Gear system dynamics calculation method considering transverse vibration influence

Abstract

This invention belongs to the field of gear transmission system dynamics technology and discloses a method for calculating the dynamic characteristics of a gear system considering the influence of lateral vibration. It includes the following steps: S1, obtaining the parameters of the gear transmission system to be calculated; S2, calculating the time-varying meshing angle and overlap ratio of the gear pair considering the influence of lateral vibration; S3, obtaining the total time-varying meshing stiffness based on the time-varying meshing angle and overlap ratio in S2; S4, calculating the dynamic transmission error considering the influence of lateral vibration; S5, calculating the dynamic meshing force considering the influence of lateral vibration; S6, calculating the elastic deformation of the transmission shaft; S7, calculating the support reaction force of the bearing; S8, determining the dynamic meshing force. This invention provides a method for calculating the time-varying center distance caused by lateral vibration in a gear transmission system under conditions of eccentricity and bearing deformation during operation.

Description

Technical Field

[0001] This invention relates to the field of gear dynamics technology, and in particular to a method for calculating the dynamics of a gear system that takes into account the effects of lateral vibration. Background Technology

[0002] Gear mechanisms, as a major component of mechanical systems, are widely used in industries such as machinery manufacturing, energy and power, and aerospace due to their stable transmission ratio, compact structure, and high transmission efficiency. However, due to gear eccentricity and bearing deformation, gears generate lateral vibrations during operation. These lateral vibrations cause constant changes in the center distance and the direction of the meshing line of the gear pair, leading to time-varying meshing angles and contact ratios. This alters the internal excitation of the gear transmission system, exacerbating vibration and noise. Therefore, understanding the impact of lateral vibration on the vibration characteristics of gear systems is crucial for suppressing vibration and extending the service life of gear transmission systems.

[0003] Existing technologies rarely consider the influence of lateral vibration caused by gear eccentricity and bearing deformation in the study of the dynamic characteristics of gear systems. They also fail to consider the time-varying nature of gear meshing angle and contact ratio caused by lateral vibration. Furthermore, no relevant studies have been found on the changes in gear meshing stiffness that take into account the influence of lateral vibration. This results in a large deviation between the calculated dynamic characteristics and the actual results. Summary of the Invention

[0004] To address the problem in existing technologies that fail to consider lateral vibrations caused by eccentricity and bearing deformation, resulting in significant deviations between the obtained dynamic characteristics and actual values, this invention provides a gear system dynamics calculation method that considers the influence of lateral vibrations. This method fully considers the impact of lateral vibrations caused by gear eccentricity and bearing deformation on the meshing angle, contact ratio, and meshing stiffness, ensuring high computational efficiency while minimizing the error between the obtained dynamic characteristics and actual values, thus improving accuracy.

[0005] To achieve the above objectives, the present invention provides the following technical solution: A dynamic calculation method for a gear transmission system considering lateral vibration specifically includes the following steps: S1. Obtain the parameters of the gear transmission system to be calculated. The gear transmission system parameters include basic parameters, operating condition parameters, initial conditions and initial meshing position. The gear transmission system includes gear pairs, transmission shaft and bearings. S2. Based on the basic parameters of the gear transmission system, calculate the time-varying meshing angle and overlap ratio of the gear pair considering the influence of lateral vibration. S3. Based on the basic parameters of the gear transmission system, according to the time-varying meshing angle and time-varying overlap in S2, the Hertzian contact stiffness, bending stiffness, shear stiffness, compressive stiffness and matrix deformation stiffness of the gear teeth considering the influence of lateral vibration are calculated using the potential energy method, and the total time-varying meshing stiffness is obtained. S4. Based on the basic parameters of the gear transmission system, and according to the time-varying meshing angle in S2, calculate the dynamic transmission error considering the influence of lateral vibration; S5. Based on the total time-varying stiffness determined in S3 and the dynamic transmission error determined in S5, calculate the dynamic meshing force considering the influence of lateral vibration. S6. Based on the deformation compatibility conditions of the shaft, calculate the elastic deformation of the transmission shaft; S7. Based on the equilibrium condition of the forces in the radial plane, and according to the elastic deformation of the transmission shaft determined in S6, calculate the support reaction force of the bearing. S8. Based on Newton's second law, and according to the dynamic meshing force determined in S5 and the bearing support reaction force determined in S7, construct a dynamic model of a multi-degree-of-freedom bending-torsion coupled gear-shaft-bearing transmission system, and output the dynamic characteristics of the gear at each moment. Preferably, in S1, the basic parameters of the gear transmission system include: the number of teeth, module, tooth width, pitch circle pressure angle, elastic modulus, Poisson's ratio, mass, moment of inertia, rotational speed, and concentrated mass of the bearings; the operating parameters of the gear transmission system include: input torque; input speed; the initial conditions and initial meshing position include: initial center distance of the gears; initial phase of the gears; initial half-tooth backlash; gear eccentricity; gear position coefficient; and the magnitude of the meshing error; Preferably, step S2 includes the following steps: S201, Time-varying center distance under transverse vibration as follows:

[0006] In formula (1), This is the initial center distance; , These represent the vibration displacements of the gear in the horizontal and vertical directions, respectively. As the driving gear, It is a driven gear; Runtime; S202. Time-varying meshing angle of gear pair considering the influence of lateral vibration The calculation is as follows:

[0007] In formula (2), The initial engagement angle; This is the initial center distance; Time-varying center distance; S203. The time-varying overlap ratio of the gear pair considering the influence of lateral vibration is calculated as follows:

[0008] In formula (3), As the driving gear, It is a driven gear; This refers to the number of teeth on the gear. Let be the radius of the gear's addendum circle; Tooth tip circle pressure angle; The engagement angle is time-varying; Preferably, step S3 includes the following steps: S301. Based on the basic parameters of the gear transmission system, calculate the Hertzian contact stiffness of the gear teeth using the potential energy method. Bending stiffness Compression stiffness Shear stiffness The corresponding expression is as follows:

[0009] In formula (4), These are the elastic modulus, Poisson's ratio, and tooth width of the gear material, respectively. =1 indicates the driving wheel; =2 indicates the driving wheel; , The expressions are as follows:

[0010] In formula (5), This refers to the number of teeth on the driving gear. Initial engagement angle

[0011] formula middle, and The expressions are as follows:

[0012] formula middle, It is the remainder function; Let be the base circle radius of the driving wheel; These are the tip circle radius and tip circle pressure angle of the driven gear, respectively. The time-varying center distance; This refers to the dynamic engagement angle; , These represent the number of teeth on the driving and driven gears, respectively. Input rotational speed; The gear meshing period is expressed as follows:

[0013] formula middle, This refers to the number of teeth on the driving gear. Input rotational speed; S402. Calculate the stiffness of the gear base according to the elastic ring principle proposed by P. Sainsot. The calculation expression is as follows:

[0014] In formula (9), Same as in formula (6); It is the distance from the intersection of the line of action and the line of symmetry of the gear teeth to the root circle; It is the arc length corresponding to the entire tooth profile curve of the gear; These are four parameters related to gear module and number of teeth; S403, based on the tooth bending stiffness in S401 Shear stiffness Axial compressive stiffness and matrix stiffness Obtain the single-tooth meshing stiffness The calculation is as follows:

[0015] formula middle, =1 indicates that it is the driving wheel; =2 driven wheels; S303. Because the overlap ratio of standard spur gears is between 1 and 2, there is a single-tooth meshing zone. and double-tooth meshing area The calculation formula is as follows:

[0016]

[0017] formula middle, This refers to the overlap ratio of the gears; This refers to the meshing cycle of the gears; S304, based on the single-tooth meshing stiffness in S303 According to the single-tooth meshing area in S304 and double-tooth meshing area Obtain the total time-varying meshing stiffness The calculation is as follows:

[0018] formula middle, =1 indicates the single-tooth meshing stiffness; =2 double-tooth meshing stiffness; Preferably, in step S4, the error is dynamically transmitted. The calculation expression is as follows:

[0019] formula middle , These are the base circle radii of the driving gear and the driven gear, respectively. , These are the mass eccentricities of the driving gear and the driven gear, respectively. Rotational displacement of the driving and driven gears; This is the initial phase; , Let be the position angle and meshing error at any given time, respectively, and their expressions are as follows:

[0020] In formula (13), in formula (13), This is the initial center distance; , These represent the vibration displacements of the gear in the horizontal and vertical directions, respectively. As the driving gear, It is a driven gear;

[0021] In formula (15), This represents the magnitude of the meshing error; The meshing angular frequency is expressed as follows:

[0022] In formula (16), This refers to the number of teeth on the driving gear. Input rotational speed; Preferably, step S5 includes the following steps: S501, Calculate the gap nonlinear function respectively. and velocity nonlinear function The calculation methods are as follows:

[0023]

[0024] In formula (17), · represents the first derivative; To dynamically transmit errors; The dynamic half-tooth backlash is expressed as follows:

[0025] In formula (18), This represents the initial half-tooth side clearance; It is an involute function. ; S502, Calculation of meshing damping The calculation method is as follows:

[0026] In formula (19), For meshing phase Nibi; This represents the total time-varying meshing stiffness; ; They are respectively S503, based on the gap nonlinear function in S501 and velocity nonlinear function According to the meshing damping in S502 Obtain the dynamic meshing force between gears. The calculation method is as follows:

[0027] In formula (20), This represents the total time-varying meshing stiffness; Preferably, in step S6, the calculation expression for the elastic deformation of the transmission shaft is as follows:

[0028] In formula (21), 4 represents four supporting bearings; The input axes are respectively at The amount of elastic deformation in the direction; The output shaft is respectively at The amount of elastic deformation in the direction; This is the gear position coefficient; , These represent the vibration displacements of the supporting bearing in the horizontal and vertical directions, respectively. These represent the vibration displacements of the main gear in the horizontal and vertical directions, respectively. These represent the vibration displacements from the horizontal and vertical directions of the gear, respectively. Preferably, in step S7, the calculation expression for the elastic deformation of the support shaft is as follows:

[0029] In formula (22), 4 represents four supporting bearings; This is the gear position coefficient; , These are the bending damping components for the input and output shafts, respectively. These are the bending stiffnesses of the input and output shafts, respectively. The input axes are respectively at The amount of elastic deformation in the direction; The output shaft is respectively at The amount of elastic deformation in the direction; Preferably, step S8 includes the following steps: S801. Based on Newton's second law, establish the dynamic equation for the active wheel's bending-torsional coupling, which takes the following form:

[0030] In formula (23), · and ·· represent the first derivative and the second derivative, respectively; For the mass of the driving gear; For input torque; Input shaft bending damping; Input shaft bending stiffness; For the dynamic meshing force of gears; The radius of the base circle of the driving wheel; The eccentricity of the driving wheel mass; This refers to the rotational displacement of the driving gear; , These represent the vibration displacements of the main gear in the horizontal and vertical directions, respectively. This refers to the dynamic engagement angle; The input axes are respectively at Elastic deformation in the direction For dynamic position angle; This is the initial phase; S802. Based on Newton's second law, establish the bending-torsional coupling dynamic equation of the driven wheel, which takes the following form:

[0031] In formula (24), · and ·· represent the first derivative and the second derivative, respectively; The mass of the driven gear; This is the output torque; For output shaft bending damping; For the output shaft bending stiffness; This refers to the dynamic meshing force of the gears. The radius of the driven wheel's base circle; The mass eccentricity of the driven wheel; This represents the rotational displacement of the driven gear. , These represent the vibration displacements from the horizontal and vertical directions of the gear, respectively. The output shaft is respectively at The amount of elastic deformation in the direction; This refers to the dynamic engagement angle; For dynamic position angle; This is the initial phase; S803. Based on Newton's second law, establish the bending-torsional coupling dynamic equation of the support bearing, which takes the following form:

[0032] In formula (25), 4 represents the four supporting bearings; · and ·· represent the first and second derivatives, respectively; To support the concentrated mass of the bearing; They are respectively along the support bearing. Radial support damping in the direction; They are respectively along the support bearing. Radial support stiffness in the direction; , These represent the vibration displacements of the supporting bearing in the horizontal and vertical directions, respectively. , The input axes are respectively at The amount of elastic deformation in the direction; , The output shaft is respectively at The amount of elastic deformation in the direction; , They are respectively along the support bearing. Radial support force in the direction; The Runge-Kutta method is used to perform iterative calculations within time t on the following dynamic equations: S804, S801 (active wheel bending-torsion coupling dynamic equation), S802 (driven wheel bending-torsion coupling dynamic equation), and S803 (support bearing bending-torsion coupling dynamic equation), and output the dynamic characteristics.

[0033] In summary, by adopting the above technical solution, the present invention has at least the following beneficial effects compared with the prior art: This invention provides a method for calculating the time-varying center distance caused by lateral vibration in a gear transmission system under conditions of eccentricity and bearing deformation during operation, as well as an iterative calculation method for the time-varying meshing angle and time-varying overlap of the gear pair at each moment. The present invention also provides a method for solving the total time-varying meshing stiffness of a gear pair using the energy method when there is lateral vibration in a gear transmission system; The present invention also provides a method for calculating the dynamic transmission error of a gear pair in the case of lateral vibration in a gear transmission system; The present invention also provides a method for calculating the dynamic meshing force of a gear pair in the case of lateral vibration in a gear transmission system; The present invention also provides a method for calculating the elastic deformation of the drive shaft in a gear transmission system; This invention also provides a method for calculating the support reaction force of the support bearing in a gear transmission system; The present invention also provides an iterative calculation method for a multi-degree-of-freedom dynamic model of a gear transmission system with bending-torsional coupling, which determines the dynamic characteristics of the gear transmission system at each moment, ensuring high calculation efficiency while minimizing the error between the obtained dynamic characteristics and the actual values, thereby improving the accuracy of the obtained dynamic characteristics. Attached Figure Description

[0034] Figure 1 A schematic diagram illustrating a dynamic calculation method for a gear transmission system considering the influence of lateral vibration, as exemplarily implemented according to the present invention; Figure 2 This is a schematic diagram of the relative positions of a gear pair according to an exemplary embodiment of the present invention.

[0035] Figure 3 This is a schematic diagram of the involute tooth profile and its stress according to an exemplary embodiment of the present invention.

[0036] Figure 4 This is a schematic diagram of a single-double tooth meshing process according to an exemplary implementation of the present invention.

[0037] Figure 3 This is a schematic diagram of the dynamic model of a bending-torsional coupled gear transmission system according to an exemplary implementation of the present invention. Detailed Implementation

[0038] To make the objectives, technical solutions, and advantages / features of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0039] like Figure 1 As shown, this invention provides a dynamic calculation method for a gear transmission system considering the influence of lateral vibration, comprising the following steps: S1. Obtain the parameters of the gear transmission system to be calculated. The gear transmission system parameters include basic parameters, operating condition parameters, initial conditions and initial meshing position. The gear transmission system includes gear pairs, transmission shaft and bearings. In this embodiment, the basic parameters of the gear transmission system include: the number of teeth, module, tooth width, pitch circle pressure angle, elastic modulus, Poisson's ratio, mass, moment of inertia, rotational speed, and concentrated mass of the bearings; the operating parameters of the gear transmission system include: input torque; input speed; the initial conditions and initial meshing position include: initial center distance of the gears; initial phase of the gears; initial half-tooth backlash; gear eccentricity; gear position coefficient; and the magnitude of the meshing error. S2. Based on the basic parameters of the gear transmission system, calculate the time-varying meshing angle and overlap ratio of the gear pair considering the influence of lateral vibration. S201, such as Figure 1 As shown in the figure, during gear meshing transmission, the center coordinates change over time due to lateral vibration. , Points 1 and 2 are the axes of the driving wheel and driven wheel at the initial moment, respectively. , These are the axes of the driving wheel and the driven wheel at any given time. arrive Distance and time-varying center distance The expression is as follows: Based on the basic parameters of the gear transmission system,

[0040] In formula (1), This is the initial center distance; , These represent the vibration displacements of the gear in the horizontal and vertical directions, respectively. As the driving gear, It is a driven gear; Runtime; S202. Time-varying meshing angle of gear pair considering the influence of lateral vibration The calculation is as follows:

[0041] In formula (2), The initial engagement angle; This is the initial center distance; Time-varying center distance; S203. The time-varying overlap ratio of the gear pair considering the influence of lateral vibration is calculated as follows:

[0042] In formula (3), As the driving gear, It is a driven gear; This refers to the number of teeth on the gear. Let be the radius of the gear's addendum circle; Tooth tip circle pressure angle; The engagement angle is time-varying; S3. Based on the basic parameters of the gear transmission system, according to the time-varying meshing angle and time-varying overlap in S2, the Hertzian contact stiffness, bending stiffness, shear stiffness, compression stiffness and base flexible deformation stiffness of the gear considering the influence of lateral vibration are calculated using the potential energy method, and the total time-varying meshing stiffness is obtained. S301, such as Figure 3 As shown, in the potential energy method for calculating meshing stiffness, the gear teeth are equivalent to a cantilever beam fixed at the base circle of the gear. Based on the force... Calculate the Hertzian contact stiffness of the gear teeth by considering the contact deformation, bending deformation, shear deformation, and compressive deformation generated under the action of the gear. Bending stiffness Shear stiffness Compression stiffness The corresponding expression is as follows:

[0043] In formula (4), These are the elastic modulus, Poisson's ratio, and tooth width of the gear material, respectively. =1 indicates the driving wheel; =2 indicates the driving wheel; , The expressions are as follows:

[0044] In formula (5), This refers to the number of teeth on the driving gear. Initial engagement angle

[0045] formula middle, and The expressions are as follows:

[0046] formula middle, It is the remainder function; Let be the base circle radius of the driving wheel; These are the tip circle radius and tip circle pressure angle of the driven gear, respectively. The time-varying center distance; This refers to the dynamic engagement angle; , These represent the number of teeth on the driving and driven gears, respectively. The gear meshing period is expressed as follows:

[0047] formula middle, This refers to the number of teeth on the driving gear. Input rotational speed; S302. Calculate the stiffness of the gear base according to the elastic ring principle proposed by P. Sainsot. The calculation expression is as follows:

[0048] In formula (9), For formula (6); It is the distance from the intersection of the line of action and the line of symmetry of the gear teeth to the root circle; It is the arc length corresponding to the entire tooth profile curve of the gear; These are four parameters related to gear module and number of teeth; S303, based on the tooth bending stiffness in S401 Shear stiffness Axial compressive stiffness and matrix stiffness Obtain the single-tooth meshing stiffness The calculation is as follows:

[0049] formula middle, =1 indicates that it is the driving wheel; =2 driven wheels; S304, such as Figure 4 As shown, since the overlap ratio of standard spur gears is between 1 and 2, there exists a single-tooth meshing zone. and double-tooth meshing area The calculation formula is as follows:

[0050] formula middle, This refers to the overlap ratio of the gears; This refers to the meshing cycle of the gears; S305, based on the single-tooth meshing stiffness in S303 According to the single-tooth meshing area in S304 and double-tooth meshing area Obtain the total time-varying meshing stiffness The calculation is as follows:

[0051] formula middle, =1 indicates the single-tooth meshing stiffness; =2 double-tooth meshing stiffness; S4. Based on the time-varying meshing angle in S2, calculate the dynamic transmission error considering the influence of lateral vibration; In this embodiment, the expression for dynamic error propagation is as follows:

[0052] formula middle , These are the base circle radii of the driving gear and the driven gear, respectively. , These are the mass eccentricities of the driving gear and the driven gear, respectively. Rotational displacement of the driving and driven gears; This is the initial phase; , Let be the position angle and meshing error at any given time, respectively, and their expressions are as follows:

[0053] In formula (14), This is the initial center distance; , These represent the vibration displacements of the gear in the horizontal and vertical directions, respectively. As the driving gear, It is a driven gear;

[0054] In formula (15), This represents the magnitude of the meshing error; The meshing angular frequency is expressed as follows:

[0055] In formula (16), This refers to the number of teeth on the driving gear. Input rotational speed; S5. Based on the total time-varying stiffness determined in S3 and the dynamic transmission error determined in S4, calculate the dynamic meshing force considering the influence of lateral vibration. S501, Calculate the gap nonlinear function respectively. and velocity nonlinear function The calculation methods are as follows:

[0056] In formula (17), · represents the first derivative; To dynamically transmit errors; The dynamic half-tooth backlash is expressed as follows:

[0057] In formula (18), This represents the initial half-tooth side clearance; It is an involute function. ; S502, Calculation of meshing damping The calculation method is as follows:

[0058] In formula (19), For meshing phase Nibi; This represents the total time-varying meshing stiffness; ; They are respectively S503, based on the gap nonlinear function in S501 and velocity nonlinear function According to the meshing damping in S502 Obtain the dynamic meshing force between gears. The calculation method is as follows:

[0059] In formula (20), This represents the total time-varying meshing stiffness; S6. Based on the deformation compatibility conditions of the shaft, calculate the elastic deformation of the transmission shaft; In this embodiment, the formula for calculating the elastic deformation of the drive shaft is as follows:

[0060] In formula (21), 4 represents four supporting bearings; The input axes are respectively at The amount of elastic deformation in the direction; The output shaft is respectively at The amount of elastic deformation in the direction; This is the gear position coefficient; , These represent the vibration displacements of the supporting bearing in the horizontal and vertical directions, respectively. These represent the vibration displacements of the main gear in the horizontal and vertical directions, respectively. These represent the vibration displacements from the horizontal and vertical directions of the gear, respectively. S7. Based on the equilibrium condition of the forces in the radial plane, and according to the elastic deformation of the transmission shaft determined in S6, calculate the support reaction force of the bearing. In this embodiment, the expression for the support bearing reaction force is as follows:

[0061] In formula (22), 4 represents four supporting bearings; This is the gear position coefficient; , These are the bending damping components for the input and output shafts, respectively. These are the bending stiffnesses of the input and output shafts, respectively. The input axes are respectively at The amount of elastic deformation in the direction; The output shaft is respectively at The amount of elastic deformation in the direction; S8, such as Figure 5 As shown, based on the dynamic meshing force determined in S5 and the bearing support reaction force determined in S7, a dynamic model of a multi-degree-of-freedom bending-torsion coupled gear-shaft-bearing transmission system is constructed based on Newton's second law. S801. Establish the dynamic equations for the active wheel's bending-torsional coupling, which are as follows:

[0062] In formula (23), · and ·· represent the first derivative and the second derivative, respectively; For the mass of the driving gear; For input torque; Input shaft bending damping; Input shaft bending stiffness; This refers to the dynamic meshing force of the gears. The radius of the base circle of the driving wheel; The eccentricity of the driving wheel mass; This refers to the rotational displacement of the driving gear; , These represent the vibration displacements of the main gear in the horizontal and vertical directions, respectively. This refers to the dynamic engagement angle; The input axes are respectively at Elastic deformation in the direction For dynamic position angle; This is the initial phase; S802. Establish the bending-torsional coupling dynamic equations for the driven wheel, which are as follows:

[0063] In formula (24), · and ·· represent the first derivative and the second derivative, respectively; The mass of the driven gear; This is the output torque; For output shaft bending damping; For the output shaft bending stiffness; This refers to the dynamic meshing force of the gears. The radius of the driven wheel's base circle; The mass eccentricity of the driven wheel; This represents the rotational displacement of the driven gear. , These represent the vibration displacements from the horizontal and vertical directions of the gear, respectively. The output shaft is respectively at The amount of elastic deformation in the direction; This refers to the dynamic engagement angle; For dynamic position angle; This is the initial phase; S803. Establish the bending-torsional coupling dynamic equation of the support bearing, which is as follows:

[0064] In formula (25), 4 represents the four supporting bearings; · and ·· represent the first and second derivatives, respectively; To support the concentrated mass of the bearing; They are respectively along the support bearing. Radial support damping in the direction; They are respectively along the support bearing. Radial support stiffness in the direction; , These represent the vibration displacements of the supporting bearing in the horizontal and vertical directions, respectively. , The input axes are respectively at The amount of elastic deformation in the direction; , The output shaft is respectively at The amount of elastic deformation in the direction; , They are respectively along the support bearing. Radial support force in the direction; S804. Using the Longge method, iterative calculations are performed within time t on the active wheel bending-torsion coupling dynamic equations constructed in S801, the driven wheel bending-torsion coupling dynamic equations constructed in S802, and the support bearing bending-torsion coupling dynamic equations constructed in S803. If t has not reached the set time, the calculations are updated. , And update the time-varying meshing angle and time-varying overlap, and Repeat steps S2-S7; if t reaches the set time, end the calculation and output the dynamic characteristics of the gear transmission system.

Claims

1. A method for dynamic modeling of a gear system considering the effects of meshing stiffness and lateral vibration, characterized in that, include: S1. Obtain the parameters of the gear transmission system to be calculated. The gear transmission system parameters include basic parameters, operating condition parameters, initial conditions and initial meshing position. The gear transmission system includes gear pairs, transmission shaft and bearings. S2. Based on the basic parameters of the gear transmission system, calculate the time-varying meshing angle and overlap ratio of the gear pair considering the influence of lateral vibration. S3. Based on the basic parameters of the gear transmission system, according to the time-varying meshing angle and time-varying overlap in S2, the Hertzian contact stiffness, bending stiffness, shear stiffness, compression stiffness and flexible deformation stiffness of the gear tooth matrix considering the influence of lateral vibration are calculated using the potential energy method, and the total time-varying meshing stiffness is obtained. S4. Based on the basic parameters of the gear transmission system, and according to the time-varying meshing angle in S2, calculate the dynamic transmission error considering the influence of lateral vibration; S5. Based on the total time-varying stiffness determined in S3 and the dynamic transmission error determined in S5, calculate the dynamic meshing force considering the influence of lateral vibration. S6. Based on the deformation compatibility conditions of the shaft, calculate the elastic deformation of the transmission shaft; S7. Based on the equilibrium condition of the forces in the radial plane, and according to the elastic deformation of the transmission shaft determined in S6, calculate the support reaction force of the bearing. S8. Based on Newton's second law, and according to the dynamic meshing force determined in S5 and the bearing support reaction force determined in S7, construct a dynamic model of a multi-degree-of-freedom bending-torsion coupled gear-shaft-bearing transmission system, and output the dynamic characteristics of the gear at each moment.

2. The gear system dynamics calculation method considering the influence of lateral vibration as described in claim 1, characterized in that, In S1, the basic parameters of the gear transmission system include: the number of teeth, module, tooth width, pitch circle pressure angle, elastic modulus, Poisson's ratio, mass, moment of inertia, rotational speed, and concentrated mass of the bearings; the operating parameters of the gear transmission system include: input torque; input speed; the initial conditions and initial meshing position include: initial center distance of the gears; initial phase of the gears; initial half-tooth flank clearance; gear eccentricity; gear position coefficient; and the magnitude of the meshing error.

3. The method for calculating the dynamics of a gear system considering the influence of lateral vibration as described in claim 1, characterized in that, S2 includes the following steps: S201, Time-varying center distance under transverse vibration as follows: In formula (1), This is the initial center distance; , These represent the vibration displacements of the gear in the horizontal and vertical directions, respectively. As the driving gear, For driven gear; Runtime; S202. Time-varying meshing angle of gear pair considering the influence of lateral vibration. The calculation is as follows: In formula (2), The initial engagement angle; This is the initial center distance; Time-varying center distance; S203. The time-varying overlap ratio of the gear pair considering the influence of lateral vibration is calculated as follows: In formula (3), As the driving gear, For driven gear; This refers to the number of teeth on the gear. Let be the radius of the gear's tip circle; Tooth tip circle pressure angle; It is a time-varying engagement angle.

4. The gear system dynamics calculation method considering the influence of lateral vibration as described in claim 1, wherein step S3 includes the following steps: S301. Based on the basic parameters of the gear transmission system, calculate the Hertzian contact stiffness of the gear teeth using the potential energy method. Bending stiffness Shear stiffness Compression stiffness The corresponding expression is as follows: In formula (4), These are the elastic modulus, Poisson's ratio, and tooth width of the gear material, respectively. =1 indicates the driving wheel; =2 indicates the driving wheel; , The expressions are as follows: In formula (5), This refers to the number of teeth on the driving gear. Initial engagement angle formula middle, and The expressions are as follows: formula middle, It is the remainder function; Let be the base circle radius of the driving wheel; These are the tip circle radius and tip circle pressure angle of the driven gear, respectively. The time-varying center distance; This refers to the dynamic engagement angle; , These represent the number of teeth on the driving and driven gears, respectively. Input rotational speed; The gear meshing period is expressed as follows: formula middle, This refers to the number of teeth on the driving gear. Input rotational speed; S302. Calculate the stiffness of the gear base according to the elastic ring principle proposed by P. Sainsot. The calculation expression is as follows: In formula (9), Same as in formula (6); It is the distance from the intersection of the line of action and the line of symmetry of the gear teeth to the root circle; It is the arc length corresponding to the entire tooth profile curve of the gear; These are four parameters related to gear module and number of teeth; S303, based on the tooth bending stiffness in S201 Shear stiffness Axial compressive stiffness and matrix stiffness Obtain the single-tooth meshing stiffness The calculation is as follows: formula middle, =1 indicates that it is the driving wheel; =2 driven wheels; S304. Because the overlap ratio of standard spur gears is between 1 and 2, there is a single-tooth meshing zone. and double-tooth meshing area The calculation formula is as follows: formula middle, This refers to the overlap ratio of the gears; This refers to the gear meshing cycle; S305, based on the single-tooth meshing stiffness in S303 According to the single-tooth meshing area in S304 and double-tooth meshing area Obtain the total time-varying meshing stiffness The calculation is as follows: formula middle, =1 indicates the single-tooth meshing stiffness; =2 double-tooth meshing stiffness.

5. The gear system dynamics calculation method considering the influence of lateral vibration as described in claim 1, characterized in that, In S4, the error is dynamically transmitted. The calculation expression is as follows: formula middle , These are the base circle radii of the driving gear and the driven gear, respectively. , These are the mass eccentricities of the driving gear and the driven gear, respectively. Rotational displacement of the driving and driven gears; This is the initial phase; , Let be the position angle and meshing error at any given time, respectively, and their expressions are as follows: In formula (14), This is the initial center distance; , These represent the vibration displacements of the gear in the horizontal and vertical directions, respectively. As the driving gear, For driven gear; In formula (15), This represents the magnitude of the meshing error; The meshing angular frequency is expressed as follows: In formula (16), This refers to the number of teeth on the driving gear. Input rotational speed.

6. The method for calculating the dynamics of a gear system considering the influence of lateral vibration as described in claim 1, characterized in that, S5 includes the following steps: S501, Calculate the gap nonlinear function respectively. and velocity nonlinear function The calculation methods are as follows: In formula (17), · represents the first derivative; To dynamically transmit errors; The dynamic half-tooth backlash is expressed as follows: In formula (18), This represents the initial half-tooth side clearance. It is an involute function. ; S502, Calculate meshing damping The calculation method is as follows: In formula (19), For meshing phase Nibi; This represents the total time-varying meshing stiffness; ; These are the masses of the driving gear and the driven gear, respectively. S503, based on the gap nonlinear function in S501 and velocity nonlinear function According to the meshing damping in S502 Obtain the dynamic meshing force between gears. The calculation method is as follows: 。 7. The method for calculating the dynamics of a gear system considering the influence of lateral vibration as described in claim 1, characterized in that, In step S6, the calculation expression for the elastic deformation of the transmission shaft is as follows: In formula (21), 4 represents four supporting bearings; The input axes are respectively at The amount of elastic deformation in the direction; The output shaft is respectively at The amount of elastic deformation in the direction; This is the gear position coefficient; , These represent the vibration displacements of the supporting bearing in the horizontal and vertical directions, respectively. These represent the vibration displacements of the main gear in the horizontal and vertical directions, respectively. These represent the vibration displacements in the horizontal and vertical directions of the gear, respectively.

8. The method for calculating the dynamics of a gear system considering the influence of lateral vibration according to claim 1, characterized in that, In step S7, the calculation expression for the elastic deformation of the support shaft is as follows: In formula (22), 4 represents four supporting bearings; This is the gear position coefficient; , These are the bending damping components for the input and output shafts, respectively. These are the bending stiffnesses of the input and output shafts, respectively. The input axes are respectively at The amount of elastic deformation in the direction; The output shaft is respectively at The amount of elastic deformation in the direction.

9. The method for calculating the dynamics of a gear system considering the influence of lateral vibration as described in claim 1, characterized in that, S8 includes the following steps: S801. Based on Newton's second law, establish the dynamic equation for the active wheel's bending-torsional coupling, which takes the following form: In formula (23), · and ·· represent the first derivative and the second derivative, respectively; For the mass of the driving gear; For input torque; Input shaft bending damping; Input shaft bending stiffness; For the dynamic meshing force of gears; The radius of the base circle of the driving wheel; The eccentricity of the driving wheel mass; This refers to the rotational displacement of the driving gear; , These represent the vibration displacements of the main gear in the horizontal and vertical directions, respectively. This refers to the dynamic engagement angle; The input axes are respectively at Elastic deformation in the direction For dynamic position angle; This is the initial phase; S802. Based on Newton's second law, establish the bending-torsional coupling dynamic equation of the driven wheel, which takes the following form: （24） In formula (24), · and ·· represent the first derivative and the second derivative, respectively; The mass of the driven gear; This is the output torque; For output shaft bending damping; For the output shaft bending stiffness; For the dynamic meshing force of gears; The radius of the driven wheel's base circle; The mass eccentricity of the driven wheel; This represents the rotational displacement of the driven gear. , These represent the vibration displacements from the horizontal and vertical directions of the gear, respectively. The output shaft is respectively at The amount of elastic deformation in the direction; This refers to the dynamic engagement angle; For dynamic position angle; This is the initial phase; S803. Based on Newton's second law, establish the bending-torsional coupling dynamic equation of the support bearing, which takes the following form: In formula (25), 4 represents the four supporting bearings; · and ·· represent the first and second derivatives, respectively; To support the concentrated mass of the bearing; They are respectively along the support bearing. Radial support damping in the direction; They are respectively along the support bearing. Radial support stiffness in the direction; , These represent the vibration displacements of the supporting bearing in the horizontal and vertical directions, respectively. , The input axes are respectively at The amount of elastic deformation in the direction; , The output shaft is respectively at The amount of elastic deformation in the direction; , They are respectively along the support bearing. Radial support force in the direction; The Runge-Kutta method is used to perform iterative calculations within time t on the following dynamic equations: S804, S801 (active wheel bending-torsion coupling dynamic equation), S802 (driven wheel bending-torsion coupling dynamic equation), and S803 (support bearing bending-torsion coupling dynamic equation), and output the dynamic characteristics.

Images