according to the collected input shaft torsional stiffness coefficient and output shaft torsional stiffness coefficient, process the average meshing stiffness, input shaft torsional damping coefficient, output shaft torsional damping coefficient, base circle radius of the driving gear, base circle radius of the driven gear, moment of inertia of the driving gear and moment of inertia of the driven gear to obtain meshing viscoelastic damping, construct a static transmission error function according to the motor rotation frequency of the driving gear, and construct a nonlinear tooth side clearance function according to the relative displacement of the meshing point between the driving gear and the driven gear;

[0073] S3. Process the average mesh stiffness and the amplitude of the fault-free stiffness change to generate the time-varying mesh stiffness of the fault-free spur gear pair, process the amplitude of the single-break tooth stiffness change and the number of teeth of the driving gear to generate the time-varying mesh stiffness of the single-break tooth spur gear pair, process the time-varying mesh stiffness of the fault-free spur gear pair and the time-varying mesh stiffness of the single-break tooth spur gear pair to generate the comprehensive time-varying mesh stiffness of the spur gear pair;

[0074] S4. Process the torsion angle of the driving gear, the torsion angle of the driven gear, the base circle radius of the driving gear, the base circle radius of the driven gear and the static transmission error function to generate the torsion displacement of the gear pair. According to the comprehensive time-varying meshing stiffness of the spur gear pair, the torsion displacement of the gear pair, the meshing viscoelastic damping and the nonlinear tooth side clearance function, construct the single-degree-of-freedom nonlinear dynamic equation of the torsion of the spur gear pair. According to the interval range of the torsion displacement of the gear pair, generate the dynamic load when the spur gear teeth are meshed. According to the dynamic load when the spur gear teeth are meshed, generate the time-varying bending stress of the tooth root and the time-varying contact stress of the tooth surface.

[0075] S5.According to the time-varying bending stress at the tooth root and the time-varying contact stress at the tooth surface, the time-varying bending fatigue damage amount at the tooth root and the time-varying contact fatigue damage amount at the tooth surface are obtained. According to the time-varying bending fatigue damage amount at the tooth root and the time-varying contact fatigue damage amount at the tooth surface, a fault coefficient is constructed. The comprehensive time-varying meshing stiffness of the spur gear pair, the single-degree-of-freedom nonlinear dynamic equation of the torsion of the spur gear pair and the fault coefficient are coupled to construct a dynamic and gradual coupling torsional vibration model of the spur gear pair. The fault coefficient is compared with the preset fault coefficient threshold to predict the fault condition of the spur gear pair.

[0076] On the basis of the above embodiments, the number of teeth of the driving gear, the number of teeth of the driven gear, the base circle radius of the driving gear, the base circle radius of the driven gear, the rotational inertia of the driving gear, the rotational inertia of the driven gear, the input shaft torsional damping coefficient, the output shaft torsional damping coefficient, the input shaft torsional stiffness coefficient and the output shaft torsional stiffness coefficient, or the changes in their combination, to a certain extent, explain the reasons why the spur gear pair has a fault as follows:

[0077] Changes in the number of teeth of the driving gear or the driven gear usually mean wear or breakage of the gear teeth; changes in the base circle radius reflect the wear of the gear profile, which is usually caused by friction or poor lubrication during long-term operation; changes in the moment of inertia of the driving gear or the driven gear indicate damage or looseness of the gear or related components (such as bearings, shafts), thereby changing the mass distribution of the system; changes in the torsional damping coefficient of the input shaft or the output shaft usually reflect changes in the lubrication condition, such as deterioration of the lubricating oil or failure of the lubrication system; changes in the torsional stiffness coefficient of the input shaft or the output shaft indicate structural defects or damage to the shaft or gear, such as cracks or fatigue damage.

[0078] In summary, the changes in the number of teeth on the active gear, the number of teeth on the passive gear, the base circle radius, the moment of inertia, the torsional damping coefficient and the torsional stiffness coefficient or their combined changes can reflect the different causes of failure of the spur gear pair. By monitoring the changes in these parameters, potential failures of the gear transmission system can be identified and diagnosed early, so that preventive maintenance measures can be taken to improve the operating reliability and service life of the equipment.

[0079] Based on the above embodiments, the changes in the torsion angle of the driving gear, the torsion angle of the driven gear, the input torque, the load torque, the motor frequency, and the relative displacement of the meshing point of the driving gear and the driven gear, or their combined changes, to a certain extent, explain the reasons why the spur gear pair has a fault as follows:

[0080] Inconsistent or abnormal fluctuations in the torsion angle of the driving gear and the driven gear, or periodic or random abnormal fluctuations in the torsion angle indicate gear wear or broken teeth or bearing failure; abnormal increase or violent fluctuations in the input torque and load torque indicate mechanical looseness or wear; unstable or sudden changes in the motor frequency indicate an electrical fault; an increase or irregular change in the relative displacement of the meshing point or periodic fluctuations in the relative displacement over time indicate gear looseness or wear or changes in meshing stiffness.

[0081] In summary, the changes in parameters such as the torsion angle of the driving gear, the torsion angle of the driven gear, the input torque, the load torque, the motor frequency and the relative displacement of the meshing point or their combined changes can reveal various causes of failure of spur gear pairs. By monitoring the dynamic changes of these parameters, potential problems of gear pairs can be identified early, and corresponding maintenance measures can be taken to ensure the stable operation of the equipment and extend its service life.

[0082] In this embodiment, the driving wheel and the driven wheel are both involute spur gears; the gear tooth pair damage only includes root cracks, tooth surface pitting, spalling and broken teeth; the meshing relationship of the gear pair is equivalent to the parallel connection of the spring and the damper.

[0083] On the basis of the above embodiment, the average meshing stiffness is obtained according to the collected input shaft torsional stiffness coefficient and output shaft torsional stiffness coefficient, and the formula is as follows:

[0084]

[0085] in, is the average meshing stiffness, k1 is the input shaft torsional stiffness coefficient, k2 is the output shaft torsional stiffness coefficient;

[0086] The meshing viscoelastic damping is obtained by processing the average meshing stiffness, input shaft torsional damping coefficient, output shaft torsional damping coefficient, base circle radius of the driving gear, base circle radius of the driven gear, moment of inertia of the driving gear and moment of inertia of the driven gear. The formula is as follows:

[0087]

[0088] Where c is the meshing viscoelastic damping, c1 is the input shaft torsional damping coefficient, c2 is the output shaft torsional damping coefficient, r b1 is the base circle radius of the driving gear, r b2 is the base circle radius of the passive gear, I1 is the moment of inertia of the active gear, and I2 is the moment of inertia of the passive gear;

[0089] According to the motor rotation frequency and error amplitude of the driving gear, the static transmission error function is constructed based on the following formula:

[0090] e(t)=e a sin(2πft)

[0091] Among them, e(t) is the transmission error value at time t, e a is the error amplitude, f is the motor speed frequency of the driving gear, and t is the time variable;

[0092] According to the relative displacement of the meshing point between the active gear and the passive gear, a nonlinear tooth backlash function is constructed based on the following formula:

[0093]

[0094] Wherein, g(x) is a function of the nonlinear side clearance changing with the relative displacement x of the meshing point between the active gear and the passive gear, indicating the side clearance value at the displacement x, x is the relative displacement of the meshing point between the active gear and the passive gear, b is the side clearance threshold, and 2b is the sum of the negative side clearance threshold to the positive side clearance threshold;

[0095] When the input displacement x exceeds the positive tooth side clearance threshold b, the tooth side clearance function g(x) = xb, indicating that the gear has exceeded the preset positive tooth side clearance and has generated additional displacement;

[0096] When the input displacement x is between positive and negative b, the tooth backlash function g(x) is zero, indicating that the gear system is in normal working condition, there is no tooth backlash, or the tooth backlash is within the allowable range, and no relative motion occurs;

[0097] When the input displacement x exceeds the negative tooth side clearance threshold -b, the tooth side clearance function g(x)=x+b, indicating that the gear has exceeded the preset negative tooth side clearance threshold -b and has generated additional displacement.

[0098] On the basis of the above embodiment, the Fourier series is used to process the average meshing stiffness and the amplitude of the fault-free stiffness change to generate the time-varying meshing stiffness of the fault-free spur gear pair, according to the following formula:

[0099]

[0100] Among them, K m0 (t) is the time-varying mesh stiffness of the fault-free spur gear pair, k a0 is the amplitude of the fault-free stiffness change, n is the term index of the Fourier series, indicating the component number of the sine term, n = 1, 3, 5, ..., ∞, n is an odd number;

[0101] like Figure 2 As shown, T0 is the reference period, is the average value of the time-varying mesh stiffness of the fault-free spur gear pair.

[0102] The Fourier series is used to process the amplitude of the change in the single-break tooth stiffness and the number of teeth of the driving gear to generate the time-varying meshing stiffness of the single-break tooth spur gear pair, based on the following formula:

[0103]

[0104] Among them, K m1 (t) is the time-varying mesh stiffness of the single-broken-tooth spur gear pair, k a1 is the amplitude of the change in stiffness of a single broken tooth, z1 is the number of teeth of the driving gear, and n is the term index of the Fourier series, indicating the component number of the sine term;

[0105] like Figure 3 As shown, T0 is the reference period, is the average value of the time-varying meshing stiffness of the single-broken-tooth spur gear pair, and T1 is the period of the waveform of the meshing stiffness curve of the spur gear pair with a single broken tooth.

[0106] The time-varying mesh stiffness of the fault-free spur gear pair and the time-varying mesh stiffness of the single-broken-tooth spur gear pair are processed to generate the comprehensive time-varying mesh stiffness of the spur gear pair, based on the following formula:

[0107] K(t)=(1-ε)K m0 (t)+εK m1 (t)

[0108] Where K(t) is the comprehensive time-varying mesh stiffness of the spur gear pair, and ε is the fault factor;

[0109] The fault factor ε∈[0,1] is defined as: when there is no torsional vibration of the gear, ε=0; when there is a broken tooth fault, ε=1; when there are minor faults such as early cracks, pitting, and spalling of the gear, 0<ε<1.

[0110] The fault factor ε is a value between 0 and 1, indicating the degree of fault in the gear pair. When ε = 0, it means that the gear pair has no fault at all, and the comprehensive time-varying mesh stiffness K(t) of the spur gear pair is equal to the time-varying mesh stiffness K of the fault-free spur gear pair. m0 (t); When ε = 1, it means that the gear pair is completely in a single broken tooth fault state, and the comprehensive time-varying meshing stiffness K(t) of the spur gear pair is equal to the time-varying meshing stiffness K of the single broken tooth spur gear pair m1 (t).

[0111] On the basis of the above embodiment, the torsion angle of the driving gear, the torsion angle of the driven gear, the base circle radius of the driving gear, the base circle radius of the driven gear and the static transmission error function are processed to generate the torsional displacement of the gear pair, according to the following formula:

[0112] q(t)=r b1 θ1-r b2 θ2-e(t)

[0113] Among them, q(t) is the torsional displacement of the gear pair, θ1 is the torsional angle of the driving gear, and θ2 is the torsional angle of the driven gear.

[0114] On the basis of the above embodiments, according to the comprehensive time-varying meshing stiffness of the spur gear pair, the torsional displacement of the gear pair, the meshing viscoelastic damping and the nonlinear tooth side clearance function, a single-degree-of-freedom nonlinear dynamic equation of the torsion of the spur gear pair is constructed, and its formula is as follows:

[0115]

[0116] in, is the second-order derivative of the torsional displacement, is the first-order derivative of the torsional displacement, g(q(t)) is the nonlinear tooth backlash, m e is the equivalent mass, F m is the total normal force, F m =J1 / r b1 =J2 / r b2 , J1 is the input torque of the driving gear, J2 is the load torque of the passive gear, is the second-order derivative of the auxiliary variable transmission error.

[0117] This equation provides a comprehensive nonlinear dynamic model by considering time-varying mesh stiffness, nonlinear tooth side clearance and viscoelastic damping, which can more accurately describe the dynamic behavior of the gear system and plays an important role in gear design, control and fault diagnosis.

[0118] On the basis of the above embodiment, according to the interval range of the torsional displacement of the gear pair, the dynamic load when the spur gear is meshing is generated according to the following formula:

[0119]

[0120] Among them, W d is the dynamic load when the spur gear is meshing, k3 is the nonlinear stiffness coefficient, E2 is the elastic modulus of the second material, h2 is the width of the passive gear tooth, χ is the displacement coefficient, λ is the loading position coefficient, λ=(r n2 -r h2 ) / τ,r n2 is the pitch radius of the passive gear at the meshing point, r h2 is the radius of the top circle of the passive gear, τ is the gear module of the passive gear, A2, B2, C2 are all nonlinear adjustment coefficients;

[0121] According to the dynamic load when the spur gear is meshing, the time-varying bending stress at the tooth root and the time-varying contact stress on the tooth surface are generated according to the following formula:

[0122]

[0123] Among them, σ F is the time-varying bending stress at the tooth root, σ H is the time-varying bending stress on the tooth surface, K is the load distribution coefficient K = K a K α K β , K a is the amplitude of stiffness change, K α is the inter-tooth load distribution coefficient, K β is the tooth load distribution coefficient, Y Fa is the tooth form factor, Y Sa is the stress correction factor, α n is the pressure angle of the gear pair, h max is the large gear tooth width, is the gear module, Z H is the contact stress coefficient, Z E is the elastic influence coefficient, U is the gear ratio, that is, the ratio of the number of teeth on the driven gear to the number of teeth on the driving gear, d1 is the pitch diameter of the driving gear, E1 is the elastic modulus of the first material μ1, μ2 are the first Poisson's ratio and the second Poisson's ratio respectively, and z2 is the number of teeth on the passive gear.

[0124] On the basis of the above embodiment, the tooth root time-varying bending stress and the tooth surface time-varying contact stress are used to obtain the tooth root time-varying bending fatigue damage amount and the tooth surface time-varying contact fatigue damage amount, and the formula used is as follows:

[0125]

[0126] Among them, D F (t) is the time-varying bending fatigue damage amount of the tooth root, D H (t) is the time-varying bending fatigue damage amount of the tooth surface, ρ is the fatigue life parameter of the material, P is the cumulative strength limit value of the material, s is the stress amplitude, u F is the mean value of the time-varying bending stress of the tooth root, u H is the mean value of the time-varying bending stress of the tooth surface, is the average value of the time-varying bending stress of the tooth root, is the average value of the time-varying contact stress of the tooth surface.

[0127] Based on the above embodiments, according to the time-varying bending fatigue damage amount of the tooth root and the time-varying contact fatigue damage amount of the tooth surface, a fault coefficient is constructed, and the formula is as follows:

[0128]

[0129] Among them, gxε(t) is the fault coefficient, ξ fH is the damage weight coefficient;

[0130] The damage weight coefficient ξ fH ∈[0, 1]. When it is full tooth root bending fatigue, ξ fH = 1. When it is full tooth surface contact fatigue, ξ fH = 0. When there is both tooth root bending fatigue and tooth surface contact fatigue, 0 < ξ fH < 1;

[0131] Couple the comprehensive time-varying meshing stiffness of the spur gear pair, the single-degree-of-freedom nonlinear dynamic equation of the spur gear pair torsion and the fault coefficient to construct a dynamic and gradual coupling torsional vibration model of the spur gear pair. The model is as follows:

[0132]

[0133]

[0134] Among them, is the angular velocity of the driving gear, is the angular velocity of the driven gear, is the angular acceleration of the driving gear, is the angular acceleration of the driven gear;

[0135] Compare the fault coefficient gx with the preset fault coefficient threshold yz to predict the fault condition of the spur gear pair:

[0136] Compare the fault factor function ε(t) with the preset fault factor threshold yz to predict the fault condition of the spur gear pair:

[0137] When gx < yz, the spur gear pair is in a normal working state;

[0138] When gx≥yz, the spur gear pair fails.

[0139] The dynamic and gradually coupled torsional vibration model of the spur gear pair is used to truly reflect the actual working state of the gear, and the fault coefficient is used to accurately update the damage status and timely predict and warn of faults.

[0140] k1, k2, k3, c1, c2, χ, λ, A2, B2, C2, K, K in the formula a , K α , K β , Y Fa , Y Sa , Z H , Z E and fH The specific value of is generally determined by those skilled in the art according to actual conditions. The essence of this formula is a comprehensive analysis by weighted summation. The technicians in this field collect multiple groups of sample data, and set corresponding preset proportional coefficients for each group of sample data. The preset proportional coefficients and the collected sample data are substituted into the formula. Through repeated experiments and parameter adjustments, the accuracy of the model output and the rationality of the results are observed, and these factor coefficients are gradually adjusted. The performance and effect of the model under different parameter settings are compared to find the optimal coefficient combination. The calculated factor coefficients are screened and averaged to obtain k1, k2, k3, c1, c2, χ, λ, A2, B2, C2, K, K a , K α , K β , Y Fa , Y Sa , Z H , Z E and fH The value of .

[0141] In addition, the size of the preset factor coefficient is to quantify each parameter to obtain a specific value. In order to facilitate subsequent comparison, the size of the coefficient depends on the amount of sample data and the preset proportional coefficient initially set by technical personnel in this field for each set of sample data. It is not unique as long as it does not affect the proportional relationship between the parameter and the quantized value.

[0142] The above formulas are all dimensionless and numerical calculations. The formula is a formula for the most recent real situation obtained by collecting a large amount of data and performing software simulation. The preset parameters in the formula are set by technicians in this field according to actual conditions.

[0143] The above embodiments may be implemented in whole or in part by software, hardware, firmware or any other combination thereof. When implemented by software, the above embodiments may be implemented in whole or in part in the form of a computer program product. Those skilled in the art may appreciate that the units and algorithm steps of each example described in conjunction with the embodiments disclosed herein may be implemented by electronic hardware, or a combination of computer software and electronic hardware. Whether these functions are performed by hardware or software methods depends on the specific application and design constraints of the technical solution.

[0144] The units described as separate components may or may not be physically separated, and the components shown as units may or may not be physical units, and may be located in one place or distributed on multiple network units. Some or all of the units may be selected according to actual needs to achieve the purpose of the solution of this embodiment.

[0145] The above description is only a specific implementation manner of the present application, but the protection scope of the present application is not limited thereto. Any technician familiar with the technical field can easily think of changes or substitutions within the technical scope disclosed in the present application, which should be included in the protection scope of the present application.

Claims

1. A fault prediction method for spur gear pairs based on the dynamic and gradual coupled torsional vibration model of spur gear pairs, characterized in that: The specific steps include: S1. Collecting physical parameters and operating parameters of the spur gear pair, the physical parameters include the number of teeth of the driving gear and the passive gear, the base circle radius of the driving gear and the passive gear, the moment of inertia of the driving gear and the passive gear, the input shaft torsional damping coefficient, the output shaft torsional damping coefficient, the input shaft torsional stiffness coefficient and the output shaft torsional stiffness coefficient, the operating parameters include the torsion angle of the driving gear, the torsion angle of the passive gear, the input torque of the driving gear, the load torque of the passive gear, the motor speed frequency of the driving gear and the relative displacement of the meshing point of the driving gear and the passive gear; S2. Obtain the average meshing stiffness according to the collected input shaft torsional stiffness coefficient and output shaft torsional stiffness coefficient, process the average meshing stiffness, input shaft torsional damping coefficient, output shaft torsional damping coefficient, base circle radius of the driving gear, base circle radius of the driven gear, moment of inertia of the driving gear and moment of inertia of the driven gear to obtain meshing viscoelastic damping, construct a static transmission error function according to the motor rotation frequency of the driving gear, and construct a nonlinear tooth side clearance function according to the relative displacement of the meshing point between the driving gear and the driven gear; S3. Process the average mesh stiffness and the amplitude of the fault-free stiffness change to generate the time-varying mesh stiffness of the fault-free spur gear pair, process the amplitude of the single-break tooth stiffness change and the number of teeth of the driving gear to generate the time-varying mesh stiffness of the single-break tooth spur gear pair, process the time-varying mesh stiffness of the fault-free spur gear pair and the time-varying mesh stiffness of the single-break tooth spur gear pair to generate the comprehensive time-varying mesh stiffness of the spur gear pair; S4. Process the torsion angle of the driving gear, the torsion angle of the driven gear, the base circle radius of the driving gear, the base circle radius of the driven gear and the static transmission error function to generate the torsion displacement of the gear pair. According to the comprehensive time-varying meshing stiffness of the spur gear pair, the torsion displacement of the gear pair, the meshing viscoelastic damping and the nonlinear tooth side clearance function, construct the single-degree-of-freedom nonlinear dynamic equation of the torsion of the spur gear pair. According to the interval range of the torsion displacement of the gear pair, generate the dynamic load when the spur gear teeth are meshed. According to the dynamic load when the spur gear teeth are meshed, generate the time-varying bending stress of the tooth root and the time-varying contact stress of the tooth surface. S5. Obtain the time-varying bending fatigue damage of the tooth root and the time-varying contact stress of the tooth surface according to the time-varying bending fatigue damage of the tooth root and the time-varying contact fatigue damage of the tooth surface, construct the fault coefficient according to the time-varying bending fatigue damage of the tooth root and the time-varying contact fatigue damage of the tooth surface, couple the comprehensive time-varying meshing stiffness of the spur gear pair, the single-degree-of-freedom nonlinear dynamic equation of the torsion of the spur gear pair and the fault coefficient, construct the dynamic and gradual coupling torsional vibration model of the spur gear pair, compare the fault coefficient with the preset fault coefficient threshold, and predict the fault condition of the spur gear pair; According to the time-varying bending stress at the tooth root and the time-varying contact stress at the tooth surface, the time-varying bending fatigue damage amount at the tooth root and the time-varying contact fatigue damage amount at the tooth surface are obtained according to the following formula: Among them, D F (t) is the time-varying bending fatigue damage of the tooth root, D H (t) is the time-varying bending fatigue damage of the tooth surface, ρ is the fatigue life parameter of the material, P is the cumulative strength limit of the material, s is the stress amplitude, u F is the mean value of the time-varying bending stress at the tooth root, u H is the mean value of the time-varying bending stress on the tooth surface, is the average value of the time-varying bending stress at the tooth root, is the average value of the time-varying contact stress on the tooth surface; According to the time-varying bending fatigue damage of the tooth root and the time-varying contact fatigue damage of the tooth surface, the failure coefficient is constructed according to the following formula: Where gx is the failure coefficient, ξ fH is the damage weight coefficient; Damage weight coefficient ξ fH ∈[0,1], when the whole tooth root is bending fatigue, ξ fH =1, when the whole tooth surface is in contact fatigue, ξ fH =0, when there is both tooth root bending fatigue and tooth surface contact fatigue, 0<ξ fH <1; Couple the comprehensive time-varying meshing stiffness of the spur gear pair, the single-degree-of-freedom nonlinear dynamic equation of the spur gear pair torsion, and the fault coefficient to construct a dynamic and gradually changing coupled torsional vibration model of the spur gear pair. The model is as follows: in, is the angular velocity of the driving gear, is the angular velocity of the driven gear, is the angular acceleration of the driving gear, is the angular acceleration of the driven gear; Compare the fault coefficient gx with the preset fault coefficient threshold yz to predict the fault condition of the spur gear pair: When gx < yz, the spur gear pair is in a normal working state; When gx ≥ yz, the spur gear pair has a fault.

2. The fault prediction method for spur gear pairs based on the dynamic and gradual coupling torsional vibration model of spur gear pairs according to claim 1 is characterized by: According to the collected torsional stiffness coefficients of the input shaft and the output shaft, obtain the average meshing stiffness. The formula is as follows: in, is the average meshing stiffness, k1 is the input shaft torsional stiffness coefficient, k2 is the output shaft torsional stiffness coefficient; Process the average meshing stiffness, the torsional damping coefficient of the input shaft, the torsional damping coefficient of the output shaft, the base circle radius of the driving gear, the base circle radius of the driven gear, the moment of inertia of the driving gear, and the moment of inertia of the driven gear to obtain the meshing viscoelastic damping. The formula is as follows: Where c is the meshing viscoelastic damping, c1 is the input shaft torsional damping coefficient, c2 is the output shaft torsional damping coefficient, r b1 is the base circle radius of the driving gear, r b2 is the base circle radius of the passive gear, I1 is the moment of inertia of the active gear, and I2 is the moment of inertia of the passive gear.

3. The fault prediction method for spur gear pairs based on the dynamic and gradual coupling torsional vibration model of spur gear pairs according to claim 2 is characterized by: Construct a static transmission error function according to the motor rotation frequency of the driving gear. The formula is as follows: e(t)=e a sin(2πft) Among them, e(t) is the transmission error value at time t, e a is the error amplitude, f is the motor speed frequency of the driving gear, and t is the time variable; Construct a nonlinear backlash function according to the relative displacement of the meshing points of the driving gear and the driven gear. The formula is as follows: Among them, g(x) is a function of the nonlinear backlash changing with the relative displacement of the meshing points of the driving gear and the driven gear, representing the backlash value at the displacement x. x is the relative displacement of the meshing points of the driving gear and the driven gear, and b is the backlash threshold.

4. The fault prediction method for spur gear pairs based on the dynamic and gradual coupling torsional vibration model of spur gear pairs according to claim 3 is characterized by: Use the Fourier series to process the average meshing stiffness and the amplitude of the fault-free stiffness change to generate the time-varying meshing stiffness of the fault-free spur gear pair. The formula is as follows: Among them, K m0 (t) is the time-varying mesh stiffness of the fault-free spur gear pair, k a0 is the amplitude of the fault-free stiffness change, n is the term index of the Fourier series, indicating the component number of the sine term, n = 1, 3, 5, ..., ∞, n is an odd number; Use the Fourier series to process the amplitude of the single tooth breakage stiffness change and the number of teeth of the driving gear to generate the time-varying meshing stiffness of the single tooth breakage spur gear pair. The formula is as follows: Among them, K m1 (t) is the time-varying mesh stiffness of the single-broken-tooth spur gear pair, k a1 is the amplitude of the change in stiffness of a single broken tooth, z1 is the number of teeth of the driving gear, and n is the index of the number of terms in the Fourier series; Process the time-varying meshing stiffness of the fault-free spur gear pair and the time-varying meshing stiffness of the single tooth breakage spur gear pair to generate the comprehensive time-varying meshing stiffness of the spur gear pair. The formula is as follows: K(t)=(1-ε)K m0 (t)+εK m1 (t) Among them, K(t) is the comprehensive time-varying meshing stiffness of the spur gear pair, and ε is the fault factor.

5. The fault prediction method for spur gear pairs based on the dynamic and gradual coupling torsional vibration model of spur gear pairs according to claim 4 is characterized by: Process the torsional angle of the driving gear, the torsional angle of the driven gear, the base circle radius of the driving gear, the base circle radius of the driven gear, and the static transmission error function to generate the torsional displacement of the gear pair. The formula is as follows: q(t)=r b1 θ1-r b2 θ2-e(t) Among them, q(t) is the torsional displacement of the gear pair, θ1 is the torsional angle of the driving gear, and θ2 is the torsional angle of the driven gear.

6. The fault prediction method for spur gear pairs based on the dynamic and gradual coupling torsional vibration model of spur gear pairs according to claim 5 is characterized by: According to the comprehensive time-varying meshing stiffness of the spur gear pair, the torsional displacement of the gear pair, the meshing viscoelastic damping, and the nonlinear backlash function, construct a single-degree-of-freedom nonlinear dynamic equation of the spur gear pair torsion. The formula is as follows: in, is the second-order derivative of the torsional displacement, is the first-order derivative of the torsional displacement, g(q(t)) is the nonlinear tooth backlash, m e is the equivalent mass, F m is the total normal force, F m =J1 / r b1 =J2 / r b2 , J1 is the input torque of the driving gear, J2 is the load torque of the passive gear, is the second-order derivative of the auxiliary variable transmission error.

7. The fault prediction method for spur gear pairs based on the dynamic and gradual coupling torsional vibration model of spur gear pairs according to claim 6 is characterized by: Generate the dynamic load during the meshing of the straight gear teeth according to the interval range of the torsional displacement of the gear pair. The formula is as follows: Among them, W d is the dynamic load when the spur gear is meshing, k3 is the nonlinear stiffness coefficient, E2 is the elastic modulus of the second material, h2 is the width of the passive gear tooth, χ is the displacement coefficient, λ is the loading position coefficient, λ=(r n2 -r h2 ) / τ,r n2 is the pitch radius of the passive gear at the meshing point, r h2 is the radius of the top circle of the passive gear, τ is the gear module of the passive gear, A2, B2, C2 are all nonlinear adjustment coefficients; Generate the time-varying bending stress at the tooth root and the time-varying contact stress on the tooth surface according to the dynamic load during the meshing of the straight gear teeth. The formula is as follows: Among them, σ F is the time-varying bending stress at the tooth root, σ H is the time-varying bending stress on the tooth surface, K is the load distribution coefficient K = K a K α K β , K a is the amplitude of stiffness change, K α is the inter-tooth load distribution coefficient, K β is the tooth load distribution coefficient, Y Fa is the tooth form factor, Y Sa is the stress correction factor, α n is the pressure angle of the gear pair, h max is the tooth width of the large gear, θ is the gear module, Z H is the contact stress coefficient, Z E is the elastic influence coefficient, U is the gear ratio, that is, the ratio of the number of teeth on the driven gear to the number of teeth on the driving gear, d1 is the pitch diameter of the driving gear, E1 is the elastic modulus of the first material, μ1 and μ2 are the first Poisson's ratio and the second Poisson's ratio respectively, and z2 is the number of teeth on the passive gear.

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