Nonlinear dynamic modeling method for spiral bevel gear transmission system

Abstract

The invention discloses a nonlinear dynamic modeling method for a spiral bevel gear transmission system. The nonlinear dynamic modeling method comprises the following steps: (1) constructing a bearing dynamic model and calculating bearing supporting force; (2) establishing a time-varying meshing stiffness model of the spiral bevel gear transmission system; (3) establishing a meshing displacement equation of the spiral bevel gear pair and solving a meshing force; (4) establishing a nonlinear dynamic equation set of the spiral bevel gear transmission system; (5) solving nonlinear response characteristics of the system; the method has the beneficial effects that the nonlinear dynamic response characteristics of the spiral bevel gear transmission system can be solved, and the blank of dynamic research on the spiral bevel 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 nonlinear dynamics modeling method for a spiral bevel gear transmission system. Background Art

[0002] As a highly efficient staggered-axis transmission, spiral bevel gear transmission offers significant advantages in complex power transmission applications due to its unique spatial meshing characteristics. Its tooth surface features a spiral involute distribution, enabling simultaneous force decomposition and synthesis in both the axial and radial directions. This allows for strong load-bearing capacity, smooth transmission, and low noise, leading to its widespread application in key sectors such as the automotive industry, aerospace, and heavy machinery. Due to the complex operating conditions of spiral bevel gears during meshing, such as multiple tooth contact surfaces, time-varying mesh stiffness, and load fluctuations, the system is susceptible to nonlinear vibrations, bifurcations, and chaos under the coupling of internal excitation and external disturbances, directly impacting transmission accuracy and service reliability. Therefore, analyzing the nonlinear response characteristics of the system and identifying the conditions that maintain system stability can provide theoretical support for the optimal design of spiral bevel gear transmission systems. However, conventional research often provides a crude definition of the factors influencing the stability of spiral bevel gear transmission systems, failing to meet the demands for higher power density and higher reliability in these transmission systems.

[0003] To address these issues, this paper proposes a nonlinear dynamic modeling method for spiral bevel gear transmission systems. This method comprehensively considers key factors such as bearings, time-varying meshing stiffness, and meshing damping. It constructs a dynamic model of the spiral bevel gear pair transmission system and numerically solves the equations using the Runge-Kutta integration method. This method accurately and effectively studies the nonlinear characteristics of spiral bevel gear transmission systems during meshing, filling a technical gap in the calculation of nonlinear characteristics of spiral bevel gear transmission systems, promoting engineering development, and potentially generating significant social and economic benefits. Summary of the Invention

[0004] In order to overcome the shortcomings of the existing technology and fill the gaps in the relevant technology, the present invention provides a nonlinear dynamic modeling method for a spiral bevel gear transmission system. This method establishes a nonlinear dynamic model of a spiral bevel gear transmission under the premise of fully considering multiple factors such as bearings, time-varying meshing stiffness, tooth side clearance, static transmission error, meshing damping, etc., and derives a set of vibration differential equations of the spiral bevel gear transmission system by combining the bearing dynamics model. Then, the Runge-Kutta integration method is used to solve the vibration differential dynamics equations of the system to obtain the nonlinear response characteristics of the system under the influence of external excitation and internal factors. This method has certain universality.

[0005] The technical solution adopted by the present invention to solve the technical problem is as follows: a nonlinear dynamic modeling method for a spiral bevel gear transmission system, characterized by comprising the following steps:

[0006] Step (1): Construct the bearing dynamics model and calculate the bearing support force and the rolling element rotational angular velocity ω b Expressed as:

[0007]

[0008] Among them, ω b is the angular velocity of the bearing rolling element, ω i is the angular velocity of the bearing shaft, r bn 、r bw are the radii of the inner and outer rings of the bearing respectively;

[0009] Bearing support force F on the three coordinate axes X, Y, and Z bx 、F by 、F bz Respectively expressed as:

[0010]

[0011] Among them, N b is the number of bearing rolling elements, K b is the bearing support stiffness, l' is the center distance of the inner and outer diameter curvature centers of the bearing after deformation, l is the center distance of the inner and outer diameter curvature centers of the bearing before deformation, c is the bearing clearance, is the contact angle between the rolling element and the outer ring of the bearing after deformation (i=1,2,…,N b ), H(δ) is the Heaviside function, δ is the deformation of the rolling element, and t represents time;

[0012] Step (2): Establish a time-varying meshing stiffness model for the spiral bevel gear transmission system and use the potential energy method to solve the time-varying meshing stiffness of the spiral bevel gear pair. The subscript j represents the jth slice, and k Bj is the bending deformation stiffness of the jth slice, k Aj is the axial compression deformation stiffness of the jth slice, k Sj is the shear deformation stiffness of the jth slice, k Fj is the matrix deformation stiffness of the jth slice, k H is the Hertz contact stiffness of the spiral bevel gear, which can be expressed as:

[0013]

[0014] Among them, α1 is the angle between the perpendicular line of the spiral bevel gear center line and the direction of meshing force, α2 is the semi-tooth angle on the base circle, α3 is the semi-tooth angle on the root circle, α is the rotation angle of the spiral bevel gear corresponding to any point on the tooth profile, α n is the pressure angle of the spiral bevel gear, α m is the pressure angle at the contact point of the gear teeth, subscripts f=1 and 2 represent the driving gear and the driven gear respectively, z f is the number of teeth of the corresponding spiral bevel gear, E is the Young's modulus of the spiral bevel gear, Δw is the width of a single slice, ν is the Poisson's ratio, S F is the tooth root arc length, μ F L is the distance between the tooth root circle and the center of the action point, * 、M * 、P * , Q * are all polynomial coefficients related to the ratio of the root circle radius to the hole diameter and the semi-arc angle corresponding to the root of a single tooth;

[0015] Time-varying mesh stiffness K h (t) can be expressed as:

[0016]

[0017] Where u is the number of teeth in contact at the same time when the gears are meshing, n is the number of slices, and β m is the helix angle at the midpoint of the tooth width, subscript a is the ath pair of meshing gears in multi-tooth meshing, k Hj is the Hertzian contact stiffness of the jth slice, k Fj1 、k Fj2 are the matrix deformation stiffness of the jth slice of the driving wheel and the driven wheel, respectively, Bj1 、k Bj2 are the bending deformation stiffness of the jth slice of the driving wheel and the driven wheel, respectively, k Aj1 、k Aj2 are the axial compression deformation stiffness of the jth slice of the driving wheel and the driven wheel, respectively, k Sj1 、k Sj2 are the shear deformation stiffness of the jth slice of the driving wheel and the driven wheel respectively;

[0018] Step (3): Establish the meshing displacement equation of the spiral bevel gear pair and solve the meshing force. The relative displacement equation X of the meshing point of the spiral bevel gear pair due to vibration and error along the meshing line is: ml It can be expressed as:

[0019]

[0020] Among them, α n1 is the pressure angle of the driving wheel, δ z is the pitch angle of the driving wheel, r m1 、rm2 are the meshing radius of the driving wheel and the driven wheel, θ1 and θ2 are the vibration angular displacements of the driving wheel and the driven wheel, X1, Y1, and Z1 are the vibration displacements of the driving wheel in the x, y, and z directions, respectively; X2, Y2, and Z2 are the vibration displacements of the driven wheel in the x, y, and z directions, respectively; e n (t) is the static transmission error;

[0021] The meshing force F of spiral bevel gear along the meshing line n It can be expressed as:

[0022]

[0023] Among them, K h (t) is the time-varying meshing stiffness, X ml is the relative displacement between teeth, r m1 、r m2 are the meshing radius of the driving wheel and the driven wheel, b m is half of the tooth side clearance, ξ m is the meshing damping ratio of the spiral bevel gear pair, I1 and I2 are the moments of inertia of the driving and driven gears respectively;

[0024] Step (4): Establish the nonlinear dynamic equations of the spiral bevel gear transmission system and consider the nonlinear dynamic model of the spiral bevel gear transmission system with 20 degrees of freedom:

[0025]

[0026] Among them, X1, Y1, and Z1 are the vibration displacements of the driving wheel in the x, y, and z directions respectively. b11 、Y b11 、Z b11 They are the vibration displacements of the first bearing at the input end in the x, y, and z directions, respectively. b12 、Y b12 、Z b12 are the vibration displacements of the second bearing at the input end in the x, y, and z directions respectively; X2, Y2, and Z2 are the vibration displacements of the passive wheel in the x, y, and z directions respectively; X b21 、Y b21 、Z b21 They are the vibration displacements of the first bearing at the output end in the x, y, and z directions, respectively. b22 、Y b22 、Z b22 are the vibration displacements of the second bearing at the output end in the x, y, and z directions, respectively; θ1 and θ2 are the vibration angular displacements of the driving wheel and the driven wheel, respectively;

[0027] From this, the vibration differential equation of the spiral bevel gear transmission system can be derived:

[0028] Driving wheel vibration differential equations:

[0029]

[0030] Among them, m1 is the mass of the driving wheel, X1, Y1, and Z1 are the vibration displacements of the driving wheel in the x, y, and z directions respectively, and X b11 、Y b11 、Z b11 are the vibration displacements of the first bearing at the input end in the x, y, and z directions, θ1 is the vibration angular displacement of the driving wheel, and C a1x 、C a1y 、C a1z are the support damping of the driving wheel in the x, y, and z directions, K a1x , K a1y , K a1z are the support stiffness of the driving wheel in the x, y, and z directions, respectively, and F x 、F y 、F z The meshing force F n The components of force in the x, y, and z directions, I1 is the moment of inertia of the driving wheel, r m1 is the meshing radius of the driving wheel, T1 is the input torque of the driving wheel;

[0031] Input end bearing b11 vibration differential equations:

[0032]

[0033] Among them, m b11 is the mass of the first bearing at the input end, X1, Y1, and Z1 are the vibration displacements of the driving wheel in the x, y, and z directions respectively, and X b11 、Y b11 、Z b11 They are the vibration displacements of the first bearing at the input end in the x, y, and z directions, respectively. b12 、Y b12 、Z b12 are the vibration displacements of the second bearing at the input end in the x, y, and z directions, respectively, and C a1x 、C a1y 、C a1z are the support damping of the driving wheel in the x, y, and z directions, respectively, C b11 is the support damping of the first bearing at the input end, K a1x , K a1y , K a1z are the support stiffness of the driving wheel in the x, y, and z directions, K b11 is the support stiffness of the first bearing at the input end, F b11x 、F b11y 、Fb11z are the bearing support forces of the first bearing at the input end in the x, y, and z directions, respectively, and g is the acceleration due to gravity;

[0034] Input end bearing b12 vibration differential equations:

[0035]

[0036] Among them, m b12 is the mass of the second bearing at the input end, X b11 、Y b11 、Z b11 They are the vibration displacements of the first bearing at the input end in the x, y, and z directions, respectively. b12 、Y b12 、Z b12 are the vibration displacements of the second bearing at the input end in the x, y, and z directions, respectively, and C a1x 、C a1y 、C a1z are the support damping of the driving wheel in the x, y, and z directions, respectively, C b12 is the support damping of the second bearing at the input end, K a1x , K a1y , K a1z are the support stiffness of the driving wheel in the x, y, and z directions, K b12 is the support stiffness of the second bearing at the input end, F b12x 、F b12y 、F b12z are the bearing support forces of the second bearing at the input end in the x, y, and z directions, respectively, and g is the acceleration due to gravity;

[0037] The passive wheel vibration differential equations:

[0038]

[0039] Among them, m2 is the mass of the passive wheel, X2, Y2, and Z2 are the vibration displacements of the passive wheel in the x, y, and z directions respectively, and X b21 、Y b21 、Z b21 They are the vibration displacements of the first bearing at the output end in the x, y, and z directions, respectively. b22 、Y b22 、Z b22 are the vibration displacements of the second bearing at the output end in the x, y, and z directions, θ2 is the vibration angular displacement of the passive wheel, and C a2x 、C a2y 、C a2z are the support damping of the passive wheel in the x, y, and z directions, K a2x , K a2y , K a2zare the support stiffness of the passive wheel in the x, y, and z directions, respectively, and F x 、F y 、F z The meshing force F n The components of force in the x, y, and z directions, r m2 is the meshing radius of the driven wheel, I2 is the moment of inertia of the driven wheel, and T2 is the output torque of the driven wheel;

[0040] The vibration differential equations of the output end bearing b21 are:

[0041]

[0042] Among them, m b21 is the mass of the first bearing at the output end, X2, Y2, and Z2 are the vibration displacements of the passive wheel in the x, y, and z directions respectively, and X b21 、Y b21 、Z b21 are the vibration displacements of the first bearing at the output end in the x, y, and z directions, respectively, and C a2x 、C a2y 、C a2z are the support damping of the passive wheel in the x, y, and z directions, C b21 K is the support damping of the first bearing at the output end, a2x , K a2y , K a2z are the support stiffness of the passive wheel in the x, y, and z directions, K b21 is the support stiffness of the first bearing at the output end, F b21x 、F b21y 、F b21z are the bearing support forces of the first bearing at the output end in the x, y, and z directions, respectively, and g is the acceleration due to gravity;

[0043] The vibration differential equations of the output end bearing b22 are:

[0044]

[0045] Among them, m b22 is the mass of the second bearing at the output end, X2, Y2, and Z2 are the vibration displacements of the passive wheel in the x, y, and z directions respectively. b22 、Y b22 、Z b22 are the vibration displacements of the second bearing at the output end in the x, y, and z directions, respectively, and C a2x 、C a2y 、C a2z are the support damping of the passive wheel in the x, y, and z directions, C b22 K is the support damping of the second bearing at the output end, a2x , Ka2y , K a2z are the support stiffness of the passive wheel in the x, y, and z directions, K b22 F is the support stiffness of the second bearing at the output end, b22x 、F b22y 、F b22z are the bearing support forces of the second bearing at the output end in the x, y, and z directions, respectively, and g is the acceleration due to gravity;

[0046] Combine the vibration angular displacements θ1 and θ2 of the spiral bevel gear and use the displacement X along the meshing line of the spiral bevel gear to calculate the value of the vibration angular displacements θ1 and θ2. ml As a new degree of freedom for force analysis, the new vibration differential equation can be obtained as follows:

[0047]

[0048] Among them, m e is the equivalent mass of the spiral bevel gear pair, β m is the helix angle at the midpoint of the tooth width, α n1 is the pressure angle of the driving wheel, δ z is the pitch angle of the driving wheel, C m is the meshing damping, K h (t) is the time-varying meshing stiffness, X ml is the relative displacement of inter-tooth vibration, f(X ml ) is the tooth side clearance function, F 1m is the average value of the driving force in the circumferential direction on the driving wheel, F 1v is the fluctuation value of the driving force, is the second-order derivative of the static transmission error;

[0049] Step (5): solve the nonlinear response characteristics of the system;

[0050] By comprehensively considering the influence of parameters such as bearing force, time-varying meshing stiffness, static transmission error, driving force, external excitation frequency, etc. in the system's nonlinear vibration differential equations, the vibration displacement of the spiral bevel gear pair under different external excitation frequencies and different driving forces is solved. The influence of external excitation frequency and driving wheel driving force on the dynamic response of the system is revealed, and the conditions required to maintain system stability are obtained.

[0051] Compared with the existing technology, the beneficial effects of the present invention are as follows: by integrating key parameters such as gear support stiffness, time-varying meshing stiffness, meshing damping and static transmission error, a nonlinear dynamic model of the spiral bevel gear transmission system under the coupling of multiple factors is constructed, and the vibration differential equations of the spiral bevel gear transmission system are derived. Then, the Runge-Kutta integration method is used to solve the system vibration differential equations, and the nonlinear response characteristics of the system under the coupling of external excitation and internal factors are obtained, filling the technical gap in the calculation of nonlinear characteristics of spiral bevel gear transmission systems, and providing key theoretical support for vibration suppression, noise control and dynamic optimization design of spiral bevel gear transmission systems. BRIEF DESCRIPTION OF THE DRAWINGS

[0052] Figure 1 It is a flow chart of the nonlinear dynamics method of spiral bevel gear transmission system;

[0053] Figure 2 is the bearing dynamics model;

[0054] Figure 3 It is a single tooth slice model of spiral bevel gear;

[0055] Figure 4 It is a nonlinear dynamic model and three-dimensional model of spiral bevel gear transmission system;

[0056] Figure 5 It is the time domain diagram of the nonlinear response characteristics of the system dimension-frequency and dimension-driving force. DETAILED DESCRIPTION

[0057] 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.

[0058] like Figure 1 The figure shows a flow chart of the nonlinear dynamics calculation method for the spiral bevel gear transmission system, which includes the following steps:

[0059] Step (1): Construct the bearing dynamics model and calculate the bearing support force. Figure 2 It is a bearing dynamics model. The inner ring of the bearing is rigidly connected to the shaft, and the outer ring is fixed to the bearing seat. The bearing coordinate system is established with its origin at the center of the bearing width, the Z axis coincides with the axis of the rotary shaft, and the rolling element rotation angular velocity ω b Expressed as:

[0060]

[0061] Among them, ω b is the angular velocity of the bearing rolling element, ω i is the angular velocity of the bearing shaft, r bn 、r bware the radii of the inner and outer rings of the bearing respectively;

[0062] Bearing support force F on the three coordinate axes X, Y, and Z bx 、F by 、F bz Respectively expressed as:

[0063]

[0064] Among them, N b is the number of bearing rolling elements, K b is the bearing support stiffness, l' is the center distance of the inner and outer diameter curvature centers of the bearing after deformation, l is the center distance of the inner and outer diameter curvature centers of the bearing before deformation, c is the bearing clearance, is the contact angle between the rolling element and the outer ring of the bearing after deformation (i=1,2,…,N b ), H(δ) is the Heaviside function, δ is the deformation of the rolling element, and t represents time;

[0065] Step (2): Establish a time-varying meshing stiffness model for the spiral bevel gear transmission system and use the potential energy method to solve the time-varying meshing stiffness of the spiral bevel gear pair. Figure 3 This is a single tooth slice model of a spiral bevel gear. A single tooth is evenly cut into multiple pieces along the normal direction of the tooth width. Assuming that adjacent slices are independent of each other, each part can be approximately regarded as a spur gear. The ellipse in the figure represents the actual contact area when the spiral bevel gear is meshing, and the dot represents the equivalent meshing point of a single slice. The width of a single slice is Δw. The subscript j represents the jth slice, and k Bj is the bending deformation stiffness of the jth slice, k Aj is the axial compression deformation stiffness of the jth slice, k Sj is the shear deformation stiffness of the jth slice, k Fj is the matrix deformation stiffness of the jth slice, k H is the Hertz contact stiffness, which can be expressed as:

[0066]

[0067] Among them, α1 is the angle between the perpendicular line of the spiral bevel gear centerline and the direction of meshing force, α2 is the semi-tooth angle on the base circle, α3 is the semi-tooth angle on the root circle, α is the spiral bevel gear angle corresponding to any point on the tooth profile, α n is the pressure angle of the spiral bevel gear, α m is the pressure angle at the contact point of the gear teeth, subscripts f=1 and 2 represent the driving gear and the driven gear respectively, z f is the number of teeth of the corresponding spiral bevel gear, E is the Young's modulus of the gear, ν is the Poisson's ratio, S F is the tooth root arc length, μ FL is the distance between the tooth root circle and the center of the action point, * 、M * 、P * , Q * are all polynomial coefficients related to the ratio of the root circle radius to the hole diameter and the semi-arc angle corresponding to the root of a single tooth;

[0068] Time-varying mesh stiffness K h (t) can be expressed as:

[0069]

[0070] Where u is the number of teeth in contact at the same time when the gears are meshing, n is the number of slices, and β m is the helix angle at the midpoint of the tooth width, subscript j is the jth slice, subscript a is the ath pair of meshing gears in multi-tooth meshing, k Hj is the Hertzian contact stiffness of the jth slice, k Fj1 、k Fj2 are the matrix deformation stiffness of the jth slice of the driving wheel and the driven wheel, respectively, Bj1 、k Bj2 are the bending deformation stiffness of the jth slice of the driving wheel and the driven wheel, respectively, k Aj1 、k Aj2 are the axial compression deformation stiffness of the jth slice of the driving wheel and the driven wheel, respectively, k Sj1 、k Sj2 are the shear deformation stiffness of the jth slice of the driving wheel and the driven wheel respectively;

[0071] Step (3): Establish the meshing displacement equation of the spiral bevel gear pair and solve the meshing force. Figure 4 It is a nonlinear dynamic model and three-dimensional model of the spiral bevel gear transmission system. The coordinate system is established with the intersection of the input shaft and the output shaft midline as the center of the circle. The concentrated mass method is used to simplify the mass of the shaft to the center of the gear tooth width. The stiffness and damping of the shaft are equivalent to the stiffness and damping acting at the center of the tooth width. The gear meshing is regarded as a pair of meshing stiffness and meshing damping. The torsional vibration of the gear is ignored. The static transmission error e is expressed by Fourier series. n (t):

[0072]

[0073] Where, e0 is the error constant, subscript d is the d-order harmonic amplitude of the transmission error, ω m is the gear meshing frequency, Φ ed The initial phase, the relative displacement equation of the meshing point of the spiral bevel gear pair along the meshing line due to vibration and error is X ml It can be expressed as:

[0074]

[0075] Among them, α n1 is the pressure angle of the driving wheel, δ z is the pitch angle of the driving wheel, r m1 、r m2 are the meshing radius of the driving wheel and the driven wheel, θ1 and θ2 are the vibration angular displacements of the driving wheel and the driven wheel, X1, Y1, and Z1 are the vibration displacements of the driving wheel in the x, y, and z directions, respectively; X2, Y2, and Z2 are the vibration displacements of the driven wheel in the x, y, and z directions, respectively; e n (t) is the static transmission error;

[0076] The meshing force F of spiral bevel gear along the meshing line n It can be expressed as:

[0077]

[0078] Among them, K h (t) is the time-varying meshing stiffness, X ml is the relative displacement between teeth, r m1 、r m2 are the meshing radius of the driving wheel and the driven wheel, b m is half of the tooth side clearance, ξ m is the meshing damping ratio of the spiral bevel gear pair, I1 and I2 are the moments of inertia of the driving and driven gears respectively;

[0079] Meshing force F n Component force F in the x, y, and z directions x 、F y 、F z Respectively expressed as

[0080]

[0081] Among them, α n1 is the pressure angle of the driving wheel, δ z is the pitch angle of the driving wheel, β m is the helix angle at the midpoint of the tooth width;

[0082] Step (4): Establish the nonlinear dynamic equations of the spiral bevel gear transmission system and consider the nonlinear dynamic model of the spiral bevel gear transmission system with 20 degrees of freedom:

[0083]

[0084] Among them, X1, Y1, and Z1 are the vibration displacements of the driving wheel in the x, y, and z directions respectively. b11 、Y b11 、Z b11 They are the vibration displacements of the first bearing at the input end in the x, y, and z directions, respectively.b12 、Y b12 、Z b12 are the vibration displacements of the second bearing at the input end in the x, y, and z directions respectively; X2, Y2, and Z2 are the vibration displacements of the passive wheel in the x, y, and z directions respectively; X b21 、Y b21 、Z b21 They are the vibration displacements of the first bearing at the output end in the x, y, and z directions, respectively. b22 、Y b22 、Z b22 are the vibration displacements of the second bearing at the output end in the x, y, and z directions, respectively; θ1 and θ2 are the vibration angular displacements of the driving wheel and the driven wheel, respectively;

[0085] From this, the vibration differential equation of the spiral bevel gear transmission system can be derived:

[0086] Driving wheel vibration differential equations:

[0087]

[0088] Among them, m1 is the mass of the driving wheel, X1, Y1, and Z1 are the vibration displacements of the driving wheel in the x, y, and z directions respectively, and X b11 、Y b11 、Z b11 are the vibration displacements of the first bearing at the input end in the x, y, and z directions, θ1 is the vibration angular displacement of the driving wheel, and C a1x 、C a1y 、C a1z are the support damping of the driving wheel in the x, y, and z directions, K a1x , K a1y , K a1z are the support stiffness of the driving wheel in the x, y, and z directions, respectively, and F x 、F y 、F z The meshing force F n The components of force in the x, y, and z directions, I1 is the moment of inertia of the driving wheel, r m1 is the meshing radius of the driving wheel, T1 is the input torque of the driving wheel;

[0089] Input end bearing b11 vibration differential equations:

[0090]

[0091] Among them, m b11 is the mass of the first bearing at the input end, X1, Y1, and Z1 are the vibration displacements of the driving wheel in the x, y, and z directions respectively, and X b11 、Y b11 、Z b11They are the vibration displacements of the first bearing at the input end in the x, y, and z directions, respectively. b12 、Y b12 、Z b12 are the vibration displacements of the second bearing at the input end in the x, y, and z directions, respectively, and C a1x 、C a1y 、C a1z are the support damping of the driving wheel in the x, y, and z directions, respectively, C b11 is the support damping of the first bearing at the input end, K a1x , K a1y , K a1z are the support stiffness of the driving wheel in the x, y, and z directions, K b11 is the support stiffness of the first bearing at the input end, F b11x 、F b11y 、F b11z are the bearing support forces of the first bearing at the input end in the x, y, and z directions, respectively, and g is the acceleration due to gravity;

[0092] Input end bearing b12 vibration differential equations:

[0093]

[0094] Among them, m b12 is the mass of the second bearing at the input end, X b11 、Y b11 、Z b11 They are the vibration displacements of the first bearing at the input end in the x, y, and z directions, respectively. b12 、Y b12 、Z b12 are the vibration displacements of the second bearing at the input end in the x, y, and z directions, respectively, and C a1x 、C a1y 、C a1z are the support damping of the driving wheel in the x, y, and z directions, respectively, C b12 is the support damping of the second bearing at the input end, K a1x , K a1y , K a1z are the support stiffness of the driving wheel in the x, y, and z directions, K b12 is the support stiffness of the second bearing at the input end, F b12x 、F b12y 、F b12z are the bearing support forces of the second bearing at the input end in the x, y, and z directions, respectively, and g is the acceleration due to gravity;

[0095] The passive wheel vibration differential equations:

[0096]

[0097] Among them, m2 is the mass of the passive wheel, X2, Y2, and Z2 are the vibration displacements of the passive wheel in the x, y, and z directions respectively, and X b21 、Y b21 、Z b21 They are the vibration displacements of the first bearing at the output end in the x, y, and z directions, respectively. b22 、Y b22 、Z b22 are the vibration displacements of the second bearing at the output end in the x, y, and z directions, θ2 is the vibration angular displacement of the passive wheel, and C a2x 、C a2y 、C a2z are the support damping of the passive wheel in the x, y, and z directions, K a2x , K a2y , K a2z are the support stiffness of the passive wheel in the x, y, and z directions, respectively, and F x 、F y 、F z The meshing force F n The components of force in the x, y, and z directions, r m2 is the meshing radius of the driven wheel, I2 is the moment of inertia of the driven wheel, and T2 is the output torque of the driven wheel;

[0098] The vibration differential equations of the output end bearing b21 are:

[0099]

[0100] Among them, m b21 is the mass of the first bearing at the output end, X2, Y2, and Z2 are the vibration displacements of the passive wheel in the x, y, and z directions respectively, and X b21 、Y b21 、Z b21 are the vibration displacements of the first bearing at the output end in the x, y, and z directions, respectively, and C a2x 、C a2y 、C a2z are the support damping of the passive wheel in the x, y, and z directions, C b21 K is the support damping of the first bearing at the output end, a2x , K a2y , K a2z are the support stiffness of the passive wheel in the x, y, and z directions, K b21 is the support stiffness of the first bearing at the output end, F b21x 、F b21y 、F b21z are the bearing support forces of the first bearing at the output end in the x, y, and z directions, respectively, and g is the acceleration due to gravity;

[0101] The vibration differential equations of the output end bearing b22 are:

[0102]

[0103] Among them, m b22 is the mass of the second bearing at the output end, X2, Y2, and Z2 are the vibration displacements of the passive wheel in the x, y, and z directions respectively. b22 、Y b22 、Z b22 are the vibration displacements of the second bearing at the output end in the x, y, and z directions, respectively, and C a2x 、C a2y 、C a2z are the support damping of the passive wheel in the x, y, and z directions, C b22 K is the support damping of the second bearing at the output end, a2x , K a2y , K a2z are the support stiffness of the passive wheel in the x, y, and z directions, K b22 F is the support stiffness of the second bearing at the output end, b22x 、F b22y 、F b22z are the bearing support forces of the second bearing at the output end in the x, y, and z directions, respectively, and g is the acceleration due to gravity;

[0104] Combine the vibration angular displacements θ1 and θ2 of the spiral bevel gear and use the displacement X along the meshing line of the spiral bevel gear ml As a new degree of freedom for force analysis, the new vibration differential equation can be obtained as follows:

[0105]

[0106] Among them, m e is the equivalent mass of the spiral bevel gear pair, β m is the helix angle at the midpoint of the tooth width, α n1 is the pressure angle of the driving wheel, δ z is the pitch angle of the driving wheel, C m is the meshing damping, K h (t) is the time-varying meshing stiffness, X ml is the relative displacement of inter-tooth vibration, f(X ml ) is the tooth side clearance function, F 1m is the average value of the driving force in the circumferential direction on the driving wheel, F 1v is the fluctuation value of driving force, t is time, C m is the meshing damping of the spiral bevel gear pair, which can be expressed as:

[0107]

[0108] Among them, Kh (t) is the time-varying meshing stiffness of the spiral bevel gear pair, r m1 、r m2 are the meshing radius of the driving wheel and the driven wheel, I1 and I2 are the moments of inertia of the driving wheel and the driven wheel, and t is the time;

[0109] Step (5): solve the nonlinear response characteristics of the system;

[0110] By comprehensively considering the influence of parameters such as bearing force, time-varying meshing stiffness, static transmission error, driving force, external excitation frequency, etc. in the system's nonlinear vibration differential equations, the relative vibration displacement of the spiral bevel gear pair under different external excitation frequencies and different driving wheel driving forces is solved. The influence of external excitation frequency and driving wheel driving force on the dynamic response of the spiral bevel gear transmission system is revealed, and the conditions required to maintain the stability of the spiral bevel gear transmission system are obtained.

[0111] In this example, the parameters selected for the spiral bevel gear transmission system are shown in Tables 1 and 2. The nonlinear dynamic response of the spiral bevel gear transmission system is obtained through programming calculation, and the response state of the spiral bevel gear transmission system is displayed using a time domain diagram.

[0112] Table 1 Basic parameters of spiral bevel gears

[0113]

[0114] Table 2 Basic parameters of bearings

[0115]

[0116] Figure 5 These time-domain plots show the system's nonlinear response characteristics, comparing dimension to frequency and dimension to driving force. These plots show the time-varying vibration displacement of the spiral bevel gear meshing pair when the dimensionless external excitation frequency is 0.4 and when the dimensionless driving force of the driving wheel is 0.1. The time-domain plots exhibit clear regularity, indicating a stable system state. This state improves system performance and reduces vibration and noise.

[0117] 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 nonlinear dynamic modeling method for a spiral bevel gear transmission system, characterized in that: The following steps are involved: Step (1): Construct the bearing dynamics model and calculate the bearing support force. b 、Y b , Z b Bearing support force F on the three coordinate axes bx 、F by 、F bz Respectively expressed as: Among them, N b is the number of bearing rolling elements, K b is the bearing support stiffness, l' is the center distance of the inner and outer diameter curvature centers of the bearing after deformation, l is the center distance of the inner and outer diameter curvature centers of the bearing before deformation, c is the bearing clearance, is the contact angle between the rolling element and the outer ring of the bearing after deformation (i=1,2,…,N b ), H(δ) is the Heaviside function, δ is the deformation of the rolling element, ω b is the angular velocity of the rolling element, and t is the time; Step (2): Establish the time-varying meshing stiffness model of the spiral bevel gear transmission system, and use the potential energy method to solve the time-varying meshing stiffness of the spiral bevel gear pair. The time-varying meshing stiffness K h (t) can be expressed as: Where u is the number of teeth in contact at the same time when the gears are meshing, n is the number of slices, and β m is the helix angle at the midpoint of the tooth width, subscript j is the jth slice, subscript a is the ath pair of meshing gears in multi-tooth meshing, k Hj is the Hertzian contact stiffness of the jth slice, k Fj1 、k Fj2 are the matrix deformation stiffness of the jth slice of the driving wheel and the driven wheel, respectively, Bj1 、k Bj2 are the bending deformation stiffness of the jth slice of the driving wheel and the driven wheel, respectively, k Aj1 、k Aj2 are the axial compression deformation stiffness of the jth slice of the driving wheel and the driven wheel, respectively, k Sj1 、k Sj2 are the shear deformation stiffness of the jth slice of the driving wheel and the driven wheel respectively; Step (3): Establish the meshing displacement equation of the spiral bevel gear pair and solve the meshing force. The relative displacement equation X generated by the meshing point of the spiral bevel gear pair along the meshing line due to vibration and error is: ml It can be expressed as: Among them, α n1 is the pressure angle of the driving wheel, δ z is the pitch angle of the driving wheel, r m1 、r m2 are the meshing radius of the driving wheel and the driven wheel respectively, θ1 and θ2 are the vibration angular displacements of the driving wheel and the driven wheel respectively, X1 and X2 are the vibration displacements of the driving wheel and the driven wheel respectively on the x-axis, Y1 and Y2 are the vibration displacements of the driving wheel and the driven wheel respectively on the y-axis, Z1 and Z2 are the vibration displacements of the driving wheel and the driven wheel respectively on the z-axis, e n (t) is the static transmission error, t is time; The meshing force F of spiral bevel gear along the meshing line n It can be expressed as: Among them, K h (t) is the time-varying meshing stiffness, X ml is the relative displacement between teeth, r m1 、r m2 are the meshing radius of the driving wheel and the driven wheel, b m is half of the tooth side clearance, ξ m is the meshing damping ratio of the spiral bevel gear pair, I1 and I2 are the moments of inertia of the driving wheel and the driven wheel respectively, and t represents time; Step (4): Establish the nonlinear dynamic equations of the spiral bevel gear transmission system and consider the nonlinear dynamic model of the spiral bevel gear transmission system with 20 degrees of freedom: Among them, X1, Y1, and Z1 are the vibration displacements of the driving wheel in the x, y, and z directions respectively. b11 、Y b11 , Z b11 They are the vibration displacements of the first bearing at the input end in the x, y, and z directions, respectively. b12 、Y b12 , Z b12 are the vibration displacements of the second bearing at the input end in the x, y, and z directions, respectively; X2, Y2, and Z2 are the vibration displacements of the passive wheel in the x, y, and z directions, respectively; b21 、Y b21 , Z b21 They are the vibration displacements of the first bearing at the output end in the x, y, and z directions, respectively. b22 、Y b22 , Z b22 are the vibration displacements of the second bearing at the output end in the x, y, and z directions, respectively; θ1 and θ2 are the vibration angular displacements of the driving wheel and the driven wheel, respectively; From this, the vibration differential equation of the spiral bevel gear transmission system can be derived: Driving wheel vibration differential equations: Among them, m1 is the mass of the driving wheel, X1, Y1, and Z1 are the vibration displacements of the driving wheel in the x, y, and z directions respectively, and X b11 、Y b11 , Z b11 are the vibration displacements of the first bearing at the input end in the x, y, and z directions, θ1 is the vibration angular displacement of the driving wheel, and C a1x 、C a1y 、C a1z are the support damping of the driving wheel in the x, y, and z directions, K a1x , K a1y , K a1z are the support stiffness of the driving wheel in the x, y, and z directions, respectively, and F x 、F y 、F z The meshing force F n The components of force in the x, y, and z directions, I1 is the moment of inertia of the driving wheel, r m1 is the meshing radius of the driving wheel, T1 is the input torque of the driving wheel; Input end bearing b11 vibration differential equations: Among them, m b11 is the mass of the first bearing at the input end, X1, Y1, and Z1 are the vibration displacements of the driving wheel in the x, y, and z directions respectively, and X b11 、Y b11 , Z b11 They are the vibration displacements of the first bearing at the input end in the x, y, and z directions, respectively. b12 、Y b12 , Z b12 are the vibration displacements of the second bearing at the input end in the x, y, and z directions, respectively, and C a1x 、C a1y 、C a1z are the support damping of the driving wheel in the x, y, and z directions, respectively, C b11 is the support damping of the first bearing at the input end, K a1x , K a1y , K a1z are the support stiffness of the driving wheel in the x, y, and z directions, K b11 is the support stiffness of the first bearing at the input end, F b11x 、F b11y 、F b11z are the bearing support forces of the first bearing at the input end in the x, y, and z directions, respectively, and g is the acceleration due to gravity; Input end bearing b12 vibration differential equations: Among them, m b12 is the mass of the second bearing at the input end, X b11 、Y b11 , Z b11 They are the vibration displacements of the first bearing at the input end in the x, y, and z directions, respectively. b12 、Y b12 , Z b12 are the vibration displacements of the second bearing at the input end in the x, y, and z directions, respectively, and C a1x 、C a1y 、C a1z are the support damping of the driving wheel in the x, y, and z directions, respectively, C b12 is the support damping of the second bearing at the input end, K a1x , K a1y , K a1z are the support stiffness of the driving wheel in the x, y, and z directions, K b12 is the support stiffness of the second bearing at the input end, F b12x 、F b12y 、F b12z are the bearing support forces of the second bearing at the input end in the x, y, and z directions, respectively, and g is the acceleration due to gravity; The passive wheel vibration differential equations: Among them, m2 is the mass of the passive wheel, X2, Y2, and Z2 are the vibration displacements of the passive wheel in the x, y, and z directions respectively, and X b21 、Y b21 , Z b21 They are the vibration displacements of the first bearing at the output end in the x, y, and z directions, respectively. b22 、Y b22 , Z b22 are the vibration displacements of the second bearing at the output end in the x, y, and z directions, θ2 is the vibration angular displacement of the passive wheel, and C a2x 、C a2y 、C a2z are the support damping of the passive wheel in the x, y, and z directions, K a2x , K a2y , K a2z are the support stiffness of the passive wheel in the x, y, and z directions, respectively, and F x 、F y 、F z The meshing force F n The components of force in the x, y, and z directions, r m2 is the meshing radius of the driven wheel, I2 is the moment of inertia of the driven wheel, and T2 is the output torque of the driven wheel; The vibration differential equations of the output end bearing b21 are: Among them, m b21 is the mass of the first bearing at the output end, X2, Y2, and Z2 are the vibration displacements of the passive wheel in the x, y, and z directions respectively, and X b21 、Y b21 , Z b21 are the vibration displacements of the first bearing at the output end in the x, y, and z directions, respectively, and C a2x 、C a2y 、C a2z are the support damping of the passive wheel in the x, y, and z directions, C b21 K is the support damping of the first bearing at the output end, a2x , K a2y , K a2z are the support stiffness of the passive wheel in the x, y, and z directions, K b21 is the support stiffness of the first bearing at the output end, F b21x 、F b21y 、F b21z are the bearing support forces of the first bearing at the output end in the x, y, and z directions, respectively, and g is the acceleration due to gravity; The vibration differential equations of the output end bearing b22 are: Among them, m b22 is the mass of the second bearing at the output end, X2, Y2, and Z2 are the vibration displacements of the passive wheel in the x, y, and z directions respectively. b22 、Y b22 , Z b22 are the vibration displacements of the second bearing at the output end in the x, y, and z directions, respectively, and C a2x 、C a2y 、C a2z are the support damping of the passive wheel in the x, y, and z directions, C b22 K is the support damping of the second bearing at the output end, a2x , K a2y , K a2z are the support stiffness of the passive wheel in the x, y, and z directions, K b22 F is the support stiffness of the second bearing at the output end, b22x 、F b22y 、F b22z are the bearing support forces of the second bearing at the output end in the x, y, and z directions, respectively, and g is the acceleration due to gravity; Combine the vibration angular displacements θ1 and θ2 of the spiral bevel gear and use the displacement X ml As a new degree of freedom for force analysis, the new vibration differential equation can be obtained as follows: Among them, m e is the equivalent mass of the spiral bevel gear pair, β m is the helix angle at the midpoint of the tooth width, α n1 is the pressure angle of the driving wheel, δ z is the pitch angle of the driving wheel, C m is the meshing damping, K h (t) is the time-varying meshing stiffness, X ml is the relative displacement between teeth, f(X ml ) is the tooth side clearance function, F 1m is the average value of the driving force in the circumferential direction on the driving wheel, F 1v is the fluctuation value of the driving force, is the second-order derivative of the static transmission error; Step (5): Solve the nonlinear response characteristics of the system.

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