A method for calculating multi-state meshing characteristics and fatigue strength of a precision helical gear

Abstract

This invention discloses a method for calculating the multi-state meshing characteristics and fatigue strength of precision helical gears. The specific steps are as follows: Step 1: Establish a multi-state meshing model of the precision helical gear; Step 2: Calculate the time-varying meshing stiffness and load distribution coefficient of the precision helical gear; Step 3: Calculate the friction coefficient of the precision helical gear under mixed elastohydrodynamic lubrication; Step 4: Establish a multi-state meshing dynamic model of the precision helical gear; Step 5: Verify the fatigue failure of the multi-state meshing of the precision helical gear. The beneficial effect is that it proposes a method for calculating the multi-state meshing characteristics and fatigue strength of precision helical gears, reveals the influence mechanism of excitation frequency on the multi-state meshing characteristics and fatigue strength of precision helical gears, fills the relevant technical gap in the calculation of the multi-state meshing characteristics and fatigue strength of precision helical gears, and promotes the development of engineering technology.

Description

Technical Field

[0001] This invention relates to the field of gear dynamics technology, and in particular to a method for calculating the multi-state meshing characteristics and fatigue strength of precision helical gears. Background Technology

[0002] Precision helical gears, due to their smooth transmission and high load-bearing capacity, are widely used in high-end equipment such as aerospace. Their meshing characteristics and fatigue strength directly affect the reliability of the transmission system. Traditional calculation methods are mostly based on ideal meshing states and employ simplified assumptions of a single state, making it difficult to accurately reflect the actual characteristics of helical gears under different loads and phases, including the alternating meshing of multiple teeth and dynamic state switching. This results in insufficient accuracy in calculating time-varying meshing stiffness, load distribution, and stress. Fatigue strength calculations often use standard simplified formulas, relying on nominal loads and failing to incorporate dynamic load spectra, leading to low accuracy in life prediction. Currently, there is a lack of calculation methods that integrate multi-state meshing and fatigue strength, making it difficult to meet the requirements of high-precision design. Therefore, there is an urgent need to develop a method for calculating the multi-state meshing characteristics and fatigue strength of precision helical gears to improve the accuracy and reliability of transmission design.

[0003] To address the aforementioned problems, this invention proposes a method for calculating the multi-state meshing characteristics and fatigue strength of precision helical gears. This method fully considers the actual working conditions of alternating meshing of multiple tooth pairs under different meshing states of precision helical gears. It calculates the time-varying meshing stiffness, load distribution, and time-varying friction coefficient under mixed elastohydrodynamic lubrication conditions, establishes a multi-state meshing dynamic model and dynamic equations for precision helical gears, and formulates a corresponding fatigue strength calculation method. This method can accurately and effectively study the multi-state meshing characteristics and fatigue strength of precision helical gears, filling the relevant technical gap in the calculation of multi-state meshing characteristics and fatigue strength of precision helical gears, and promoting the development of engineering technology. Summary of the Invention

[0004] To overcome the shortcomings of existing technologies and fill the gaps in related technologies, this invention provides a method for calculating the multi-state meshing characteristics and fatigue strength of precision helical gears. The method fully considers the actual working conditions of multiple tooth pairs alternating meshing and dynamic switching of meshing states under different meshing states of precision helical gears. It calculates the time-varying meshing stiffness, load distribution, and time-varying friction coefficient under mixed elastohydrodynamic lubrication conditions of the gears, establishes a multi-state meshing dynamic model and dynamic equations of precision helical gears, and then solves the dynamic equations to obtain the multi-state meshing characteristics of precision helical gears and form a corresponding fatigue strength calculation method.

[0005] The technical solution adopted by this invention to solve its technical problem is as follows: A method for calculating the multi-state meshing characteristics and fatigue strength of precision helical gears, characterized by comprising the following steps:

[0006] step Establish a multi-state meshing model for precision helical gears; during the meshing process, helical gears are affected by tooth flank clearance, resulting in tooth surface meshing and tooth back meshing. Based on this phenomenon, a multi-state meshing model for gears is established.

[0007] step Calculate the time-varying meshing stiffness and load distribution coefficient of a precision helical gear; maximum contact line length of the driving gear. Represented as:

[0008] ;

[0009] in, The tooth width of the driving gear. The helix angle of the driving wheel. The pitch of the teeth on the end face of the driving gear. The end face overlap of the gears. This refers to the axial overlap ratio of the gears;

[0010] Contact wire length over time changing function for:

[0011] ;

[0012] in, For a single tooth meshing cycle of the driving gear, The number of engagement cycles. , , The end face overlap of the gears. This refers to the axial overlap ratio of the gears;

[0013] Establish a calculation model for the time-varying meshing stiffness of a helical gear pair, and the bending stiffness of the driving gear. Shear stiffness Axial compressive stiffness for:

[0014] ,

[0015] ,

[0016] ;

[0017] in, , These are the elastic modulus and Poisson's ratio of the gear material, respectively. It is the distance from the current driving gear spur gear element to the front face of the gear. For the present The distance to the root circle plane of the tooth. For the present pressure angle, It is the distance from the point of meshing to the plane formed by the gear on the root circle. The distance from the meshing point to the gear's line of symmetry. This is the pressure angle corresponding to the meshing force on the current driving gear spur gear element. It is half the angle corresponding to a single tooth of the driving gear on the base circle. For the number of slices of the helical gear, This is the function for calculating bending stiffness from the base circle to the root circle. The approximate spur gear micro-element is obtained by slicing the helical gear along the tooth width direction. The slice width, , , , , The expression is:

[0018] ,

[0019] ,

[0020] ;

[0021] in, For the width of the driving gear teeth, This represents the angular displacement of the driving wheel. The number of teeth on the driving gear. Divided into the number of teeth of the driven gear, and These are the base circle radius and root circle radius of the driving gear, respectively. The pressure angle of the helical gear end face. This is the tooth tip height coefficient;

[0022] Bending stiffness of the driven wheel Shear stiffness Axial compressive stiffness for:

[0023] ,

[0024] ,

[0025] ;

[0026] in, The pressure angle corresponding to the meshing force on the current driven gear spur gear element is given. It is half the angle corresponding to a single tooth of the driven gear on the base circle. , The expression is:

[0027] ;

[0028] Single-tooth time-varying meshing stiffness Represented as:

[0029] ;

[0030] in, For Hertzian contact stiffness, For the base stiffness of the drive wheel, The base stiffness of the driven wheel is given by the subscript. The number of teeth involved in meshing; the time-varying meshing stiffness of precision helical gears. for:

[0031] ;

[0032] in, The overlap ratio of helical gears. It is a rounding function;

[0033] The ratio of the time-varying meshing stiffness of a thin-plate gear to the time-varying meshing stiffness of the entire helical gear is defined as the load distribution factor of the gear. :

[0034] ;

[0035] step : Calculate the friction coefficient of a precision helical gear under mixed elastohydrodynamic lubrication conditions; at any given time The coefficient of friction in both tooth surface meshing and tooth back meshing cases and for:

[0036] ;

[0037] in, The average surface roughness of the tooth surface. For normal unit load, For lubricating oil viscosity, The entrainment speed during tooth surface meshing. The entrainment speed during tooth back meshing. This represents the sliding speed of the tooth surfaces during meshing. This is the sliding speed of the tooth surface during back-to-back meshing;

[0038] step Establish the multi-state meshing dynamics equations for precision helical gears and solve for the multi-state meshing characteristics;

[0039] The torsional dynamics equations for a gear system under multiple meshing conditions are established. The torsional equation under tooth surface meshing conditions is as follows:

[0040] ;

[0041] in, The moment of inertia of the driving wheel. The torsional vibration acceleration of the driving wheel, Let be the moment of inertia of the driven wheel. The torsional vibration acceleration of the driven wheel, Let the base circle radius of the driving wheel be . Let be the base circle radius of the driven wheel. This refers to the friction arm during the meshing of the driving gear teeth. The friction arm is the force arm when the teeth of the driven gear mesh. This refers to the normal force when the teeth of the driving gear mesh. The normal force when the teeth of the driven gear are engaged. This refers to the frictional force generated when the teeth of the driving gear mesh. The frictional force is the force generated when the teeth of the driven gear mesh. The torque of the drive wheel, The torque of the driven wheel;

[0042] The torsional equation under back-to-back meshing conditions is:

[0043] ;

[0044] in, This refers to the friction arm during back-to-back meshing of the driving gear teeth. The friction arm is the force arm when the driven gear teeth mesh. This refers to the normal force when the driving gear teeth mesh back to back. The normal force when the driven gear teeth mesh. This refers to the frictional force during back-to-back meshing of the driving gear teeth. The frictional force when the driven gear teeth mesh. The torque of the drive wheel, The torque of the driven wheel;

[0045] The torsional equation under the de-toothed state is:

[0046] ;

[0047] Introducing equivalent displacement on the meshing line As a new degree of freedom, it replaces the degrees of freedom in tooth surface meshing, tooth back meshing, and tooth disengagement states. and The transition conditions of the meshing state can be determined by the equivalent displacement on the meshing line. With half tooth side clearance value To describe it, the dynamic model considering the transformation of the helical gear meshing state is as follows:

[0048] ;

[0049] in, The equivalent mass of the driving wheel and the driven wheel, For equivalent load, To propagate the error function, For time-varying meshing stiffness of gears, For gear meshing damping, for:

[0050] ;

[0051] in, This is a direction function used to distinguish between gear engagement and gear disengagement. This is the equivalent frictional force arm during tooth surface meshing. This is the equivalent frictional force arm during tooth back meshing;

[0052] step Fatigue failure check of precision helical gears under multi-state meshing; the gear meshing force during multi-state meshing is:

[0053] ;

[0054] in, This represents the average gear meshing stiffness. The dynamic meshing force of a single tooth at any given time. The total meshing force of the gears at any given moment;

[0055] According to Hertz's contact theory, the maximum contact stress between meshing teeth pairs is... for:

[0056] ;

[0057] in The contact line length varies over time. Pi It is half the width of the contact plane formed by the meshing of the tooth surfaces. It is half the width of the contact plane formed by the meshing of the tooth backs;

[0058] The maximum contact stress of the meshing teeth pair The condition of being less than the allowable contact stress as a condition to prevent tooth surface contact fatigue is expressed as:

[0059] ;

[0060] in, This represents the ultimate contact stress of the gear material. This is the working condition coefficient for the gear. Safety factor for gear contact fatigue; bending stress on weak sections. for:

[0061] ;

[0062] in, The tooth thickness is for the weakest section. The dynamic meshing force of a single tooth at any given time. The angle between the friction direction and the centerline of the driving gear teeth during meshing. The angle between the friction direction of the driving gear and the centerline of the driving gear teeth when the driving gear teeth are engaged. This refers to the bending lever arm of the weak section of the meshing tooth under the dynamic meshing force during the meshing of the driving gear tooth surfaces. This refers to the bending lever arm of the weak section of the meshing tooth under the dynamic meshing force during back-meshing of the driving gear teeth. for;

[0063] ;

[0064] in, This refers to the bending lever arm of the weak section of the meshing tooth under the dynamic meshing force during the meshing of the driving gear tooth surfaces. This refers to the bending lever arm of the weak section of the meshing tooth under the dynamic meshing force during back-meshing of the driving gear teeth. This refers to the bending lever arm of the weakest section of the meshing gear teeth under dynamic friction during meshing of the active gear teeth. The bending lever arm of the weak section of the meshing tooth under dynamic friction force during back meshing of the active gear teeth;

[0065] To ensure the gear pair operates in a safe state at the tooth root, bending stress on the weak section is required. It should meet the following requirements:

[0066] ;

[0067] in, This represents the ultimate bending stress of the gear material. This is the working condition coefficient for the gear. The safety factor for gear bending fatigue;

[0068] When the tooth surface contact stress exceeds When the gear teeth enter the plastic deformation stage, it damages the gear's lifespan. The deformation of the driving gear teeth is considered to be less than the maximum elastic deformation as a check condition for plastic deformation. Taking into account the bending, shearing, and compressive deformation components along the meshing line, the deformation at the meshing point is obtained. for:

[0069] ;

[0070] in, This refers to the tangential deformation caused by bending deformation. This refers to the tangential deformation caused by shear deformation. Radial deformation caused by compressive deformation. The angle between the normal direction of the meshing force and the center line of the gear teeth; the maximum elastic deformation of the meshing teeth. As the allowable deformation, it is expressed as:

[0071] ;

[0072] in, For tooth surface contact stress, It is the angle between the normal direction of the meshing force and the center line of the gear teeth during meshing. The meshing angle is the angle at which the teeth of the driving gear mesh. The meshing angle is the angle when the driving gear teeth are engaged back to back. For elastic modulus, The angle between the normal direction of the meshing force and the center line of the gear teeth during back-to-back meshing is expressed as:

[0073] ;

[0074] in, It is half of the central angle corresponding to the meshing teeth of the driving gear;

[0075] To ensure that the gears do not fail due to plastic deformation, the deformation amount should meet the following requirements:

[0076] ;

[0077] Complete the multi-state meshing characteristics and fatigue strength analysis of precision helical gears. Attached Figure Description

[0078] Figure 1 This is a flowchart of the calculation method for multi-state meshing characteristics and fatigue strength of precision helical gears;

[0079] Figure 2 It is a precision helical gear dynamics model that takes into account the transition of meshing states;

[0080] Figure 3 It is a global bifurcation diagram of the coexisting response under the influence of excitation frequency;

[0081] Figure 4 It is a contact fatigue check curve. Detailed Implementation

[0082] Embodiments of the present invention will be described below with reference to the accompanying drawings. -picture The specific embodiments of the present invention will be described in detail below.

[0083] As shown in the figure The diagram shows a flowchart of a method for calculating the multi-state meshing characteristics and fatigue strength of precision helical gears. The method is characterized by the following steps:

[0084] step Establish a multi-state meshing model for precision helical gears; during the meshing process, helical gears are affected by tooth flank clearance, resulting in tooth surface meshing and tooth back meshing. Based on this phenomenon, a multi-state meshing model for gears is established.

[0085] step Calculate the time-varying meshing stiffness and load distribution coefficient of a precision helical gear; maximum contact line length of the driving gear. Represented as:

[0086] ;

[0087] in, The tooth width of the driving gear. The helix angle of the driving wheel. The pitch of the teeth on the end face of the driving gear. The end face overlap of the gears. This refers to the axial overlap ratio of the gears;

[0088] Contact wire length over time changing function for:

[0089] ;

[0090] in, For a single tooth meshing cycle of the driving gear, The number of engagement cycles. , , The end face overlap of the gears. This refers to the axial overlap ratio of the gears;

[0091] Establish a calculation model for the time-varying meshing stiffness of a helical gear pair, and the bending stiffness of the driving gear. Shear stiffness Axial compressive stiffness for:

[0092] ,

[0093] ,

[0094] ;

[0095] in, , These are the elastic modulus and Poisson's ratio of the gear material, respectively. It is the distance from the current driving gear spur gear element to the front face of the gear. For the present The distance to the root circle plane of the tooth. For the present pressure angle, It is the distance from the point of meshing to the plane formed by the gear on the root circle. The distance from the meshing point to the gear's line of symmetry. This is the pressure angle corresponding to the meshing force on the current driving gear spur gear element. It is half the angle corresponding to a single tooth of the driving gear on the base circle. For the number of slices of the helical gear, This is the function for calculating bending stiffness from the base circle to the root circle. The approximate spur gear micro-element is obtained by slicing the helical gear along the tooth width direction. The slice width, , , , , The expression is:

[0096] ,

[0097] ,

[0098] ;

[0099] in, For the width of the driving gear teeth, This represents the angular displacement of the driving wheel. The number of teeth on the driving gear. Divided into the number of teeth of the driven gear, and These are the base circle radius and root circle radius of the driving gear, respectively. The pressure angle of the helical gear end face. This is the tooth tip height coefficient;

[0100] Bending stiffness of the driven wheel Shear stiffness Axial compressive stiffness for:

[0101] ,

[0102] ,

[0103] ;

[0104] in, The pressure angle corresponding to the meshing force on the current driven gear spur gear element is given. It is half the angle corresponding to a single tooth of the driven gear on the base circle. , The expression is:

[0105] ;

[0106] Single-tooth time-varying meshing stiffness Represented as:

[0107] ;

[0108] in, For Hertzian contact stiffness, For the base stiffness of the drive wheel, The base stiffness of the driven wheel is given by the subscript. The number of teeth involved in meshing;

[0109] Precision helical gear composite time-varying meshing stiffness for:

[0110] ;

[0111] in, The overlap ratio of helical gears. It is a rounding function;

[0112] The ratio of the time-varying meshing stiffness of a thin-plate gear to the time-varying meshing stiffness of the entire helical gear is defined as the load distribution factor of the gear. :

[0113] ;

[0114] Step (3): Calculate the friction coefficient of the precision helical gear under mixed elastohydrodynamic lubrication; at any time The coefficient of friction in both tooth surface meshing and tooth back meshing cases and for:

[0115] ;

[0116] in, The average surface roughness of the tooth surface. For normal unit load, For lubricating oil viscosity, The entrainment speed during tooth surface meshing. The entrainment speed during tooth back meshing. This represents the sliding speed of the tooth surfaces during meshing. This is the sliding speed of the tooth surface during back-to-back meshing;

[0117] step :picture It is a precision helical gear dynamic model that considers the transition of meshing state, establishes the multi-state meshing dynamic equation of precision helical gear, and solves the multi-state meshing characteristics;

[0118] The torsional dynamics equations for a gear system under multiple meshing conditions are established. The torsional equation under tooth surface meshing conditions is as follows:

[0119] ;

[0120] in, The moment of inertia of the driving wheel. The torsional vibration acceleration of the driving wheel, Let be the moment of inertia of the driven wheel. The torsional vibration acceleration of the driven wheel, Let the base circle radius of the driving wheel be . Let be the base circle radius of the driven wheel. This refers to the friction arm during the meshing of the driving gear teeth. The friction arm is the force arm when the teeth of the driven gear mesh. This refers to the normal force when the teeth of the driving gear mesh. The normal force when the teeth of the driven gear are engaged. This refers to the frictional force generated when the teeth of the driving gear mesh. The frictional force is the force generated when the teeth of the driven gear mesh. The torque of the drive wheel, The torque of the driven wheel;

[0121] The torsional equation under back-to-back meshing conditions is:

[0122] ;

[0123] in, This refers to the friction arm during back-to-back meshing of the driving gear teeth. The friction arm is the force arm when the driven gear teeth mesh. This refers to the normal force when the driving gear teeth mesh back to back. The normal force when the driven gear teeth mesh. This refers to the frictional force during back-to-back meshing of the driving gear teeth. The frictional force when the driven gear teeth mesh. The torque of the drive wheel, The torque of the driven wheel;

[0124] The torsional equation under the de-toothed state is:

[0125] ;

[0126] Introducing equivalent displacement on the meshing line As a new degree of freedom, it replaces the degrees of freedom in tooth surface meshing, tooth back meshing, and tooth disengagement states. and The transition conditions of the meshing state can be determined by the equivalent displacement on the meshing line. With half tooth side clearance value To describe it, the dynamic model considering the transformation of the helical gear meshing state is as follows:

[0127] ;

[0128] in, The equivalent mass of the driving wheel and the driven wheel, For equivalent load, To propagate the error function, For time-varying meshing stiffness of gears, For gear meshing damping, for:

[0129] ;

[0130] in, This is a direction function used to distinguish between gear engagement and gear disengagement. This is the equivalent frictional force arm during tooth surface meshing. This is the equivalent frictional force arm during tooth back meshing;

[0131] step Fatigue failure check of precision helical gears under multi-state meshing; the gear meshing force during multi-state meshing is:

[0132] ;

[0133] in, This represents the average gear meshing stiffness. The dynamic meshing force of a single tooth at any given time. The total meshing force of the gears at any given moment;

[0134] According to Hertz's contact theory, the maximum contact stress between meshing teeth pairs is... for:

[0135] ;

[0136] in The contact line length varies over time. Pi It is half the width of the contact plane formed by the meshing of the tooth surfaces. It is half the width of the contact plane formed by the meshing of the tooth backs;

[0137] The maximum contact stress of the meshing teeth pair The condition of being less than the allowable contact stress as a condition to prevent tooth surface contact fatigue is expressed as:

[0138] ;

[0139] in, This represents the ultimate contact stress of the gear material. This is the working condition coefficient for the gear. Safety factor for gear contact fatigue; bending stress on weak sections. for:

[0140] ;

[0141] in, The tooth thickness is for the weakest section. The dynamic meshing force of a single tooth at any given time. The angle between the friction direction and the centerline of the driving gear teeth during meshing. The angle between the friction direction of the driving gear and the centerline of the driving gear teeth when the driving gear teeth are engaged. This refers to the bending lever arm of the weak section of the meshing tooth under the dynamic meshing force during the meshing of the driving gear tooth surfaces. This refers to the bending lever arm of the weak section of the meshing tooth under the dynamic meshing force during back-meshing of the driving gear teeth. for;

[0142] ;

[0143] in, This refers to the bending lever arm of the weak section of the meshing tooth under the dynamic meshing force during the meshing of the driving gear tooth surfaces. This refers to the bending lever arm of the weak section of the meshing tooth under the dynamic meshing force during back-meshing of the driving gear teeth. This refers to the bending lever arm of the weakest section of the meshing gear teeth under dynamic friction during meshing of the active gear teeth. The bending lever arm of the weak section of the meshing tooth under dynamic friction force during back meshing of the active gear teeth;

[0144] To ensure the gear pair operates in a safe state at the tooth root, bending stress on the weak section is required. It should meet the following requirements:

[0145] ;

[0146] in, This represents the ultimate bending stress of the gear material. This is the working condition coefficient for the gear. The safety factor for gear bending fatigue;

[0147] When the tooth surface contact stress exceeds When the gear teeth enter the plastic deformation stage, it damages the gear's lifespan. The deformation of the driving gear teeth is considered to be less than the maximum elastic deformation as a check condition for plastic deformation. Taking into account the bending, shearing, and compressive deformation components along the meshing line, the deformation at the meshing point is obtained. for:

[0148] ;

[0149] in, This refers to the tangential deformation caused by bending deformation. This refers to the tangential deformation caused by shear deformation. Radial deformation caused by compressive deformation. The angle between the normal direction of the meshing force and the center line of the gear teeth; the maximum elastic deformation of the meshing teeth. As the allowable deformation, it is expressed as:

[0150] ;

[0151] in, For tooth surface contact stress, It is the angle between the normal direction of the meshing force and the center line of the gear teeth during meshing. The meshing angle is the angle at which the teeth of the driving gear mesh. The meshing angle is the angle when the driving gear teeth are engaged back to back. For elastic modulus, The angle between the normal direction of the meshing force and the center line of the gear teeth during back-to-back meshing is expressed as:

[0152] ;

[0153] in, It is half of the central angle corresponding to the meshing teeth of the driving gear;

[0154] To ensure that the gears do not fail due to plastic deformation, the deformation amount should meet the following requirements:

[0155] ;

[0156] Complete the multi-state meshing characteristics and fatigue strength analysis of precision helical gears.

[0157] In the example, using the method described above, through The nonlinear response of the system is obtained by the method.

[0158] picture To be at the meshing frequency Global bifurcation diagram under influence, diagram Meshing frequency Verification curve.

[0159] The above description is merely a preferred embodiment of the invention and does not constitute any limitation on the invention. Any modifications, alterations, or equivalent changes made to the above embodiments based on the essence of the invention shall still fall within the protection scope of the invention.

Claims

1. A method for calculating the multi-state meshing characteristics and fatigue strength of precision helical gears, characterized in that, Includes the following steps: step Establish a multi-state meshing model for precision helical gears; during the meshing process, helical gears are affected by tooth flank clearance, resulting in tooth surface meshing and tooth back meshing. Based on this phenomenon, a multi-state meshing model for gears is established. Step : Calculation of time-varying mesh stiffness and load distribution factor of precision helical gears; maximum contact line length of driving gear is expressed as: ； wherein is the tooth width of the drive wheel, is the helix angle of the drive wheel, is the face pitch of the drive wheel, is the face overlap of the gear, is the axial overlap of the gear; The length of the contact line as a function of time The length of the contact line as a function of time is: ； wherein, is the single tooth engagement period of the driving wheel, is the number of engagements, , , is the face coincidence of the gear, is the axial coincidence of the gear; The time-varying meshing stiffness calculation model of helical gear pair is established, and the bending stiffness, shear stiffness and axial compression deformation stiffness of the driving gear are , , ​ ， ， ； in, , These are the elastic modulus and Poisson's ratio of the gear material, respectively. It is the distance from the current driving gear spur gear element to the front face of the gear. For the present The distance to the root circle plane of the tooth. For the present pressure angle, It is the distance from the point of meshing to the plane formed by the gear on the root circle. The distance from the meshing point to the gear's line of symmetry. This is the pressure angle corresponding to the meshing force on the current driving gear spur gear element. It is half the angle corresponding to a single tooth of the driving gear on the base circle. For the number of slices of the helical gear, This is the function for calculating bending stiffness from the base circle to the root circle. The approximate spur gear micro-element is obtained by slicing the helical gear along the tooth width direction. The slice width, , , , , The expression is: ， ， ； in, For the width of the driving gear teeth, This represents the angular displacement of the driving wheel. The number of teeth on the driving gear. Divided into the number of teeth of the driven gear, and These are the base circle radius and root circle radius of the driving gear, respectively. The pressure angle of the helical gear end face. This is the tooth tip height coefficient; Bending stiffness of the driven wheel Shear stiffness Axial compressive stiffness for: ， ， ； in, The pressure angle corresponding to the meshing force on the current driven gear spur gear element is given. It is half the angle corresponding to a single tooth of the driven gear on the base circle. , The expression is: ； Single-tooth time-varying mesh stiffness is expressed as: ； wherein, is the Hertz contact stiffness, is the base stiffness of the driving wheel, is the base stiffness of the driven wheel, subscript is the number of teeth involved in the engagement; precise helical gear synthesis time-varying engagement stiffness is: ； wherein is the overlap of the helical gear, is a ceiling function; The ratio of the time-varying meshing stiffness of a thin-plate gear to the time-varying meshing stiffness of the entire helical gear is defined as the load distribution factor of the gear. : ； step : Calculate the friction coefficient of a precision helical gear under mixed elastohydrodynamic lubrication conditions; at any given time The coefficient of friction in both tooth surface meshing and tooth back meshing cases and for: ； in, The average surface roughness of the tooth surface. For normal unit load, For lubricating oil viscosity, The entrainment speed during tooth surface meshing. The entrainment speed during tooth back meshing. This represents the sliding speed of the tooth surfaces during meshing. This is the sliding speed of the tooth surface during back-to-back meshing; Steps : Establishing the multi-state engagement dynamics equation of precision helical gear, and solving the multi-state engagement characteristics The torsional dynamics equations for a gear system under multiple meshing conditions are established. The torsional equation under tooth surface meshing conditions is as follows: ； in, The moment of inertia of the driving wheel. The torsional vibration acceleration of the driving wheel, Let the moment of inertia of the driven wheel be _____. The torsional vibration acceleration of the driven wheel, Let the base circle radius of the driving wheel be . Let be the base circle radius of the driven wheel. This refers to the friction arm during the meshing of the driving gear teeth. The friction arm is the force arm when the teeth of the driven gear mesh. This refers to the normal force when the teeth of the driving gear mesh. The normal force when the teeth of the driven gear are engaged. This refers to the frictional force generated when the teeth of the driving gear mesh. The frictional force when the teeth of the driven gear mesh. The torque of the drive wheel, The torque of the driven wheel; The torsional equation under back-to-back meshing conditions is: ； in, This refers to the friction arm during back-to-back meshing of the driving gear teeth. The friction arm is the force arm when the driven gear teeth mesh. This refers to the normal force when the driving gear teeth mesh back to back. The normal force when the driven gear teeth mesh. This refers to the frictional force during back-to-back meshing of the driving gear teeth. The frictional force when the driven gear teeth mesh. The torque of the drive wheel, The torque of the driven wheel; The torsional equation under the de-toothed state is: ； Introducing equivalent displacement on the meshing line As a new degree of freedom, it replaces the degrees of freedom in tooth surface meshing, tooth back meshing, and tooth disengagement states. and The transition conditions of the meshing state can be determined by the equivalent displacement on the meshing line. With half tooth side clearance value To describe it, the dynamic model considering the transformation of the helical gear meshing state is as follows: ； in, The equivalent mass of the driving wheel and the driven wheel, For equivalent load, To propagate the error function, For time-varying meshing stiffness of gears, For gear meshing damping, for: ； in, This is a direction function used to distinguish between gear engagement and gear disengagement. This is the equivalent frictional force arm during tooth surface meshing. This is the equivalent frictional force arm during tooth back meshing; Step (5): Multi-state meshing fatigue failure check of precision helical gears; the gear meshing force during multi-state meshing is: ； wherein, is the average value of the gear mesh stiffness, is the dynamic mesh force of a single tooth at any instant, is the total mesh force of the gear at any instant; According to the Hertz contact theory, the maximum contact stress between the meshing tooth pairs is: ； wherein is the length of the contact line as a function of time, is the ratio of the circumference of a circle to its diameter, is half the width of the contact plane formed by the tooth flanks in mesh, is half the width of the contact plane formed by the tooth flanks in mesh, Maximum contact stress of a pair of meshing teeth The condition for preventing tooth surface contact fatigue is expressed as: ； wherein, is the limit contact stress for the gear material, is the service factor for the gear, is the safety factor for gear contact fatigue; Bending stress on weak cross section is: ； in, The tooth thickness is for the weakest section. The dynamic meshing force of a single tooth at any given time. The angle between the friction direction and the centerline of the driving gear teeth during meshing. The angle between the friction direction of the driving gear and the centerline of the driving gear teeth when the driving gear teeth are engaged. This refers to the bending lever arm of the weak section of the meshing tooth under the dynamic meshing force during the meshing of the driving gear tooth surfaces. This refers to the bending lever arm of the weak section of the meshing tooth under the dynamic meshing force during back-meshing of the driving gear teeth. for; ； in, This refers to the bending lever arm of the weak section of the meshing tooth under the dynamic meshing force during the meshing of the driving gear tooth surfaces. This refers to the bending lever arm of the weak section of the meshing tooth under the dynamic meshing force during back-meshing of the driving gear teeth. This refers to the bending lever arm of the weakest section of the meshing gear teeth under dynamic friction during meshing of the active gear teeth. The bending lever arm of the weak section of the meshing tooth under dynamic friction force during back meshing of the active gear teeth; To ensure the gear pair operates in a safe state at the tooth root, bending stress on the weak section is required. Should meet: ； wherein, is the limit bending stress of the gear material, is the service factor of the gear, is the safety factor of the bending fatigue of the gear; When the tooth surface contact stress exceeds When the gear teeth enter the plastic deformation stage, it damages the gear's lifespan. The deformation of the driving gear teeth is considered to be less than the maximum elastic deformation as a check condition for plastic deformation. Taking into account the bending, shearing, and compressive deformation components along the meshing line, the deformation at the meshing point is obtained. for: ； in, This refers to the tangential deformation caused by bending deformation. This refers to the tangential deformation caused by shear deformation. Radial deformation caused by compressive deformation. The angle between the normal direction of the meshing force and the center line of the gear teeth; the maximum elastic deformation of the meshing teeth. As the allowable deformation, it is expressed as: ； in, For tooth surface contact stress, It is the angle between the normal direction of the meshing force and the center line of the gear teeth during meshing. The meshing angle is the angle at which the teeth of the driving gear mesh. The meshing angle is the angle when the driving gear teeth are engaged back to back. For elastic modulus, The angle between the normal direction of the meshing force and the center line of the gear teeth during back-to-back meshing is expressed as: ； in, It is half of the central angle corresponding to the meshing teeth of the driving gear; To ensure that the gears do not fail due to plastic deformation, the deformation amount should meet the following requirements: ； Complete the multi-state meshing characteristics and fatigue strength analysis of precision helical gears.

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