A method for establishing a dynamic simulation model of a parallel axis gear system under tooth breakage failure

Abstract

The application provides a method for establishing a dynamic simulation model of a parallel shaft gear system under tooth breakage failure, and comprises the following steps: according to the generation method of rack machining gears, the machining coordinates of the rack and the gears are converted to determine the tooth profile coordinates of the gears; the gear engagement state is determined, the tooth breakage failure of the gears in the tooth pair is judged through the gear engagement state, the potential contact points of the tooth pair in the tooth surface contact process are calculated respectively, and the engagement point distribution data containing abnormal engagement states are obtained; the initial gap data between each tooth pair in the gear pair is calculated, and the initial gap data is the engagement gap data in the initial state; the initial gap data is substituted into a tooth bearing contact analysis model to obtain the no-load transmission error, the load distribution coefficient and the time-varying engagement stiffness of the gear pair; and the dynamic simulation model of the parallel shaft gear system under tooth breakage failure is established. The abnormal engagement caused by the tooth breakage failure is considered, and the simulation response is closer to the experimental result than that of the simulation response in which the abnormal engagement is ignored.

Description

Technical Field

[0001] This invention relates to the field of meshing characteristic analysis and dynamic analysis of gear pair transmission systems, specifically to a method for establishing a dynamic simulation model of a parallel shaft gear system under tooth breakage fault.

[0002] Background Introduction

[0003] As one of the most commonly used power transmission systems, the working condition of spur gear pairs is crucial to the safety and stable operation of mechanical systems. Due to manufacturing processes and sudden load changes, spur gear transmission systems are highly susceptible to tooth breakage. A broken tooth wedge rapidly deteriorates the gear meshing between the normal teeth, causing other teeth to bend and break. This can lead to mechanical failure and even serious hazards threatening human life. Therefore, accurately identifying tooth breakage faults in spur gear transmission systems and taking timely remedial measures is of paramount practical importance.

[0004] Regarding methods for calculating the time-varying meshing stiffness of gears, patent (CN201911120877.3) introduces a method for calculating the time-varying meshing stiffness of gear pairs using the energy method, which considers the influence of gear errors. Patent (CN201811449533.2) also introduces a method for calculating the time-varying meshing stiffness of internal meshing gear pairs using the energy method. Patent (CN201910303911.4) introduces a method for calculating the meshing stiffness of internal meshing gear pairs considering extended meshing and gear ring flexibility. From the above, it can be seen that most methods for calculating time-varying meshing stiffness are based on the energy method, and there is little research on time-varying meshing stiffness under tooth breakage faults. Furthermore, current simulation responses to abnormal meshing mostly ignore tooth breakage faults, resulting in simulation results that do not closely approximate actual experimental results. To address the aforementioned problems, it is essential to research and design a novel method for establishing a dynamic simulation model of a parallel shaft gear system under tooth breakage faults. This method is crucial for resolving the issues present in existing techniques for analyzing the meshing characteristics and dynamics of gear transmission systems. Summary of the Invention

[0005] This invention addresses the shortcomings of existing gear transmission system meshing characteristic analysis and dynamic analysis techniques, such as the lack of research on time-varying meshing stiffness under broken tooth faults and the neglect of broken tooth faults in simulation responses of abnormal meshing. It proposes a method for establishing a dynamic simulation model of a parallel shaft gear system under broken tooth faults. This method considers the abnormal situation where meshing occurs outside the theoretical line of action, accurately calculates the initial clearance between the teeth, and obtains the no-load transmission error and load distribution coefficient of the gear pair under broken tooth fault conditions based on the gear tooth bearing contact analysis model. Furthermore, it calculates the time-varying meshing stiffness of the gear pair, uses Timoshenko beam theory to simulate the support shaft of the gear pair, and combines the time-varying meshing stiffness and no-load transmission error of the gear pair to establish a dynamic simulation model of the parallel shaft gear system under broken tooth faults.

[0006] This invention provides a method for calculating the meshing clearance of a spur gear transmission system and a fault analysis model, comprising the following steps:

[0007] S1. Based on the generating method of rack and pinion gear machining, the machining coordinates of the rack and gear are transformed to determine the tooth profile coordinates of the gear;

[0008] S2. Determine the gear meshing state based on the gear tooth profile coordinates obtained in step S1, judge the gear tooth breakage fault in the gear pair through the gear meshing state, calculate the potential contact points of the gear pair during the tooth surface contact process, and obtain the meshing point distribution data including abnormal meshing state.

[0009] S3. Based on the meshing point distribution data with abnormal meshing state obtained in step S2, calculate the initial clearance data between each tooth pair in the gear pair, wherein the initial clearance data is the meshing clearance data in the initial state;

[0010] S4. Substitute the initial clearance data obtained in step S3 into the gear tooth bearing contact analysis model to obtain the no-load transmission error and load distribution coefficient of the gear pair, and further calculate the time-varying meshing stiffness of the gear pair.

[0011] S5. Using Timoshenko beam theory to simulate the support shaft of the gear pair, and combining the time-varying meshing stiffness and no-load transmission error of the gear pair obtained in step S4, a dynamic simulation model of the parallel shaft gear system under tooth breakage fault is established.

[0012] Preferably, in step S1, the tangent point C between the rack tooth surface and the tooth tip arc is defined as the machining start point, an x-axis is established along the rack pitch line, and a y-axis is established along the rack tooth groove centerline, thus obtaining the rack's coordinate system S. t The tooth tip center (b, a) is in S t The coordinate system can be represented as follows:

[0013]

[0014] In the formula, r t h is the radius of the rack tooth tip arc. a Here, c is the addendum coefficient, m is the clearance, and α is the module. t The tooth profile angle of the rack;

[0015] When the rack moves along x t When moving forward, the gear pitch circle rolls tangentially along the rack pitch line, and the gear center O... g In coordinate system S t x in g0 and the vertical coordinate y g0 They are respectively:

[0016]

[0017] In the formula, r b Let r be the base circle radius of the gear, r be the pitch circle radius of the gear, χ be the gear displacement coefficient, and α be the displacement coefficient of the gear. c Let α be the pressure angle at point C. t1 =acos[r b [ / (r+χ·m)] is the nodal pressure angle;

[0018] When the rack moves along x t When the gear moves Δx in the positive direction, it rotates -Δx / y around its center. g0 The initial position coordinates S of the rack t The current coordinates S of the rack t1 The initial position coordinates S of the gear g The current position coordinates of the gear S g1 The conversion relationship is as follows:

[0019]

[0020] In the formula:

[0021]

[0022]

[0023]

[0024]

[0025] In coordinate system S t In the middle, the tooth profile r of the rack tla and the tooth tip radius r of the rack tlb Defined as:

[0026]

[0027]

[0028] For any point i on the gear tooth profile, the meshing pressure angle The corresponding Δx is obtained by the following formula:

[0029]

[0030] In the formula, Let C be the meshing pressure angle.

[0031] Pressure angle α at point i on tooth flank i With meshing pressure angle The relationship is as follows:

[0032]

[0033] In the formula, θ b The central angle from the point where the extension of the involute intersects the base circle to the center line of the gear tooth is half of the angle between the two points.

[0034]

[0035] In the formula, z represents the number of teeth.

[0036] For the transition curve, θ is the angular position of the rack tooth tip radius, and its range is (α...). t1 The relationship between θ and Δx is: (π / 2)

[0037] Δx=x g0 -ba / tanθ (13)

[0038] After gear generating by rack and pinion machining, the gear tooth profile coordinates are:

[0039]

[0040] Preferably, in step S2, the gear state is assumed before the tooth surface contact occurs:

[0041] (1) Assume that all gears are perfectly machined, i.e., the tooth surface has no error;

[0042] (2) Assume that the driven gear is completely fixed and only consider the rotation of the driving gear;

[0043] (3) Assume that the gear pair is well lubricated, there is no friction between the teeth, and the contact force is always along the tooth flank normal.

[0044] Preferably, after determining the broken tooth fault in the gear pair in step S2, the gear pair preceding the broken tooth fault in the driven gear is defined as gear pair 1, the gear pair with the broken tooth fault in the driven gear is defined as gear pair 2, and the two subsequent gear pairs with the broken tooth fault in the driven gear are defined as gear pair 3 and gear pair 4, respectively; the initial state is the state before the application of external force, and the initial clearance data is 2πε / z in the initial state. p During the cycle, the meshing clearance data between each tooth pair in the gear pair, where ε is the contact ratio and z is the contact clearance. p Given the number of teeth on the driving gear, calculate the potential contact point A of tooth pair 1. 1p and A 1g The potential contact point A of the tooth pair 2 2p and A 2g The tooth pair has 3 potential contact points A 3p and A 3g And the potential contact point A of the tooth pair 4 4p and A 4g The specific calculation method is as follows:

[0045] The state of any tooth pair during meshing can be divided into three categories: (a) premature engagement, (b) engagement, and (c) prolonged disengagement. First, the driving gear is given a rotation angle Δθ. p The corresponding driven gear rotation angle Δθ g =Δθ p z p / z g , where z g The number of teeth on the driven gear;

[0046] In the initial state Δθ p <2π / z p The tooth pair 1 is in a pre-engaged state, and the potential contact point A of the tooth pair 1 is... 1p and A 1g The expression is:

[0047]

[0048] In the formula, O g O is the center coordinate of the driven gear. p Let α be the coordinates of the center of the driving gear, and α be the pressure angle. ag The pressure angle at the tip of the driven gear tooth, r ag Let r be the radius of the addendum circle of the driven gear. bp Let be the radius of the base circle of the driving gear. The meshing pressure angle at the initial engagement point of the driven gear. The meshing pressure angle at the tooth tip of the driven gear is... This refers to the potential contact point of the driving gear of the gear pair 1. α is the potential contact point of the driven gear of the gear pair 1. A1p Potential contact points The pressure angle at the point;

[0049] After one meshing cycle, the tooth pair 1 begins to mesh, when 2π / z p ≤Δθ p ≤2πε / z p The tooth pair 1 potential contact point A 1p and A 1g The expression is:

[0050]

[0051] The broken tooth height of the tooth pair 2 is h, and the meshing pressure angle of the broken tooth tip is... for:

[0052]

[0053] In the formula, r bg The base circle radius of the driven gear;

[0054] The pressure angle α′ at the tip of the broken tooth of the driven gear ag and the radius r′ of the broken tooth tip ag The expression for is:

[0055]

[0056] The rotation angle Δθ of the driving gear of the gear pair 2 p <tan[arctan(r) bg / r ag )]–tan[arctan(r bg / r′ ag If the tooth pair 2 is in a pre-engaged state, then the potential contact point A of the tooth pair 2 is... 2p and A 2g The expression is:

[0057]

[0058] When the tooth pair 2 is in the meshing state, tan[arctan(r)] bg / r ag )]–tan[arctan(r bg / r′ ag )]≤Δθ p ≤2πε / z p Potential contact point A 2p and A 2g The expression is:

[0059]

[0060] The gear pair 3 will experience an engaged state and an extended disengaged state, depending on the rotation angle of the driving gear of the gear pair 3. When the tooth pair 3 is in the meshing state, the contact point changes along the meshing line, with potential contact point A. 3p and A 3g The expression is:

[0061]

[0062] When the tooth pair 3 is in the extended engagement state At that time, the meshing of the tooth tip of the driving gear and the tooth profile of the driven gear of the tooth pair 3 occurs outside the contact line, and the potential contact point A of the tooth pair 3 is... 3p and A 3g The expression is:

[0063]

[0064] In the formula, the expression for Δθ3 of the tooth pair 3 is:

[0065]

[0066] Since the tooth pair 4 is far from the theoretical line of engagement, it can only be in an extended disengagement state. Assuming the driving gear rotates by an angle Δθ4 and contacts the driven gear, the potential contact point A of the tooth pair 4 is... 4p and A 4g The expression is:

[0067]

[0068] In the formula, the expression for Δθ4 of the tooth pair 4 is:

[0069]

[0070] Preferably, in step S2, the initial clearance of each pair of teeth is the arc length of the two potential contact points about the rotation center of the driven gear, expressed as:

[0071]

[0072] In the formula, ε i This represents the initial clearance for each pair of teeth in the corresponding gear meshing state.

[0073] Preferably, in step S3,

[0074] The compatibility equation for the gear pair is:

[0075]

[0076] In the formula, λ b For structural flexibility, λ c For contact flexibility; I 1×Np For an array of dimension 1×Np, where all elements are 1; Np×1 is an array of Np×1 dimensions, all elements being 1; f is the total meshing force applied to the gear pair; LSR is the load distribution coefficient matrix; STE is the static transmission error; ε is a vector containing the initial backlash of all tooth pairs.

[0077] Time-varying meshing stiffness k of a gear pair pg The expression is:

[0078]

[0079] Where NLTE is the no-load transmission error, which is the minimum initial clearance of all potential meshing tooth pairs under the target gear tooth meshing state.

[0080] Preferably, in step S4, the relative projected displacement p of the driving gear and the driven gear pg The expression is:

[0081] p pg =[sinα cosα 0 0 0 r bp -sinα -cosα 0 0 0 r bg ] T (29)

[0082] In the formula, α is the pressure angle of the gear pair; r bp and r bg Let K be the base circle radius of the driving gear and the driven gear, respectively, and K be the meshing stiffness matrix of the driving gear and the driven gear. pg for:

[0083]

[0084] Where k pg This represents the time-varying meshing stiffness of the gear pair.

[0085] Preferably, in step S4, the no-load transmission error caused by the broken tooth fault will cause the gear pair to produce a certain rigid motion. For ease of calculation, according to d'Alembert's principle, this rigid body motion is equivalent to a virtual force, F, the meshing virtual force between the driving gear and the driven gear. pg for:

[0086]

[0087] The virtual force derived from the above equation acts on the node between the driving gear and the driven gear, and the expression for the external force vector F is:

[0088]

[0089] In the formula T in For input torque, T out This is the output torque.

[0090] Preferably, in step S5, the mass matrix M of the driving gear-input shaft structure is solved based on the Timoshenko beam theory. i and stiffness matrix K i The mass matrix M of the driven gear-output shaft structure o and stiffness matrix K o .

[0091] Preferably, in step S5, the expression for the broken tooth fault analysis model is:

[0092]

[0093] In the formula, q is the system displacement vector, M is the mass matrix, K is the stiffness matrix, C is the damping matrix, and F is the external force vector.

[0094] The beneficial effects of this invention are as follows: This invention considers the abnormal meshing caused by tooth breakage, calculates the initial clearance data between each tooth pair in the gear pair, substitutes the initial clearance data into the gear tooth bearing contact analysis model, obtains the no-load transmission error and load distribution coefficient of the gear pair, further calculates the time-varying meshing stiffness of the gear pair, and then uses Timoshenko beam theory to simulate the support shaft of the gear pair, establishes a dynamic simulation model of the parallel shaft gear system under tooth breakage, and obtains the vibration response under the combined action of time-varying meshing stiffness and no-load transmission error. The simulation response considering the abnormal meshing caused by tooth breakage is closer to the experimental results than the simulation response ignoring abnormal meshing, providing a more accurate basis for the diagnosis of tooth breakage. Attached Figure Description

[0095] Figure 1 This is a flowchart of the method of the present invention;

[0096] Figure 2 This is a schematic diagram of the gear generated by the rack and pinion method;

[0097] Figure 3 It is a finite element model of a gear pair;

[0098] Figure 4 This is the initial meshing state of the gear pair;

[0099] Figure 5 It refers to the meshing state of a pair of teeth: (a) pre-engagement state; (b) extended engagement state;

[0100] Figure 6 It is a gear that has experienced a broken tooth failure;

[0101] Figure 7 The initial clearance and potential contact point of the gear pair within one solution cycle (taking a 1mm broken tooth height as an example): (a) initial clearance; (b) potential contact point;

[0102] Figure 8 The time-varying meshing stiffness of spur gear pairs considering different driven gear tooth breakage heights is: (a) healthy; (b) 0.5 mm; (c) 1 mm; (d) 1.5 mm; (e) 2 mm; (f) 2.5 mm;

[0103] Figure 9 The time-varying meshing stiffness obtained by the method proposed in this invention is compared with that obtained by the method in reference [Journal of Mechanical Design.141(2019)].

[0104] Figure 10 This takes into account the no-load transmission error for different broken tooth heights: (a) 1.5mm; (b) 2mm; (c) 2.5mm;

[0105] Figure 11 This takes into account the variation in load distribution coefficients for different tooth break heights: (a) 1.5mm; (b) 2mm; (c) 2.5mm;

[0106] Figure 12 This refers to the contact state of a gear pair with a broken tooth height of 1.5mm at position P1.

[0107] Figure 13 This refers to the contact state of a gear pair with a broken tooth height of 2mm at position P2.

[0108] Figure 14 This refers to the contact state of a gear pair with a broken tooth height of 2.5mm at position P3;

[0109] Figure 15 It considers the distribution of no-load transmission error and time-varying meshing stiffness under different input torques for a 1.5mm tooth breakage fault.

[0110] Figure 16 It considers the distribution of no-load transmission error and time-varying meshing stiffness under different input torques for a 2.5mm tooth breakage fault.

[0111] Figure 17 This takes into account the variation of tooth tip contact percentage with torque at different tooth break heights;

[0112] Figure 18 This is a schematic diagram of a coupled gear pair-shaft coupling system.

[0113] Figure 19 This is a schematic diagram of the system stiffness matrix assembly;

[0114] Figure 20 The waveforms are in the time domain under healthy conditions: (a) simulation results; (b) experimental results;

[0115] Figure 21 The frequency domain amplitude diagrams under healthy conditions are shown in the following images: (a) Simulation results; (b) Experimental results.

[0116] Figure 22 The time-domain waveforms under a 1.5mm broken tooth fault state are: (a) Simulation results considering external meshing of the meshing line; (b) Simulation results without considering external meshing of the meshing line; (c) Experimental results.

[0117] Figure 23 The frequency domain amplitude under a 1.5mm broken tooth fault condition is as follows: (a) Simulation results considering external meshing of the meshing line; (b) Simulation results without considering external meshing of the meshing line; (c) Experimental results. Detailed Implementation

[0118] The embodiments of the present invention will be described in further detail below with reference to the accompanying drawings and examples. The following examples are for illustrative purposes only and should not be construed as limiting the scope of the invention.

[0119] Example 1

[0120] Step 1: Based on the generating method for gear machining using a rack, transform the machining coordinates of the rack and gear to determine the gear tooth profile coordinates.

[0121] Table 1 Parameters of the Gear Pair

[0122]

[0123] Establish the x-axis along the pitch line of the rack and the y-axis along the center line of the rack tooth groove to obtain the coordinate system S of the rack. t Then the tooth tip center (b, a) lies in coordinate system S t The coordinates in the matrix can be represented as:

[0124]

[0125] In the formula, r t h is the radius of the rack tooth tip arc. a Here, c is the addendum coefficient, m is the clearance, and α is the module. t The tooth profile angle of the rack.

[0126] When the rack moves along x t When moving forward, the gear pitch circle rolls tangentially along the rack pitch line, and the gear center O... g In coordinate system S t x ing0 and the vertical coordinate y g0 They are respectively:

[0127]

[0128] In the formula, r b Let r be the base circle radius of the gear, r be the pitch circle radius of the gear, χ be the gear displacement coefficient, and α be the displacement coefficient of the gear. c Let α be the pressure angle at point C. t1 =acos[r b [ / (r+χ·m)] is the nodal pressure angle;

[0129] When the rack moves along x t When the gear moves Δx in the positive direction, it rotates -Δx / y around its center. g0 The initial position coordinates S of the rack t The current coordinates S of the rack t1 The initial position coordinates S of the gear g The current position coordinates of the gear S g1 The conversion relationship is as follows:

[0130] S t1 →S t :M tt1 =TRSX(Δx)

[0131]

[0132] S g →S g1 :M g1g =ROTZ(-Δx / y) g0 )

[0133] In the formula:

[0134]

[0135]

[0136]

[0137]

[0138] In coordinate system S t In the middle, the tooth profile r of the rack tla and the tooth tip radius r of the rack tlb Defined as:

[0139]

[0140]

[0141] For any point i on the gear tooth profile, the meshing pressure angle The corresponding Δx is obtained by the following formula:

[0142]

[0143] In the formula, Let C be the meshing pressure angle.

[0144] Pressure angle α at point i on tooth flank i With meshing pressure angle The relationship is as follows:

[0145]

[0146] In the formula, θ b The central angle from the point where the extension of the involute intersects the base circle to the center line of the gear tooth is half of the angle between the two points.

[0147]

[0148] In the formula, z is the number of teeth.

[0149] For the transition curve, θ is the angular position of the rack tooth tip radius, and its range is (α...). t1 The relationship between θ and Δx is: (π / 2)

[0150] Δx=x g0 -ba / tanθ

[0151] After gear generating by rack and pinion machining, the gear tooth profile coordinates are:

[0152]

[0153] At this point, the tooth profile coordinates of the gear have been obtained, and a gear model can be built based on the obtained tooth surface coordinates.

[0154] Discrete gear meshing model based on finite element method, such as Figure 3 As shown, the parameters of the gear pair are shown in Table 1.

[0155] Step 2: Determine the gear meshing state by using the gear tooth profile coordinates obtained in Step 1. Judge the gear tooth breakage fault in the gear pair by the gear meshing state. Calculate the potential contact points of the gear pair during the tooth surface contact process and obtain the distribution data of meshing points with abnormal meshing state.

[0156] Three states of gear meshing are defined: premature engagement, normal engagement, and prolonged disengagement. Based on each gear meshing state and considering the possibility of broken teeth on the driven gear, the potential meshing points in the gear process are calculated. Abnormal meshing phenomena are considered in the simulation process of this invention.

[0157] Based on the aforementioned derivation, the tooth profile coordinates of the gear were obtained. The tooth surface contact analysis will then be performed based on these coordinates. Before proceeding with the analysis, some reasonable assumptions will be made:

[0158] (1) Assume that all gears are perfectly machined, i.e., the tooth surface has no error.

[0159] (2) Assuming the driven gear is completely fixed, only the rotation of the driving gear is considered.

[0160] (3) Since the gear pair is well lubricated, the friction of the teeth is ignored, that is, the contact force is always along the tooth side normal.

[0161] After determining that the gear in the gear pair has a broken tooth fault, the gear pair preceding the one with the broken tooth fault in the driven gear is defined as gear pair 1, the gear pair with the broken tooth fault in the driven gear is defined as gear pair 2, and the two gear pairs following the one with the broken tooth fault in the driven gear are defined as gear pair 3 and gear pair 4, respectively. The initial state is the state before the application of external force, and the initial clearance data is 2πε / z in the initial state. p During the cycle, the meshing clearance data between each tooth pair in the gear pair, where ε is the contact ratio and z is the contact clearance. p Given the number of teeth on the driving gear, calculate the potential contact point A for tooth pair 1. 1p and A 1g 2 potential contact points A of the teeth 2p and A 2g 3 potential contact points A of the teeth 3p and A 3g Potential contact point A of tooth pair 4 4p and A 4g The specific calculation method is as follows:

[0162] The states of any gear pair can be divided into three categories: (a) pre-engagement state; (b) engagement state; and (c) extended disengagement state. First, give the driving gear a rotation angle Δθ. p The corresponding driven gear rotation angle Δθ g =Δθ p z p / z g , where z p Z represents the number of teeth on the driving gear. g The number of teeth on the driven gear.

[0163] In the initial state Δθ p <2π / z p Tooth pair 1 is in a pre-engaged state, and the potential contact point A of tooth pair 1 is... 1p and A 1g The expression is:

[0164]

[0165] In the formula, Og O is the center coordinate of the driven gear. p Let α be the coordinates of the center of the driving gear, and α be the pressure angle. ag The pressure angle at the tip of the driven gear tooth, r ag Let r be the radius of the addendum circle of the driven gear. bp Let be the radius of the base circle of the driving gear. The meshing pressure angle at the initial engagement point of the driven gear. For the meshing pressure angle at the tooth tip of the driven gear, A 1p A is the potential contact point of the driving gear in gear pair 1. 1g For the potential contact point of the driven gear of gear pair 1, α A1p Potential contact point A 1p The pressure angle at the point;

[0166] After one meshing cycle, tooth pair 1 begins to mesh, when 2π / z p ≤Δθ p ≤2πε / z p Tooth to potential contact point A 1p and A 1g The expression is:

[0167]

[0168] Considering the broken tooth fault in the driven gear of gear pair 2, the determination of the potential contact point location needs special discussion. Assume the driven gear of gear pair 2 experiences a broken tooth fault, with a broken tooth height of h and a broken tooth tip meshing pressure angle... for:

[0169]

[0170] In the formula, r bg The base circle radius of the driven gear;

[0171] The pressure angle α′ at the tip of the broken tooth of the driven gear ag and the radius r′ of the broken tooth tip ag The expression for is:

[0172]

[0173] Under ideal conditions, gear pair 2 should engage from its initial position. However, a broken tooth in gear pair 2 delays engagement. The rotation angle Δθ of the driving gear in gear pair 2... p ≤tan[arctan(r bg / r ag )]–tan[arctan(r bg / r′ agIf tooth pair 2 is in an early engagement state, then the potential contact point A of tooth pair 2 is... 2p and A 2g The expression is:

[0174]

[0175] When tooth pair 2 is in the meshing state, tan[arctan(r)] bg / r ag )]–tan[arctan(r bg / r′ ag )]≤Δθ p ≤2πε / z p Potential contact point A 2p and A 2g The expression is:

[0176]

[0177] During the solution process, gear pair 3 will experience a meshing state and an extended disengagement state. When the rotation angle of the driving gear in gear pair 3... When tooth pair 3 is in meshing, the contact point changes along the meshing line, with potential contact point A. 3p and A 3g The expression is:

[0178]

[0179] When tooth pair 3 is in the extended engagement state At that time, the meshing of the tooth tip of the driving gear and the tooth profile of the driven gear in gear pair 3 occurs outside the contact line, and the potential contact point A of gear pair 3 is... 3p and A 3g The expression is:

[0180]

[0181] In the formula, the expression for Δθ3 of tooth pair 3 is:

[0182]

[0183] Since tooth pair 4 is far from the theoretical line of engagement, it can only be in an extended disengagement state. Assuming the driving gear rotates by an angle Δθ4 and contacts the driven gear, the potential contact point A of tooth pair 4 is... 4p and A 4g The expression is:

[0184]

[0185] In the formula, the expression for Δθ4 of tooth pair 4 is:

[0186]

[0187] The initial clearance of each gear pair is the arc length of the two potential contact points about the center of rotation of the driven gear, expressed as:

[0188]

[0189] In the formula, ε i This represents the initial clearance for each pair of teeth in the corresponding gear meshing state.

[0190] Step 3: Based on the distribution data of meshing points with abnormal meshing states obtained in Step 2, calculate the initial clearance data between each tooth pair in the gear pair. The initial clearance data is the meshing clearance data in the initial state.

[0191] The compatibility equation for the gear pair is:

[0192]

[0193] In the formula, λ b For structural flexibility, λ c For contact flexibility; I 1×Np For an array of dimension 1×Np, where all elements are 1; Np×1 is an array of Np×1 dimensions, all elements being 1; f is the total meshing force applied to the gear pair; LSR is the load distribution coefficient matrix; STE is the static transmission error; ε is a vector containing the initial backlash of all tooth pairs.

[0194] Time-varying meshing stiffness k of a gear pair pg The expression is:

[0195]

[0196] Where NLTE is the no-load transmission error, which is the minimum initial clearance of all potential meshing tooth pairs under the target gear tooth meshing state.

[0197] Step 4: Substitute the initial clearance data obtained in Step 3 into the gear tooth bearing contact analysis model to obtain the no-load transmission error and load distribution coefficient of the gear pair, and further calculate the time-varying meshing stiffness of the gear pair.

[0198] Considering the time-varying meshing stiffness curves with different tooth breakage heights, as shown in the figure below. Figure 8 As shown, the proposed gear tooth bearing contact analysis method is verified using the finite element method. Based on an input torque of 200 Nm, it can be seen that the time-varying meshing stiffness curve obtained by the proposed method is very close to the time-varying meshing stiffness curve obtained by the finite element method. Therefore, the effectiveness of this method is verified.

[0199] In this example, the contact ratio of the gear pair is 1.712, indicating that one tooth undergoes an alternating process of double-tooth contact, single-tooth contact, and a second double-tooth contact within one solution cycle. By analyzing the stiffness curve and the no-load transmission error curve, the following patterns can be summarized:

[0200] (1) If the broken tooth height only damages the first double-tooth zone ( Figure 8 b、 Figure 8 c) In this case, the reduction in time-varying meshing stiffness caused by tooth breakage is only reflected in the first double-tooth zone, while the single-tooth zone and the second double-tooth zone remain unchanged. In the first double-tooth zone, due to the presence of tooth breakage, only one pair of teeth participates in meshing, and the load-bearing capacity of the gear pair is weakened.

[0201] (2) As the height of the broken tooth increases, the single-tooth area is destroyed. Figure 8 d) or the second bidental region is damaged ( Figure 8 (e, 8f). The gear meshing state undergoes a qualitative change, and no-load transmission error begins to appear. If a gear pair with a broken tooth fault rotates strictly according to the transmission ratio, it will experience an abnormal meshing state of "disengagement" in the single-tooth region due to the influence of the no-load transmission error. The time-varying meshing stiffness in the single-tooth region under the influence of the broken tooth fault is 0 (see...). Figure 9 The curve marked with "o" in the middle), but if the rigid motion of the gear pair closes the initial clearance, the gear pair will eventually mesh together, resulting in a non-zero time-varying meshing stiffness value at that location (see Figure 9 (Solid line).

[0202] (3) When the single-tooth zone or the second double-tooth zone is damaged (see...) Figure 8 (df). The contact ratio of the gear pair is disrupted. The greater the tooth breakage height, the smaller the actual contact ratio of the gear pair. When the actual contact ratio of the gear pair is less than 1, the no-load transmission error between the gear pairs is no longer 0 (see...). Figure 10 ).

[0203] If the no-load transmission error is not zero, the meshing of the gear pair will inevitably occur outside the theoretical line of action, leading to an abnormal meshing state. Simultaneously, abnormal meshing states caused by tooth breakage (such as tooth pair 1 and tooth pair 3 meshing simultaneously, while tooth pair 2 is completely non-meshing) may also occur. After a tooth breakage occurs, the load distribution coefficient between the gear teeth also changes. Figure 11 The load distribution coefficients for tooth breakage heights of 1.5mm, 2mm, and 2.5mm are shown. To further illustrate the contact state of the gear pair under tooth breakage conditions, positions P1, P2, and P3 are selected at different tooth breakage heights, and their load-bearing state diagrams are plotted, as shown below. Figure 12-14 As shown.

[0204] For a tooth breakage height of 1.5mm, the tooth breakage failure damages the entire single tooth area. Based on ANSYS software, Figure 12 The contact state of the gear pair after static solution at position P1 is given. As can be seen from the figure, all three tooth pairs are meshing simultaneously, with tooth pair 2 exhibiting the largest contact force, followed by tooth pair 3, and tooth pair 1 exhibiting the smallest contact force. This is consistent with... Figure 11 The load distribution coefficients at position P1 in section a are uniformly distributed. It is worth noting that due to the broken tooth fault, all three teeth are in edge contact, which significantly impacts the meshing stability of the gear pair and could potentially lead to complete failure of the gear transmission system.

[0205] For a 2mm broken tooth height, at position P2, tooth pair 2 has just entered the meshing state, forming a three-tooth meshing state together with tooth pairs 1 and 3 (e.g., Figure 13 (As shown). Similarly, at this position, all three pairs of teeth are in edge contact. Its load distribution factor is as follows: Figure 11 As shown in position P2 in diagram b, at this point, tooth pair 1 experiences the greatest contact force. Tooth pair 2 has just entered the meshing state and shares the least contact force.

[0206] For a broken tooth height of 2.5mm, the contact force state at P3 is as follows: Figure 14 As shown. Due to the large height of the broken tooth, the load on tooth pair 2 is only borne at the end of the second double-tooth zone (see...). Figure 11 c). Gear pairs exhibit intermittent meshing near the theoretical single-tooth region. Figure 14 (Taking position P3 as an example). At position P3, tooth pair 1 and tooth pair 3 are meshing, both in an edge contact state. Because the meshing position of the two tooth pairs deviates significantly from the theoretical line of action, the meshing force deviates from the base circle tangent, generating redundant non-tangential components. These non-tangential components can disrupt the stability of the gear transmission system, and interleaved tooth meshing should be avoided.

[0207] In summary, it can be seen that the meshing characteristics of the gear pair change when a broken tooth occurs. When the actual contact ratio is less than 1 due to the broken tooth, a non-zero no-load transmission error begins to appear. This non-zero no-load transmission error induces the contact point of the gear teeth to no longer change along the theoretical meshing line. At the same time, due to the missing tooth profile, the contact state of the gear pair deteriorates, making it easy for three pairs of edge contact teeth to mesh simultaneously, or even for teeth to mesh alternately, ultimately compromising the stability of the gear transmission system.

[0208] Gear transmission systems frequently experience tooth breakage under harsh operating conditions. Therefore, it is worthwhile to explore the changes in meshing characteristics caused by tooth breakage under different input torque conditions. The following analysis is based on operating conditions with 1.5mm and 2.5mm tooth breakage.

[0209] Considering the input torque variation range T = (20, 60, 100, 140, 180) Nm, Figure 15 and Figure 16The figures show the load distribution coefficient curves and time-varying meshing stiffness curves for tooth breakage faults of 1.5mm and 2.5mm, respectively. As can be seen from the figures, the time-varying meshing stiffness increases with increasing input torque. This is because the contact compliance of the gear pair is related to the meshing force. Simultaneously, with increasing input torque, the prolonged meshing phenomenon becomes more pronounced, and the load distribution coefficient curve becomes smoother.

[0210] In the load distribution coefficient curve and time-varying meshing stiffness curve under a 1.5mm broken tooth fault ( Figure 15 In the case of a single tooth, the single-tooth region is locally damaged. Taking the working condition of T=20Nm as an example, before the shaded area "a", tooth pair 3 (TP-3) meshes alone; in area "a", TP-3 gradually disengages, and tooth pair 2 (TP-2) gradually engages, but at this time both teeth are in abnormal meshing, so the time-varying meshing stiffness curve shows a fluctuating peak. After passing through area "a", the gear pair enters the normal working state.

[0211] In the load distribution coefficient curve and time-varying meshing stiffness curve under a 2.5mm broken tooth fault ( Figure 15 The second double-tooth region was partially damaged. Figure 15 At the location of shaded area c, TP-3 disengages, while TP-1 engages prematurely. Similar to the 1.5mm broken tooth height condition, the time-varying meshing stiffness also exhibits fluctuation peaks. This is also a result of the combined action of two abnormally meshing teeth. Figure 15 In the region between "b" and "c", only TP-1 engages in meshing, and the time-varying meshing stiffness in this region is reduced compared to the healthy state. In region "c", TP-2 gradually engages in meshing, forming a double-tooth region with TP-1. After region "c", the gear pair leaves the area affected by the broken tooth fault and begins to recover its normal meshing state.

[0212] A significant characteristic of tooth breakage is an increased proportion of tooth tip engagement. Under different input torque conditions, within a time range of overlap multiplied by the engagement cycle, the percentage of tooth tip contact is as follows: Figure 17 As shown.

[0213] Under healthy conditions, the tooth tip contact percentage increases from 7% to 18% with increasing torque. When a broken tooth occurs, the tooth tip contact percentage increases significantly. At a broken tooth height of 2.5mm, under low torque conditions (T=20Nm), the tooth tip contact percentage is 26%, 19% higher than under healthy conditions. Under high torque conditions (T=180Nm), the tooth tip contact percentage is 39%, 21% higher than normal conditions. Such a large increase in the tooth tip contact percentage leads to an increased time of abnormal meshing after a broken tooth, deteriorating the operating condition and potentially increasing the uncertainty of the gear transmission system.

[0214] Step 5: Using Timoshenko beam theory to simulate the support shaft of the gear pair, and combining the time-varying meshing stiffness and no-load transmission error of the gear pair obtained in Step 4, a dynamic simulation model of the parallel shaft gear system under tooth breakage fault is established.

[0215] Figure 18 The image shows a test bench for a straight gear pair. A broken tooth fault, 1.5 mm high, was intentionally implanted into one tooth of the driven gear. The sensor is mounted on the housing near bearing B3. Based on the actual parameters of the test bench, a simulation dynamic model is established below.

[0216] The dynamic model description of the shaft and gear coupling system is as follows: Figure 18 As shown, the driving gear is part of the input shaft, and the driven gear is part of the output shaft. Four bearings are used to limit the movement of the input and output shafts. The mass matrix and stiffness matrix M of the driving gear-input shaft structure are solved based on Timoshenko beam theory. i ,K i The mass matrix and stiffness matrix M of the driven gear-output shaft structure o ,K o The parameters for each axis are shown in Table 2.

[0217] Table 2 Parameters of the Axis

[0218]

[0219] The bearing support stiffness values ​​are listed in Table 3.

[0220] Table 3 Support Stiffness Values

[0221]

[0222] The relative projected displacement p of the driving gear and the driven gear pg The expression is:

[0223] p pg =[sinα cosα 0 0 0 r bp -sinα -cosα 0 0 0 r bg ] T

[0224] In the formula, α is the pressure angle of the gear pair; r bp and r bg Let K be the base circle radius of the driving gear and the driven gear, respectively, and K be the meshing stiffness matrix of the driving gear and the driven gear. pg for:

[0225]

[0226] Where k pgThis represents the time-varying meshing stiffness of the gear pair.

[0227] System stiffness assembly scheme as follows Figure 19 As shown.

[0228] For the input shaft structure and output shaft structure, the Rayleigh damping method is used to simulate the internal damping:

[0229] C j =α j M j +β j K j

[0230] Where j = I and O represent the input axis and output axis, respectively. α j and β j These are the damping coefficients. They can be obtained from the following formula:

[0231]

[0232] Among them, f 1j and f 2j Let ξ be the first two non-zero natural frequencies of the shaft structure j. 1j and ξ 2j Let f be the natural frequency of structure j. 1j and f 2j The corresponding modal damping values. Here, the modal damping values ​​are all taken as 0.03.

[0233] The formula for calculating the meshing damping of a gear pair is:

[0234]

[0235] Where m p and m g These are approximations of the masses of the driving gear and the driven gear, respectively.

[0236] The no-load transmission error caused by a broken tooth will result in a certain degree of rigid motion in the gear pair. For ease of calculation, according to d'Alembert's principle, the virtual force F of the meshing between the driving and driven gears is... pg for:

[0237]

[0238] The system's input torque acts on the rotational degree of freedom of the driving shaft's force-applying node, and the output torque acts on the rotational degree of freedom of the driven shaft's drive node. The virtual force derived from the above equation acts on the node between the driving and driven gears. The expression for F is:

[0239]

[0240] In the formula T in For input torque, Tout This is the output torque.

[0241] Thus, the mass matrix M, stiffness matrix K, damping matrix C, and external force vector F are obtained. The expression for the tooth breakage fault analysis model is:

[0242]

[0243] In the formula, q is the system displacement vector, M is the mass matrix, K is the stiffness matrix, C is the damping matrix, and F is the external force vector.

[0244] Comparative Example 1

[0245] Based on the simulation results of the broken tooth fault analysis model in step 5, and through experimental verification, it is proposed that for a pair of spur gear transmission systems, when a broken tooth fault occurs on the driven gear, a cluster of side frequencies appears around the meshing frequency and its harmonics, with the driven gear rotation frequency as the interval.

[0246] Under the operating conditions of an input torque of 20 Nm and an input speed of 1000 r / min, the simulation results and experimental results under healthy conditions are compared as follows: Figure 20 As shown in the figure, the time-domain waveform under healthy conditions exhibits periodic fluctuations, with the fluctuation period being the gear pair meshing period, and the fluctuation range being ±20m / s. 2 Within. Regarding the amplitude spectrum, it can be seen that the peak amplitude in the simulation results is mainly the meshing frequency f. m And its harmonics, the amplitude spectrum has no sidebands (see Figure 21 a). However, in the amplitude spectrum of the experimental results (see...) Figure 21 b), some f appeared p =f m / z p The intermittent sidebands. This is because during the installation of the test bench, there is a certain coaxiality error between the motor shaft and the input shaft. This misalignment will cause the input shaft's rotational frequency f to decrease. p The modulation system engagement frequency. Nevertheless, it can still be seen that the simulated amplitude spectrum and the experimental amplitude spectrum are similar at 5f... m ,6f m and 7f m The frequency amplitude is relatively high.

[0247] Considering a broken tooth fault in the driven gear with a height of 1.5mm, the simulation results and experimental waveforms for abnormal meshing conditions occurring outside the line of engagement are as follows: Figure 22 As shown in the figure, the peak amplitudes of the time-domain waveforms caused by tooth breakage in both cases are very similar. To illustrate the necessity of considering abnormal meshing conditions, the figure also shows the time-domain waveforms without considering abnormal meshing conditions (e.g., Figure 22(b) In other words, when the no-load transmission error is not zero, the gear is considered not to be meshing. It can be seen that in the time-domain waveform without considering abnormal meshing conditions, the peak amplitude of the time-domain waveform caused by the broken tooth fault is up to 4 times higher than the experimental results because the time-varying meshing stiffness suddenly becomes 0. This means that considering abnormal meshing conditions in the proposed method is necessary and accurate.

[0248] At the same time, the amplitude spectra obtained from the proposed model, the finite element model, and experiments were compared, such as... Figure 23 As shown. It's worth noting that, to more clearly describe the sideband distribution caused by the broken tooth fault, we used 5f... m The amplitude spectrum has been magnified horizontally. Generally speaking, the simulation and experimental results show good agreement on the amplitude trends near the meshing frequency and its harmonics. (In the magnified graph of the experimental results...) Figure 23 c) It can be seen that at 5f m The surrounding area showed a rotational frequency f of the driving gear. p The spaced sidebands, as explained in the health amplitude spectrum analysis, are caused by misalignment between the input shaft and the motor shaft. However, this phenomenon was not considered in the simulation, therefore, this pattern was not observed in the simulation results. A broken tooth fault in the driven wheel led to the generation of sidebands, which was observed in the simulation results ( Figure 23 a, b), in 5f m There exists nearby with f g =f m / z g The sidebands are spaced out. This is consistent with the sideband distribution pattern caused by the broken tooth fault in the experimental results. (Comparison) Figure 23 As can be seen from a and b, when abnormal meshing conditions are not considered, the amplitude of the characteristic sideband of the driven gear tooth breakage fault is significantly higher than both the experimental results and the simulation results considering abnormal meshing phenomena. This indicates that the simulation results considering abnormal meshing conditions are more accurate.

[0249] 1. Torque variation analysis revealed that as the input torque increases, the load distribution coefficient curve of the gear gradually flattens, and the time-varying meshing stiffness gradually increases. Under the condition of a broken tooth, when the input torque is sufficiently large, the gear pair may exhibit three-tooth contact, with all three pairs of teeth in edge contact. If the broken tooth height continues to increase, the gear pair will exhibit intermittent meshing (two non-adjacent tooth pairs meshing simultaneously). This abnormal meshing state means that the meshing force is no longer along the tangential direction of the base circle, and the stability of the gear transmission is severely compromised.

[0250] 2. Within a solution cycle (overlap ratio × meshing cycle), the percentage of tooth tip engagement is affected by the input torque and the tooth breakage height. It can be observed that as the torque increases, the percentage of tooth tip engagement increases regardless of whether a tooth breakage fault occurs. When a tooth breakage fault occurs, the higher the tooth breakage height, the higher the percentage of tooth tip engagement. An excessively high percentage of tooth tip engagement can cause instability in the gear pair's operation, easily leading to more serious problems.

[0251] 3. To verify the superiority of the proposed method, a shaft-gear pair coupled dynamic model was established. This model considers the no-load transmission error and time-varying meshing stiffness under tooth breakage faults. Verification revealed that the simulation results considering abnormal meshing conditions outside the meshing line, in both the time-domain waveform and amplitude spectrum, show strong consistency with experimental results. Without considering abnormal meshing conditions, the peak value of the time-domain waveform caused by tooth breakage is very high, and the sideband amplitude is significantly higher than the experimental results. Comparative analysis of the amplitude spectra of the simulation and experimental results shows that when a tooth breakage fault occurs in the driven gear, sidebands with intervals equal to the driven gear's rotational frequency appear near the meshing frequency and its harmonics.

[0252] The embodiments of the present invention are given for illustrative and descriptive purposes only, and are not intended to be exhaustive or to limit the invention to the forms disclosed. Many modifications and variations will be apparent to those skilled in the art. The embodiments were chosen and described in order to better illustrate the principles and practical application of the invention, and to enable those skilled in the art to understand the invention and to design various embodiments with various modifications suitable for a particular purpose.

Claims

1. A method for establishing a dynamic simulation model of a parallel shaft gear system under tooth breakage fault, characterized in that, Includes the following steps: S1. Based on the generating method of rack and pinion gear machining, the machining coordinates of the rack and gear are transformed to determine the tooth profile coordinates of the gear; S2. Determine the gear meshing state based on the gear tooth profile coordinates obtained in step S1, judge the gear tooth breakage fault in the gear pair through the gear meshing state, calculate the potential contact points of the gear pair during the tooth surface contact process, and obtain the meshing point distribution data including abnormal meshing state. S3. Based on the meshing point distribution data with abnormal meshing state obtained in step S2, calculate the initial clearance data between each tooth pair in the gear pair, wherein the initial clearance data is the meshing clearance data in the initial state. S4. Substitute the initial clearance data obtained in step S3 into the gear tooth bearing contact analysis model to obtain the no-load transmission error and load distribution coefficient of the gear pair, and further calculate the time-varying meshing stiffness of the gear pair. S5. Using Timoshenko beam theory to simulate the support shaft of the gear pair, and combining the time-varying meshing stiffness and no-load transmission error of the gear pair obtained in step S4, a dynamic simulation model of the parallel shaft gear system under tooth breakage fault is established.

2. The method for establishing a dynamic simulation model of a parallel shaft gear system under tooth breakage fault as described in claim 1, characterized in that, In step S1, the tangent point C between the rack tooth surface and the tooth tip arc is defined as the machining start point. An x-axis is established along the rack pitch line, and a y-axis is established along the rack tooth groove centerline, thus obtaining the rack's coordinate system S. t The tooth tip center (b, a) is in S t The coordinate system can be represented as follows: In the formula, r t h is the radius of the rack tooth tip arc. a Here, c is the addendum coefficient, m is the clearance, and α is the module. t The tooth profile angle of the rack; When the rack moves along x t When moving forward, the gear pitch circle rolls tangentially along the rack pitch line, and the gear center O... g In coordinate system S t x in g0 and the vertical coordinate y g0 They are respectively: In the formula, r b Let r be the base circle radius of the gear, r be the pitch circle radius of the gear, χ be the gear displacement coefficient, and α be the displacement coefficient of the gear. c Let α be the pressure angle at point C. t1 =acos[r b [ / (r+χ·m)] is the nodal pressure angle; When the rack moves along x t When the gear moves Δx in the positive direction, it rotates -Δx / y around its center. g0 The initial position coordinates S of the rack t The current coordinates S of the rack t1 The initial position coordinates S of the gear g The current position coordinates of the gear S g1 The conversion relationship is as follows: In the formula: In coordinate system S t In the middle, the tooth profile r of the rack tla and the tooth tip radius r of the rack tlb Defined as: For any point i on the gear tooth profile, the meshing pressure angle The corresponding Δx is obtained by the following formula: In the formula, Let C be the meshing pressure angle. Pressure angle α at point i on tooth flank i With meshing pressure angle The relationship is as follows: In the formula, θ b The central angle from the point where the extension of the involute intersects the base circle to the center line of the gear tooth is half of the angle between the two points. In the formula, z is the number of teeth. For the transition curve, θ is the angular position of the rack tooth tip radius, and its range is (α...). t1 The relationship between θ and Δx is: (π / 2) Δx=x g0 -b-a / tanθ (13) After gear generating by rack and pinion machining, the gear tooth profile coordinates are:

3. The method for establishing a dynamic simulation model of a parallel shaft gear system under tooth breakage fault as described in claim 1, characterized in that, In step S2, the gear state is assumed before the tooth surface contact occurs: (1) Assume that all gears are perfectly machined, i.e., the tooth surface has no error; (2) Assume that the driven gear is completely fixed and only consider the rotation of the driving gear; (3) Assume that the gear pair is well lubricated, there is no friction between the teeth, and the contact force is always along the tooth flank normal.

4. The method for establishing a dynamic simulation model of a parallel shaft gear system under tooth breakage fault as described in claim 3, characterized in that, After determining the broken tooth fault in the gear pair in step S2, the gear pair preceding the broken tooth fault in the driven gear is defined as gear pair 1, the gear pair with the broken tooth fault in the driven gear is defined as gear pair 2, and the two subsequent gear pairs with the broken tooth fault in the driven gear are defined as gear pair 3 and gear pair 4, respectively. The initial state is the state before the application of external force, and the initial clearance data is 2πε / z in the initial state. p During the cycle, the meshing clearance data between each tooth pair in the gear pair, where ε is the contact ratio and z is the contact clearance. p The number of teeth on the driving gear; calculate the potential contact point A of each tooth pair 1. 1p and A 1g The potential contact point A of the tooth pair 2 2p and A 2g The tooth pair has 3 potential contact points A 3p and A 3g And the potential contact point A of the tooth pair 4 4p and A 4g The specific calculation method is as follows: First, give the driving gear a rotation angle Δθ p The corresponding driven gear rotation angle Δθ g =Δθ p z p / z g , where z g The number of teeth on the driven gear; In the initial state Δθ p <2π / z p The tooth pair 1 is in a pre-engaged state, and the potential contact point A of the tooth pair 1 is... 1p and A 1g The expression is: In the formula, O g O is the center coordinate of the driven gear. p Let α be the coordinates of the center of the driving gear, and α be the pressure angle. ag The pressure angle at the tip of the driven gear tooth, r ag Let r be the radius of the addendum circle of the driven gear. bp Let be the radius of the base circle of the driving gear. The meshing pressure angle at the initial engagement point of the driven gear. For the meshing pressure angle at the tooth tip of the driven gear, A 1p A is the potential contact point of the driving gear of the gear pair 1. 1g α is the potential contact point of the driven gear of the gear pair 1. A1p Potential contact point A 1p The pressure angle at the point; After one meshing cycle, the tooth pair 1 begins to mesh, when 2π / z p ≤Δθ p ≤2πε / z p The tooth pair 1 potential contact point A 1p and A 1g The expression is: The broken tooth height of the tooth pair 2 is h, and the meshing pressure angle of the broken tooth tip is... for: In the formula, r bg The base circle radius of the driven gear; The pressure angle α′ at the tip of the broken tooth of the driven gear ag and the radius r′ of the broken tooth tip ag The expression for is: The rotation angle Δθ of the driving gear of the gear pair 2 p <tan[arctan(r) bg / r ag )]–tan[arctan(r bg / r′ ag If the tooth pair 2 is in a pre-engaged state, then the potential contact point A of the tooth pair 2 is... 2p and A 2g The expression is: When the tooth pair 2 is in the meshing state, tan[arctan(r)] bg / r ag )]–tan[arctan(r bg / r′ ag )]≤Δθ p ≤2πε / z p Potential contact point A 2p and A 2g The expression is: The gear pair 3 will experience an engaged state and an extended disengaged state, depending on the rotation angle of the driving gear of the gear pair 3. When the tooth pair 3 is in the meshing state, the contact point changes along the meshing line, with potential contact point A. 3p and A 3g The expression is: When the tooth pair 3 is in the extended engagement state At that time, the meshing of the tooth tip of the driving gear and the tooth profile of the driven gear of the tooth pair 3 occurs outside the contact line, and the potential contact point A of the tooth pair 3 is... 3p and A 3g The expression is: In the formula, the expression for Δθ3 of the tooth pair 3 is: Since the tooth pair 4 is far from the theoretical line of engagement, it can only be in an extended disengagement state. Assuming the driving gear rotates by an angle Δθ4 and contacts the driven gear, the potential contact point A of the tooth pair 4 is... 4p and A 4g The expression is: In the formula, the expression for Δθ4 of the tooth pair 4 is:

5. The method for establishing a dynamic simulation model of a parallel shaft gear system under tooth breakage fault according to claim 4, characterized in that, In step S2, the initial clearance of each pair of teeth is the arc length of the two potential contact points about the rotation center of the driven gear, expressed as: In the formula, ε i This represents the initial clearance for each pair of teeth in the corresponding gear meshing state.

6. The method for establishing a dynamic simulation model of a parallel shaft gear system under tooth breakage fault as described in claim 5, characterized in that, In step S3 The compatibility equation for the gear pair is: In the formula, λ b For structural flexibility, λ c For contact flexibility; I 1×Np For an array of dimension 1×Np, where all elements are 1; Np×1 It is an array with Np×1 dimensions and all elements being 1; f is the total meshing force applied to the gear pair; LSR is the load distribution coefficient matrix; STE is the static transfer error; ε is a vector containing the initial clearance of all tooth pairs. Time-varying meshing stiffness k of a gear pair pg The expression is: Where NLTE is the no-load transmission error, which is the minimum initial clearance of all potential meshing tooth pairs under the target gear tooth meshing state.

7. The method for establishing a dynamic simulation model of a parallel shaft gear system under tooth breakage fault as described in claim 6, characterized in that, In step S4, the relative projected displacement p of the driving gear and the driven gear pg The expression is: p pg =[sinα cosα 0 0 0 r bp -sinα -cosα 0 0 0 r bg ] T In equation (29), α is the pressure angle of the gear pair; r bp and r bg Let K be the base circle radius of the driving gear and the driven gear, respectively, and K be the meshing stiffness matrix of the driving gear and the driven gear. pg for: Where k pg This represents the time-varying meshing stiffness of the gear pair.

8. The method for establishing a dynamic simulation model of a parallel shaft gear system under tooth breakage fault according to claim 7, characterized in that, In step S4, the no-load transmission error causes rigid motion in the gear pair. According to d'Alembert's principle, the virtual force F of the meshing between the driving gear and the driven gear... pg for: The virtual force derived from the above equation acts on the node between the driving gear and the driven gear, and the expression for the external force vector F is: In the formula, T in For input torque, T out This is the output torque.

9. The method for establishing a dynamic simulation model of a parallel shaft gear system under tooth breakage fault as described in claim 8, characterized in that, In step S5, the mass matrix M of the driving gear-input shaft structure is solved based on Timoshenko beam theory. i and stiffness matrix K i The mass matrix M of the driven gear-output shaft structure o and stiffness matrix K o .

10. The method for establishing a dynamic simulation model of a parallel shaft gear system under tooth breakage fault according to claim 9, characterized in that, In step S5, the expression for the broken tooth fault analysis model is: In the formula, q is the system displacement vector, M is the mass matrix, K is the stiffness matrix, C is the damping matrix, and F is the external force vector.

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