Fault dynamics modeling method for Ravigneaux planetary gear transmission system

Abstract

The invention discloses a Ravigneaux type planetary gear transmission system fault dynamics modeling method. The method comprises the following steps: step (1), establishing a Ravigneaux type planetary gear pair time-varying meshing stiffness calculation formula containing a crack fault; (2) establishing a relative displacement formula of the Ravigneaux planetary gear pair containing the crack fault; (3) introducing a gear backlash nonlinear function under an elastic lubrication condition, and determining dynamic friction force on a meshing line of each gear pair; (4) establishing an oscillatory differential equation set of the Ravigneaux planetary gear transmission system containing the crack fault; (5) selecting different crack fault parameters of the Ravigneaux planetary gear, and inputting torque to solve nonlinear response characteristics of the system; the method has the beneficial effects that the nonlinear fault dynamics model of the Ravigneaux type planetary gear transmission system is established, and the related technical blank of fault dynamics modeling of the Ravigneaux type planetary gear transmission system at present is filled.

Description

Technical Field

[0001] The present invention relates to the technical field of gear dynamics, and in particular to a fault dynamics modeling method for a Lavinia-type planetary gear transmission system. Background Art

[0002] Ravina-type planetary gear transmissions, with their diverse ratios, high load capacity, and adaptability across multiple fields, demonstrate significant application potential and industrialization prospects in the automotive industry, particularly in new energy electric drive transmissions. However, these systems exhibit complex structures, multiple meshing relationships, and harsh operating environments, and their dynamic characteristics are affected by multiple factors. Gear root cracks are a primary factor contributing to the unstable dynamic response of Ravina-type planetary gear transmissions. Therefore, to ensure the safe operation of cracked Ravina-type planetary gear transmissions, their dynamic characteristics must be studied. Understanding the mechanism by which cracks influence the system's dynamic behavior is crucial for ensuring transmission reliability and life prediction.

[0003] In order to solve the above problems, the present invention proposes a fault dynamics modeling method for a Ravinia-type planetary gear transmission system. This method can accurately and effectively reveal the nonlinear characteristics of a Ravinia-type planetary gear set during transmission engagement when a crack fault occurs, and can also promote the development of engineering technology. Summary of the Invention

[0004] To overcome the shortcomings of existing technologies and fill related technical gaps, this paper proposes a method for modeling the fault dynamics of a Lavinia-type planetary gear transmission system. This model not only considers the crack coupling effect but also multiple nonlinear factors, such as the time-varying mesh stiffness of the cracked mesh, bearing clearance, time-varying support stiffness, and friction under EHL conditions. By leveraging these complex nonlinear factors and the crack coupling mechanism, a set of dynamic differential equations for a crack-fault Lavinia-type planetary gear transmission system is established. The Runge-Kutta integration method is used to solve the system's dynamic differential equations, revealing the nonlinear vibration characteristics of the system under external excitation and varying crack parameters.

[0005] The technical solution adopted by the present invention to solve the technical problem is as follows: a method for modeling fault dynamics of a Lavinia-type planetary gear transmission system, characterized by comprising the following steps:

[0006] Step (1) establishes a calculation formula for the time-varying meshing stiffness of a Lavinia-type planetary gear pair with a crack fault; the time-varying meshing stiffness of a gear pair with a crack can be expressed as:

[0007]

[0008] Where K ij (t) represents the time-varying meshing stiffness of the Lavinia planetary gear pair with cracks, K hIndicates the Hertzian stiffness of the gear pair, K a1i , K a2i Respectively represent the axial compression deformation stiffness of the driving and driven wheels, K B1i , K B2i Represents the bending stiffness of the driving and driven wheels, K S1i , K S2i Represents the shear stiffness of the driving and driven wheels, K f1i , K f2i Represent the matrix stiffness of the driving and driven wheel gears respectively.

[0009] When a Lavinia planetary gear pair has a crack fault, the Hertz stiffness, gear matrix stiffness, and axial compression deformation stiffness of the gear pair are calculated according to the normal gear stiffness. The calculation formula is as follows:

[0010]

[0011] Where K h represents the Hertzian stiffness of the gear pair, V represents the Poisson's ratio of the gear teeth, E represents the elastic modulus of the gear teeth, and B represents the width of the gear teeth; K a1i represents the axial compression deformation stiffness of the driving wheel, a2 represents the upper half tooth angle of the driving wheel base circle, a represents the current meshing angle, a1 represents the angle corresponding to the contact point of each slice of the driving wheel, K f1i Indicates the base stiffness of the driving wheel gear, u f1 Indicates the distance from the meshing point to the intersection of the driving gear tooth root arc and the gear center line, S f1 Indicates the arc length of the driving gear tooth root, L * 、M * 、P * , Q * Represents the coefficients of each item, K a2i represents the axial compression deformation stiffness of the driven wheel, a4 represents the upper half tooth angle of the driven wheel base circle, a3 represents the angle corresponding to the contact point of each slice of the driven wheel, K f2i Indicates the rigidity of the driven wheel gear matrix, u f2 Indicates the distance from the meshing point to the intersection of the driven gear tooth root arc and the gear center line, S f2 Indicates the arc length of the driven gear tooth root.

[0012] When a Lavinia planetary gear has a crack fault, the bending stiffness and shear stiffness are affected by the crack. The calculation method is divided into two cases. The first case is that part of the meshing position is affected by the crack. The time-varying meshing stiffness of the gear pair can be expressed as:

[0013]

[0014] Where K B1i , K B2iThey represent the bending stiffness of the cracked driving and driven wheels, a2 represents the upper half tooth angle of the driving wheel base circle, and a q represents the critical angle of single and double tooth meshing, a1 represents the angle corresponding to the contact point of each slice of the driving wheel, E represents the elastic modulus of the gear tooth, B represents the width of the gear tooth, V represents the Poisson's ratio of the gear tooth, q represents the crack depth, R b1 Indicates the base circle radius of the driving wheel crack gear, a L represents the crack angle, a represents the current meshing angle, K S1i , K S2i They represent the shear stiffness of the cracked master and driven wheel teeth, a4 represents the upper half tooth angle of the base circle of the driven wheel, a3 represents the angle corresponding to the contact point of each slice of the driven wheel, and R b2 Indicates the base circle radius of the cracked gear.

[0015] The second case is when all meshing positions are affected by cracks. The bending stiffness and shear stiffness are calculated as follows:

[0016]

[0017] Where K B1i , K B2i They represent the bending stiffness of the cracked driving and driven wheels, a2 represents the upper half tooth angle of the driving wheel base circle, and a q represents the critical angle of single and double tooth meshing, a1 represents the angle corresponding to the contact point of each slice of the driving wheel, E represents the elastic modulus of the gear tooth, B represents the width of the gear tooth, V represents the Poisson's ratio of the gear tooth, q represents the crack depth, R b1 Indicates the base circle radius of the driving wheel crack gear, a L represents the crack angle, a represents the current meshing angle, K S1i , K S2i They represent the shear stiffness of the cracked master and driven wheel teeth, a4 represents the upper half tooth angle of the base circle of the driven wheel, a3 represents the angle corresponding to the contact point of each slice of the driven wheel, and R b2 Indicates the base circle radius of the cracked gear.

[0018] Step (2) establishes the relative displacement formula of the Lavinia planetary gear pair with crack fault; the relative displacement of the first-stage large sun gear-planet gear a gear pair and the second-stage small sun gear-planet gear b gear pair is:

[0019]

[0020] Where x m1 、x m2 They represent the relative displacements of the first-stage large sun gear-planet gear a gear pair and the second-stage small sun gear-planet gear b gear pair, respectively. represents the large sun gear pressure angle, represents the pressure angle of planet gear b, Respectively represent the vibration displacement of planetary gear a along the x-axis and y-axis directions, They represent the torsional displacement of planet gear a, Respectively represent the vibration displacement of the large sun gear along the x-axis and y-axis directions, They represent the torsional displacement of the large sun gear, r a They represent the pitch circle radius of the large sun gear and planet gear a respectively, e1(t) represents the comprehensive error of the first-stage large sun gear-planet gear a gear pair, Respectively represent the vibration displacement of planetary gear b along the x-axis and y-axis directions, They represent the torsional displacement of planetary gear b, Respectively represent the vibration displacement of the small sun gear along the x-axis and y-axis directions, They represent the torsional displacement of the small sun gear, They represent the pitch circle radii of the small sun gear and planet gear b respectively, and e2(t) represents the comprehensive error of the second-stage small sun gear-planet gear b gear pair.

[0021] Step (3) considers the elastic lubrication condition, introduces the nonlinear function of the tooth side clearance, and determines the dynamic friction force on the meshing line of each gear pair; the calculation method of the dynamic friction force is:

[0022]

[0023] Where, F f represents dynamic friction, ν represents directional coefficient, μ(t) represents friction coefficient, C ij represents the meshing damping of each gear, f(x ml ,D) represents the gap function, K ij (t) represents the time-varying mesh stiffness of the Lavinia planetary gear pair with cracks, and D represents half of the tooth side clearance.

[0024] Step (4) establishes a set of vibration differential equations for the Lavinia planetary gear transmission system with crack fault; the vibration differential equation of the large sun gear is:

[0025]

[0026] Where, is the mass of the large sun gear, ω c is the theoretical angular velocity of the planet carrier, are the dynamic meshing force and friction force between the large sun gear and planet gear a, respectively. Respectively represent the vibration displacement of the large sun gear bearing along the x-axis and y-axis directions, are the support stiffness of the large sun gear along the x-axis and y-axis directions, are the support damping of the large sun gear along the x and y axis directions, are the torsional stiffness and torsional damping of the large sun gear, is the input torque of the large sun gear, is the large sun gear helix angle, is the moment of inertia of the large sun gear, and g is the acceleration due to gravity.

[0027] Vibration differential equation of planetary gear a:

[0028]

[0029] Where, represents the mass of planet gear a, represents the pressure angle of the ring gear, represents the pressure angle of planet gear b, They represent the vibration displacement of the planetary gear a bearing along the x-axis and y-axis directions, Respectively represent the dynamic meshing force and friction force between the large sun gear and planetary gear a, They represent the dynamic meshing force and friction force between the ring gear and the planetary gear a, They represent the dynamic meshing force and friction force between the planetary gear b and the planetary gear a, respectively. Respectively represent the support stiffness of planetary gear a along the x-axis and y-axis directions, Respectively represent the support damping of planetary gear a along the x-axis and y-axis directions, Respectively represent the torsional stiffness and torsional damping of planetary gear a, represents the helix angle of planet gear a, represents the moment of inertia of planet gear a.

[0030] The differential equation of the large sun gear bearing vibration is:

[0031]

[0032] Where, Indicates the mass of the large sun gear bearing, Respectively represent the vibration displacement of the large sun gear bearing along the x-axis and y-axis directions, Represent the components of force on the large sun gear and the large sun gear bearing along the x-axis and y-axis directions respectively, They represent the support damping of the large sun gear bearing along the x-axis and y-axis directions respectively.

[0033] The differential equation of vibration of planetary gear a bearing is:

[0034]

[0035] Where, Indicates the mass of the planetary gear a bearing, are the components of the bearing force of the planetary gear a and the bearing along the x-axis and y-axis directions, are the support damping of the planetary gear a bearing along the x and y axis directions respectively.

[0036] Differential equation of ring gear vibration:

[0037]

[0038] Where m r represents the mass of the ring gear, x r 、y r Respectively represent the vibration displacement of the gear ring along the x-axis and y-axis directions, u r They represent the torsional displacement of the ring gear, They represent the dynamic meshing force and friction force between the ring gear and the planetary gear a, respectively, and K rx , K ry Respectively represent the support stiffness of the gear ring along the x-axis and y-axis directions, C rx 、C ry Respectively represent the support damping of the gear ring along the x and y axis directions, K ru 、C ru Respectively represent the torsional stiffness and torsional damping of the ring gear, T r represents the output torque of the ring gear, β r represents the helix angle of the gear ring, r r Indicates the pitch circle radius of the gear ring, I r Indicates the moment of inertia of the ring gear.

[0039] Differential equation for planet carrier vibration:

[0040]

[0041] Where m c represents the mass of the planet carrier, ω c represents the theoretical angular velocity of the planet carrier, x c 、y c Respectively represent the vibration displacement of the planet carrier along the x-axis and y-axis, u c They represent the torsional displacement of the planet carrier, Respectively represent the vibration displacement of planetary gear a along the x-axis and y-axis directions, They represent the torsional displacement of planet gear a, Respectively represent the vibration displacement of planetary gear b along the x-axis and y-axis directions, They represent the torsional displacement of planetary gear b, K cx , K cy Respectively represent the support stiffness of the planet carrier along the x and y axis directions, C cx 、Ccy Respectively represent the support damping of the planet carrier along the x and y axis directions, K cu 、C cu They represent the torsional stiffness and torsional damping of the planet carrier respectively. Respectively represent the support stiffness of planetary gear a along the x-axis and y-axis directions, Respectively represent the support damping of planetary gear a along the x-axis and y-axis directions, Respectively represent the torsional stiffness and torsional damping of planetary gear a, Respectively represent the support stiffness of planetary gear b along the x-axis and y-axis directions, Respectively represent the support damping of planetary gear b along the x-axis and y-axis directions, They represent the torsional stiffness and torsional damping of the planetary gear b, r c Indicates the outer radius of the planet carrier, I c Indicates the moment of inertia of the planet carrier.

[0042] The differential equation of planetary gear b bearing vibration is:

[0043]

[0044] Where, represents the mass of the planetary gear b bearing, They represent the vibration displacement of the planetary gear b bearing along the x-axis and y-axis directions respectively, Respectively represent the vibration displacement of planetary gear b along the x-axis and y-axis directions, Represent the components of force on the planetary gear b and the bearing along the x-axis and y-axis directions, Respectively represent the support stiffness of planetary gear b along the x-axis and y-axis directions, Respectively represent the support damping of planetary gear b along the x-axis and y-axis directions, They represent the support damping of the planetary gear b bearing along the x-axis and y-axis directions respectively.

[0045] Small sun gear bearing vibration differential equation:

[0046]

[0047] Where, Indicates the mass of the small sun gear bearing, Respectively represent the vibration displacement of the small sun gear bearing along the x-axis and y-axis directions, Respectively represent the vibration displacement of the small sun gear along the x-axis and y-axis directions, Respectively represent the component forces of the small sun gear and the bearing along the x-axis and y-axis directions, Respectively represent the support damping of the small sun gear bearing along the x-axis and y-axis directions, Respectively represent the support stiffness of the small sun gear along the x-axis and y-axis directions, Respectively represent the support damping of the small sun gear along the x-axis and y-axis directions.

[0048] Vibration differential equation of planetary gear b:

[0049]

[0050] Where, represents the mass of planet gear b, represents the pressure angle of planet gear b, Respectively represent the vibration displacement of planetary gear b along the x-axis and y-axis directions, They represent the torsional displacement of planetary gear b, They represent the vibration displacement of the planetary gear b bearing along the x-axis and y-axis directions, They represent the dynamic meshing force and friction force between planetary gear a and planetary gear b, respectively. They represent the dynamic meshing force and friction force between the planetary gear b and the small sun gear respectively, Respectively represent the support stiffness of planetary gear b along the x-axis and y-axis directions, Respectively represent the support damping of planetary gear b along the x-axis and y-axis directions, They represent the torsional stiffness and torsional damping of the planetary gear b, represents the helix angle of planet gear b, represents the pitch circle radius of planetary gear b, represents the moment of inertia of planet gear b.

[0051] Small sun gear vibration differential equation:

[0052]

[0053] Where, represents the mass of the small sun gear, represents the pressure angle of the small sun gear, Respectively represent the vibration displacement of the small sun gear along the x-axis and y-axis directions, They represent the torsional displacement of the small sun gear, Respectively represent the vibration displacement of the small sun gear bearing along the x-axis and y-axis directions, They represent the dynamic meshing force and friction force between the small sun gear and the planet gear b, respectively. Respectively represent the support stiffness of the small sun gear along the x-axis and y-axis directions, Respectively represent the support damping of the small sun gear along the x-axis and y-axis directions, Respectively represent the torsional stiffness and torsional damping of the small sun gear, represents the helix angle of the small sun gear, Indicates the pitch circle radius of the small sun gear, is the moment of inertia of the small sun gear.

[0054] Step (5) selects different crack fault parameters of the Ravinia planetary gear and inputs the torque to solve the nonlinear response characteristics of the system; the global characteristic behavior of the Ravinia planetary gear transmission system is analyzed by the cell mapping method, revealing the global dynamic response of the system in the healthy state and the cracked state.

[0055] Compared with the existing technology, the beneficial effects of the present invention are as follows: the proposed nonlinear fault dynamics method of the Ravinia-type planetary gear transmission system can accurately reflect the dynamic characteristics of the system gear pair, improve the nonlinear dynamics theoretical system of the Ravinia-type planetary gear transmission system under crack faults, and provide support for vibration reduction, noise reduction and performance improvement of the Ravinia-type planetary gear transmission system. BRIEF DESCRIPTION OF THE DRAWINGS

[0056] Figure 1 It is a flow chart of the fault dynamics modeling method of the Lavinia planetary gear transmission system;

[0057] Figure 2 It is the meshing stiffness model of the Ravina planetary gear transmission system;

[0058] Figure 3 It is the nonlinear fault dynamic response characteristic of the Ravinia planetary gear transmission system. DETAILED DESCRIPTION

[0059] The embodiments of the present invention are described below with reference to the accompanying drawings. Figure 1-Figure 3 The specific embodiments of the present invention are described in detail.

[0060] like Figure 1 The figure shows a flow chart of the fault dynamics modeling method of the Lavinia planetary gear transmission system, which includes the following steps:

[0061] Step (1) establishing a calculation formula for the time-varying meshing stiffness of a Lavinia planetary gear pair with a crack fault; Figure 2 is the meshing stiffness model of the Ravinia planetary gear transmission system. The time-varying meshing stiffness of the gear pair with cracks can be expressed as:

[0062]

[0063] Where K ij (t) represents the time-varying meshing stiffness of the Lavinia planetary gear pair with cracks, K h Indicates the Hertzian stiffness of the gear pair, K a1i , K a2i Respectively represent the axial compression deformation stiffness of the driving and driven wheels, K B1i , K B2iRepresents the bending stiffness of the driving and driven wheels, K S1i , K S2i Represents the shear stiffness of the driving and driven wheels, K f1i , K f2i Represent the matrix stiffness of the driving and driven wheel gears respectively.

[0064] When a Lavinia planetary gear pair has a crack fault, the Hertz stiffness, gear matrix stiffness, and axial compression deformation stiffness of the gear pair are calculated according to the normal gear stiffness. The calculation formula is as follows:

[0065]

[0066] Where K h represents the Hertzian stiffness of the gear pair, V represents the Poisson's ratio of the gear teeth, E represents the elastic modulus of the gear teeth, and B represents the width of the gear teeth; K a1i represents the axial compression deformation stiffness of the driving wheel, a2 represents the upper half tooth angle of the driving wheel base circle, a represents the current meshing angle, a1 represents the angle corresponding to the contact point of each slice of the driving wheel, K f1i Indicates the base stiffness of the driving wheel gear, u f1 Indicates the distance from the meshing point to the intersection of the driving gear tooth root arc and the gear center line, S f1 Indicates the arc length of the driving gear tooth root, L * 、M * 、P * , Q * Represents the coefficients of each item, K a2i represents the axial compression deformation stiffness of the driven wheel, a4 represents the upper half tooth angle of the driven wheel base circle, a3 represents the angle corresponding to the contact point of each slice of the driven wheel, K f2i Indicates the rigidity of the driven wheel gear matrix, u f2 Indicates the distance from the meshing point to the intersection of the driven gear tooth root arc and the gear center line, S f2 Indicates the arc length of the driven gear tooth root.

[0067] When a Lavinia planetary gear has a crack fault, the bending stiffness and shear stiffness are affected by the crack. The calculation method is divided into two cases. The first case is that part of the meshing position is affected by the crack. The time-varying meshing stiffness of the gear pair can be expressed as:

[0068]

[0069] Where K B1i , K B2i They represent the bending stiffness of the cracked driving and driven wheels, a2 represents the upper half tooth angle of the driving wheel base circle, and a qrepresents the critical angle of single and double tooth meshing, a1 represents the angle corresponding to the contact point of each slice of the driving wheel, E represents the elastic modulus of the gear tooth, B represents the width of the gear tooth, V represents the Poisson's ratio of the gear tooth, q represents the crack depth, R b1 Indicates the base circle radius of the driving wheel crack gear, a L represents the crack angle, a represents the current meshing angle, K S1i , K S2i They represent the shear stiffness of the cracked master and driven wheel teeth, a4 represents the upper half tooth angle of the base circle of the driven wheel, a3 represents the angle corresponding to the contact point of each slice of the driven wheel, and R b2 Indicates the base circle radius of the cracked gear.

[0070] The second case is when all meshing positions are affected by cracks. The bending stiffness and shear stiffness are calculated as follows:

[0071]

[0072] Where K B1i , K B2i They represent the bending stiffness of the cracked driving and driven wheels, a2 represents the upper half tooth angle of the driving wheel base circle, and a q represents the critical angle of single and double tooth meshing, a1 represents the angle corresponding to the contact point of each slice of the driving wheel, E represents the elastic modulus of the gear tooth, B represents the width of the gear tooth, V represents the Poisson's ratio of the gear tooth, q represents the crack depth, R b1 Indicates the base circle radius of the driving wheel crack gear, a L represents the crack angle, a represents the current meshing angle, K S1i , K S2i They represent the shear stiffness of the cracked master and driven wheel teeth, a4 represents the upper half tooth angle of the base circle of the driven wheel, a3 represents the angle corresponding to the contact point of each slice of the driven wheel, and R b2 Indicates the base circle radius of the cracked gear.

[0073] Step (2) establishes the relative displacement formula of the Lavinia planetary gear pair with crack fault; the relative displacement of the first-stage large sun gear-planet gear a gear pair and the second-stage small sun gear-planet gear b gear pair is:

[0074]

[0075] Where x m1 、x m2 They represent the relative displacements of the first-stage large sun gear-planet gear a gear pair and the second-stage small sun gear-planet gear b gear pair, respectively. represents the large sun gear pressure angle, represents the pressure angle of planet gear b, Respectively represent the vibration displacement of planetary gear a along the x-axis and y-axis directions, They represent the torsional displacement of planet gear a, Respectively represent the vibration displacement of the large sun gear along the x-axis and y-axis directions, They represent the torsional displacement of the large sun gear, r a They represent the pitch circle radius of the large sun gear and planet gear a respectively, e1(t) represents the comprehensive error of the first-stage large sun gear-planet gear a gear pair, Respectively represent the vibration displacement of planetary gear b along the x-axis and y-axis directions, They represent the torsional displacement of planetary gear b, Respectively represent the vibration displacement of the small sun gear along the x-axis and y-axis directions, They represent the torsional displacement of the small sun gear, They represent the pitch circle radii of the small sun gear and planet gear b respectively, and e2(t) represents the comprehensive error of the second-stage small sun gear-planet gear b gear pair.

[0076] Step (3) considers the elastic lubrication condition, introduces the nonlinear function of the tooth side clearance, and determines the dynamic friction force on the meshing line of each gear pair; the calculation method of the dynamic friction force is:

[0077]

[0078] Where, F f represents dynamic friction, ν represents directional coefficient, μ(t) represents friction coefficient, C ij represents the meshing damping of each gear, f(x ml ,D) represents the gap function, K ij (t) represents the time-varying mesh stiffness of the Lavinia planetary gear pair with cracks, and D represents half of the tooth side clearance;

[0079] Step (4) establishes a set of vibration differential equations of a Ravina-type planetary gear transmission system with a crack fault; the Ravina-type planetary gear set is divided into two stages, the first stage consists of a large sun gear and a planet gear a, and the second stage consists of a small sun gear, a planet gear a and a planet gear b, and the two stages share a ring gear.

[0080] The large sun gear vibration differential equation is:

[0081]

[0082] Where, is the mass of the large sun gear, ω c is the theoretical angular velocity of the planet carrier, are the dynamic meshing force and friction force between the large sun gear and planet gear a, respectively. Respectively represent the vibration displacement of the large sun gear bearing along the x-axis and y-axis directions, are the support stiffness of the large sun gear along the x-axis and y-axis directions, are the support damping of the large sun gear along the x and y axis directions, are the torsional stiffness and torsional damping of the large sun gear, is the input torque of the large sun gear, is the large sun gear helix angle, is the moment of inertia of the large sun gear, and g is the acceleration due to gravity.

[0083] Vibration differential equation of planetary gear a:

[0084]

[0085] Where, represents the mass of planetary gear a, represents the pressure angle of the ring gear, represents the pressure angle of planet gear b, They represent the vibration displacement of the planetary gear a bearing along the x-axis and y-axis directions, Respectively represent the dynamic meshing force and friction force between the large sun gear and planetary gear a, They represent the dynamic meshing force and friction force between the ring gear and the planetary gear a, They represent the dynamic meshing force and friction force between the planetary gear b and the planetary gear a, respectively. Respectively represent the support stiffness of planetary gear a along the x-axis and y-axis directions, Respectively represent the support damping of planetary gear a along the x-axis and y-axis directions, Respectively represent the torsional stiffness and torsional damping of planetary gear a, represents the helix angle of planet gear a, represents the moment of inertia of planet gear a.

[0086] The differential equation of the large sun gear bearing vibration is:

[0087]

[0088] Where, Indicates the mass of the large sun gear bearing, Respectively represent the vibration displacement of the large sun gear bearing along the x-axis and y-axis directions, Represent the components of force on the large sun gear and the large sun gear bearing along the x-axis and y-axis directions respectively, They represent the support damping of the large sun gear bearing along the x-axis and y-axis directions respectively.

[0089] The differential equation of vibration of planetary gear a bearing is:

[0090]

[0091] Where, Indicates the mass of the planetary gear a bearing, are the components of the bearing force of the planetary gear a and the bearing along the x-axis and y-axis directions, are the support damping of the planetary gear a bearing along the x and y axis directions respectively.

[0092] Differential equation of ring gear vibration:

[0093]

[0094] Where m r represents the mass of the ring gear, x r 、y r Respectively represent the vibration displacement of the gear ring along the x-axis and y-axis directions, u r They represent the torsional displacement of the ring gear, They represent the dynamic meshing force and friction force between the ring gear and the planetary gear a, respectively, K rx , K ry Respectively represent the support stiffness of the gear ring along the x and y axis directions, C rx 、C ry Respectively represent the support damping of the gear ring along the x and y axis directions, K ru 、C ru Respectively represent the torsional stiffness and torsional damping of the ring gear, T r represents the output torque of the ring gear, β r represents the helix angle of the gear ring, r r Indicates the pitch circle radius of the gear ring, I r Indicates the moment of inertia of the ring gear.

[0095] Differential equation for planet carrier vibration:

[0096]

[0097] Where m c represents the mass of the planet carrier, ω c represents the theoretical angular velocity of the planet carrier, x c 、y c Respectively represent the vibration displacement of the planet carrier along the x-axis and y-axis, u c They represent the torsional displacement of the planet carrier, Respectively represent the vibration displacement of planetary gear a along the x-axis and y-axis directions, They represent the torsional displacement of planetary gear a, Respectively represent the vibration displacement of planetary gear b along the x-axis and y-axis directions, They represent the torsional displacement of planetary gear b, K cx , K cyRespectively represent the support stiffness of the planet carrier along the x and y axis directions, C cx 、C cy Respectively represent the support damping of the planet carrier along the x and y axis, K cu 、C cu They represent the torsional stiffness and torsional damping of the planet carrier respectively. Respectively represent the support stiffness of planetary gear a along the x-axis and y-axis directions, Respectively represent the support damping of planetary gear a along the x-axis and y-axis directions, Respectively represent the torsional stiffness and torsional damping of planetary gear a, Respectively represent the support stiffness of planetary gear b along the x-axis and y-axis directions, Respectively represent the support damping of planetary gear b along the x-axis and y-axis directions, They represent the torsional stiffness and torsional damping of the planetary gear b, r c Indicates the outer radius of the planet carrier, I c Indicates the moment of inertia of the planet carrier.

[0098] The differential equation of planetary gear b bearing vibration is:

[0099]

[0100] Where, represents the mass of the planetary gear b bearing, They represent the vibration displacement of the planetary gear b bearing along the x-axis and y-axis directions, Respectively represent the vibration displacement of planetary gear b along the x-axis and y-axis directions, Represent the components of force on the planetary gear b and the bearing along the x and y axes, respectively. Respectively represent the support stiffness of planetary gear b along the x-axis and y-axis directions, Respectively represent the support damping of planetary gear b along the x-axis and y-axis directions, They represent the support damping of the planetary gear b bearing along the x-axis and y-axis directions respectively.

[0101] Small sun gear bearing vibration differential equation:

[0102]

[0103] Where, Indicates the mass of the small sun gear bearing, Respectively represent the vibration displacement of the small sun gear bearing along the x-axis and y-axis directions, Respectively represent the vibration displacement of the small sun gear along the x-axis and y-axis directions, Respectively represent the component forces of the small sun gear and the bearing along the x-axis and y-axis directions, Respectively represent the support damping of the small sun gear bearing along the x-axis and y-axis directions, Respectively represent the support stiffness of the small sun gear along the x-axis and y-axis directions, Respectively represent the support damping of the small sun gear along the x-axis and y-axis directions.

[0104] Vibration differential equation of planetary gear b:

[0105]

[0106] Where, represents the mass of planet gear b, represents the pressure angle of planet gear b, Respectively represent the vibration displacement of planetary gear b along the x-axis and y-axis directions, They represent the torsional displacement of planet gear b, They represent the vibration displacement of the planetary gear b bearing along the x-axis and y-axis directions respectively, They represent the dynamic meshing force and friction force between planetary gear a and planetary gear b, respectively. They represent the dynamic meshing force and friction force between the planetary gear b and the small sun gear respectively, Respectively represent the support stiffness of planetary gear b along the x-axis and y-axis directions, Respectively represent the support damping of planetary gear b along the x-axis and y-axis directions, They represent the torsional stiffness and torsional damping of the planetary gear b, represents the helix angle of planet gear b, represents the pitch circle radius of planetary gear b, represents the moment of inertia of planet gear b.

[0107] Small sun gear vibration differential equation:

[0108]

[0109] Where, represents the mass of the small sun gear, represents the pressure angle of the small sun gear, Respectively represent the vibration displacement of the small sun gear along the x-axis and y-axis directions, They represent the torsional displacement of the small sun gear, Respectively represent the vibration displacement of the small sun gear bearing along the x-axis and y-axis directions, They represent the dynamic meshing force and friction force between the small sun gear and the planet gear b, respectively. Respectively represent the support stiffness of the small sun gear along the x-axis and y-axis directions, Respectively represent the support damping of the small sun gear along the x-axis and y-axis directions, Respectively represent the torsional stiffness and torsional damping of the small sun gear, represents the helix angle of the small sun gear, Indicates the pitch circle radius of the small sun gear, is the moment of inertia of the small sun gear.

[0110] Step (5) selects different crack fault parameters of the Ravinia planetary gear and inputs the torque to solve the nonlinear response characteristics of the system; the global characteristic behavior of the Ravinia planetary gear transmission system is analyzed by the cell mapping method, revealing the global dynamic response of the system in the healthy state and the cracked state.

[0111] Solve the vibration differential equations by Runge-Kutta integration method, Figure 3 It is the nonlinear fault dynamic response characteristic of the Ravina planetary gear transmission system;

[0112] Figure 3 (a) is the global dynamic response of the large sun-planet gear a gear pair, Figure 3 (b) is the global dynamic response of the large sun-planet gear a gear pair with cracks. Figure 3 Where P1 and PN represent single cycle and chaotic attractor. Figure 3 (a) shows two coexisting states, chaos and single cycle coexistence; and when there is a crack fault, Figure 3 As shown in (b), the original single-cycle state transforms into a chaotic state, indicating that the presence of the crack fault has caused the system's spatial domain to become unstable. This phenomenon has implications for engineering practice by deepening our understanding of Lavinia-type planetary gear transmission systems with crack faults. It provides engineers with a basis for optimizing fault diagnosis, stability analysis, performance improvement, and maintenance management, thereby ensuring the long-term stability and efficiency of the transmission system.

[0113] The above description is only a preferred embodiment of the invention and does not limit the invention in any way. Any modifications, changes and relative changes made to the above embodiments based on the essence of the present invention are still within the scope of protection of the technology of the present invention.

Claims

1. A fault dynamics modeling method for a Lavinia planetary gear transmission system, characterized in that: The following steps are involved: Step (1) establishes a calculation formula for the time-varying meshing stiffness of a Lavinia-type planetary gear pair with a crack fault; the time-varying meshing stiffness of a gear pair with a crack can be expressed as: Where K ij (t) represents the time-varying meshing stiffness of the Lavinia planetary gear pair with cracks, K h Indicates the Hertzian stiffness of the gear pair, K a1i , K a2i Respectively represent the axial compression deformation stiffness of the driving and driven wheels, K B1i , K B2i Represents the bending stiffness of the driving and driven wheels, K S1i , K S2i Represents the shear stiffness of the driving and driven wheels, K f1i , K f2i Respectively represent the matrix stiffness of the driving and driven gears; When a Lavinia planetary gear pair has a crack fault, the Hertz stiffness, gear matrix stiffness, and axial compression deformation stiffness of the gear pair are calculated according to the normal gear stiffness. The calculation formula is as follows: Where K h represents the Hertzian stiffness of the gear pair, V represents the Poisson's ratio of the gear teeth, E represents the elastic modulus of the gear teeth, and B represents the width of the gear teeth; K a1i represents the axial compression deformation stiffness of the driving wheel, a2 represents the upper half tooth angle of the driving wheel base circle, a represents the current meshing angle, a1 represents the angle corresponding to the contact point of each slice of the driving wheel, K f1i Indicates the base stiffness of the driving wheel gear, u f1 Indicates the distance from the meshing point to the intersection of the driving gear tooth root arc and the gear center line, S f1 Indicates the arc length of the driving gear tooth root, L * 、M * 、P * , Q * Represents the coefficients of each item, K a2i represents the axial compression deformation stiffness of the driven wheel, a4 represents the upper half tooth angle of the driven wheel base circle, a3 represents the angle corresponding to the contact point of each slice of the driven wheel, K f2i Indicates the rigidity of the driven wheel gear matrix, u f2 Indicates the distance from the meshing point to the intersection of the driven gear tooth root arc and the gear center line, S f2 Indicates the arc length of the driven gear tooth root; When a Lavinia planetary gear has a crack fault, the bending stiffness and shear stiffness are affected by the crack. The calculation method is divided into two cases. The first case is that part of the meshing position is affected by the crack. The time-varying meshing stiffness of the gear pair can be expressed as: Where K B1i , K B2i They represent the bending stiffness of the cracked driving and driven wheels, a2 represents the upper half tooth angle of the driving wheel base circle, and a q represents the critical angle of single and double tooth meshing, a1 represents the angle corresponding to the contact point of each slice of the driving wheel, E represents the elastic modulus of the gear tooth, B represents the width of the gear tooth, V represents the Poisson's ratio of the gear tooth, q represents the crack depth, R b1 Indicates the base circle radius of the driving wheel crack gear, a L represents the crack angle, a represents the current meshing angle, K S1i , K S2i They represent the shear stiffness of the cracked master and driven wheel teeth, a4 represents the upper half tooth angle of the base circle of the driven wheel, a3 represents the angle corresponding to the contact point of each slice of the driven wheel, and R b2 Indicates the base circle radius of the cracked gear; The second case is when all meshing positions are affected by cracks. The bending stiffness and shear stiffness are calculated as follows: Where K B1i , K B2i They represent the bending stiffness of the cracked driving and driven wheels, a2 represents the upper half tooth angle of the driving wheel base circle, and a q represents the critical angle of single and double tooth meshing, a1 represents the angle corresponding to the contact point of each slice of the driving wheel, E represents the elastic modulus of the gear tooth, B represents the width of the gear tooth, V represents the Poisson's ratio of the gear tooth, q represents the crack depth, R b1 Indicates the base circle radius of the driving wheel crack gear, a L represents the crack angle, a represents the current meshing angle, K S1i , K S2i They represent the shear stiffness of the cracked master and driven wheel teeth, a4 represents the upper half tooth angle of the base circle of the driven wheel, a3 represents the angle corresponding to the contact point of each slice of the driven wheel, and R b2 Indicates the base circle radius of the cracked gear; Step (2) establishes the relative displacement formula of the Lavinia planetary gear pair with crack fault; the relative displacement of the first-stage large sun gear-planet gear a gear pair and the second-stage small sun gear-planet gear b gear pair is: Where x m1 、x m2 They represent the relative displacements of the first-stage large sun gear-planet gear a gear pair and the second-stage small sun gear-planet gear b gear pair, respectively. represents the large sun gear pressure angle, represents the pressure angle of planet gear b, Respectively represent the vibration displacement of planetary gear a along the x-axis and y-axis directions, They represent the torsional displacement of planetary gear a, Respectively represent the vibration displacement of the large sun gear along the x-axis and y-axis directions, They represent the torsional displacement of the large sun gear, r a They represent the pitch circle radius of the large sun gear and planet gear a respectively, e1(t) represents the comprehensive error of the first-stage large sun gear-planet gear a gear pair, Respectively represent the vibration displacement of planetary gear b along the x-axis and y-axis directions, They represent the torsional displacement of planetary gear b, Respectively represent the vibration displacement of the small sun gear along the x-axis and y-axis directions, They represent the torsional displacement of the small sun gear, They represent the pitch circle radii of the small sun gear and planet gear b respectively, and e2(t) represents the comprehensive error of the second-stage small sun gear-planet gear b gear pair; Step (3) considers the elastic lubrication condition, introduces the nonlinear function of the tooth side clearance, and determines the dynamic friction force on the meshing line of each gear pair; the calculation method of the dynamic friction force is: Where, F f represents dynamic friction, ν represents directional coefficient, μ(t) represents friction coefficient, C ij represents the meshing damping of each gear, f(x ml ,D) represents the gap function, K ij (t) represents the time-varying mesh stiffness of the Lavinia planetary gear pair with cracks, and D represents half of the tooth side clearance; Step (4) establishes a set of vibration differential equations for the Lavinia planetary gear transmission system with crack fault; the vibration differential equation of the large sun gear is: Where, is the mass of the large sun gear, ω c is the theoretical angular velocity of the planet carrier, are the dynamic meshing force and friction force between the large sun gear and planet gear a, respectively. Respectively represent the vibration displacement of the large sun gear bearing along the x-axis and y-axis directions, are the support stiffness of the large sun gear along the x-axis and y-axis directions, are the support damping of the large sun gear along the x and y axis directions, are the torsional stiffness and torsional damping of the large sun gear, is the input torque of the large sun gear, is the large sun gear helix angle, is the moment of inertia of the large sun gear, g is the acceleration due to gravity; Vibration differential equation of planetary gear a: Where, represents the mass of planetary gear a, represents the pressure angle of the ring gear, represents the pressure angle of planet gear b, They represent the vibration displacement of the planetary gear a bearing along the x-axis and y-axis directions, Respectively represent the dynamic meshing force and friction force between the large sun gear and planetary gear a, They represent the dynamic meshing force and friction force between the ring gear and the planetary gear a, They represent the dynamic meshing force and friction force between the planetary gear b and the planetary gear a, respectively. Respectively represent the support stiffness of planetary gear a along the x-axis and y-axis directions, Respectively represent the support damping of planetary gear a along the x-axis and y-axis directions, Respectively represent the torsional stiffness and torsional damping of planetary gear a, represents the helix angle of planet gear a, represents the moment of inertia of planet gear a; The differential equation of the large sun gear bearing vibration is: Where, Indicates the mass of the large sun gear bearing, Respectively represent the vibration displacement of the large sun gear bearing along the x-axis and y-axis directions, Represent the components of force on the large sun gear and the large sun gear bearing along the x-axis and y-axis directions respectively, Respectively represent the support damping of the large sun gear bearing along the x-axis and y-axis directions; The differential equation of vibration of planetary gear a bearing is: Where, Indicates the mass of the planetary gear a bearing, are the components of the bearing force of the planetary gear a and the bearing along the x-axis and y-axis directions, are the support damping of the planetary gear a bearing along the x-axis and y-axis directions respectively; Differential equation of ring gear vibration: Where m r represents the mass of the ring gear, x r 、y r Respectively represent the vibration displacement of the gear ring along the x-axis and y-axis directions, u r They represent the torsional displacement of the ring gear, They represent the dynamic meshing force and friction force between the ring gear and the planetary gear a, respectively, and K rx , K ry Respectively represent the support stiffness of the gear ring along the x-axis and y-axis directions, C rx 、C ry Respectively represent the support damping of the gear ring along the x and y axis directions, K ru 、C ru Respectively represent the torsional stiffness and torsional damping of the ring gear, T r represents the output torque of the ring gear, β r represents the helix angle of the gear ring, r r Indicates the pitch circle radius of the gear ring, I r represents the moment of inertia of the ring gear; Differential equation for planet carrier vibration: Where m c represents the mass of the planet carrier, ω c represents the theoretical angular velocity of the planet carrier, x c 、y c Respectively represent the vibration displacement of the planet carrier along the x-axis and y-axis, u c They represent the torsional displacement of the planet carrier, Respectively represent the vibration displacement of planetary gear a along the x-axis and y-axis directions, They represent the torsional displacement of planetary gear a, Respectively represent the vibration displacement of planetary gear b along the x-axis and y-axis directions, They represent the torsional displacement of planetary gear b, K cx , K cy Respectively represent the support stiffness of the planet carrier along the x and y axis directions, C cx 、C cy Respectively represent the support damping of the planet carrier along the x and y axis directions, K cu 、C cu They represent the torsional stiffness and torsional damping of the planet carrier respectively. Respectively represent the support stiffness of planetary gear a along the x-axis and y-axis directions, Respectively represent the support damping of planetary gear a along the x-axis and y-axis directions, Respectively represent the torsional stiffness and torsional damping of planetary gear a, Respectively represent the support stiffness of planetary gear b along the x-axis and y-axis directions, Respectively represent the support damping of planetary gear b along the x-axis and y-axis directions, They represent the torsional stiffness and torsional damping of the planetary gear b, r c Indicates the outer radius of the planet carrier, I c represents the moment of inertia of the planet carrier; The differential equation of vibration of planetary gear b bearing is: Where, represents the mass of the planetary gear b bearing, They represent the vibration displacement of the planetary gear b bearing along the x-axis and y-axis directions, Respectively represent the vibration displacement of planetary gear b along the x-axis and y-axis directions, Represent the components of force on the planetary gear b and the bearing along the x and y axes, respectively. Respectively represent the support stiffness of planetary gear b along the x-axis and y-axis directions, Respectively represent the support damping of planetary gear b along the x-axis and y-axis directions, They represent the support damping of the planetary gear b bearing along the x-axis and y-axis directions respectively; Small sun gear bearing vibration differential equation: Where, Indicates the mass of the small sun gear bearing, Respectively represent the vibration displacement of the small sun gear bearing along the x-axis and y-axis directions, Respectively represent the vibration displacement of the small sun gear along the x-axis and y-axis directions, Respectively represent the component forces of the small sun gear and the bearing along the x-axis and y-axis directions, Respectively represent the support damping of the small sun gear bearing along the x-axis and y-axis directions, Respectively represent the support stiffness of the small sun gear along the x-axis and y-axis directions, Respectively represent the support damping of the small sun gear along the x-axis and y-axis directions; Vibration differential equation of planetary gear b: Where, represents the mass of planet gear b, represents the pressure angle of planet gear b, Respectively represent the vibration displacement of planetary gear b along the x-axis and y-axis directions, They represent the torsional displacement of planetary gear b, They represent the vibration displacement of the planetary gear b bearing along the x-axis and y-axis directions respectively, They represent the dynamic meshing force and friction force between planetary gear a and planetary gear b, respectively. They represent the dynamic meshing force and friction force between the planetary gear b and the small sun gear respectively, Respectively represent the support stiffness of planetary gear b along the x-axis and y-axis directions, Respectively represent the support damping of planetary gear b along the x-axis and y-axis directions, They represent the torsional stiffness and torsional damping of the planetary gear b, represents the helix angle of planet gear b, represents the pitch circle radius of planetary gear b, represents the moment of inertia of planet gear b; Small sun gear vibration differential equation: Where, represents the mass of the small sun gear, represents the pressure angle of the small sun gear, Respectively represent the vibration displacement of the small sun gear along the x-axis and y-axis directions, They represent the torsional displacement of the small sun gear, Respectively represent the vibration displacement of the small sun gear bearing along the x-axis and y-axis directions, They represent the dynamic meshing force and friction force between the small sun gear and the planet gear b, respectively. Respectively represent the support stiffness of the small sun gear along the x-axis and y-axis directions, Respectively represent the support damping of the small sun gear along the x-axis and y-axis directions, Respectively represent the torsional stiffness and torsional damping of the small sun gear, represents the helix angle of the small sun gear, Indicates the pitch circle radius of the small sun gear, is the moment of inertia of the small sun gear; Step (5) selects different crack fault parameters of the Ravinia planetary gear and inputs the torque to solve the nonlinear response characteristics of the system; the global characteristic behavior of the Ravinia planetary gear transmission system is analyzed by the cell mapping method, revealing the global dynamic response of the system in the healthy state and the cracked state.

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