Planetary gear transmission system tooth surface three-dimensional dynamic contact characteristic analysis method considering tooth surface error

Abstract

The invention provides a planetary gear transmission system tooth surface three-dimensional dynamic contact characteristic analysis method considering a tooth surface error, which comprises the following steps of: firstly, combining a finite element substructure method, establishing an internal / external meshing gear tooth bearing contact analysis agent model considering the tooth surface error, and obtaining internal / external meshing dynamic internal excitation under any gear pair meshing load through interpolation; then establishing a planetary gear transmission system dynamic model considering a shaft unit, a bearing-casing unit, a casing-basic unit, an internal meshing unit, an external meshing unit and a planet carrier unit based on a generalized finite element method; and finally, changing the input torque of the system for iterative calculation to obtain the dynamic internal excitation of each internal / external meshing gear pair considering the tooth surface error under different input torques, and establishing an internal / external meshing gear tooth bearing contact analysis proxy model. According to the method, the influence of shafting deformation and tooth surface errors on the three-dimensional dynamic contact characteristics of the tooth surface of the planetary gear transmission system is considered, the method is more practical, and the calculation result is more accurate.

Description

Technical Field

[0001] This invention belongs to the field of dynamics technology and relates to a method for analyzing the three-dimensional dynamic contact characteristics of the tooth surface of a planetary gear transmission system considering tooth surface errors. Background Technology

[0002] As a multi-degree-of-freedom complex gear train, a planetary gear transmission system is simultaneously subjected to internal excitation forces caused by the time-varying meshing internal excitation of the internal / external meshing gear pairs during operation. Furthermore, due to the different meshing phases of each meshing gear pair, the dynamic load borne by each meshing gear pair in the system varies significantly. Therefore, the dynamic response of a planetary gear transmission system exhibits more complex nonlinear characteristics than that of a fixed-axis gear train.

[0003] Currently, most dynamic studies of planetary gear transmission systems employ lumped mass and generalized finite element methods for dynamic modeling. The sun gear, planet gears, and planet carrier are modeled using the lumped mass method, while the internal gear ring is modeled using the Timoshenko beam element theory and the generalized finite element method. The Fourier series method is then used to solve the overall system motion differential equations, yielding the dynamic response parameters of each meshing gear pair node in the planetary gear train. However, for planetary gear trains, the structural flexibility of the internal gear ring has a limited impact on its overall dynamic response. Meanwhile, shaft deformation (shaft-bearing-casing) and tooth surface errors caused by gear manufacturing / modification directly affect the instantaneous contact characteristics of the gear pair tooth surfaces. Under specific operating conditions, local disengagement may even occur on the meshing tooth surfaces, significantly altering the system's dynamic response characteristics. Summary of the Invention

[0004] To address the problems existing in the prior art, this invention proposes a three-dimensional dynamic contact characteristic analysis method for the tooth surface of a planetary gear transmission system considering tooth surface errors. A three-dimensional dynamic contact analysis model of the planetary gear transmission system is established, coupling the instantaneous load-bearing contact analysis of the tooth surface with the system dynamics model, thereby obtaining the dynamic response parameters of the planetary gear system under different tooth surface error distributions and operating conditions. Furthermore, considering computational efficiency, the method reduces the overall number of degrees of freedom of the system. Moreover, given that the flexibility of the internal gear ring structure has a relatively small impact on the system's dynamic response, the method employs the lumped mass method for modeling to improve computational efficiency, thereby reducing the computational load while ensuring the model's computational accuracy.

[0005] The technical solution of this invention is as follows:

[0006] A method for analyzing the three-dimensional dynamic contact characteristics of the tooth surface of a planetary gear transmission system considering tooth surface errors, characterized by the following steps:

[0007] Step 1: Combine the finite element substructure method to establish a proxy model for the contact analysis of internal / external meshing gear teeth considering tooth surface errors, and obtain the dynamic internal excitation of internal / external meshing under meshing load of any gear pair by interpolation;

[0008] Step 2: Combining Timoshenko beam element theory, a dynamic model of the planetary gear transmission system considering shaft elements, bearing-casing elements, casing-foundation elements, internal meshing elements, external meshing elements, and planet carrier elements is established based on the generalized finite element method.

[0009] Step 3: Combining the Newmark numerical integration method, the proxy model for the load-bearing contact analysis of internal / external meshing gear teeth considering tooth surface error is coupled with the dynamic model of the planetary gear transmission system to establish a three-dimensional dynamic contact characteristic analysis model of the planetary gear transmission system tooth surface considering tooth surface error and system flexibility, thereby realizing the analysis of the three-dimensional dynamic contact characteristics of the planetary gear transmission system tooth surface.

[0010] Furthermore, step 1 includes the following processes:

[0011] Step 1.1: Based on the finite element substructure method, extract the stiffness matrix at the casing-foundation position on the central wheel axle, and assemble it with the stiffness / mass matrices of the gear, shaft, and bearing to solve the initial meshing static load of the gear pair;

[0012] Step 1.2: Based on the finite element substructure method, obtain the tooth surface compliance matrix of the internal / external meshing gear pair considering the deformation of the gear body. Combine the initial meshing static load of the gear pair obtained in Step 1.1 and the tooth surface error terms caused by manufacturing, modification, and shaft deformation, use the tooth surface bearing contact equation to solve the dynamic internal excitation of the internal / external meshing gear pair, and obtain the dynamic internal excitation of each meshing internal / external gear pair by interpolation based on the meshing phase.

[0013] Step 1.3: Change the system input torque and repeat steps 1.1 and 1.2 to obtain the dynamic internal excitation of each internal / external meshing gear pair considering tooth surface error under different input torques, and establish a proxy model for the load-bearing contact analysis of internal / external meshing gear teeth.

[0014] Furthermore, in step 1.2, the tooth surface bearing contact equation is:

[0015]

[0016] In the formula: Let P be the macroscopic deformation compliance matrix of the possible contact points, and P be the external load vector. For local contact deformation at possible contact points, The initial gap for possible contact points. The remaining clearance at possible contact points is represented by STE, where STE is the static transmission error and i is the contact point number. The dynamic internal excitation of the gear pair can be obtained by iteratively solving the tooth surface bearing contact equation.

[0017] Furthermore, step 2 includes the following processes:

[0018] Step 2.1: Treat the shaft element as a Timoshenko beam element and establish the stiffness / mass / damping matrix of the shaft element;

[0019] Step 2.2: Treat the bearing-casing unit, casing-foundation unit, and planetary carrier unit as spring-damping units, and establish the motion differential equations based on the stiffness, mass, and damping matrix of the bearing-casing unit, casing-foundation unit, and planetary carrier unit, respectively.

[0020] Step 2.3: Establish the kinematic differential equations for the internal / external meshing gear pair unit;

[0021] Step 2.4: Assemble the overall stiffness / mass / damping matrix, establish the overall motion differential equation of the planetary gear transmission system based on the motion relationship, and thus establish the dynamic model of the planetary gear transmission system.

[0022] Furthermore, in step 2.1, the differential equation of motion for the shaft element is constructed based on the Timoshenko beam element theory to simultaneously consider the effects of shear deformation and rotational inertia on the shaft deflection. Its differential equation of motion is:

[0023]

[0024] Where: M s The mass matrix of the shaft element, C s The damping matrix of the shaft element, K s Let be the stiffness matrix of the shaft element. , , These represent the displacement, velocity, and acceleration at the node of the axis element at the current moment;

[0025] In step 2.2, the bearing-casing unit is treated as a spring-damping unit, and its equation of motion is:

[0026]

[0027] Where: M b For the mass matrix of the bearing-casing unit, C b For the damping matrix of the bearing-casing unit, K b Here is the stiffness matrix of the bearing-casing unit. , , These represent the displacement, velocity, and acceleration of the bearing-casing unit node at the current moment;

[0028] The casing-base unit is considered as a spring-damped unit, and its equation of motion is:

[0029]

[0030] Where: M h For the mass matrix of the casing-basic unit, C h For the damping matrix of the casing-base unit, K h Here is the stiffness matrix of the casing-base unit. , , These are the displacement, velocity, and acceleration at the casing-base unit node at the current moment;

[0031] The planetary carrier element is considered as a spring-damped element, and its differential equation of motion is:

[0032]

[0033] Where: M c The mass matrix of the planetary carrier unit, K c Here is the stiffness matrix of the planetary carrier element. , These represent the displacement and acceleration at the node of the planetary carrier unit at the current moment.

[0034] Furthermore, in step 2.3, the coordinates of the gear pair node of the gear meshing unit are defined as follows:

[0035]

[0036] The relative displacement of the gear pair along the normal line of meshing can be expressed as:

[0037]

[0038] Where V is the projection vector of displacement in each direction towards the end face meshing line direction, expressed by the following formula.

[0039]

[0040] In the formula: r p r g The base circle radii of the master and driven gears; β b φ is the base circle helix angle, which is defined as positive when the gear is right-handed and negative when it is left-handed; φ is the angle between the end face meshing line and the positive direction of the y-axis;

[0041] According to Newton's second law, the kinematic differential equations of the internal / external meshing gear pair unit considering tooth surface errors are as follows:

[0042]

[0043]

[0044] The upper part corresponds to external meshing, and the lower part corresponds to internal meshing; where: m i (i = p, g) represent the masses of the driving and driven gears, respectively; I zi (i = p, g) are the moments of inertia of the driving and driven wheels about the z-axis, respectively; These represent the relative displacement and velocity at the meshing unit node along the normal meshing line at that moment; φ is the angle between the end face meshing line and the positive y-axis. This represents the normal composite meshing stiffness of the gear pair at that moment; c represents the combined normal meshing error of the gear pair at that moment; m The meshing damping of the gear pair is calculated using the following formula:

[0045]

[0046] In the formula: ζ is the meshing damping ratio; The mean normal meshing stiffness of the gear pair; m eq,i = I zi / r i 2 Let r be the equivalent mass of gear i (i=p, g). i (i = p, g) are the pitch circle radii of the driving and driven wheels, respectively.

[0047] Furthermore, step 3 includes the following processes:

[0048] Step 3.1: Combining the dynamic internal excitation of each internal / external meshing gear pair obtained in Step 1, the Newmark numerical integration method is used to solve the overall motion differential equation of the system, thereby obtaining the displacement, velocity and acceleration of each internal / external meshing gear pair along the end face meshing line direction, and based on this, the dynamic meshing force of each internal / external meshing gear pair is obtained.

[0049] Step 3.2: Based on the dynamic meshing force of each internal / external meshing gear pair obtained in Step 3.1, determine the instantaneous contact state of each meshing tooth surface. Combined with the contact analysis proxy model of each internal / external meshing gear tooth established in Step 1, the dynamic internal excitation of the internal / external meshing gear pair corresponding to the dynamic meshing force is obtained by interpolation.

[0050] Step 3.3: Based on the fixed-point iteration method, repeat steps 3.1 and 3.2 until the system response at the current meshing position converges, thereby obtaining the displacement, velocity and acceleration of each internal / external meshing gear pair along the end face meshing line direction at that meshing position;

[0051] Step 3.4: Repeat steps 3.1 to 3.3, traversing all meshing positions of each internal / external meshing gear pair in step 3.1, until the system response at all meshing positions converges, obtaining the displacement, velocity, and acceleration of each internal / external meshing gear pair along the end face meshing line direction in the system under a complete meshing cycle, and simultaneously obtaining its tooth surface dynamic contact stress and dynamic load sharing coefficient.

[0052] Furthermore, the specific calculation process for step 3 is as follows:

[0053] The overall kinematic differential equation of the planetary gear transmission system can be rewritten as follows:

[0054]

[0055] in, .

[0056] Given the initial displacement of the system Initial velocity value The initial acceleration values ​​are obtained by solving:

[0057]

[0058] Select integration step size ,parameter , Calculate the integration constant:

[0059]

[0060] Formed based on Newmark numerical integration method Given the system's mass matrix, stiffness matrix, and damping matrix at time t, the effective load at that time is calculated as follows:

[0061]

[0062] The effective stiffness matrix at this moment is as follows:

[0063]

[0064] The displacement at that moment is:

[0065]

[0066] The meshing torque at the gear pair node at that moment is:

[0067]

[0068] in, Given the base circle radius of the driving gear, the meshing torque of the gear pair node obtained by solving is substituted into the surrogate model established in step 1. The dynamic internal excitation of the internal / external meshing gear pair at that moment can be interpolated. Based on the meshing phase, the time-varying internal excitation is interpolated to solve the dynamic internal excitation of each meshing gear pair in the system. This is then substituted back into the overall motion differential equation of the planetary gear transmission system. The displacement, velocity, and acceleration at the node of each meshing gear pair are repeatedly calculated until the dynamic response of each meshing gear pair in the system converges. By traversing all meshing positions, the displacement, velocity, and acceleration of each meshing gear pair node in the system along the end face meshing line direction in a complete meshing cycle are obtained. At the same time, the dynamic contact stress and dynamic load sharing coefficient of each meshing gear pair are obtained.

[0069] Beneficial effects

[0070] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0071] This invention considers the impact of shaft deformation and tooth surface errors caused by gear manufacturing / modification on the three-dimensional dynamic contact characteristics of planetary gear transmission systems. Compared to previous calculation methods that only considered the flexibility of the internal gear ring, this approach is more realistic and yields more accurate results. This invention establishes a three-dimensional dynamic contact characteristic analysis model for planetary gear transmission systems that considers tooth surface errors. Based on this model, the dynamic response parameters of each meshing gear pair in the planetary gear transmission system under different operating conditions can be quickly calculated, taking into account the distribution of tooth surface errors. Furthermore, through extensive pre-calculation, a proxy model for the load-bearing contact analysis of internal / external meshing gear teeth considering tooth surface errors is established, significantly improving computational efficiency.

[0072] Additional aspects and advantages of the invention will be set forth in part in the description which follows, and in part will be obvious from the description, or may be learned by practice of the invention. Attached Figure Description

[0073] The above and / or additional aspects and advantages of the present invention will become apparent and readily understood from the description of the embodiments taken in conjunction with the following drawings, in which:

[0074] Figure 1 Flowchart for calculating the three-dimensional dynamic contact characteristics of the tooth surface of a planetary gear transmission system considering tooth surface errors;

[0075] Figure 2 This is a 3D model of a gear-shaft-bearing transmission system.

[0076] Figure 3 This is a dynamic model diagram of a planetary gear transmission system;

[0077] Figure 4 Extraction of the equivalent stiffness matrix and mass matrix of the casing;

[0078] Figure 5 This is a schematic diagram of the meshing dynamics model of a gear pair. Detailed Implementation

[0079] The embodiments of the present invention are described in detail below. These embodiments are exemplary and intended to explain the present invention, and should not be construed as limiting the present invention.

[0080] This embodiment proposes a method for analyzing the three-dimensional dynamic contact characteristics of the tooth surface of a planetary gear transmission system considering tooth surface errors, which specifically includes the following steps:

[0081] Step 1: Using the finite element substructure method, establish a proxy model for the contact analysis of internal / external meshing gear teeth considering tooth surface errors. Interpolation is then used to obtain the dynamic internal excitation of internal / external meshing under any gear pair meshing load. Specifically:

[0082] Step 1.1: Based on the finite element substructure method, extract the stiffness matrix at the casing-foundation position on the central wheel axle, and assemble it with the stiffness / mass matrices of the gear, shaft, and bearing to solve the initial meshing static load of the gear pair.

[0083] For each internal / external meshing gear pair in a planetary transmission system, it can be considered as a gear-shaft-bearing system, and the three-dimensional model is as follows: Figure 2 As shown, a finite element model of the shaft segment is established using Timoshenko beam elements. The sun gear, planet gears, internal gear ring, and planet carrier are modeled using the lumped mass method (see...). Figure 3 The mass / stiffness matrices of the above components are established and assembled, and the equivalent stiffness / mass matrix of the casing obtained by the finite element substructure method is combined (see...). Figure 4 Solve the static displacement equation to obtain the initial meshing static load on the tooth surface.

[0084] Step 1.2: Based on the finite element substructure method, obtain the tooth surface compliance matrix of the internal / external meshing gear pair considering wheel deformation. Combined with the initial meshing static load of the gear pair obtained in Step 1.1 and the tooth surface error terms caused by manufacturing / modification / shaft deformation, solve the dynamic internal excitation (time-varying meshing stiffness, comprehensive meshing error) of the internal / external meshing gear pair using the tooth surface bearing contact equation. Then, based on the meshing phase, obtain the dynamic internal excitation of each meshing internal / external gear pair through interpolation. The bearing contact analysis solution process is as follows:

[0085] Gear meshing unit such as Figure 5As shown, the tooth surface is unfolded into a two-dimensional plane with the meshing line direction and the tooth width direction as dimensions, and contact points are arranged. Based on the deformation coordination condition, load balance condition and contact condition at a certain contact point, the tooth surface bearing contact equation of the internal / external meshing gear pair at a certain meshing position considering system flexibility (casing and gear body flexibility) and tooth surface error (tooth surface error caused by manufacturing and tooth surface modification) is obtained as shown in Equation (1):

[0086] (1)

[0087] In the formula: Here, P represents the macroscopic deformation compliance matrix (i.e., wheel compliance, obtained using the finite element substructure method) of the possible contact points, and P is the external load vector. The local contact deformation at possible contact points is obtained using the line contact deformation formula. This represents the initial clearance (i.e., tooth surface error) at possible contact points. Let STE represent the remaining clearance at possible contact points, STE represent the static transmission error, and i represent the contact point number. The dynamic internal excitation of the gear pair (time-varying meshing stiffness and comprehensive meshing error) can be obtained by iteratively solving the above equations.

[0088] Step 1.3: Change the system input torque and repeat steps 1.1 and 1.2 to obtain the dynamic internal excitation of each internal / external meshing gear pair considering tooth surface error under different input torques, and establish a proxy model for the load-bearing contact analysis of internal / external meshing gear teeth.

[0089] By changing the external load vector P (i.e., the system input torque) and performing pre-calculation using load contact analysis, a proxy model for load contact analysis of internal / external meshing gear teeth considering tooth surface errors can be obtained. In other words, the established proxy model for load contact analysis can be used to quickly obtain the variation law of dynamic internal excitation (time-varying meshing stiffness, comprehensive meshing error) corresponding to any external load vector P through interpolation.

[0090] Step 2: Combining Timoshenko beam element theory, a dynamic model of the planetary gear transmission system considering shaft elements, bearing-casing elements, casing-foundation elements, internal meshing elements, external meshing elements, and planet carrier elements is established based on the generalized finite element method.

[0091] The general idea of ​​the generalized finite element method is to divide the planetary gear system into elements (shaft segment elements, bearing-casing elements, casing-foundation elements, planet carrier elements, and meshing elements), and combine the motion and mechanical characteristics of each element to first construct differential equations describing the dynamic behavior of each element. Then, the dynamic differential equations of the above elements are assembled to achieve accurate modeling and analysis of the dynamic response of the planetary gear system.

[0092] The specific implementation method is as follows:

[0093] Step 2.1: Treat the shaft element as a Timoshenko beam element and establish the stiffness / mass / damping matrix of the shaft element;

[0094] The differential equations of motion for the shaft element are constructed based on the Timoshenko beam element theory to simultaneously consider the effects of shear deformation and rotational inertia on the shaft deflection. The differential equations of motion are as follows:

[0095] (2)

[0096] Where: M s The mass matrix of the shaft element, C s The damping matrix of the shaft element, K s Let be the stiffness matrix of the shaft element. , , These represent the displacement, velocity, and acceleration at the node of the axis element at the current moment.

[0097] Step 2.2: Treat the bearing-casing unit, casing-foundation unit, and planetary carrier unit as spring-damping units, and establish the motion differential equations based on the stiffness, mass, and damping matrices of the bearing-casing unit, casing-foundation unit, and planetary carrier unit, respectively.

[0098] The bearing-casing unit is considered as a spring-damping unit, and its equation of motion is:

[0099] (3)

[0100] Where: M b For the mass matrix of the bearing-casing unit, C b For the damping matrix of the bearing-casing unit, K b Here is the stiffness matrix of the bearing-casing unit. , , These represent the displacement, velocity, and acceleration of the bearing-casing unit node at the current moment.

[0101] The casing-base unit is considered as a spring-damped unit, and its equation of motion is:

[0102] (4)

[0103] Where: M h For the mass matrix of the casing-basic unit, C h For the damping matrix of the casing-base unit, K h Here is the stiffness matrix of the casing-base element. The dynamic matrix of the aforementioned casing-base element can be obtained using the finite element substructure method. , , These represent the displacement, velocity, and acceleration at the casing-base unit node at the current moment.

[0104] The planetary carrier element is considered as a spring-damped element, and its differential equation of motion is:

[0105] (5)

[0106] Where: M c The mass matrix of the planetary carrier unit, K c Here is the stiffness matrix of the planetary carrier element. , These represent the displacement and acceleration at the node of the planetary carrier unit at the current moment.

[0107] Step 2.3: Establish the kinematic differential equations for the internal / external meshing gear pair unit;

[0108] Gear meshing unit such as Figure 5 As shown, the coordinates of the gear pair nodes are defined as follows:

[0109] (6)

[0110] The relative displacement of the gear pair along the normal line of meshing can be expressed as:

[0111] (7)

[0112] Where V is the projection vector of displacement in each direction towards the end face meshing line direction, which can be expressed by the following formula.

[0113] (8)

[0114] In the formula: r p r g The base circle radii of the master and driven gears; β b φ is the base circle helix angle, which is defined as positive when the gear is right-handed and negative when it is left-handed; φ is the angle between the end face meshing line and the positive direction of the y-axis.

[0115] According to Newton's second law, the kinematic differential equations of the internal / external meshing gear pair unit considering tooth surface errors are as follows:

[0116] (9)

[0117] (10)

[0118] The upper part corresponds to external meshing, and the lower part corresponds to internal meshing; where: m i (i = p, g) represent the masses of the driving and driven gears, respectively; Izi (i = p, g) are the moments of inertia of the driving and driven wheels about the z-axis, respectively; These represent the relative displacement and velocity at the meshing unit node along the normal meshing line at that moment; φ is the angle between the end face meshing line and the positive y-axis. This represents the normal composite meshing stiffness of the gear pair at that moment; c represents the combined normal meshing error of the gear pair at that moment; m The meshing damping of the gear pair can be calculated using the following formula:

[0119] (11)

[0120] In the formula: ζ is the meshing damping ratio, which is generally taken as 0.03~0.17, and is taken as 0.07 in this paper; The mean normal meshing stiffness of the gear pair; m eq,i = I zi / r i 2 Let r be the equivalent mass of gear i (i=p, g). i (i = p, g) are the pitch circle radii of the driving and driven wheels, respectively.

[0121] The matrix forms of equations (9) and (10) are as follows:

[0122] (12)

[0123] (13)

[0124] Where: M n (n = i, o) are the mass matrices of the internal / external meshing units, respectively; C n (n = i, o) are the damping matrices of the internal / external meshing units, respectively; K n (n = i, o) are the stiffness matrices of the internal / external meshing elements, respectively; e n (n = i, o) are the equivalent displacement column vectors of the comprehensive meshing error of the internal / external meshing units, respectively.

[0125] Step 2.4: Assemble the overall stiffness / mass / damping matrix, establish the overall motion differential equation of the planetary gear transmission system based on the motion relationship, and thus establish the dynamic model of the planetary gear transmission system.

[0126] By assembling the stiffness / damping / mass of the above units, the dynamic model of the planetary transmission system can be obtained, and its matrix form is as follows:

[0127] (14)

[0128] In the formula: M is the overall system mass matrix; C is the overall system damping matrix; Consider the overall stiffness matrix of the system during transient contact; , , is the column vector of displacement, velocity and acceleration of all nodes; e(t) is the column vector of equivalent displacement of comprehensive meshing error, which has a value only in each degree of freedom of the meshing element, and the elements in other degrees of freedom are all zero; F is the system external load vector.

[0129] Step 3: Combining the Newmark numerical integration method, the proxy model for the load-bearing contact analysis of internal / external meshing gear teeth considering tooth surface errors is coupled with the dynamic model of the planetary gear transmission system to establish a three-dimensional dynamic contact characteristic analysis model of the planetary gear transmission system tooth surface considering tooth surface errors and system flexibility, thereby realizing the analysis of the three-dimensional dynamic contact characteristics of the planetary gear transmission system tooth surface; including the following steps:

[0130] Step 3.1: Combining the dynamic internal excitation of each internal / external meshing gear pair obtained in Step 1, the Newmark numerical integration method is used to solve the overall motion differential equation of the system, thereby obtaining the displacement, velocity and acceleration of each internal / external meshing gear pair along the end face meshing line direction, and based on this, the dynamic meshing force of each internal / external meshing gear pair is obtained.

[0131] Step 3.2: Based on the dynamic meshing force of each internal / external meshing gear pair obtained in Step 3.1, determine the instantaneous contact state of each meshing tooth surface. Combined with the contact analysis proxy model of each internal / external meshing gear tooth established in Step 1, the dynamic internal excitation of the internal / external meshing gear pair corresponding to the dynamic meshing force is quickly obtained by interpolation.

[0132] Step 3.3: Based on the fixed-point iteration method, repeat steps 3.1 and 3.2 until the system response at the current meshing position converges, thereby obtaining the displacement, velocity and acceleration of each internal / external meshing gear pair along the end face meshing line direction at that meshing position;

[0133] Step 3.4: Repeat steps 3.1 to 3.3, traversing all meshing positions of each internal / external meshing gear pair in step 3.1, until the system response at all meshing positions converges, thereby obtaining the displacement, velocity, and acceleration of each internal / external meshing gear pair along the end face meshing line direction in the system under a complete meshing cycle, and simultaneously obtaining its tooth surface dynamic contact stress and dynamic load sharing coefficient.

[0134] The specific calculation process is as follows:

[0135] The system of equations (14) can be rewritten as follows:

[0136] (15)

[0137] in, .

[0138] Given the initial displacement of the system Initial velocity value The initial acceleration values ​​are obtained by solving:

[0139] (16)

[0140] Select integration step size ,parameter , Calculate the integration constant:

[0141]

[0142] Formed based on Newmark numerical integration method Given the system's mass matrix, stiffness matrix, and damping matrix at time t, the effective load at that time is calculated as follows:

[0143] (17)

[0144] The effective stiffness matrix at this moment is as follows:

[0145] (18)

[0146] The displacement at that moment is:

[0147] (19)

[0148] The meshing torque at the gear pair node at that moment is:

[0149] (20)

[0150] in, Let the base circle radius of the driving wheel be denoted as . Substitute the obtained meshing torque of the gear pair node into the surrogate model established in step 1. The dynamic internal excitation of the internal / external meshing gear pair at that moment can be obtained by interpolation. Based on the meshing phase, interpolate the time-varying internal excitation to solve the dynamic internal excitation of each meshing gear pair in the system. Substitute this back into equation (14) and repeat the calculation of the displacement, velocity, and acceleration at the node of each meshing gear pair until the dynamic response of each meshing gear pair in the system converges. By traversing all meshing positions, the displacement, velocity, and acceleration of each meshing gear pair node in the system along the end face meshing line direction in a complete meshing cycle can be obtained. At the same time, the dynamic contact stress and dynamic load sharing coefficient of each meshing gear pair tooth surface can be obtained.

[0151] Although embodiments of the present invention have been shown and described above, it is understood that the above embodiments are exemplary and should not be construed as limiting the present invention. Those skilled in the art can make changes, modifications, substitutions and variations to the above embodiments within the scope of the present invention without departing from the principles and spirit of the present invention.

Claims

1. A method for analyzing the three-dimensional dynamic contact characteristics of the tooth surface of a planetary gear transmission system considering tooth surface errors, characterized in that: Includes the following steps: Step 1: Combine the finite element substructure method to establish a proxy model for the contact analysis of internal / external meshing gear teeth considering tooth surface errors, and obtain the dynamic internal excitation of internal / external meshing under meshing load of any gear pair by interpolation; Step 2: Combining Timoshenko beam element theory, a dynamic model of the planetary gear transmission system considering shaft elements, bearing-casing elements, casing-foundation elements, internal meshing elements, external meshing elements, and planet carrier elements is established based on the generalized finite element method. Step 3: Combining the Newmark numerical integration method, the proxy model for the load-bearing contact analysis of internal / external meshing gear teeth considering tooth surface error is coupled with the dynamic model of the planetary gear transmission system to establish a three-dimensional dynamic contact characteristic analysis model of the planetary gear transmission system tooth surface considering tooth surface error and system flexibility, thereby realizing the analysis of the three-dimensional dynamic contact characteristics of the planetary gear transmission system tooth surface.

2. The method for analyzing the three-dimensional dynamic contact characteristics of the tooth surface of a planetary gear transmission system considering tooth surface errors, as described in claim 1, is characterized in that: Step 1 includes the following processes: Step 1.1: Based on the finite element substructure method, extract the stiffness matrix at the casing-foundation position on the central wheel axle, and assemble it with the stiffness / mass matrices of the gear, shaft, and bearing to solve the initial meshing static load of the gear pair; Step 1.2: Based on the finite element substructure method, obtain the tooth surface compliance matrix of the internal / external meshing gear pair considering the deformation of the gear body. Combine the initial meshing static load of the gear pair obtained in Step 1.1 and the tooth surface error terms caused by manufacturing, modification, and shaft deformation, use the tooth surface bearing contact equation to solve the dynamic internal excitation of the internal / external meshing gear pair, and obtain the dynamic internal excitation of each meshing internal / external gear pair by interpolation based on the meshing phase. Step 1.3: Change the system input torque and repeat steps 1.1 and 1.2 to obtain the dynamic internal excitation of each internal / external meshing gear pair considering tooth surface error under different input torques, and establish a proxy model for the load-bearing contact analysis of internal / external meshing gear teeth.

3. The method for analyzing the three-dimensional dynamic contact characteristics of the tooth surface of a planetary gear transmission system considering tooth surface errors, as described in claim 2, is characterized in that: In step 1.2, the tooth surface bearing contact equation is: In the formula: Let P be the macroscopic deformation compliance matrix of the possible contact points, and P be the external load vector. For local contact deformation at possible contact points, The initial gap for possible contact points. The remaining clearance at possible contact points is denoted by STE, the static transmission error is denoted by i, and the contact point number is denoted by i. The dynamic internal excitation of the gear pair can be obtained by iteratively solving the tooth surface bearing contact equation.

4. The method for analyzing the three-dimensional dynamic contact characteristics of the tooth surface of a planetary gear transmission system considering tooth surface errors according to claim 1, characterized in that: Step 2 includes the following processes: Step 2.1: Treat the shaft element as a Timoshenko beam element and establish the stiffness / mass / damping matrix of the shaft element; Step 2.2: Treat the bearing-casing unit, casing-foundation unit, and planetary carrier unit as spring-damping units, and establish the motion differential equations based on the stiffness, mass, and damping matrix of the bearing-casing unit, casing-foundation unit, and planetary carrier unit, respectively. Step 2.3: Establish the kinematic differential equations for the internal / external meshing gear pair unit; Step 2.4: Assemble the overall stiffness / mass / damping matrix, establish the overall motion differential equation of the planetary gear transmission system based on the motion relationship, and thus establish the dynamic model of the planetary gear transmission system.

5. The method for analyzing the three-dimensional dynamic contact characteristics of the tooth surface of a planetary gear transmission system considering tooth surface errors according to claim 1, characterized in that: In step 2.1, the differential equation of motion for the shaft element is constructed based on the Timoshenko beam element theory to simultaneously consider the effects of shear deformation and rotational inertia on the shaft deflection. Its differential equation of motion is: Where: M s The mass matrix of the shaft element, C s The damping matrix of the shaft element, K s Let be the stiffness matrix of the shaft element. , , These represent the displacement, velocity, and acceleration at the node of the axis element at the current moment; In step 2.2, the bearing-casing unit is treated as a spring-damping unit, and its equation of motion is: Where: M b For the mass matrix of the bearing-casing unit, C b For the damping matrix of the bearing-casing unit, K b Here is the stiffness matrix of the bearing-casing unit. , , These represent the displacement, velocity, and acceleration of the bearing-casing unit node at the current moment; The casing-base unit is considered as a spring-damped unit, and its equation of motion is: Where: M h For the mass matrix of the casing-basic unit, C h For the damping matrix of the casing-base unit, K h Here is the stiffness matrix of the casing-base unit. , , These are the displacement, velocity, and acceleration at the casing-base unit node at the current moment; The planetary carrier element is considered as a spring-damped element, and its differential equation of motion is: Where: M c The mass matrix of the planetary carrier unit, K c Here is the stiffness matrix of the planetary carrier element. , These represent the displacement and acceleration at the node of the planetary carrier unit at the current moment.

6. The method for analyzing the three-dimensional dynamic contact characteristics of the tooth surface of a planetary gear transmission system considering tooth surface errors according to claim 1, characterized in that: In step 2.3, the coordinates of the gear pair node of the gear meshing unit are defined as follows: The relative displacement of the gear pair along the normal line of meshing can be expressed as: Where V is the projection vector of displacement in each direction towards the end face meshing line direction, expressed by the following formula. In the formula: r p r g The base circle radii of the master and driven gears; β b φ is the base circle helix angle, which is defined as positive when the gear is right-handed and negative when it is left-handed; φ is the angle between the end face meshing line and the positive direction of the y-axis; According to Newton's second law, the kinematic differential equations of the internal / external meshing gear pair unit considering tooth surface errors are as follows: The upper part corresponds to external meshing, and the lower part corresponds to internal meshing; where: m i (i = p, g) represent the masses of the driving and driven gears, respectively; I zi (i = p, g) are the moments of inertia of the driving and driven wheels about the z-axis, respectively; These represent the relative displacement and velocity at the meshing unit node along the normal meshing line at that moment; φ is the angle between the end face meshing line and the positive y-axis; This represents the normal composite meshing stiffness of the gear pair at that moment; c represents the combined normal meshing error of the gear pair at that moment; m The meshing damping of the gear pair is calculated using the following formula: In the formula: ζ is the meshing damping ratio; The mean normal meshing stiffness of the gear pair; m eq,i = I zi / r i 2 Let r be the equivalent mass of gear i (i=p,g). i (i = p, g) are the pitch circle radii of the driving and driven wheels, respectively.

7. The method for analyzing the three-dimensional dynamic contact characteristics of the tooth surface of a planetary gear transmission system considering tooth surface errors according to claim 1, characterized in that: Step 3 includes the following processes: Step 3.1: Combining the dynamic internal excitation of each internal / external meshing gear pair obtained in Step 1, the Newmark numerical integration method is used to solve the overall motion differential equation of the system, thereby obtaining the displacement, velocity and acceleration of each internal / external meshing gear pair along the end face meshing line direction, and based on this, the dynamic meshing force of each internal / external meshing gear pair is obtained. Step 3.2: Based on the dynamic meshing force of each internal / external meshing gear pair obtained in Step 3.1, determine the instantaneous contact state of each meshing tooth surface. Combined with the contact analysis proxy model of each internal / external meshing gear tooth established in Step 1, the dynamic internal excitation of the internal / external meshing gear pair corresponding to the dynamic meshing force is obtained by interpolation. Step 3.3: Based on the fixed-point iteration method, repeat steps 3.1 and 3.2 until the system response at the current meshing position converges, thereby obtaining the displacement, velocity and acceleration of each internal / external meshing gear pair along the end face meshing line direction at that meshing position; Step 3.4: Repeat steps 3.1 to 3.3, traversing all meshing positions of each internal / external meshing gear pair in step 3.1, until the system response at all meshing positions converges, obtaining the displacement, velocity, and acceleration of each internal / external meshing gear pair along the end face meshing line direction in the system under a complete meshing cycle, and simultaneously obtaining its tooth surface dynamic contact stress and dynamic load sharing coefficient.

8. The method for analyzing the three-dimensional dynamic contact characteristics of the tooth surface of a planetary gear transmission system considering tooth surface errors according to claim 7, characterized in that: The specific calculation process for step 3 is as follows: The overall kinematic differential equation of the planetary gear transmission system can be rewritten as follows: in, , Given the initial displacement of the system Initial velocity value The initial acceleration values ​​are obtained by solving: Select integration step size ,parameter , Calculate the integration constant: Formed based on Newmark numerical integration method Given the system's mass matrix, stiffness matrix, and damping matrix at time t, the effective load at that time is calculated as follows: The effective stiffness matrix at this moment is as follows: The displacement at that moment is: The meshing torque at the gear pair node at that moment is: in, Given the base circle radius of the driving gear, the meshing torque of the gear pair node obtained by solving is substituted into the surrogate model established in step 1. The dynamic internal excitation of the internal / external meshing gear pair at that moment can be interpolated. Based on the meshing phase, the time-varying internal excitation is interpolated to solve the dynamic internal excitation of each meshing gear pair in the system. This is then substituted back into the overall motion differential equation of the planetary gear transmission system. The displacement, velocity, and acceleration at the node of each meshing gear pair are repeatedly calculated until the dynamic response of each meshing gear pair in the system converges. By traversing all meshing positions, the displacement, velocity, and acceleration of each meshing gear pair node in the system along the end face meshing line direction in a complete meshing cycle are obtained. At the same time, the dynamic contact stress and dynamic load sharing coefficient of each meshing gear pair are obtained.

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