A dynamic modeling method of differential drive non-circular gear transmission system

Abstract

The application relates to the field of non-circular gear transmission non-linear dynamics, and discloses a dynamic modeling method of a differential driving non-circular gear transmission system, which comprises the following steps: firstly, acquiring master and slave wheel pitch curves and determining the curvature radius of each point, establishing the corresponding relationship between the rotation angle and the arc length to determine the actual meshing point position under any input rotation angle, then calculating the equivalent number of teeth based on the point curvature radius, respectively calculating the stiffness components corresponding to four deformations of Hertz contact, bending, shearing and axial compression by using the potential energy method, and synthesizing the single-tooth meshing stiffness according to the series flexibility relationship; finally, introducing the real time-varying meshing stiffness obtained by point-by-point calculation into the system dynamics model, the application can establish the time-varying meshing stiffness reflecting the actual geometric meshing process of the non-circular gear, and introduce the time-varying meshing stiffness into the non-linear dynamics model of the differential driving non-circular gear transmission system, so that the system stability and the prediction accuracy of the beating characteristics are improved.

Description

Technical Field

[0001] This invention relates to the field of nonlinear dynamics of non-circular gear transmissions, and in particular to a dynamic modeling method for a differential drive non-circular gear transmission system. Background Technology

[0002] Non-circular gears can achieve predetermined instantaneous transmission ratio changes through non-uniform pitch curves, making them suitable for mechanical systems requiring differential, variable speed, or periodic motion modulation. For differential drive devices such as rotary opposed piston engines, two sets of non-circular gear pairs typically need to achieve continuous variable transmission ratio output within a limited space, and their operating state directly affects transmission accuracy, vibration levels, and service reliability. Compared to ordinary cylindrical gears, non-circular gears exhibit variations in pitch curve curvature radius, instantaneous meshing position, and transmission radius with rotational angle during meshing, leading to periodic changes in meshing stiffness and damping. Simultaneously, tooth backlash, machining and assembly errors, and load fluctuations further introduce piecewise nonlinearity and periodic excitation, potentially causing complex dynamic behaviors such as bifurcation, chaos, disengagement, and impact shocks.

[0003] In related technologies, for example, Liu Dawei, Tan Wanxin, Jin Xin and Ren Tingzhi, in their article "Nonlinear Vibration Model and Impact Characteristics of Variable Speed ​​Elliptical Gear Pump", established a nonlinear impact dynamics model that considers tooth elastic deformation, tooth backlash, static transmission error, meshing damping and cyclic load for a transmission system containing two-stage non-circular gear meshing pairs.

[0004] However, to simplify calculations, the time-varying meshing stiffness of both stages of non-circular gear meshing pairs is expressed as a simple harmonic function containing only the average component and the fundamental frequency component. For differential drive non-circular gear transmission systems, the radius of curvature, meshing position, and number of meshing teeth of the two sets of non-circular gear pairs change periodically under different input angles and installation phases. If the empirical simple harmonic function is still used to characterize the time-varying meshing stiffness, it is difficult to accurately reflect the actual meshing excitation and its influence on the nonlinear response of the system, thus leading to insufficient accuracy in predicting the system's stable operating range, bifurcation behavior, and impact threshold. Summary of the Invention

[0005] To address the aforementioned technical problems, this invention provides a dynamic modeling method for differential drive non-circular gear transmission systems. This method can establish a time-varying meshing stiffness that reflects the actual geometric meshing process of non-circular gears and introduce it into the nonlinear dynamic model of the differential drive non-circular gear transmission system to improve system stability and the accuracy of impact characteristic prediction.

[0006] This invention provides a dynamic modeling method for a differential drive non-circular gear transmission system, comprising the following steps: Obtain the pitch curves of the driving and driven gears in a differential drive non-circular gear transmission system, and determine the radius of curvature of each point on the driving and driven gears based on the pitch curves; Based on the pitch curves of the driving and driven gears, a correspondence between the rotation angle of the driving / driven gear and the arc length of the pitch curve is established. Based on this relationship, the current meshing point position is determined, and the curvature radii of the driving and driven gears at the current meshing point are obtained. The equivalent number of teeth of the two gears at the meshing point is calculated respectively. The total elastic deformation of the individual teeth involved in meshing under force is decomposed into Hertzian contact deformation, bending deformation, shear deformation, and axial compression deformation. Using the equivalent number of teeth and the tooth profile geometric parameters of the current meshing point, the stiffness components corresponding to the four deformations are calculated respectively. The components are synthesized according to the series compliance relationship to obtain the time-varying meshing stiffness of the gear pair at the current meshing position. Based on the pitch curve and time-varying meshing stiffness of the gears, a dynamic model of the non-circular gear transmission system of the differential drive mechanism is established.

[0007] Optionally, establish the correspondence between the rotation angle of the driving / driven wheel and the arc length of the pitch curve, including the following steps: Based on the non-slip conjugate meshing condition, the instantaneous relationship between the small rotation angle d1 of the driving wheel and the small rotation angle d2 of the driven wheel is established; Integrating the instantaneous relationship yields the global kinematic mapping between the polar angle 1 of the active wheel pitch curve and the polar angle 2 of the driven wheel pitch curve. The meshing point position and corresponding pitch curve arc length of the two gears under any input rotation angle are determined by global kinematic mapping. The meshing point position and corresponding pitch curve arc length of the driven gear are expressed as follows: , in, m n The normal module of the driven wheel is expressed in mm. r 2 represents the radial direction of the driven wheel pitch curve, in mm; the driving wheel is the same as the driven wheel.

[0008] Optionally, calculate the equivalent number of teeth for each gear at the meshing point, including the following steps: Obtain the radius of curvature of the driving gear pitch curve at the current meshing point. r 1. Radius of curvature of the driven gear pitch curve r 2. Calculate the equivalent number of teeth according to the following relationship: , in, Z r1 This represents the number of teeth on the driving gear under the equivalent circular gear condition. Z r2 This represents the number of teeth on the driven gear under the equivalent circular gear condition. m nThe normal module of the driven wheel is expressed in mm. The non-circular gear in its current meshing position is equivalent to having a number of teeth. Z r1 and Z r2 The hypothetical circular gear is used to determine the subsequent tooth profile geometry parameters.

[0009] Optionally, the components are synthesized according to the series compliance relationship to obtain the time-varying meshing stiffness of the tooth pair at the current meshing position, including the following steps: The time-varying meshing stiffness of a single tooth is calculated according to the following compliance series relationship: , in, k h For Hertzian contact stiffness, k b For bending stiffness, k a For radial compressive stiffness, k s The shear stiffness is expressed in N / mm. Subscript 1 indicates the driving wheel and subscript 2 indicates the driven wheel. For the double-tooth meshing range, the stiffness of the teeth that participate in meshing simultaneously is superimposed to finally obtain the time-varying meshing stiffness of the non-circular gear that varies with rotation angle or time.

[0010] Optional, Hertzian contact stiffness k h Bending stiffness k b Radial compressive stiffness k a and shear stiffness k s Calculate using the following formulas respectively: , , , , in, α Let be the angle between the direction of the force at any contact point on the involute and the Y-axis; α 0 represents the pressure angle; α 1 represents the angle between the force direction at the meshing point and the Y-axis; α 2 is half of the base tooth angle; α 3 is half of the half-tooth angle corresponding to the root circle; the unit of included angle or angle is °; Z r2 This represents the number of teeth on the driven gear under the equivalent circular gear condition. EThis represents the elastic modulus of the material, expressed in MPa. L This refers to the tooth width, in mm. v The material's Poisson's ratio.

[0011] Optionally, a dynamic model of the non-circular gear transmission system of the differential drive mechanism is established, including the following steps: Construct a differential coupling structure with two sets of coaxial input meshing pairs; Based on the torsional angular displacement of each gear and the corresponding base circle radius, the equivalent displacement of each gear along the meshing line direction is obtained. Based on the equivalent displacement of the two gears in each meshing pair and the comprehensive error of the meshing pair, the relative displacement of the meshing pair is obtained. Based on the time-varying meshing stiffness and relative displacement of each meshing pair, the meshing force of the meshing pair is obtained. Based on the differential coupling structure, the torsional angular displacement of each gear, and the meshing force of each meshing pair, the torsional dynamic equations of each gear are established, and the dynamic model of the non-circular gear transmission system of the differential drive mechanism is obtained.

[0012] Optionally, the dynamic model of the non-circular gear transmission system of the differential drive mechanism is as follows: , In the formula, I 1 represents the moment of inertia of the first driven wheel; I 4 represents the moment of inertia of the second driven wheel; I s Input the coaxial moment of inertia; the unit of moment of inertia is kg•m. 2 ; The vibration acceleration indicates the direction of torsion of the first driven wheel; The vibration acceleration indicates the direction of torsion of the second driven wheel; The vibration acceleration represents the torsional direction of the coaxial axis; the unit of vibration acceleration is rad / s. 2 ; f This represents the fixed phase angle between the first and second driving wheels, expressed in rad. T in This indicates the torque acting on the coaxial axis; T L1 , T L4 The torque is the load torque at both ends; the unit of torque is N•m. Q 21 This represents the meshing force between the first driving wheel and the first driven wheel; Q 34 This represents the meshing force between the second driving wheel and the second driven wheel; the unit of meshing force is N. r b1 ( i 1) The first driven wheel at the turning angle iThe transient base circle radius at point 1; r b2 ( i s (The first driving wheel is at the corner) θs The transient base circle radius at that point; r b3 ( i s (The second driving wheel is at the corner) θs The transient base circle radius at that point; r b4 ( i 4) For the second driven wheel at the turning angle i 4 The transient base circle radius at the specified location; the unit for transient base circle radius is mm.

[0013] Optionally, the meshing force is expressed as: , in, C e21 This indicates the damping between the first driving wheel and the first driven wheel; C e34 This indicates the damping between the second driving wheel and the second driven wheel; the unit of damping is N•s / mm. This indicates the speed between the first driving wheel and the first driven wheel; This indicates the speed between the second driving wheel and the second driven wheel; the unit of speed is mm / s. K 21 This indicates the time-varying meshing stiffness between the first driving wheel and the first driven wheel; K 34 This indicates the time-varying meshing stiffness between the second driving wheel and the second driven wheel; the unit of time-varying meshing stiffness is N / mm. f ( x 21 () represents the displacement function of the first driving wheel and the first driven wheel; f ( x 34 ) represents the displacement function between the second driving wheel and the second driven wheel.

[0014] Optionally, the following steps may also be included: The dimensionless relative displacement of each meshing pair is obtained by considering the relative displacement of each meshing pair and its corresponding tooth flank clearance. The dimensionless time is obtained based on the time variable and the system's natural frequency; Based on dimensionless relative displacement, dimensionless time, and system parameters, the dimensionless damping ratio, dimensionless time-varying stiffness, and dimensionless excitation force are obtained, and the torsional dynamics equation is then transformed into a dimensionless vibration differential equation.

[0015] Optionally, after establishing the dynamic model of the non-circular gear transmission system of the differential drive mechanism, the following steps are also included: Input the various parameters of the transmission system into the dynamic model, and solve the dynamic model using the numerical integration method to obtain the relative displacement, dynamic meshing force and nonlinear dynamic response of the two meshing pairs; Based on the relationship between relative displacement and tooth flank clearance, the striking state and striking threshold are determined.

[0016] The technical solution provided by the embodiments of the present invention has the following advantages compared with the prior art: This invention provides a dynamic modeling method for a differential drive non-circular gear transmission system. First, the pitch curves of the driving and driven gears are obtained, and the radius of curvature at each point is determined. Based on this, a precise correspondence between the rotation angle and the arc length of the pitch curve is established to determine the actual meshing point position under any input rotation angle. Then, based on the actual radius of curvature of the two gears at the meshing point, the equivalent number of teeth is calculated. This equivalent number of teeth is then substituted into the potential energy method to calculate the stiffness components corresponding to four deformations: Hertzian contact, bending, shearing, and axial compression. The single-tooth pair meshing stiffness at this meshing position is synthesized according to the series compliance relationship, thereby directly mapping the geometric characteristic of the non-circular gear's radius of curvature continuously changing with the rotation angle to the true time-varying law of meshing stiffness. Finally, this time-varying meshing stiffness obtained based on instantaneous curvature equivalence is introduced into the dynamic model of the differential drive mechanism's non-circular gear transmission system, enabling the dynamic equations to truly reflect the influence of the periodic changes in the radius of curvature, meshing position, and number of meshing teeth of the two sets of non-circular gear pairs on the system excitation under different input rotation angles and installation phases. Based on the conjugate meshing relationship of non-circular gears and the curvature variation law of pitch curves, this invention uses the instantaneous curvature equivalent circular gear method to calculate the time-varying meshing stiffness of non-circular gears during meshing. This avoids the excitation distortion caused by simplifying the meshing stiffness to a constant or empirical periodic function, and improves the accuracy of non-circular gear dynamic modeling. Attached Figure Description

[0017] To more clearly illustrate the embodiments and design schemes of the present invention, the accompanying drawings required for this embodiment will be briefly described below. The drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0018] Figure 1 A flowchart illustrating the overall process of a dynamic modeling method for a differential drive non-circular gear transmission system provided in this embodiment of the invention; Figure 2 This is a schematic diagram of the structural configuration of a differential drive non-circular gear transmission system provided in an embodiment of the present invention; Figure 3This is a schematic diagram of the calculation results of the time-varying meshing stiffness of non-circular gears provided in an embodiment of the present invention; Figure 4 Figure 1 is a schematic diagram of the nonlinear response results of the differential drive non-circular gear transmission system provided in the embodiment of the present invention. Figure 2(a) is a diagram of the nonlinear influence of rotational speed on the system, and Figure 3(b) is a diagram of the nonlinear influence of load torque on the system. Figure 5 This is a schematic diagram of the impact state discrimination result of the differential drive non-circular gear transmission system provided in the embodiment of the present invention. In the figure, (a) is the impact state; (b) is the effect of tooth backlash on the impact threshold speed; and (c) is the effect of load torque on the impact threshold speed. Detailed Implementation

[0019] To enable those skilled in the art to better understand and implement the technical solutions of the present invention, the present invention will be described in detail below with reference to the accompanying drawings and specific embodiments. The following embodiments are only used to more clearly illustrate the technical solutions of the present invention and should not be construed as limiting the scope of protection of the present invention.

[0020] The technical solutions provided by the various embodiments of the present invention will be described in detail below with reference to the accompanying drawings.

[0021] First, embodiments of the present invention provide a dynamic modeling method for a differential drive non-circular gear transmission system, specifically as follows: Figure 1 As shown, it includes the following steps: Step (1): Establish the pitch curve model of the differential drive non-circular gear transmission system. The differential mechanism is equivalent to four non-circular gears, and the parameters of the four non-circular gears are exactly the same. Therefore, this section only introduces the calculation of the pitch curve. The polar coordinate equation of the pitch curve of the non-circular driving wheel of the differential drive mechanism is:

[0022] (1) In the formula: A The length of the drive wheel's half-shaft is in mm; k 1 represents the eccentricity of the drive wheel; n 1 represents the order of the driving wheel; 1 represents the pitch curve polar angle of the driving wheel, in rad. r 1 represents the radial direction of the driving wheel pitch curve, in mm; p 1 represents the semi-groove diameter of the drive wheel joint curve, in mm.

[0023] Using the pitch curve of the driving wheel as a known condition, the pitch curve expression of the meshing non-circular driven wheel of the same order can be further obtained. When the driving wheel completes one full rotation cycle, i.e., when 1 changes from 0 to 2π, the cumulative rotation angle of the driven wheel should be exactly equal to one fundamental pitch curve cycle 2π / n2. To ensure that the starting point and ending point of the driven wheel pitch curve coincide and form a continuous closed profile, the closure constraint condition shown in equation (2) needs to be satisfied:

[0024] (2) In the formula: a This is the center distance between the driving wheel and the driven wheel, in mm. n 2 represents the order of the driven gear; k 2 represents the eccentricity of the driven wheel. r 2 represents the radial direction of the driven wheel pitch curve, in mm. p 2 is the semi-nominal diameter of the driven wheel pitch curve in mm, and 2 is the polar angle of the driven wheel pitch curve in rad.

[0025] In the design of non-circular gear pitch curves, ensuring a uniform tooth distribution along the pitch curve is crucial. This guarantees smooth gear meshing and reduces vibration and stress. To ensure a consistent tooth pitch throughout the entire pitch curve cycle, non-circular gears employ a tooth distribution method with equal arc lengths along the pitch curve. During a complete closed loop of the pitch curve, the total arc length of the pitch curve should be exactly equal to the number of teeth z2 of the driven gear and the single normal tooth pitch π. m n The product of . Therefore, the basic condition for uniform tooth distribution is:

[0026] (3) In the formula: m n Z1 is the normal module of the driven gear, in mm, and Z2 is the number of teeth on the driven gear.

[0027] The center distance between the driving and driven gears in the meshing theory a for: (4) In the formula: a The unit is mm; n 1 represents the order of the driving wheel; n 2 represents the order of the driven gear; k 2 represents the eccentricity of the driven wheel. A This refers to the long half-shaft of the drive wheel, measured in mm.

[0028] The equation of the pitch curve of the driven wheel is: (5) Equation (5) reflects the correspondence between the small rotation angles of the driving gear and the driven gear under the condition of non-slip conjugate meshing. That is, at any instantaneous meshing position, the rotation angle increment of the driven gear corresponding to the small angle rotated by the driving gear is jointly determined by the radial directions of the instantaneous pitch curves of the two gears. Integrating Equation (5) further yields the overall kinematic relationship between the polar angle 1 of the driving gear pitch curve and the polar angle 2 of the driven gear pitch curve.

[0029] Integrating equation (5) yields the following equation: (6) Step (2): Establish the time-varying meshing stiffness model of the instantaneous curvature equivalent circular gear, specifically as follows: Based on the pitch curves of the driving and driven gears, a correspondence is established between the rotation angle of the driving / driven gear and the arc length of the pitch curve. This relationship determines the current meshing point position, and the radii of curvature of the driving and driven gears at the current meshing point are obtained. The equivalent number of teeth for each gear at this meshing point is then calculated. The total elastic deformation of a single meshing tooth under load is decomposed into Hertzian contact deformation, bending deformation, shear deformation, and axial compression deformation. Using the equivalent number of teeth and the tooth profile geometry parameters at the current meshing point, the stiffness components corresponding to the four deformations are calculated. These components are then synthesized according to a series compliance relationship to obtain the time-varying meshing stiffness of the gear pair at the current meshing position.

[0030] Step (3): Establish a nonlinear dynamic model of the differential drive non-circular gear transmission system. Specifically, based on the pitch curve of the gear and the time-varying meshing stiffness, establish a dynamic model of the differential drive mechanism non-circular gear transmission system.

[0031] This invention provides a dynamic modeling method for a differential drive non-circular gear transmission system. First, the pitch curves of the driving and driven gears are obtained, and the radius of curvature at each point is determined. Based on this, a precise correspondence between the rotation angle and the arc length of the pitch curve is established to determine the actual meshing point position under any input rotation angle. Then, based on the actual radius of curvature of the two gears at the meshing point, the equivalent number of teeth is calculated. This equivalent number of teeth is then substituted into the potential energy method to calculate the stiffness components corresponding to four deformations: Hertzian contact, bending, shearing, and axial compression. The single-tooth pair meshing stiffness at this meshing position is synthesized according to the series compliance relationship, thereby directly mapping the geometric characteristic of the non-circular gear's radius of curvature continuously changing with the rotation angle to the true time-varying law of meshing stiffness. Finally, this time-varying meshing stiffness obtained based on instantaneous curvature equivalence is introduced into the dynamic model of the differential drive mechanism's non-circular gear transmission system, enabling the dynamic equations to truly reflect the influence of the periodic changes in the radius of curvature, meshing position, and number of meshing teeth of the two sets of non-circular gear pairs on the system excitation under different input rotation angles and installation phases. Based on the conjugate meshing relationship of non-circular gears and the curvature variation law of pitch curves, this invention uses the instantaneous curvature equivalent circular gear method to calculate the time-varying meshing stiffness of non-circular gears during meshing. This avoids the excitation distortion caused by simplifying the meshing stiffness to a constant or empirical periodic function, and improves the accuracy of non-circular gear dynamic modeling.

[0032] Optionally, establish the correspondence between the rotation angle of the driving / driven wheel and the arc length of the pitch curve, including the following steps: Based on the non-slip conjugate meshing condition, establish the small rotation angle of the driving wheel. d 1. Small rotation angle with the driven wheel d The instantaneous relationship between 2; Integrating the instantaneous relationship yields the global kinematic mapping between the polar angle 1 of the active wheel pitch curve and the polar angle 2 of the driven wheel pitch curve. The meshing point position of the two gears and the corresponding pitch curve arc length under any input rotation angle are determined by global kinematic mapping.

[0033] Complete steps (2): Establishing a time-varying meshing stiffness model of an equivalent circular gear with instantaneous curvature, including the following steps: set up Z r1 This represents the number of teeth on the driving gear under the equivalent circular gear condition. Z r2 Given the number of teeth of the driven gear under the equivalent circular gear condition, then the radius of curvature of its pitch curve relative to the driving gear at the current meshing point is... r 1. Radius of curvature of the driven gear pitch curve r The following relationship exists between the two: (7) The relationship between the tooth position angle and the arc length of the non-circular pitch curve can be expressed as: (8) The total deformation of non-circular gear teeth can be decomposed into Hertzian contact deformation, bending deformation, axial compressive deformation, and shear deformation, with corresponding potential energies of Hertzian potential energy, respectively. U h The unit is N•mm, bending potential energy. U b The unit is N•mm, axial compressive potential energy. U a The units are N•mm and shear potential energy. U s The unit is N•mm.

[0034] (9) In the formula: k h This refers to Hertzian contact stiffness, expressed in N / mm. k b This refers to bending stiffness, expressed in N / mm. k a Radial compressive stiffness, in N / mm; k s Shear stiffness, in N / mm; F The meshing force is expressed in N (newtons). F a This represents the X-axis component of the meshing force, in N. F b This represents the Y-axis component of the meshing force, in N. d This is the effective distance from the gear meshing point to the tooth root, in mm. x This represents the distance from any point of contact on the involute to the tooth root, in mm. h This is the distance from the meshing point to the center line of the gear, in mm. I x Here are the moments of inertia, in mm. 4 ; A x Distance from tooth root x The cross-sectional areas, in mm², are respectively. 2 ; E This represents the elastic modulus of the material, expressed in MPa. G This indicates the shear modulus of the material, expressed in GPa.

[0035] (10) In the formula: L This refers to the tooth width, in mm. h x This represents the radial distance from any meshing point on the involute to the centerline of the gear, expressed in mm.

[0036] Hertzian contact stiffness can be expressed as: (11) In the formula, v The material's Poisson's ratio.

[0037] Calculate bending stiffness k b Shear stiffness k s and radial compressive stiffness k a : (12) (13) (14) In the formula: α The angle between the direction of the force at any contact point on the involute and the Y-axis is expressed in degrees. α 0 represents the pressure angle, measured in degrees (°). α 1 represents the angle between the force direction at the meshing point and the Y-axis, in degrees. α 2 is half of the base tooth angle, in degrees; α 3 is half of the half-tooth angle corresponding to the root circle, in degrees.

[0038] The curvature of a non-circular meshing gear pair at the instantaneous pitch curve is equivalent to the curvature of a corresponding circular gear. The comprehensive meshing stiffness of a single gear pair can be expressed as: (15) In this context, subscript '1' indicates the driving wheel and '2' indicates the driven wheel.

[0039] like Figure 3 As shown, this embodiment of the invention, based on the characteristic that the curvature of the non-circular gear pitch curve changes with rotation angle, employs the instantaneous curvature equivalent circular gear method. This method equates the non-circular gear at different meshing positions to a circular gear with the same radius of curvature, and determines the instantaneous equivalent number of teeth accordingly. Then, the potential energy method is used to calculate the Hertzian contact stiffness, bending stiffness, shear stiffness, and axial compressive stiffness, respectively, and the stiffnesses of each component are synthesized to obtain the comprehensive meshing stiffness of a single tooth pair. For the double-tooth meshing interval, the stiffnesses of the tooth pairs simultaneously participating in meshing are superimposed to finally obtain the time-varying meshing stiffness of the non-circular gear that varies with rotation angle or time.

[0040] Optionally, a dynamic model of the non-circular gear transmission system of the differential drive mechanism is established, including the following steps: like Figure 2As shown, the transmission system in this embodiment of the invention includes a first driven gear 1, a first driving gear 2, a second driving gear 3, and a second driven gear 4. The first driving gear 2 and the second driving gear 3 are mounted on the same input shaft and rotate synchronously. The first driven gear 1 and the second driven gear 4 serve as output gears on both sides. The first driving gear 2 and the first driven gear 1 form a first non-circular gear meshing pair, and the second driving gear 3 and the second driven gear 4 form a second non-circular gear meshing pair. A fixed phase difference is set between the two sets of meshing pairs to form a differential coupling transmission relationship.

[0041] Based on the torsional angular displacement of each gear and the corresponding base circle radius, the equivalent displacement of each gear along the meshing line direction is obtained. Based on the equivalent displacement of the two gears in each meshing pair and the comprehensive error of the meshing pair, the relative displacement of the meshing pair is obtained. Based on the time-varying meshing stiffness and relative displacement of each meshing pair, the meshing force of the meshing pair is obtained. Based on the differential coupling structure, the torsional angular displacement of each gear, and the meshing force of each meshing pair, the torsional dynamic equations of each gear are established, and the dynamic model of the non-circular gear transmission system of the differential drive mechanism is obtained.

[0042] Specifically, the complete steps (3) include the following steps: To analyze the nonlinear characteristics of a non-circular gear transmission system in a differential drive mechanism, its dynamic model must first be established. The relative displacement along the meshing line of the differential elliptical gear transmission system can be expressed as:

[0043] (16) In the formula: x 21 This indicates the relative displacement between the first driving wheel 2 and the first driven wheel 1, in mm. x 34 This indicates the relative displacement between the second driving wheel 3 and the second driven wheel 4, in mm. e 21 ( t ) represents the time-varying transmission error function between the first driving wheel 2 and the first driven wheel 1. e 34 ( t ) represents the time-varying transmission error function between the second driving wheel 3 and the second driven wheel 4. r b1 ( i 1) The first driven wheel 1 at the turning angle i The transient base circle radius at point 1, in mm. r b2 ( i s ) is the first driving wheel 2 at the corner θs The transient base circle radius at that location, in mm. rb3 ( i s The second driving wheel 3 is at the corner. θs The transient base circle radius at that location, in mm. r b4 ( i 4) The second driven wheel 4 at the turning angle i 4 The transient base circle radius at that point, in mm. i 1 represents the rotation angle of the output shaft of the first driving wheel 2. i 4 represents the rotation angle of the output shaft of the second driven wheel 4, in rad. The first driving wheel 2 and the second driving wheel 3 are rigidly connected coaxially, and their common angle is denoted as... i s The unit is rad.

[0044] Meshing force can be expressed as: (17) In the formula: Q 21 This represents the meshing force between the first driving wheel 2 and the first driven wheel 1, expressed in N. Q 34 This represents the meshing force between the second driving wheel 3 and the second driven wheel 4, expressed in N. C e21 This represents the damping between the first driving wheel 2 and the first driven wheel 1, expressed in N•s / mm. This indicates the speed between the first driving wheel 2 and the first driven wheel 1, in mm / s. C e34 This indicates the damping between the second driving wheel 3 and the second driven wheel 4, expressed in N•s / mm. This indicates the speed between the second driving wheel 3 and the second driven wheel 4, in mm / s. K 21 This represents the time-varying meshing stiffness between the first driving wheel 2 and the first driven wheel 1, expressed in N / mm. K 34 This indicates the time-varying meshing stiffness between the second driving wheel 3 and the second driven wheel 4, expressed in N / mm. f ( x 21 () represents the displacement function of the first driving wheel 2 and the first driven wheel 1. f ( x 34 ) represents the displacement function between the second driving wheel 3 and the second driven wheel 4.

[0045] The dynamic model of the non-circular gear transmission system of the differential drive mechanism is as follows: (18) In the formula: I 1 represents the moment of inertia of the first driven wheel 1, in kg•m. 2 . I 4 represents the moment of inertia of the second driven wheel 4, in kg•m. 2 . This represents the vibration acceleration in the torsional direction of the first driven wheel 1, expressed in rad / s². 2 . This represents the vibration acceleration in the torsional direction of the second driven wheel 4, expressed in rad / s². 2 . This represents the acceleration of vibration in the torsional direction of the coaxial axis, expressed in rad / s². 2 . f This represents the fixed phase angle between the first driving wheel 2 and the second driving wheel 3, in rad. I s Input the moment of inertia of the coaxial axis, in kg•m 2 , T in This represents the torque acting on a coaxial axis, with units of N•m. T L1 , T L4 The torque is the load torque at both ends, in N•m.

[0046] (19) In the formula: r b1 The instantaneous base circle radius of the first driven wheel 1 is in mm. r b2 The transient base circle radius of the first driving wheel 2 is in mm. r b3 The transient base circle radius of the second driving wheel 3 is in mm. r b4 The instantaneous base circle radius of the second driven wheel 4 is in mm. I 3 represents the moment of inertia of the second driving wheel 3, in kg•m. 2 , I 4 represents the moment of inertia of the second driven wheel 4, in kg•m. 2 , M eq,21 This represents the equivalent mass between the first driving wheel 2 and the first driven wheel 1, in kg. M eq,34 This indicates the equivalent mass between the second driving wheel 3 and the second driven wheel 4, expressed in kg. This represents the second derivative of the time-varying transmission error between the first driving wheel 2 and the first driven wheel 1 with respect to time, in mm / s.2 . This represents the second derivative of the time-varying transmission error between the second driving wheel 3 and the second driven wheel 4 with respect to time, in mm / s. 2 .

[0047] in: (20) The definition of a dimensionless physical quantity is as follows: (twenty one) In the formula: oh n,21 This represents the natural frequency between the first driving wheel 2 and the first driven wheel 1. oh n,34 This indicates the natural frequency between the second driving wheel 3 and the second driven wheel 4. k m,21 This represents the average meshing stiffness between the first driving wheel 2 and the first driven wheel 1. k m,34 This indicates the average meshing stiffness between the second driving wheel 3 and the second driven wheel 4.

[0048] Dimensionless variables are: (twenty two) In the formula: t 21 This represents the dimensionless time between the first driving wheel 2 and the first driven wheel 1. t 34 This represents the dimensionless time between the second driving wheel 3 and the second driven wheel 4. X 21 This represents the dimensionless displacement between the first driving wheel 2 and the first driven wheel 1. X 34 This represents the dimensionless displacement between the second driving wheel 3 and the second driven wheel 4. E 21 This represents the dimensionless transmission error between the first driving wheel 2 and the first driven wheel 1. E 34 This represents the dimensionless transmission error between the second driving wheel 3 and the second driven wheel 4. t Indicates time. b c It indicates the nominal displacement scale. e 21 This indicates the transmission error between the first driving wheel 2 and the first driven wheel 1. e 34 This indicates the transmission error between the second driving wheel 3 and the second driven wheel 4.

[0049] The dimensionless stiffness and damping are: (twenty three) In the formula: k 21 ( t 21 ) represents the dimensionless stiffness between the first driving wheel 2 and the first driven wheel 1. k 34 ( t 34 ) represents the dimensionless stiffness between the second driving wheel 3 and the second driven wheel 4. g 21 This represents the dimensionless damping between the first driving wheel 2 and the first driven wheel 1. g 34 Let represent the dimensionless damping between the second driving wheel 3 and the second driven wheel 4. The dimensionless vibration differential equation is shown below:

[0050] (twenty four) In the formula: This represents the dimensionless acceleration between the first driving wheel 2 and the first driven wheel 1. This represents the dimensionless acceleration between the second driving wheel 3 and the second driven wheel 4. This represents the dimensionless velocity between the first driving wheel 2 and the first driven wheel 1. This represents the dimensionless velocity between the second driving wheel 3 and the second driven wheel 4. f ( X 21 ) represents the dimensionless displacement function of the first driving wheel 2 and the first driven wheel 1. f ( X 34 ) represents the dimensionless displacement function between the second driving wheel 3 and the second driven wheel 4. This represents the dimensionless input torque between the first driving wheel 2 and the first driven wheel 1. This represents the dimensionless input torque between the second driving wheel 3 and the second driven wheel 4. It represents the second derivative of the dimensionless transmission error between the first driving wheel 2 and the first driven wheel 1 with respect to time. It represents the second derivative of the dimensionless transmission error between the second driving wheel 3 and the second driven wheel 4 with respect to time. This represents the dimensionless meshing force between the first driving wheel 2 and the first driven wheel 1. This represents the dimensionless meshing force between the second driving wheel 3 and the second driven wheel 4. , The torque is a dimensionless load torque at both ends. The dimensionless coupling coefficient represents the dynamic meshing force of gear pair 3-4 on the dynamic response of gear pair 2-1. The dimensionless coupling coefficient represents the dynamic meshing force of gear pair 2-1 on the dynamic response of gear pair 3-4.

[0051] in: (25) This step converts the torsional motion of the two sets of non-circular gear pairs into relative displacement along the meshing line, and couples the time-varying meshing stiffness, meshing damping, transmission error, and tooth backlash into the dynamic equations, thus providing a basis for solving and judging the nonlinear response, stability, and impact state of the system.

[0052] Optionally, after establishing the dynamic model of the non-circular gear transmission system of the differential drive mechanism, the following steps are also included: Step (4): Input the various parameters of the transmission system into the dynamic model, and solve the dynamic model using the numerical integration method to obtain the relative displacement, dynamic meshing force, and nonlinear dynamic response of the two meshing pairs. Figure 4 As shown, in this embodiment of the invention, after establishing the system dynamics equations and calculating the time-varying meshing stiffness, the fourth-order Runge-Kutta numerical integration method is used to solve the system response. Eccentricity, input speed, tooth flank clearance, damping, and load torque are used as the main control parameters to obtain the variation law of the system's relative displacement response under different parameter conditions. This invention integrates time-varying meshing stiffness, piecewise nonlinearity of tooth flank clearance, comprehensive error excitation, and load torque into the system dynamics equations to analyze the steady-state and transient responses of the differential drive non-circular gear transmission system under different eccentricity, speed, tooth flank clearance, damping, and load conditions.

[0053] Based on the relationship between relative displacement and tooth flank clearance, the striking state and striking threshold are determined.

[0054] Specifically, the process involves inputting system structural parameters, material parameters, meshing excitation parameters, and setting operating condition parameters. These operating condition parameters include at least the input rotational speed or input torque, the first output load torque, and the second output load torque. A numerical integration method is used to solve the coupled nonlinear dynamic equations, obtaining the relative displacement of the two meshing pairs, the dynamic meshing force, and the system's nonlinear dynamic response. Based on the relationship between the relative displacement and tooth flank clearance, the impact state and impact threshold are determined. This invention, by defining impact state parameters and the impact threshold rotational speed, achieves a quantitative distinction between no-impact, single-sided impact, and double-sided impact states. It can further analyze the influence of eccentricity, tooth flank clearance, and load torque on the impact threshold rotational speed, providing a direct basis for identifying impact vibrations, assessing impact risks, and selecting operating conditions in gear transmission systems.

[0055] like Figure 5 As shown, the tapping state parameters P c .in,P c =0, 1, and 2 represent three operating conditions: no impact, single-sided impact, and double-sided impact, respectively. This criterion provides a clear theoretical basis for subsequent impact interval identification and dynamic response analysis. Based on the system's nonlinear response analysis, this implementation further analyzes the impact characteristics of tooth surface separation, re-contact, and reverse impact phenomena caused by tooth flank clearance.

[0056] The above inventions are merely a few specific embodiments of the present invention. However, the embodiments of the present invention are not limited thereto, and any variations that can be conceived by those skilled in the art should fall within the protection scope of the present invention.

Claims

1. A dynamic modeling method for a differential drive non-circular gear transmission system, characterized in that, Includes the following steps: Obtain the pitch curves of the driving gear and driven gear in the differential drive non-circular gear transmission system, and determine the radius of curvature of each point on the driving gear and driven gear based on the pitch curves; Based on the pitch curves of the driving and driven gears, establish the correspondence between the rotation angle of the driving / driven gear and the arc length of the pitch curve. Based on this relationship, determine the current meshing point position, obtain the curvature radii of the driving and driven gears at the current meshing point, and calculate the equivalent number of teeth of the two gears at the meshing point. The total elastic deformation of a single meshing tooth under stress is decomposed into Hertzian contact deformation, bending deformation, shear deformation and axial compression deformation. Using the equivalent number of teeth and the tooth profile geometry parameters of the current meshing point, the stiffness components corresponding to the four deformations are calculated respectively. The components are synthesized according to the series compliance relationship to obtain the time-varying meshing stiffness of the tooth pair at the current meshing position. Based on the pitch curve and time-varying meshing stiffness of the gears, a dynamic model of the non-circular gear transmission system of the differential drive mechanism is established.

2. The dynamic modeling method for a differential drive non-circular gear transmission system as described in claim 1, characterized in that, Establishing the correspondence between the rotation angle of the driving / driven wheel and the arc length of the pitch curve includes the following steps: Based on the non-slip conjugate meshing condition, the instantaneous relationship between the small rotation angle d1 of the driving wheel and the small rotation angle d2 of the driven wheel is established; Integrating the instantaneous relationship yields the global kinematic mapping between the polar angle 1 of the driving wheel pitch curve and the polar angle 2 of the driven wheel pitch curve; The meshing point position and corresponding pitch curve arc length of the two gears under any input rotation angle are determined by the global kinematic mapping. The meshing point position and corresponding pitch curve arc length of the driven gear are expressed as follows: ， in, m n The normal module of the driven wheel is expressed in mm. r 2 represents the radial direction of the driven wheel pitch curve, in mm; the driving wheel is the same as the driven wheel.

3. The dynamic modeling method for a differential drive non-circular gear transmission system as described in claim 1, characterized in that, The calculation of the equivalent number of teeth of the two gears at the meshing point includes the following steps: Obtain the radius of curvature of the driving gear pitch curve at the current meshing point. ρ 1. Radius of curvature of the driven gear pitch curve ρ 2. Calculate the equivalent number of teeth according to the following relationship: ， in, Z r1 This represents the number of teeth on the driving gear under the equivalent circular gear condition. Z r2 This represents the number of teeth on the driven gear under the equivalent circular gear condition. m n The normal module of the driven wheel is expressed in mm. The non-circular gear in its current meshing position is equivalent to having a number of teeth. Z r1 and Z r2 The hypothetical circular gear is used to determine the subsequent tooth profile geometry parameters.

4. The dynamic modeling method for a differential drive non-circular gear transmission system as described in claim 1, characterized in that, The step of synthesizing the components according to the series compliance relationship to obtain the time-varying meshing stiffness of the tooth pair at the current meshing position includes the following steps: The time-varying meshing stiffness of a single tooth is calculated according to the following compliance series relationship: ， in, k h For Hertzian contact stiffness, k b For bending stiffness, k a For radial compressive stiffness, k s The shear stiffness is expressed in N / mm. Subscript 1 indicates the driving wheel and subscript 2 indicates the driven wheel. For the double-tooth meshing range, the stiffness of the teeth that participate in meshing simultaneously is superimposed to finally obtain the time-varying meshing stiffness of the non-circular gear that varies with rotation angle or time.

5. The dynamic modeling method for a differential drive non-circular gear transmission system as described in claim 4, characterized in that, Hertzian contact stiffness k h Bending stiffness k b Radial compressive stiffness k a and shear stiffness k s Calculate using the following formulas respectively: ， ， ， ， in, α Let be the angle between the direction of the force at any contact point on the involute and the Y-axis; α 0 represents the pressure angle; α 1 represents the angle between the force direction at the meshing point and the Y-axis; α 2 is half of the base tooth angle; α 3 is half of the half-tooth angle corresponding to the root circle; the unit of included angle or angle is °; Z r2 This represents the number of teeth on the driven gear under the equivalent circular gear condition. E This represents the elastic modulus of the material, expressed in MPa. L This refers to the tooth width, in mm. v The material's Poisson's ratio.

6. The dynamic modeling method for a differential drive non-circular gear transmission system as described in claim 1, characterized in that, The establishment of the dynamic model of the non-circular gear transmission system of the differential drive mechanism includes the following steps: A differential coupling structure with two sets of coaxial input meshing pairs is constructed. The differential coupling structure includes: a first driving gear and a first driven gear constituting a first non-circular gear meshing pair, and a second driving gear and a second driven gear constituting a second non-circular gear meshing pair. Based on the torsional angular displacement of each gear and the corresponding base circle radius, the equivalent displacement of each gear along the meshing line direction is obtained. Based on the equivalent displacement of the two gears in each meshing pair and the comprehensive error of the meshing pair, the relative displacement of the meshing pair is obtained. Based on the time-varying meshing stiffness and relative displacement of each meshing pair, the meshing force of the meshing pair is obtained. Based on the differential coupling structure, the torsional angular displacement of each gear, and the meshing force of each meshing pair, the torsional dynamic equations of each gear are established, and the dynamic model of the non-circular gear transmission system of the differential drive mechanism is obtained.

7. The dynamic modeling method for a differential drive non-circular gear transmission system as described in claim 6, characterized in that, The dynamic model of the non-circular gear transmission system of the differential drive mechanism is as follows: ， In the formula, I 1 represents the moment of inertia of the first driven wheel; I 4 represents the moment of inertia of the second driven wheel; I s Input the coaxial moment of inertia; the unit of moment of inertia is kg•m. 2 ; The vibration acceleration indicates the direction of torsion of the first driven wheel; The vibration acceleration indicates the direction of torsion of the second driven wheel; The vibration acceleration represents the torsional direction of the coaxial axis; the unit of vibration acceleration is rad / s. 2 ; φ This represents the fixed phase angle between the first and second driving wheels, expressed in rad. T in This indicates the torque acting on the coaxial axis; T L1 , T L4 The torque is the load torque at both ends; the unit of torque is N•m. Q 21 This represents the meshing force between the first driving wheel and the first driven wheel; Q 34 This represents the meshing force between the second driving wheel and the second driven wheel; the unit of meshing force is N. r b1 ( θ 1) The first driven wheel at the turning angle θ The transient base circle radius at point 1; r b2 ( θ s (The first driving wheel is at the corner) θs The transient base circle radius at that point; r b3 ( θ s (The second driving wheel is at the corner) θs The transient base circle radius at that point; r b4 ( θ 4) For the second driven wheel at the turning angle θ 4 The transient base circle radius at the specified location; the unit for transient base circle radius is mm.

8. The dynamic modeling method for a differential drive non-circular gear transmission system as described in claim 7, characterized in that, The meshing force is expressed as: ， in, C e21 This indicates the damping between the first driving wheel and the first driven wheel; C e34 This indicates the damping between the second driving wheel and the second driven wheel; the unit of damping is N•s / mm. This indicates the speed between the first driving wheel and the first driven wheel; This indicates the speed between the second driving wheel and the second driven wheel; the unit of speed is mm / s. K 21 This indicates the time-varying meshing stiffness between the first driving wheel and the first driven wheel; K 34 This indicates the time-varying meshing stiffness between the second driving wheel and the second driven wheel; the unit of time-varying meshing stiffness is N / mm. f ( x 21 () represents the displacement function of the first driving wheel and the first driven wheel; f ( x 34 ) represents the displacement function between the second driving wheel and the second driven wheel.

9. The dynamic modeling method for a differential drive non-circular gear transmission system as described in claim 6, characterized in that, It also includes the following steps: The dimensionless relative displacement of each meshing pair is obtained by considering the relative displacement of each meshing pair and its corresponding tooth flank clearance. The dimensionless time is obtained based on the time variable and the system's natural frequency; Based on the dimensionless relative displacement, dimensionless time, and system parameters, the dimensionless damping ratio, dimensionless time-varying stiffness, and dimensionless excitation force are obtained, and the torsional dynamics equation is then transformed into a dimensionless vibration differential equation.

10. The dynamic modeling method for a differential drive non-circular gear transmission system as described in claim 1, characterized in that, After establishing the dynamic model of the non-circular gear transmission system of the differential drive mechanism, the following steps are also included: Input the various parameters of the transmission system into the dynamic model, and solve the dynamic model using the numerical integration method to obtain the relative displacement, dynamic meshing force and nonlinear dynamic response of the two meshing pairs; Based on the relationship between the relative displacement and the tooth flank clearance, the striking state and striking threshold are determined.

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