Planetary gear transmission system pitting corrosion fault dynamics modeling method

Abstract

The invention discloses a kinetic modeling method for a pitting corrosion fault of a planetary gear transmission system. The method comprises the following steps: step (1), calculating time-varying meshing stiffness of a planetary gear under different pitting corrosion degrees based on a potential energy method; (2) introducing a gear backlash nonlinear function and a comprehensive meshing error, and determining relative displacement on a meshing line of each stage of gear pair; (3) calculating the time-varying friction force among all stages of gear pairs by utilizing a coulomb model; (4) establishing a planetary gear transmission system pitting corrosion fault dynamic differential equation set; (5) solving the nonlinear dynamic characteristic response of the system; the method has the beneficial effects that the time-varying friction force and the gear tooth pitting corrosion fault are considered, the nonlinear dynamic response characteristics of the planetary gear transmission system are solved, and the related technical blank of nonlinear dynamic modeling of the pitting corrosion fault of the current planetary gear transmission system is filled.

Description

Technical Field

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

[0002] Planetary gear transmissions play a crucial role in modern mechanical transmissions. Their advantages include compact structure, high transmission ratio, high transmission efficiency, and strong load-bearing capacity, making them widely used in robotics, the automotive industry, and aerospace. However, during long-term operation, planetary gears are susceptible to pitting corrosion due to various complex operating conditions such as high loads, frequent starts and stops, and poor lubrication. Under the influence of nonlinear tooth backlash, meshing error excitation, and time-varying friction, the system's vibration displacement can exhibit bifurcation and chaos when the external excitation frequency changes, increasing the probability of failure and safety accidents during mechanical operation. Therefore, studying the impact of planetary gear pitting on the dynamic characteristics of planetary gears, life prediction, and planetary gear optimization is of great significance.

[0003] To address these issues, the present invention proposes a method for modeling the dynamics of pitting failures in planetary gear transmissions. This method first uses the potential energy method to calculate the time-varying mesh stiffness of the planetary gears at different pitting levels. It then introduces a nonlinear function for tooth side clearance and a comprehensive transmission error to calculate the time-varying meshing force on each meshing line. Furthermore, a time-varying friction model for each gear pair at each stage of the planetary gear is established. Subsequently, a fault dynamics equation for the planetary gear transmission is established, solved using the fourth-order Runge-Kutta method, and its dynamic characteristics analyzed. This modeling method can fill a gap in related international technologies, promote engineering development, and generate significant socioeconomic benefits. Summary of the Invention

[0004] In order to overcome the deficiencies of the existing technology and fill the gaps in the relevant technology, the present invention provides a method for dynamic modeling of pitting failures in a planetary gear transmission system.

[0005] The present invention solves the technical problem by adopting the following technical solution: A method for dynamic modeling of pitting failure of a planetary gear transmission system, characterized by comprising the following steps:

[0006] Step (1): Calculate the time-varying meshing stiffness of the planetary gear under different pitting degrees based on the potential energy method;

[0007] A cylinder was used to simulate the pitting pits at the pitch line. 10, 15, and 20 pitting pits were defined as mild pitting, moderate pitting, and severe pitting. The potential energy method was used to calculate the time-varying meshing stiffness of the planetary gear under pitting conditions and the reduction in contact tooth width ΔL. y , the reduction in cross-sectional area ΔA y , the reduction in moment of inertia ΔIy , the calculation formula is as follows:

[0008]

[0009] Among them, δ is the pitting depth, μ is the distance from the pitting center to the tooth root, r is the pitting radius, h is the pitting depth, y For half tooth thickness, A y is the cross-sectional area of the gear teeth, y is the distance from the slice to the tooth root circle;

[0010] When there is pitting failure in the planetary gear, the Hertz contact stiffness k H , shear stiffness k S , bending stiffness k B , and axial compressive stiffness k A The calculation formula is as follows:

[0011]

[0012]

[0013] Among them, α a is the meshing pressure angle of the driving wheel, α b is the meshing pressure angle of the driven wheel, v is Poisson's ratio, E is Young's modulus, L is tooth width, R b Expressed as base circle radius, m is the number of pits, ΔL y,z is the reduction in contact tooth width caused by the zth pit, ΔA y,z is the reduction in cross-sectional area caused by the zth pit, ΔI y,z is the reduction in the moment of inertia caused by the zth pit, and α is the equivalent engagement angle;

[0014] Meshing stiffness of single tooth in meshing pair k t1 and the meshing stiffness k of the double teeth of the meshing pair t2 The calculation formula is as follows:

[0015]

[0016] Among them, k H is the Hertzian contact stiffness, k Bp 、k Bg are the bending stiffness of the driving wheel and the driven wheel, k Sp 、k Sg are the shear stiffness of the driving wheel and the driven wheel, k Ap 、k Ag are the axial compression stiffness of the driving wheel and the driven wheel, k Fp 、k Fg are the base stiffness of the driving wheel and the driven wheel, k H,j is the Hertzian contact stiffness of the j-th pair of teeth, kBp,j 、k Bg,j are the bending stiffness of the jth pair of teeth of the driving wheel and the driven wheel, k Sp,j 、k Sg,j are the shear stiffness of the jth pair of teeth of the driving wheel and the driven wheel, k Ap,j 、k Ag,j are the axial compression stiffness of the jth pair of teeth of the driving wheel and the driven wheel, k Fp,j 、k Fg,j are the base stiffness of the j-th pair of teeth of the driving wheel and the driven wheel respectively;

[0017] Step (2): Introduce the nonlinear function of tooth side clearance and the comprehensive meshing error to determine the relative displacement on the meshing line of each gear pair;

[0018] Relative displacement δ of the sun gear-planet gear pair along the normal direction of the meshing point spn Relative displacement δ along the normal direction of the meshing point with the ring gear-planet gear pair rpn The equations can be expressed as:

[0019]

[0020] in, and Respectively represent the effective meshing angles of the sun gear, the inner ring gear and the nth planet gear, u s 、u pn and u r They represent the torsional displacements of the sun gear, the nth planetary gear, and the inner ring gear caused by the system vibration, respectively. s 、x pn and x r They are respectively represented as the vibration displacement of the sun gear, the nth planet gear and the inner ring gear along the x-axis, and y s 、y pn and y r They are respectively represented as the vibration displacement of the sun gear, the nth planet gear and the inner ring gear along the y-axis, e spn (t) is the comprehensive meshing error of the sun-planet gear pair, e rpn (t) is the comprehensive meshing error of the ring gear-planetary gear pair. It is usually assumed that the comprehensive meshing error e i (t) changes according to the sinusoidal law:

[0021] e i (t) = E i sin(ω m t+φ i )(i=spn,rpn);

[0022] Among them, ω m Expressed as meshing frequency, E iRepresents the amplitude of the comprehensive meshing error of meshing pair i, φ i represents the initial phase of meshing pair i, t is the gear meshing time, spn and rpn are the sun-planet gear pair and the ring gear-planet gear pair respectively;

[0023] The projection of the displacement of the planet carrier relative to the planet gear along the x-axis, y-axis and tangent of the planet carrier:

[0024]

[0025] Among them, δ cpnx is the projection of the planet carrier relative to the planet gear along the x-axis of the planet carrier, δ cpny is the projection of the displacement of the planet carrier relative to the planet gear along the y-axis of the planet carrier, δ cpnu is the projection of the displacement of the planet carrier relative to the planet gear along the tangent line of the planet carrier, x c 、x pn They are respectively represented as the vibration displacement of the planet carrier and the nth planet gear along the x-axis, c 、y pn They are respectively represented as the vibration displacement of the planet carrier and the nth planet gear along the y-axis, u c is the torsional displacement of the planet carrier caused by system vibration, ψ pn is the angle between the line from the theoretical center of the nth planet gear to the theoretical center of the planet carrier and the positive direction of the x-axis;

[0026] Step (3): Use the Coulomb model to calculate the time-varying friction between the gear pairs at each level. The calculation method is as follows:

[0027] F f =λu|F i |(i=spn,rpn);

[0028] Among them, F f is the time-varying friction force, u is the sliding friction coefficient, λ is the direction coefficient of the friction force, F i is the meshing force of gear pair i, and the meshing force of each gear pair is calculated as follows:

[0029]

[0030] Where, k(t) is the time-varying mesh stiffness, C(t) is the time-varying mesh damping, and b m is half of the tooth side clearance, δ i is the relative displacement of gear pair i, f(δ i ,b m ) is a nonlinear function of the tooth side clearance, expressed as a piecewise function:

[0031]

[0032] Friction arm of the driving wheel l pg (t) and the friction arm l of the driven wheel gp The calculation formula of (t) is as follows:

[0033]

[0034] Among them, α0 is the pressure angle of the driving wheel, r pa 、r ga are the tooth top circle radius of the driving wheel and the driven wheel, r p 、r g are the base circle radii of the driving wheel and the driven wheel, r pb 、r gb are the root circle radius of the driving wheel and the driven wheel respectively, p is the base circle pitch, ω pc is the angular velocity of the driving wheel in the moving coordinate system, t is the gear meshing time, and mod is the remainder function;

[0035] Since the direction of the friction force changes with the relative speed during the meshing process of the gears, the direction coefficient λ of the friction force is a time-varying parameter, and its expression is as follows:

[0036]

[0037] Among them, ω cp 、ω cg are the angular velocities of the driving wheel and the driven wheel in the moving coordinate system, l pg (t) is the friction arm of the driving wheel, l gp (t) is the friction arm of the driven wheel;

[0038] The sliding friction coefficient can be calculated according to the empirical formula, as follows:

[0039]

[0040] Where u is the sliding friction coefficient, V is the relative sliding velocity, ω p is the driving wheel angular velocity, ω g is the angular velocity of the driven wheel, l pg (t) is the friction arm of the driving wheel, l gp (t) is the friction arm of the driven wheel;

[0041] Step (4): Establish a set of differential equations for the dynamics of pitting failure in the planetary gear transmission system;

[0042] Time-varying friction, nonlinear function of tooth side clearance, and time-varying meshing stiffness under the condition of tooth pitting fault are introduced to establish a set of nonlinear dynamic differential equations of the system. In the process of establishing the system vibration differential equations, the following degrees of freedom are considered:

[0043] X={x s ,y s ,u s ,x r ,y r ,u s ,x c ,y c ,u c ,x p1 ,y p1 ,u p1 ,x p2 ,y p2 ,u p2 ,x p3 ,y p3 ,u p3};

[0044] Among them, x s 、x r 、x c 、x pn (n=1,2,3) are the vibration displacements of the sun gear, the inner ring gear, the planet carrier, and the nth planet gear in the x-axis direction, s 、y r 、y c 、y pn (n=1,2,3) are the vibration displacements of the sun gear, the inner ring gear, the planet carrier, and the nth planet gear in the y-axis direction, u s 、u r 、u c 、u pn (n=1, 2, 3) are the torsional displacements of the sun gear, the ring gear, the planet carrier, and the nth planet gear respectively;

[0045] According to the above steps, the vibration differential equations of the planetary gear transmission system are derived;

[0046] Sun gear vibration differential equation:

[0047]

[0048] Among them, x s is the vibration displacement of the sun gear in the x-axis direction, y s is the vibration displacement of the sun gear in the y-axis direction, u s is the torsional displacement of the sun gear, m s is the mass of the sun gear, k s is the support stiffness of the sun gear, c s is the support damping of the sun gear, F spn is the meshing force of the sun-planet gear pair, F fspn is the friction force of the sun-planet gear pair, w c is the planet carrier angular velocity, is the effective meshing angle between the sun gear and the nth planet gear, l spn is the friction arm of the sun gear, r s is the sun gear base circle radius, T in is the input torque;

[0049] Differential equation of internal gear ring vibration:

[0050]

[0051] Among them, x r is the vibration displacement of the inner gear ring in the x-axis direction, y r is the vibration displacement of the inner gear ring in the y-axis direction, u r is the torsional displacement of the inner ring gear, m r is the mass of the inner gear ring, k r is the support stiffness of the inner ring gear, c r is the support damping of the inner ring gear, F rpn is the meshing force of the ring gear-planetary gear pair, F frpn is the friction force of the ring gear-planetary gear pair, w c is the planet carrier angular velocity, is the effective meshing angle between the inner gear ring and the nth planetary gear, l rpn is the friction arm of the inner gear ring, r r is the base circle radius of the inner gear ring;

[0052] Differential equation for planet carrier vibration:

[0053]

[0054] Among them, x c is the vibration displacement of the planet carrier in the x-axis direction, y c is the vibration displacement of the planet carrier in the y-axis direction, u c is the torsional displacement of the planet carrier, m c is the mass of the planet carrier, k c is the support stiffness of the planet carrier, k pn is the support stiffness of the nth planetary gear, c c is the support damping of the planet carrier, w c is the planet carrier angular velocity, r c is the radius of the planetary gear center distribution circle, T out is the output torque, δ cpnx is the projection of the displacement of the planet carrier relative to the planet gear along the x-axis, δ cpny is the projection of the displacement of the planet carrier relative to the planet gear along the y-axis, δ cpnu is the projection of the displacement of the planet wheel relative to the planet carrier along the tangent line of the planet carrier;

[0055] Differential equation of planetary gear vibration:

[0056]

[0057] Among them, x pn is the vibration displacement of the nth planetary gear in the x-axis direction, y pn is the vibration displacement of the nth planetary gear in the y-axis direction, u pn is the torsional displacement of the nth planetary gear, m pn is the mass of the nth planetary gear, k pn is the support stiffness of the nth planetary gear, c pn is the support damping of the nth planetary gear, F spn is the meshing force of the sun-planet gear pair, F rpn is the meshing force of the ring gear-planetary gear pair, F fspn is the friction force of the sun-planet gear pair, F frpn is the friction force of the ring gear-planetary gear pair, w c is the planet carrier angular velocity, is the effective meshing angle between the sun gear and the planet gear, is the effective meshing angle between the inner gear ring and the nth planetary gear, l spn is the friction arm of the sun gear, l rpn is the friction arm of the inner ring gear, r s is the sun gear base circle radius, r r is the base circle radius of the inner gear ring;

[0058] Step (5): solve the nonlinear response characteristics of the system;

[0059] The external excitation frequency parameters under different pitting corrosion degrees are substituted into the nonlinear vibration differential equations of the system to solve the vibration responses of each gear pair under different external excitation conditions. By comparing the vibration characteristics of the planetary gears under different pitting corrosion degrees, the influence of the external excitation frequency on the dynamic response of the system is revealed, and the influence of pitting corrosion faults on the vibration characteristics of the planetary gear transmission system is characterized. BRIEF DESCRIPTION OF THE DRAWINGS

[0060] Figure 1 It is a flow chart of the nonlinear dynamic modeling method for pitting failure of planetary gear transmission system;

[0061] Figure 2 It is the meshing stiffness analysis model for pitting failure;

[0062] Figure 3 It is the time-varying friction model of the planetary gear transmission system;

[0063] Figure 4 It is the nonlinear dynamic model of the planetary gear transmission system;

[0064] Figure 5 It is the bifurcation diagram of the planetary gear transmission system under different pitting degrees. DETAILED DESCRIPTION

[0065] The embodiments of the present invention are described below with reference to the accompanying drawings. Figure 1-Figure 5 The specific embodiments of the present invention are described in detail.

[0066] Reference Figure 1 The figure shows a flow chart of the nonlinear dynamic modeling method for pitting failure of a planetary gear transmission system, which includes the following steps:

[0067] Step (1): Calculate the time-varying meshing stiffness of the planetary gear under different pitting degrees based on the potential energy method. Figure 2 It is the meshing stiffness analysis model of pitting failure. As can be seen from the figure, the presence of pitting on the gear teeth leads to the contact tooth width L and the moment of inertia I y and effective cross-sectional area A y Reduce, the reduction in contact tooth width ΔL y , the reduction in cross-sectional area ΔA y and the reduction in moment of inertia ΔI y The calculation formula is as follows:

[0068]

[0069] Among them, δ is the pitting depth, μ is the distance from the pitting center to the tooth root, r is the pitting radius, h is the pitting depth, y For half tooth thickness, A y is the cross-sectional area of the gear teeth, y is the distance from the slice to the tooth root circle;

[0070] When there is pitting failure in the planetary gear, the Hertz contact stiffness k H , shear stiffness k S , bending stiffness k B , and axial compressive stiffness k A The calculation formula is as follows:

[0071]

[0072] Among them, α a is the meshing pressure angle of the driving wheel, α b is the meshing pressure angle of the driven wheel, v is Poisson's ratio, E is Young's modulus, L is tooth width, R b Expressed as base circle radius, m is the number of pits, ΔL y,z is the reduction in contact tooth width caused by the zth pit, ΔA y,z is the reduction in cross-sectional area caused by the zth pit, ΔI y,z is the reduction in the moment of inertia caused by the zth pit, and α is the equivalent engagement angle;

[0073] Meshing stiffness of single tooth in meshing pair k t1 and the meshing stiffness k of the double teeth of the meshing pair t2 The calculation formula is as follows:

[0074]

[0075] Among them, k H is the Hertzian contact stiffness, k Bp 、k Bg are the bending stiffness of the driving wheel and the driven wheel, k Sp 、k Sg are the shear stiffness of the driving wheel and the driven wheel, k Ap 、k Ag are the axial compression stiffness of the driving wheel and the driven wheel, k Fp 、k Fg are the base stiffness of the driving wheel and the driven wheel, k H,j is the Hertzian contact stiffness of the j-th pair of teeth, k Bp,j 、k Bg,j are the bending stiffness of the jth pair of teeth of the driving wheel and the driven wheel, k Sp,j 、k Sg,j are the shear stiffness of the jth pair of teeth of the driving wheel and the driven wheel, k Ap,j 、k Ag,j are the axial compression stiffness of the jth pair of teeth of the driving wheel and the driven wheel, k Fp,j 、k Fg,j are the base stiffness of the j-th pair of teeth of the driving wheel and the driven wheel respectively;

[0076] Step (2): Introduce the nonlinear function of tooth side clearance and the comprehensive meshing error to determine the relative displacement on the meshing line of each gear pair;

[0077] Relative displacement δ of the sun gear-planet gear pair along the normal direction of the meshing point spn Relative displacement δ along the normal direction of the meshing point with the ring gear-planet gear pair rpn The equations can be expressed as:

[0078]

[0079] in, and Respectively represent the effective meshing angles of the sun gear, the inner ring gear and the nth planet gear, u s 、u pn and u r They represent the torsional displacements of the sun gear, the nth planetary gear, and the inner ring gear caused by the system vibration, respectively. s 、x pn and x rThey are respectively represented as the vibration displacement of the sun gear, the nth planet gear and the inner ring gear along the x-axis, and y s 、y pn and y r They are respectively represented as the vibration displacement of the sun gear, the nth planet gear and the inner ring gear along the y-axis, e spn (t) is the comprehensive meshing error of the sun-planet gear pair, e rpn (t) is the comprehensive meshing error of the ring gear-planetary gear pair. It is usually assumed that the comprehensive meshing error e i (t) changes according to the sinusoidal law:

[0080] e i (t) = E i sin(ω m t+φ i )(i=spn,rpn);

[0081] Among them, ω m Expressed as meshing frequency, E i Represents the amplitude of the comprehensive meshing error of meshing pair i, φ i represents the initial phase of meshing pair i, and t is the gear meshing time;

[0082] The projection of the displacement of the planet carrier relative to the planet gear along the x-axis, y-axis and tangent of the planet carrier:

[0083]

[0084] Among them, δ cpnx is the projection of the planet carrier relative to the planet gear along the x-axis of the planet carrier, δ cpny is the projection of the displacement of the planet carrier relative to the planet gear along the y-axis of the planet carrier, δ cpnu is the projection of the displacement of the planet carrier relative to the planet gear along the tangent line of the planet carrier, x c 、x pn They are respectively represented as the vibration displacement of the planet carrier and the nth planet gear along the x-axis, c 、y pn They are respectively represented as the vibration displacement of the planet carrier and the nth planet gear along the y-axis, u c is the torsional displacement of the planet carrier caused by system vibration, ψ pn is the angle between the line from the theoretical center of the nth planet gear to the theoretical center of the planet carrier and the positive direction of the x-axis;

[0085] Step (3): Use the Coulomb model to calculate the time-varying friction between the gear pairs at each level. Figure 3To construct a time-varying friction calculation model for a planetary gear transmission system using Coulomb friction theory, a dynamic meshing force model, including the sun gear-planet gear and planet gear-ring gear meshing pairs, was first established through force analysis of the planetary gear transmission system. Secondly, a dynamic analytical model of the friction arm was established based on the geometric analysis of the relative motion trajectory of the meshing points.

[0086] Using Coulomb friction theory, we can get the calculation formula of time-varying friction force:

[0087] F f =λu|F i |(i=spn,rpn);

[0088] Among them, F f is the time-varying friction force, u is the sliding friction coefficient, λ is the direction coefficient of the friction force, F i is the meshing force of gear pair i, and the meshing force of each gear pair is calculated as follows:

[0089]

[0090] Where, k(t) is the time-varying mesh stiffness, C(t) is the time-varying mesh damping, and b m is half of the tooth side clearance, δ i is the relative displacement of gear pair i, f(δ i ,b m ) is a nonlinear function of the tooth side clearance, expressed as a piecewise function:

[0091]

[0092] Friction arm of the driving wheel l pg (t) and the friction arm l of the driven wheel gp The calculation formula of (t) is as follows:

[0093]

[0094] Among them, α0 is the pressure angle of the driving wheel, r pa 、r ga are the tooth top circle radius of the driving wheel and the driven wheel, r p 、r g are the base circle radii of the driving wheel and the driven wheel, r pb 、r gb are the root circle radius of the driving wheel and the driven wheel respectively, p is the base circle pitch, ω pc is the angular velocity of the driving wheel in the moving coordinate system, t is the gear meshing time, and mod is the remainder function;

[0095] The sliding friction coefficient can be calculated according to the empirical formula, which is as follows:

[0096]

[0097] Where u is the sliding friction coefficient, V is the relative sliding velocity, ω p is the driving wheel angular velocity, ω g is the angular velocity of the driven wheel, l pg (t) is the friction arm of the driving wheel, l gp (t) is the friction arm of the driven wheel;

[0098] Step (4): Establish a set of differential equations for the pitting fault dynamics of the planetary gear transmission system, introduce the time-varying friction force, the nonlinear function of the tooth side clearance and the time-varying meshing stiffness under the state of the tooth pitting fault, and establish a set of nonlinear dynamic differential equations for the system. Figure 4 The nonlinear dynamic model of the planetary gear transmission system is shown, considering the following degrees of freedom:

[0099] X={x s ,y s ,u s ,x r ,y r ,u s ,x c ,y c ,u c ,x p1 ,y p1 ,u p1 ,x p2 ,y p2 ,u p2 ,x p3 ,y p3 ,u p3};

[0100] Among them, x s 、x r 、x c 、x pn (n=1,2,3) are the vibration displacements of the sun gear, the inner ring gear, the planet carrier, and the nth planet gear in the x-axis direction, s 、y r 、y c 、y pn (n=1,2,3) are the vibration displacements of the sun gear, the inner ring gear, the planet carrier, and the nth planet gear in the y-axis direction, u s 、u r 、u c 、u pn (n=1, 2, 3) are the torsional displacements of the sun gear, the ring gear, the planet carrier, and the nth planet gear respectively;

[0101] According to the above steps, the vibration differential equations of the planetary gear transmission system are derived;

[0102] Sun gear vibration differential equation:

[0103]

[0104] Among them, x s is the vibration displacement of the sun gear in the x-axis direction, y s is the vibration displacement of the sun gear in the y-axis direction, u s is the torsional displacement of the sun gear, m s is the mass of the sun gear, k s is the support stiffness of the sun gear, c s is the support damping of the sun gear, F spn is the meshing force of the sun-planet gear pair, F fspn is the friction force of the sun-planet gear pair, w c is the planet carrier angular velocity, is the effective meshing angle between the sun gear and the nth planet gear, l spn is the friction arm of the sun gear, r s is the sun gear base circle radius, T in is the input torque;

[0105] Differential equation of internal gear ring vibration:

[0106]

[0107] Among them, x r is the vibration displacement of the inner gear ring in the x-axis direction, y r is the vibration displacement of the inner gear ring in the y-axis direction, u r is the torsional displacement of the inner ring gear, m r is the mass of the inner gear ring, k r is the support stiffness of the inner ring gear, c r is the support damping of the inner ring gear, F rpn is the meshing force of the ring gear-planetary gear pair, F frpn is the friction force of the ring gear-planetary gear pair, w c is the planet carrier angular velocity, is the effective meshing angle between the inner gear ring and the nth planetary gear, l rpn is the friction arm of the inner gear ring, r r is the base circle radius of the inner gear ring;

[0108] Differential equation for planet carrier vibration:

[0109]

[0110] Among them, x c is the vibration displacement of the planet carrier in the x-axis direction, y cis the vibration displacement of the planet carrier in the y-axis direction, u c is the torsional displacement of the planet carrier, m c is the mass of the planet carrier, k c is the support stiffness of the planet carrier, k pn is the support stiffness of the nth planetary gear, c c is the support damping of the planet carrier, w c is the planet carrier angular velocity, r c is the radius of the planetary gear center distribution circle, T out is the output torque, δ cpnx is the projection of the displacement of the planet carrier relative to the planet gear along the x-axis, δ cpny is the projection of the displacement of the planet carrier relative to the planet gear along the y-axis, δ cpnu is the projection of the displacement of the planet wheel relative to the planet carrier along the tangent line of the planet carrier;

[0111] Differential equation of planetary gear vibration:

[0112]

[0113] Among them, x pn is the vibration displacement of the nth planetary gear in the x-axis direction, y pn is the vibration displacement of the nth planetary gear in the y-axis direction, u pn is the torsional displacement of the nth planetary gear, m pn is the mass of the nth planetary gear, k pn is the support stiffness of the nth planetary gear, c pn is the support damping of the nth planetary gear, F spn is the meshing force of the sun-planet gear pair, F rpn is the meshing force of the ring gear-planetary gear pair, F fspn is the friction force of the sun-planet gear pair, F frpn is the friction force of the ring gear-planetary gear pair, w c is the planet carrier angular velocity, is the effective meshing angle between the sun gear and the nth planet gear, is the effective meshing angle between the inner gear ring and the nth planetary gear, l spn is the friction arm of the sun-planet gear pair meshing point relative to the sun gear, l rpn is the friction arm of the meshing point of the ring gear-planetary gear pair relative to the ring gear, r s is the sun gear base circle radius, r r is the base circle radius of the inner gear ring;

[0114] Step (5): solve the nonlinear response characteristics of the system;

[0115] The external excitation frequency parameters under different pitting corrosion degrees are substituted into the nonlinear vibration differential equations of the system to solve the vibration responses of each gear pair under different external excitation conditions. By comparing the vibration characteristics of the planetary gears under different pitting corrosion degrees, the influence of the external excitation frequency on the dynamic response of the system is revealed, and the influence of pitting corrosion faults on the vibration characteristics of the planetary gear transmission system is characterized.

[0116] During the establishment of the nonlinear dynamic model of the planetary gear transmission system, the parameters of the planetary gear transmission system are shown in Table 1:

[0117] Table 1 Basic parameters of planetary gear transmission system

[0118]

[0119] Figure 5 The following figure shows the excitation frequency-displacement bifurcation diagram for the sun-planet gear pair at different pitting corrosion levels. As shown in the figure, at the same external load excitation frequency, as the pitting corrosion level increases, the vibration amplitude of each gear pair increases, and the chaos becomes more pronounced. Furthermore, at the same pitting corrosion level, as the external load excitation frequency increases, the impact of pitting on the system's vibration increases, and the system tends to become unstable.

[0120] The above description is only a preferred embodiment of the invention and does not limit the invention in any way. Any modifications, changes and equivalent changes made to the above embodiments based on the essence of the invention shall still fall within the scope of protection of the technology of the invention.

Claims

1. A method for dynamic modeling of pitting failure in a planetary gear transmission system, characterized in that: The following steps are involved: Step (1): Calculate the time-varying meshing stiffness of the planetary gear under different pitting degrees based on the potential energy method; A cylinder was used to simulate the pitting pits at the pitch line. 10, 15, and 20 pitting pits were defined as mild pitting, moderate pitting, and severe pitting, respectively. The potential energy method was used to calculate the time-varying meshing stiffness of the planetary gear under pitting conditions and the reduction in contact tooth width ΔL. y , the reduction in cross-sectional area ΔA y and the reduction in moment of inertia ΔI y The calculation formula is as follows: Among them, δ is the pitting depth, μ is the distance from the pitting center to the tooth root, r is the pitting radius, h is the pitting depth, y For half tooth thickness, A y is the cross-sectional area of the gear teeth, y is the distance from the slice to the tooth root circle; When there is pitting failure in the planetary gear, the Hertz contact stiffness k H , shear stiffness k S , bending stiffness k B , and axial compressive stiffness k A The calculation formula is as follows: Among them, α a is the meshing pressure angle of the driving wheel, α b is the meshing pressure angle of the driven wheel, v is Poisson's ratio, E is Young's modulus, L is tooth width, R b Expressed as base circle radius, m is the number of pits, ΔL y,z is the reduction in contact tooth width caused by the zth pit, ΔA y,z is the reduction in cross-sectional area caused by the zth pit, ΔI y,z is the reduction in the moment of inertia caused by the zth pit, and α is the equivalent engagement angle; Meshing stiffness of single tooth in meshing pair k t1 and the meshing stiffness k of the double teeth of the meshing pair t2 The calculation formula is as follows: Among them, k H is the Hertzian contact stiffness, k Bp 、k Bg are the bending stiffness of the driving wheel and the driven wheel, k Sp 、k Sg are the shear stiffness of the driving wheel and the driven wheel, k Ap 、k Ag are the axial compression stiffness of the driving wheel and the driven wheel, k Fp 、k Fg are the base stiffness of the driving wheel and the driven wheel, k H,j is the Hertzian contact stiffness of the j-th pair of teeth, k Bp,j 、k Bg,j are the bending stiffness of the jth pair of teeth of the driving wheel and the driven wheel, k Sp,j 、k Sg,j are the shear stiffness of the jth pair of teeth of the driving wheel and the driven wheel, k Ap,j 、k Ag,j are the axial compression stiffness of the jth pair of teeth of the driving wheel and the driven wheel, k Fp,j 、k Fg,j are the base stiffness of the j-th pair of teeth of the driving wheel and the driven wheel respectively; Step (2): Introduce the nonlinear function of tooth side clearance and the comprehensive meshing error to determine the relative displacement on the meshing line of each gear pair; Relative displacement δ of the sun gear-planet gear pair along the normal direction of the meshing point spn Relative displacement δ along the normal direction of the meshing point with the ring gear-planet gear pair rpn The equations can be expressed as: in, and Respectively represent the effective meshing angles of the sun gear, the inner ring gear and the nth planet gear, u s 、u pn and u r They represent the torsional displacements of the sun gear, the nth planetary gear, and the inner ring gear caused by the system vibration, respectively. s 、x pn and x r They are respectively represented as the vibration displacement of the sun gear, the nth planet gear and the inner ring gear along the x-axis, and y s 、y pn and y r They are respectively represented as the vibration displacement of the sun gear, the nth planet gear and the inner ring gear along the y-axis, e spn (t) is the comprehensive meshing error of the sun-planet gear pair, e rpn (t) is the comprehensive meshing error of the ring gear-planetary gear pair; The projection of the displacement of the planet carrier relative to the planet gear along the x-axis, y-axis and tangent of the planet carrier: Among them, δ cpnx is the projection of the planet carrier relative to the planet gear along the x-axis of the planet carrier, δ cpny is the projection of the displacement of the planet carrier relative to the planet gear along the y-axis of the planet carrier, δ cpnu is the projection of the displacement of the planet carrier relative to the planet gear along the tangent line of the planet carrier, x c 、x pn They are respectively represented as the vibration displacement of the planet carrier and the nth planet gear along the x-axis, c 、y pn They are respectively represented as the vibration displacement of the planet carrier and the nth planet gear along the y-axis, u c is the torsional displacement of the planet carrier caused by system vibration, ψ pn is the angle between the line from the theoretical center of the nth planet gear to the theoretical center of the planet carrier and the positive direction of the x-axis; Step (3): Use the Coulomb model to calculate the time-varying friction between the gear pairs at each level, and the friction arm l of the driving wheel pg (t) and the friction arm l of the driven wheel gp The calculation formula of (t) is as follows: Among them, α0 is the pressure angle of the driving wheel, r pa 、r ga are the tooth top circle radius of the driving wheel and the driven wheel, r p 、r g are the base circle radii of the driving wheel and the driven wheel, r pb 、r gb are the root circle radius of the driving wheel and the driven wheel respectively, p is the base circle pitch, ω pc is the angular velocity of the driving wheel in the moving coordinate system, t is the gear meshing time, and mod is the remainder function; Step (4): Establish a set of differential equations for the dynamics of pitting failure in the planetary gear transmission system; The time-varying friction force, the nonlinear function of the tooth side clearance and the time-varying meshing stiffness under the tooth pitting fault state are introduced to establish the nonlinear dynamic differential equations of the system. Sun gear vibration differential equation: Among them, x s is the vibration displacement of the sun gear in the x-axis direction, y s is the vibration displacement of the sun gear in the y-axis direction, u s is the torsional displacement of the sun gear, m s is the mass of the sun gear, k s is the support stiffness of the sun gear, c s is the support damping of the sun gear, F spn is the meshing force of the sun-planet gear pair, F fspn is the friction force of the sun-planet gear pair, w c is the planet carrier angular velocity, is the effective meshing angle between the sun gear and the nth planet gear, l spn is the friction arm of the sun gear, r s is the sun gear base circle radius, T in is the input torque; The differential equation of the internal gear ring vibration is: Among them, x r is the vibration displacement of the inner gear ring in the x-axis direction, y r is the vibration displacement of the inner gear ring in the y-axis direction, u r is the torsional displacement of the inner ring gear, m r is the mass of the inner gear ring, k r is the support stiffness of the inner ring gear, c r is the support damping of the inner ring gear, F rpn is the meshing force of the ring gear-planetary gear pair, F frpn is the friction force of the ring gear-planetary gear pair, w c is the planet carrier angular velocity, is the effective meshing angle between the inner gear ring and the nth planetary gear, l rpn is the friction arm of the inner ring gear, r r is the base circle radius of the inner gear ring; Differential equation for planet carrier vibration: Among them, x c is the vibration displacement of the planet carrier in the x-axis direction, y c is the vibration displacement of the planet carrier in the y-axis direction, u c is the torsional displacement of the planet carrier, m c is the mass of the planet carrier, k c is the support stiffness of the planet carrier, k pn is the support stiffness of the nth planetary gear, c c is the support damping of the planet carrier, w c is the planet carrier angular velocity, r c is the radius of the planetary gear center distribution circle, T out is the output torque, δ cpnx is the projection of the displacement of the planet carrier relative to the planet gear along the x-axis, δ cpny is the projection of the displacement of the planet carrier relative to the planet gear along the y-axis, δ cpnu is the projection of the displacement of the planet wheel relative to the planet carrier along the tangent line of the planet carrier; Differential equation of planetary gear vibration: Among them, x pn is the vibration displacement of the nth planetary gear in the x-axis direction, y pn is the vibration displacement of the nth planetary gear in the y-axis direction, u pn is the torsional displacement of the nth planetary gear, m pn is the mass of the nth planetary gear, k pn is the support stiffness of the nth planetary gear, c pn is the support damping of the nth planetary gear, F spn is the meshing force of the sun-planet gear pair, F rpn is the meshing force of the ring gear-planetary gear pair, F fspn is the friction force of the sun-planet gear pair, F frpn is the friction force of the ring gear-planetary gear pair, w c is the planet carrier angular velocity, is the effective meshing angle between the sun gear and the nth planet gear, is the effective meshing angle between the inner gear ring and the nth planetary gear, l spn is the friction arm of the sun gear, l rpn is the friction arm of the inner gear ring, r s is the sun gear base circle radius, r r is the base circle radius of the inner gear ring; Step (5): solve the nonlinear response characteristics of the system; The external excitation frequency parameters under different pitting corrosion degrees are substituted into the nonlinear vibration differential equations of the system to solve the vibration responses of each gear pair under different external excitation conditions. By comparing the vibration characteristics of the planetary gears under different pitting corrosion degrees, the influence of the external excitation frequency on the dynamic response of the system is revealed, and the influence of pitting corrosion faults on the vibration characteristics of the planetary gear transmission system is characterized.

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