A method for modeling dynamics of a precision gear system coupled with bearing faults

Abstract

This invention discloses a dynamic modeling method for a precision gear system with coupled bearing faults. The specific steps are as follows: Step (1): Calculate the friction coefficient of the helical gear tooth surface under elastohydrodynamic lubrication conditions; Step (2): Establish a time-varying meshing stiffness model of the helical gear pair considering time-varying friction; Step (3): Establish a bearing dynamic model considering bearing faults; Step (4): Establish a dynamic model of the precision gear system with coupled bearing faults and list the dynamic equations of the precision gear system with coupled bearing faults; Step (5): Solve the dynamic equations of the precision gear system with coupled bearing faults and obtain the vibration response of the system. The beneficial effect is that it proposes a dynamic modeling method for a precision gear system with coupled bearing faults, fills the relevant technical gap in the calculation of vibration characteristics of precision gear systems with coupled bearing faults, 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 dynamic modeling of precision gear systems with coupled bearing faults. Background Technology

[0002] Precision gear transmission systems are core transmission components in high-end equipment such as aerospace, precision manufacturing, and rail transportation. Their operational accuracy, vibration noise, and reliability directly determine the overall performance of the equipment. Rolling bearings, as critical support components, are prone to pitting and other faults under complex conditions such as high speed, heavy load, and variable operating conditions. These faults are coupled with factors such as time-varying gear meshing stiffness, tooth flank clearance, and shaft deformation, significantly exacerbating system vibration and inducing complex nonlinear dynamic behavior. Traditional dynamic modeling often simplifies bearings as ideal linear supports, neglecting the nonlinear impacts and stiffness fluctuations caused by faults, making it difficult to reveal the dynamic coupling mechanism between bearing faults and the gear system. However, current research has not fully considered the strong coupling effect of fault excitation, support nonlinearity, and meshing excitation, failing to accurately characterize the dynamic characteristics of precision gear systems under coupled faults. This makes it difficult to meet the requirements of fault early warning, health monitoring, and reliability design. In particular, systematic research on the vibration characteristics of precision gear systems with coupled bearing faults remains relatively lacking.

[0003] To address the aforementioned issues, this invention proposes a dynamic modeling method for precision gear systems with coupled bearing faults. This method establishes a dynamic model of the precision gear system with coupled bearing faults, calculates the bearing force and the time-varying meshing force of the helical gear pair, and establishes the dynamic equations of the precision gear system with coupled bearing faults. This method can accurately and effectively study the vibration characteristics of precision gear systems with coupled bearing faults, fills the relevant technical gaps in the calculation of vibration characteristics of precision gear systems with coupled bearing faults, and promotes the development of engineering technology. Summary of the Invention

[0004] To overcome the shortcomings of existing technologies and fill related technological gaps, this invention provides a dynamic modeling method for precision gear systems with coupled bearing faults. This method considers multiple factors such as bearing support in the case of pitting faults, time-varying gear meshing stiffness, tooth flank clearance, and transmission errors. It constructs a dynamic model of the precision gear system with coupled bearing faults, establishes the mechanical equations of the system, and then solves these equations to obtain the vibration characteristics of the system. The technical solution adopted by this invention to solve its technical problem is as follows:

[0005] A method for dynamic modeling of a precision gear system with coupled bearing faults, characterized by comprising the following steps:

[0006] Step (1): Calculate the friction coefficient of the helical gear tooth surface under elastohydrodynamic lubrication conditions as follows:

[0007] ;

[0008] in, This refers to the absolute viscosity of the lubricating oil. This represents the root mean square value of the gear surface roughness. The relative sliding speed of the gears. The slip ratio of the gear pair. For the combined radius of curvature, This is the normal pressure on the tooth surface. It is an index that comprehensively affects lubrication condition and surface roughness. The friction coefficient of the helical gear tooth surface is... , , , , , , , , This is a constant coefficient for the type of lubricating oil;

[0009] Step (2): Establish a time-varying meshing stiffness model of a helical gear pair with time-varying friction. Discretize the model by slicing along the tooth width direction of the helical gear, and convert each micro-element segment into a spur gear micro-element. Integrate and sum the micro-element to obtain the comprehensive time-varying meshing stiffness of the gear under time-varying friction. for:

[0010] ;

[0011] in, For the bending stiffness of the driving wheel, For the shear stiffness of the driving wheel, The axial compressive stiffness of the drive wheel. For the base stiffness of the drive wheel, The base stiffness of the driven wheel, The bending stiffness of the driven wheel, The shear stiffness of the driven wheel, The axial compressive stiffness of the driven wheel, For Hertzian contact stiffness, subscript This represents the number of teeth engaged in a single meshing cycle.

[0012] Step (3): Establish a bearing dynamics model considering bearing failure; select one of the rollers as the analysis object and name it Roller. ; Bearing The Middle A roller in angular position at time for:

[0013] ;

[0014] in, For bearings Total number of rollers, For bearings Roller serial number, Pi For bearings angular velocity, For bearings Roller diameter, For bearings The pitch circle diameter, subscript For bearings subscript For bearings ;

[0015] Under the premise of contact, bearings in the non-defective area No. The relative contact deformation between the roller and the raceway for:

[0016] ;

[0017] in, For bearings Inner circle Vibration displacement in the direction, For bearings Inner circle Vibration displacement in the direction, For bearings outer ring Vibration displacement in the direction, For bearings Inner circle Vibration displacement in the direction, For bearings Radial clearance;

[0018] bearings No. The angular position difference between the roller and the defect for:

[0019] ;

[0020] in, A function for finding the remainder. For bearings Local defect corner location; when the corner position difference meets the condition At that time, the first The roller is located within the local defect area, and the relative contact deformation between the roller and the raceway is... for:

[0021] ;

[0022] in, For bearings The depth of local defects, For bearings The defect angle span;

[0023] bearings exist , Nonlinear support force in the direction and for:

[0024] ;

[0025] in, For bearings The rolling element stiffness, The relative contact deformation between the roller and the raceway, when hour, ,when hour, ;

[0026] bearings The Middle A roller in angular position at time for:

[0027] ;

[0028] in, For bearings Total number of rollers, For bearings Any roller, For bearings Angular velocity of the inner ring, For bearings Roller diameter, For bearings Pitch circle diameter, For bearings Angular velocity of the outer raceway, subscript For bearings subscript For bearings ;

[0029] bearings No. Contact deformation between the rolling element and the raceway for:

[0030] ;

[0031] in, For bearings Outer and inner raceways Relative displacement in the direction, For bearings Outer and inner raceways Relative displacement in the direction, For bearings Outer and inner raceways Relative displacement in the direction, For bearings The load contact angle, For bearings Radial clearance;

[0032] bearings exist , , Force in direction , , for:

[0033] ;

[0034] in, For bearings The roller stiffness, when hour, ,when hour, ;

[0035] Step (4): Establish a dynamic model of the precision gear system with coupled bearing failure; load distribution coefficient. for:

[0036] ;

[0037] in, Let be the transient contact line length at any given time. This is the total contact line length of the gear. This refers to the number of meshing teeth.

[0038] Relative meshing displacement of gears at the meshing point for:

[0039] ;

[0040] in, For the driving wheel in Vibration displacement in the direction, For the driving wheel in Vibration displacement in the direction, For the driving wheel in Vibration displacement in the direction, Let be the pitch circle radius of the driving wheel. Let be the pitch circle radius of the driven gear. For the driven wheel in Vibration displacement in the direction, For the driven wheel in Vibration displacement in the direction, For the driven wheel in Vibration displacement in the direction, This represents the angular displacement of the torsional vibration of the driving wheel. The torsional vibration angular displacement of the driven wheel. The pressure angle of the driving wheel, The helix angle of the driving wheel. This represents the transmission error of the gear pair;

[0041] Dynamic meshing forces of gears in different meshing zones for:

[0042] ;

[0043] in, For the time-varying meshing stiffness of the gear pair, For the meshing damping of the gear pair, This is a function of tooth flank clearance. The vibration velocity of the gear pair at the meshing point; the normal force exerted by the gears on the driving and driven gears at any given moment. , for:

[0044] ;

[0045] The frictional force acting between the teeth of the gear at any given moment on the driving and driven gears. , for:

[0046] ;

[0047] in, The time-varying friction coefficient of the gear pair in different meshing regions. The direction determination coefficient of gear friction force; the friction arm of any point on the tooth surface on the line of meshing. , for:

[0048] ;

[0049] in, Let the base circle radius of the driving wheel be . Let be the base circle radius of the driven wheel. Let be the radius of the tooth tip circle of the driven gear. Let ω be the angular velocity of the driving wheel. Let be the angular velocity of the driven wheel. The total overlap ratio of the gear pair. For the single-tooth meshing cycle of the driving gear; Input shaft dynamics equation:

[0050] ;

[0051] in, The torsional vibration acceleration of the motor. The torsional vibration velocity of the motor, Let be the moment of inertia of the motor. This refers to the torsional vibration angular displacement of the motor. The torsional vibration velocity of the driving wheel, This represents the angular displacement of the torsional vibration of the driving wheel. The damping is for supporting the shaft connecting the motor and the drive wheel. To provide support rigidity for the shaft connecting the motor and the drive wheel, This refers to the output torque of the motor.

[0052] bearings Dynamic equations:

[0053] ;

[0054] in, For bearings Inner ring quality, For bearings Outer ring quality, For bearings Bearing housing quality For bearings Inner ring damping, For bearings Inner ring stiffness, For bearings Roller damping, For bearings The damping of the outer ring, For bearings The stiffness of the outer ring, For bearings Damping of the bearing housing For bearings The stiffness of the bearing housing , and Bearings Inner circle Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings Inner circle Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Divided into bearings outer ring Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and For bearings outer ring Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings bearing housing in Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings bearing housing in Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. For bearings exist Component of force in direction, For bearings exist Component of force in direction, and The driving wheel is respectively Vibration velocity and vibration displacement in the direction, and The driving wheel is respectively Vibration velocity and vibration displacement in the direction;

[0055] bearings Dynamic equations:

[0056] ;

[0057] in, For bearings Inner ring quality, For bearings Outer ring quality, For bearings Bearing housing quality For bearings Inner ring damping, For bearings Inner ring stiffness, For bearings Roller damping, For bearings The damping of the outer ring, For bearings The stiffness of the outer ring, For bearings Damping of the bearing housing For bearings The stiffness of the bearing housing , and Bearings Inner circle Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings Inner circle Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Divided into bearings outer ring Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and For bearings outer ring Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings bearing housing in Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings bearing housing in Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. For bearings exist Component of force in direction, For bearings exist Component of force in direction;

[0058] The dynamic equation of the driving wheel:

[0059] ;

[0060] in, For the mass of the driving wheel, The torsional vibration acceleration of the driving wheel, The torsional vibration velocity of the driving wheel, This represents the angular displacement of the torsional vibration of the driving wheel. The moment of inertia of the driving wheel. The torque of the drive wheel, The damping is for supporting the shaft connecting the motor and the drive wheel. To provide support rigidity for the shaft connecting the motor and the drive wheel, The normal force acting on the drive wheel at any given moment. Let the base circle radius of the driving wheel be . Let be the frictional force acting on the driving wheel at any given moment. Let be the frictional force arm from any point on the tooth surface of the driving gear to the line of meshing. For the driving wheel in Vibration acceleration in the direction, For the driving wheel in Vibration acceleration in the direction, For the driving wheel in Vibration acceleration in the direction, For the driving wheel in Vibration velocity in the direction, For the driving wheel in Vibration displacement in the direction;

[0061] bearings Dynamic equations:

[0062] ;

[0063] in, For bearings Inner ring quality, For bearings Outer ring quality, For bearings Bearing housing quality For bearings Inner ring damping, For bearings Inner ring stiffness, For bearings Roller damping, For bearings The damping of the outer ring, For bearings The stiffness of the outer ring, For bearings Damping of the bearing housing For bearings The stiffness of the bearing housing , and Bearings Inner circle Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings Inner circle Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings Inner circle Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings outer ring Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings Inner circle Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings Inner circle Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and For bearings bearing housing in Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings bearing housing in Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings bearing housing in Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings exist , and Component of force in direction, and The driven wheel is respectively Vibration velocity and vibration displacement in the direction, and The driven wheel is respectively Vibration velocity and vibration displacement in the direction, and The driven wheel is respectively Vibration velocity and vibration displacement in the direction;

[0064] bearings Dynamic equations:

[0065] ;

[0066] in, For bearings Inner ring quality, For bearings Outer ring quality, For bearings Bearing housing quality For bearings Inner ring damping, For bearings Inner ring stiffness, For bearings Roller damping, For bearings The damping of the outer ring, For bearings The stiffness of the outer ring, For bearings Damping of the bearing housing For bearings The stiffness of the bearing housing , and Bearings Inner circle Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings Inner circle Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings Inner circle Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings outer ring Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings Inner circle Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings Inner circle Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and For bearings bearing housing in Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings bearing housing in Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings bearing housing in Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings exist , and Component of force in direction;

[0067] Driven wheel dynamics equation:

[0068] ;

[0069] in, Let the mass of the driven wheel be... The torsional vibration acceleration of the driven wheel, The torsional vibration velocity of the driven wheel, The torsional vibration angular displacement of the driven wheel. Let the moment of inertia of the driven wheel be _____. The torque of the driven wheel, The torsional vibration velocity of the load. The torsional vibration angular displacement of the load. The support damping for the shaft connecting the load and the driven wheel. To provide stiffness support for the shaft connecting the load and the driven wheel, The normal force acting on the driven wheel at any given moment is denoted as . Let be the base circle radius of the driven wheel. Let be the frictional force acting on the driven wheel at any given moment. Let be the frictional force arm from any point on the tooth surface of the driving gear to the line of meshing. For the driven wheel in Vibration acceleration in the direction, For the driven wheel in Vibration acceleration in the direction, For the driven wheel in Vibration acceleration in the direction;

[0070] Output shaft dynamic equations:

[0071] ;

[0072] in, Let the moment of inertia of the load be . The torsional vibration acceleration of the load, The input torque of the load;

[0073] Step (5): Solve the dynamic equation of the precision gear system with coupled bearing failure to obtain the vibration response of the system. Attached Figure Description

[0074] Figure 1 This is a flowchart of a dynamic modeling method for a precision gear system with coupled bearing faults;

[0075] Figure 2 It is a dynamic model of a precision gear system with coupled bearing failure;

[0076] Figure 3 This is a vibration spectrum diagram of a precision gear system with coupled bearing failure;

[0077] Embodiments of the present invention will be described with reference to the accompanying drawings, which will be further described below. Figure 1 — Figure 3 The specific embodiments of the present invention will be described in detail below.

[0078] like Figure 1 The diagram shows a flowchart of a dynamic modeling method for a precision gear system with coupled bearing faults, characterized by the following steps:

[0079] Step (1): Calculate the friction coefficient of the helical gear tooth surface under elastohydrodynamic lubrication conditions as follows:

[0080] ;

[0081] in, This refers to the absolute viscosity of the lubricating oil. This represents the root mean square value of the gear surface roughness. The relative sliding speed of the gears. The slip ratio of the gear pair. For the combined radius of curvature, This is the normal pressure on the tooth surface. It is an index that comprehensively affects lubrication condition and surface roughness. The friction coefficient of the helical gear tooth surface is... , , , , , , , , This is a constant coefficient for the type of lubricating oil;

[0082] Step (2): Establish a time-varying meshing stiffness model of a helical gear pair with time-varying friction. Discretize the model by slicing along the tooth width direction of the helical gear, and convert each micro-element segment into a spur gear micro-element. Integrate and sum the micro-element to obtain the comprehensive time-varying meshing stiffness of the gear under time-varying friction. for:

[0083] ;

[0084] in, For the bending stiffness of the driving wheel, For the shear stiffness of the driving wheel, The axial compressive stiffness of the drive wheel. For the base stiffness of the drive wheel, The base stiffness of the driven wheel, The bending stiffness of the driven wheel, The shear stiffness of the driven wheel, The axial compressive stiffness of the driven wheel, For Hertzian contact stiffness, subscript This represents the number of teeth engaged in a single meshing cycle.

[0085] Step (3): Establish a bearing dynamics model considering bearing failure; select one of the rollers as the analysis object and name it Roller. ; Bearing The Middle A roller in angular position at time for:

[0086] ;

[0087] in, For bearings Total number of rollers, For bearings Roller serial number, Pi For bearings angular velocity, For bearings Roller diameter, For bearings The pitch circle diameter, subscript For bearings subscript For bearings ;

[0088] Under the premise of contact, bearings in the non-defective area No. The relative contact deformation between the roller and the raceway for:

[0089] ;

[0090] in, For bearings Inner circle Vibration displacement in the direction, For bearings Inner circle Vibration displacement in the direction, For bearings outer ring Vibration displacement in the direction, For bearings Inner circle Vibration displacement in the direction, For bearings Radial clearance;

[0091] bearings No. The angular position difference between the roller and the defect for:

[0092] ;

[0093] in, A function for finding the remainder. For bearings Local defect corner location; when the corner position difference meets the condition At that time, the first The roller is located within the local defect area, and the relative contact deformation between the roller and the raceway is... for:

[0094] ;

[0095] in, For bearings The depth of local defects, For bearings The defect angle span;

[0096] bearings exist , Nonlinear support force in the direction and for:

[0097] ;

[0098] in, For bearings The rolling element stiffness, The relative contact deformation between the roller and the raceway, when hour, ,when hour, ;

[0099] bearings The Middle A roller angular position at time for:

[0100] ;

[0101] in, For bearings Total number of rollers, For bearings Any roller, For bearings Angular velocity of the inner ring, For bearings Roller diameter, For bearings Pitch circle diameter, For bearings Angular velocity of the outer raceway, subscript For bearings subscript For bearings ;

[0102] bearings No. Contact deformation between the rolling element and the raceway for:

[0103] ;

[0104] in, For bearings Outer and inner raceways Relative displacement in the direction, For bearings Outer and inner raceways Relative displacement in the direction, For bearings Outer and inner raceways Relative displacement in the direction, For bearings The load contact angle, For bearings Radial clearance;

[0105] bearings exist , , Force in direction , , for:

[0106] ;

[0107] in, For bearings The roller stiffness, when hour, ,when hour, ;

[0108] Step (4): Establish as follows Figure 2 The dynamic model of the precision gear system with coupled bearing failure is shown; load distribution coefficient. for:

[0109] ;

[0110] in, Let be the transient contact line length at any given time. This is the total contact line length of the gear. This refers to the number of meshing teeth.

[0111] Relative meshing displacement of gears at the meshing point for:

[0112] ;

[0113] in, For the active wheel in Vibration displacement in the direction, For the active wheel in Vibration displacement in the direction, For the active wheel in Vibration displacement in the direction, Let be the pitch circle radius of the driving wheel. Let be the pitch circle radius of the driven gear. For the driven wheel in Vibration displacement in the direction, For the driven wheel in Vibration displacement in the direction, For the driven wheel in Vibration displacement in the direction, This represents the angular displacement of the torsional vibration of the driving wheel. The torsional vibration angular displacement of the driven wheel. The pressure angle of the driving wheel, The helix angle of the driving wheel. This represents the transmission error of the gear pair;

[0114] Dynamic meshing forces of gears in different meshing zones for:

[0115] ;

[0116] in, For the time-varying meshing stiffness of the gear pair, For the meshing damping of the gear pair, This is a function of tooth flank clearance. The vibration velocity of the gear pair at the meshing point; the normal force exerted by the gears on the driving and driven gears at any given moment. , for:

[0117] ;

[0118] The frictional force acting between the teeth of the gear at any given moment on the driving and driven gears. , for:

[0119] ;

[0120] in, The time-varying friction coefficient of the gear pair in different meshing regions. The direction determination coefficient of gear friction force; the friction arm of any point on the tooth surface on the line of meshing. , for:

[0121] ;

[0122] in, Let the base circle radius of the driving wheel be . Let be the base circle radius of the driven wheel. Let be the radius of the tooth tip circle of the driven gear. Let ω be the angular velocity of the driving wheel. Let be the angular velocity of the driven wheel. The total overlap ratio of the gear pair. For the single-tooth meshing cycle of the driving gear; Input shaft dynamics equation:

[0123] ;

[0124] in, The torsional vibration acceleration of the motor. The torsional vibration velocity of the motor, Let be the moment of inertia of the motor. This refers to the torsional vibration angular displacement of the motor. The torsional vibration velocity of the driving wheel, This represents the angular displacement of the torsional vibration of the driving wheel. The damping is for supporting the shaft connecting the motor and the drive wheel. To provide support rigidity for the shaft connecting the motor and the drive wheel, This refers to the output torque of the motor.

[0125] bearings Dynamic equations:

[0126] ;

[0127] in, For bearings Inner ring quality, For bearings Outer ring quality, For bearings Bearing housing quality For bearings Inner ring damping, For bearings Inner ring stiffness, For bearings Roller damping, For bearings The damping of the outer ring, For bearings The stiffness of the outer ring, For bearings Damping of the bearing housing For bearings The stiffness of the bearing housing , and Bearings Inner circle Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings Inner circle Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Divided into bearings outer ring Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and For bearings outer ring Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings bearing housing in Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings bearing housing in Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. For bearings exist Component of force in direction, For bearings exist Component of force in direction, and The driving wheel is respectively Vibration velocity and vibration displacement in the direction, and The driving wheel is respectively Vibration velocity and vibration displacement in the direction;

[0128] bearings Dynamic equations:

[0129] ;

[0130] in, For bearings Inner ring quality, For bearings Outer ring quality, For bearings Bearing housing quality For bearings Inner ring damping, For bearings Inner ring stiffness, For bearings Roller damping, For bearings The damping of the outer ring, For bearings The stiffness of the outer ring, For bearings Damping of the bearing housing For bearings The stiffness of the bearing housing , and Bearings Inner circle Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings Inner circle Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Divided into bearings outer ring Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and For bearings outer ring Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings bearing housing in Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings bearing housing in Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. For bearings exist Component of force in direction, For bearings exist Component of force in direction;

[0131] The dynamic equation of the driving wheel:

[0132] ;

[0133] in, For the mass of the driving wheel, The torsional vibration acceleration of the driving wheel, The torsional vibration velocity of the driving wheel, This represents the angular displacement of the torsional vibration of the driving wheel. The moment of inertia of the driving wheel. The torque of the drive wheel, The damping is for supporting the shaft connecting the motor and the drive wheel. To provide support rigidity for the shaft connecting the motor and the drive wheel, The normal force acting on the drive wheel at any given moment. Let the base circle radius of the driving wheel be . Let be the frictional force acting on the driving wheel at any given moment. Let be the frictional force arm from any point on the tooth surface of the driving gear to the line of meshing. For the active wheel in Vibration acceleration in the direction, For the active wheel in Vibration acceleration in the direction, For the active wheel in Vibration acceleration in the direction, For the active wheel in Vibration velocity in the direction, For the active wheel in Vibration displacement in the direction;

[0134] bearings Dynamic equations:

[0135] ;

[0136] in, For bearings Inner ring quality, For bearings Outer ring quality, For bearings Bearing housing quality For bearings Inner ring damping, For bearings Inner ring stiffness, For bearings Roller damping, For bearings The damping of the outer ring, For bearings The stiffness of the outer ring, For bearings Damping of the bearing housing For bearings The stiffness of the bearing housing , and Bearings Inner circle Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings Inner circle Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings Inner circle Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings outer ring Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings Inner circle Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings Inner circle Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and For bearings bearing housing in Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings bearing housing in Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings bearing housing in Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings exist , and Component of force in direction, and The driven wheel is respectively Vibration velocity and vibration displacement in the direction, and The driven wheel is respectively Vibration velocity and vibration displacement in the direction, and The driven wheel is respectively Vibration velocity and vibration displacement in the direction;

[0137] bearings Dynamic equations:

[0138] ;

[0139] in, For bearings Inner ring quality, For bearings Outer ring quality, For bearings Bearing housing quality For bearings Inner ring damping, For bearings Inner ring stiffness, For bearings Roller damping, For bearings The damping of the outer ring, For bearings The stiffness of the outer ring, For bearings Damping of the bearing housing For bearings The stiffness of the bearing housing , and Bearings Inner circle Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings Inner circle Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings Inner circle Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings outer ring Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings Inner circle Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings Inner circle Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and For bearings bearing housing in Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings bearing housing in Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings bearing housing in Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings exist , and Component of force in direction;

[0140] Dynamic equation of the driven wheel:

[0141] ;

[0142] in, Let the mass of the driven wheel be... The torsional vibration acceleration of the driven wheel, The torsional vibration velocity of the driven wheel, The torsional vibration angular displacement of the driven wheel. Let be the moment of inertia of the driven wheel. The torque of the driven wheel, The torsional vibration velocity of the load. The torsional vibration angular displacement of the load. The support damping for the shaft connecting the load and the driven wheel. To provide stiffness support for the shaft connecting the load and the driven wheel, The normal force acting on the driven wheel at any given moment is denoted as . Let be the base circle radius of the driven wheel. Let be the frictional force acting on the driven wheel at any given moment. Let be the frictional force arm from any point on the tooth surface of the driving gear to the line of meshing. For the driven wheel in Vibration acceleration in the direction, For the driven wheel in Vibration acceleration in the direction, For the driven wheel in Vibration acceleration in the direction;

[0143] Output shaft dynamic equations:

[0144] ;

[0145] in, Let the moment of inertia of the load be . The torsional vibration acceleration of the load, The input torque of the load;

[0146] Step (5): Solve the dynamic equation of the precision gear system with coupled bearing failure to obtain the vibration response of the system.

[0147] In the example, the nonlinear response of the system is obtained by using the Runge-Kutta method described above.

[0148] Figure 3 The spectrum of vibration response of a precision gear system with coupled bearing failure at a speed of 3000 rpm is given. The spectrum shows that the bearing rotational frequency... and its second harmonic Low-frequency components can be clearly identified in the figure, including the gear meshing frequency. Modulation sidebands on both sides and The amplitude increases significantly. During the contact process between the rolling elements and the raceway, pitting defects in the rolling bearing will generate frequencies consistent with the fault characteristic frequency. Synchronous periodic shocks, thereby stimulating including , , , , and A series of modulation frequency components, including.

[0149] 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 dynamic modeling of a precision gear system with coupled bearing faults, characterized in that, Includes the following steps: Step (1): Calculate the friction coefficient of the helical gear tooth surface under elastohydrodynamic lubrication conditions as follows: ； in, This refers to the absolute viscosity of the lubricating oil. This represents the root mean square value of the gear surface roughness. The relative sliding speed of the gears. The slip ratio of the gear pair. For the combined radius of curvature, This is the normal pressure on the tooth surface. It is an index that comprehensively affects lubrication condition and surface roughness. The friction coefficient of the helical gear tooth surface is... , , , , , , , , This is a constant coefficient for the type of lubricating oil; Step (2): Establish a time-varying meshing stiffness model of a helical gear pair with time-varying friction. Discretize the model by slicing along the tooth width direction of the helical gear, and convert each micro-element segment into a spur gear micro-element. Integrate and sum the micro-element to obtain the comprehensive time-varying meshing stiffness of the gear under time-varying friction. for: ； in, For the bending stiffness of the driving wheel, For the shear stiffness of the driving wheel, The axial compressive stiffness of the drive wheel. For the base stiffness of the drive wheel, The base stiffness of the driven wheel, The bending stiffness of the driven wheel, The shear stiffness of the driven wheel, The axial compressive stiffness of the driven wheel, For Hertzian contact stiffness, subscript This represents the number of teeth engaged in a single meshing cycle. Step (3): Establish a bearing dynamics model considering bearing failure; select one of the rollers as the analysis object and name it Roller. ; Bearing The Middle A roller angular position at time for: ； in, For bearings Total number of rollers, For bearings Roller serial number, Pi For bearings angular velocity, For bearings Roller diameter, For bearings The pitch circle diameter, subscript For bearings subscript For bearings ; Under the premise of contact, bearings in the non-defective area No. The relative contact deformation between the roller and the raceway for: ； in, For bearings Inner circle Vibration displacement in the direction, For bearings Inner circle Vibration displacement in the direction, For bearings outer ring Vibration displacement in the direction, For bearings Inner circle Vibration displacement in the direction, For bearings Radial clearance; bearings No. The angular position difference between the roller and the defect for: ； in, A function for finding the remainder. For bearings Local defect corner location; when the corner position difference meets the condition At that time, the first The roller is located within the local defect area, and the relative contact deformation between the roller and the raceway is... for: ； in, For bearings The depth of local defects, For bearings The defect angle span; bearings exist , Nonlinear support force in the direction and for: ； in, For bearings The rolling element stiffness, The relative contact deformation between the roller and the raceway, when hour, ,when hour, ; bearings The Middle A roller angular position at time for: ； in, For bearings Total number of rollers, For bearings Any roller, For bearings Angular velocity of the inner ring, For bearings Roller diameter, For bearings Pitch circle diameter, For bearings Angular velocity of the outer raceway, subscript For bearings subscript For bearings ; bearings No. Contact deformation between the rolling element and the raceway for: ； in, For bearings Outer and inner raceways Relative displacement in the direction, For bearings Outer and inner raceways Relative displacement in the direction, For bearings Outer and inner raceways Relative displacement in the direction, For bearings The load contact angle, For bearings Radial clearance; bearings exist , , Force in direction , , for: ； in, For bearings The roller stiffness, when hour, ,when hour, ; Step (4): Establish a dynamic model of the precision gear system with coupled bearing failure; load distribution coefficient. for: ； in, Let be the transient contact line length at any given time. This is the total contact line length of the gear. This refers to the number of meshing teeth. Relative meshing displacement of gears at the meshing point for: ； in, For the active wheel in Vibration displacement in the direction, For the active wheel in Vibration displacement in the direction, For the driving wheel in Vibration displacement in the direction, Let be the pitch circle radius of the driving wheel. Let be the pitch circle radius of the driven gear. For the driven wheel in Vibration displacement in the direction, For the driven wheel in Vibration displacement in the direction, For the driven wheel in Vibration displacement in the direction, This represents the angular displacement of the torsional vibration of the driving wheel. The torsional vibration angular displacement of the driven wheel. The pressure angle of the driving wheel, The helix angle of the driving wheel. This represents the transmission error of the gear pair; Dynamic meshing forces of gears in different meshing zones for: ； in, For the time-varying meshing stiffness of the gear pair, For the meshing damping of the gear pair, This is a function of tooth flank clearance. The vibration velocity of the gear pair at the meshing point; the normal force exerted by the gears on the driving and driven gears at any given moment. , for: ； The frictional force acting between the teeth of the gear at any given moment on the driving and driven gears. , for: ； in, The time-varying friction coefficient of the gear pair in different meshing regions. The direction determination coefficient of gear friction force; the friction arm of any point on the tooth surface on the line of meshing. , for: ； in, Let the base circle radius of the driving wheel be . Let be the base circle radius of the driven wheel. Let be the radius of the tooth tip circle of the driven gear. Let ω be the angular velocity of the driving wheel. Let be the angular velocity of the driven wheel. The total overlap ratio of the gear pair. For the single-tooth meshing cycle of the driving gear; Input shaft dynamics equation: ； in, The torsional vibration acceleration of the motor. The torsional vibration velocity of the motor, Let be the moment of inertia of the motor. This refers to the torsional vibration angular displacement of the motor. The torsional vibration velocity of the driving wheel, This represents the angular displacement of the torsional vibration of the driving wheel. The damping is for supporting the shaft connecting the motor and the drive wheel. To provide support rigidity for the shaft connecting the motor and the drive wheel, This refers to the output torque of the motor. bearings Dynamic equations: ; in, For bearings Inner ring quality, For bearings Outer ring quality, For bearings Bearing housing quality For bearings Inner ring damping, For bearings Inner ring stiffness, For bearings Roller damping, For bearings The damping of the outer ring, For bearings The stiffness of the outer ring, For bearings Damping of the bearing housing For bearings The stiffness of the bearing housing , and Bearings Inner circle Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings Inner circle Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Divided into bearings outer ring Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and For bearings outer ring Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings bearing housing in Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings bearing housing in Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. For bearings exist Component of force in direction, For bearings exist Component of force in direction, and The driving wheel is respectively Vibration velocity and vibration displacement in the direction, and The driving wheel is respectively Vibration velocity and vibration displacement in the direction; bearings Dynamic equations: ； in, For bearings Inner ring quality, For bearings Outer ring quality, For bearings Bearing housing quality For bearings Inner ring damping, For bearings Inner ring stiffness, For bearings Roller damping, For bearings The damping of the outer ring, For bearings The stiffness of the outer ring, For bearings Damping of the bearing housing For bearings The stiffness of the bearing housing , and Bearings Inner circle Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings Inner circle Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Divided into bearings outer ring Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and For bearings outer ring Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings bearing housing in Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings bearing housing in Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. For bearings exist Component of force in direction, For bearings exist Component of force in direction; The dynamic equation of the driving wheel: ； in, For the mass of the driving wheel, The torsional vibration acceleration of the driving wheel, The torsional vibration velocity of the driving wheel, This represents the angular displacement of the torsional vibration of the driving wheel. Let the moment of inertia of the driving wheel be denoted as . The torque of the drive wheel, The damping is for supporting the shaft connecting the motor and the drive wheel. To provide support rigidity for the shaft connecting the motor and the drive wheel, The normal force acting on the drive wheel at any given moment. Let the base circle radius of the driving wheel be . Let be the frictional force acting on the driving wheel at any given moment. Let be the frictional force arm from any point on the tooth surface of the driving gear to the line of meshing. For the driving wheel in Vibration acceleration in the direction, For the driving wheel in Vibration acceleration in the direction, For the driving wheel in Vibration acceleration in the direction, For the driving wheel in Vibration velocity in the direction, For the driving wheel in Vibration displacement in the direction; bearings Dynamic equations: ； in, For bearings Inner ring quality, For bearings Outer ring quality, For bearings Bearing housing quality For bearings Inner ring damping, For bearings Inner ring stiffness, For bearings Roller damping, For bearings The damping of the outer ring, For bearings The stiffness of the outer ring, For bearings Damping of the bearing housing For bearings The stiffness of the bearing housing , and Bearings Inner circle Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings Inner circle Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings Inner circle Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings outer ring Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings Inner circle Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings Inner circle Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and For bearings bearing housing in Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings bearing housing in Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings bearing housing in Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings exist , and Component of force in direction, and The driven wheel is respectively Vibration velocity and vibration displacement in the direction, and The driven wheel is respectively Vibration velocity and vibration displacement in the direction, and The driven wheel is respectively Vibration velocity and vibration displacement in the direction; bearings Dynamic equations: ； in, For bearings Inner ring quality, For bearings Outer ring quality, For bearings Bearing housing quality For bearings Inner ring damping, For bearings Inner ring stiffness, For bearings Roller damping, For bearings The damping of the outer ring, For bearings The stiffness of the outer ring, For bearings Damping of the bearing housing For bearings The stiffness of the bearing housing , and Bearings Inner circle Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings Inner circle Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings Inner circle Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings outer ring Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings Inner circle Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings Inner circle Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and For bearings bearing housing in Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings bearing housing in Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings bearing housing in Vibration acceleration, vibration velocity, and vibration displacement in the direction of vibration. , and Bearings exist , and Component of force in direction; Driven wheel dynamics equation: ； in, Let the mass of the driven wheel be... The torsional vibration acceleration of the driven wheel, The torsional vibration velocity of the driven wheel, The torsional vibration angular displacement of the driven wheel. Let the moment of inertia of the driven wheel be _____. The torque of the driven wheel, The torsional vibration velocity of the load. The torsional vibration angular displacement of the load. The support damping for the shaft connecting the load and the driven wheel. To provide stiffness support for the shaft connecting the load and the driven wheel, The normal force acting on the driven wheel at any given moment is denoted as . Let be the base circle radius of the driven wheel. Let be the frictional force acting on the driven wheel at any given moment. Let be the frictional force arm from any point on the tooth surface of the driving gear to the line of meshing. For the driven wheel in Vibration acceleration in the direction, For the driven wheel in Vibration acceleration in the direction, For the driven wheel in Vibration acceleration in the direction; Output shaft dynamic equations: ； in, Let the moment of inertia of the load be . The torsional vibration acceleration of the load, The input torque of the load; Step (5): Solve the dynamic equation of the precision gear system with coupled bearing failure to obtain the vibration response of the system.

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