A design method for a multi-stage fixed-axis helical gear transmission system

Abstract

The present application discloses a design method for a multi-stage fixed-axis helical gear transmission system, which relates to the technical field of mechanical design and includes: establishing the internal deformation coordination equation and the rolling element force balance equation of the bearing; solving the internal deformation coordination equation and the rolling element force balance equation of the bearing to determine the bearing stiffness curve considering the influence of clearance and friction; determining the meshing stiffness curve of the helical gear pair; establishing a multi-body model; introducing the bearing stiffness curve and the meshing stiffness curve into the gear force element and the bearing force element of the multi-body model to establish a multi-body dynamics model; solving the multi-body dynamics model to obtain the dynamic characteristics of the multi-stage fixed-axis helical gear transmission system; adjusting the structural parameters, comparing the dynamic characteristics after adjusting the parameters, and obtaining the optimized structural design parameters. The present application can accurately consider the influence of bearing clearance and friction on the dynamic characteristics of the gear transmission system, and can quickly build a model according to the increase or decrease of the components of the multi-stage fixed-axis helical gear transmission system.

Description

Technical Field

[0001] This application relates to the technical field of mechanical design, and particularly to a design method for a multi-stage fixed-axis helical gear transmission system. Background Art

[0002] The gear transmission system is a key component in the mechanical system and is also one of the main sources of vibration and noise in the mechanical system. Mastering the vibration response characteristics of the gear transmission system is of crucial significance for the vibration reduction and noise reduction design and stable and reliable operation of the mechanical system. To avoid the system vibration caused by the large fluctuation of the meshing stiffness of spur gears and to meet the requirements of the mechanical system for large transmission ratios and multi-path power splitting, complex multi-stage fixed-axis helical gear transmission systems are widely used. Therefore, it is of great significance to accurately obtain the dynamic characteristics of the multi-stage fixed-axis helical gear transmission system and carry out low-vibration structural optimization design on it.

[0003] In the prior art, in the "Dynamics Modeling Method of Rigid-Flexible Coupled Planetary Gear and Rotor System" with the patent publication number CN117669329A, the mass and stiffness matrices of the input shaft and the planet carrier are obtained through the finite element analysis software Ansys and are substituted into the dynamic equation established in Matlab, thereby establishing a rigid-flexible coupled dynamic model of the planetary gear and rotor system. Although the above method solves the problem of time-consuming solution of the pure finite element modeling method to a certain extent through the modeling method combining finite element and numerical methods, it still needs to calculate and obtain the mass and stiffness matrices through the finite element software and perform data transmission across the Ansys and Matlab platforms, which is not conducive to the frequent and rapid design of the gear transmission system.

[0004] The structure of the multi-stage fixed-axis helical gear transmission system is complex. It is difficult to model and calculate its dynamic characteristics using numerical calculation methods, and it is necessary to re-model according to the increase or decrease of the components of the multi-stage fixed-axis helical gear transmission system, which is not conducive to the rapid design of the system. In addition, most of the current research on the modeling of gear transmission systems ignores the influence of bearing clearance and friction on the bearing contact stiffness, resulting in the above technologies being unable to accurately consider the influence of bearing clearance and friction on the dynamic characteristics of the gear transmission system and unable to correctly guide the low-vibration design of the multi-stage fixed-axis helical gear transmission system. Summary of the Invention

[0005] The embodiments of this application provide a design method for a multi-stage fixed-axis helical gear transmission system to solve the problems in the prior art that the simulation method is difficult to calculate the dynamic characteristics of the multi-stage fixed-axis helical gear transmission system, does not consider the influence of bearing clearance and friction on the dynamic characteristics of the gear transmission system, and cannot correctly guide the design of the multi-stage fixed-axis helical gear transmission system.

[0006] The embodiments of this application provide a design method for a multi-stage fixed-axis helical gear transmission system, including:

[0007] Establish the internal deformation coordination equation of the bearing and the force balance equation of the rolling elements;

[0008] Solve the internal deformation coordination equation of the bearing and the force balance equation of the rolling elements to obtain the contact force between the bearing and the raceway, as well as the contact force, friction force and friction torque of the rolling elements respectively;

[0009] Determine the bearing stiffness considering the influence of clearance and friction based on the contact force between the bearing and the raceway, the contact force of the rolling elements, the friction force and the friction torque;

[0010] Combine the radial deformation and clearance in different cases to fit the bearing stiffness and obtain the bearing stiffness curve;

[0011] Determine the meshing stiffness curve of the helical gear pair;

[0012] Establish the multi-body models of all components in the system according to the geometric information of the multi-stage fixed-axis helical gear transmission system;

[0013] Define the gear force element and the bearing force element according to the force transfer relationship between the multi-body models;

[0014] Introduce the bearing stiffness curve and the meshing stiffness curve into the gear force element and the bearing force element to establish the multi-body dynamics model of the multi-stage fixed-axis helical gear transmission system;

[0015] Solve the multi-body dynamics model to obtain the dynamic characteristics of the multi-stage fixed-axis helical gear transmission system;

[0016] Adjust the structural parameters of the multi-stage fixed-axis helical gear transmission system, compare the dynamic characteristics of the multi-stage fixed-axis helical gear transmission system after adjusting the structural parameters, and obtain the optimized structural design parameters.

[0017] A design method for a multi-stage fixed-axis helical gear transmission system in this application has the following advantages:

[0018] Compared with the prior art, it considers the influence of bearing clearance and friction on bearing stiffness, and can more accurately consider the influence of bearing clearance and friction on the dynamic characteristics of the gear transmission system. In addition, compared with the method of calculating the dynamic characteristics of the system by using the finite element modeling method, it takes less time, is simpler to operate than the numerical modeling calculation method, and can quickly model according to the increase or decrease of the components of the multi-stage fixed-axis helical gear transmission system, which is conducive to the rapid design of the system. Description of the Drawings

[0019] To more clearly illustrate the technical solutions in the embodiments of the present application or the prior art, the following will briefly introduce the drawings required for the description of the embodiments or the prior art. Obviously, the drawings in the following description are only some embodiments of the present application. For those of ordinary skill in the art, without creative efforts, other drawings can also be obtained based on these drawings.

[0020] Figure 1 It is a flowchart of a design method for a multi-stage fixed-axis helical gear transmission system provided by an embodiment of the present application.

[0021] Figure 2 It is a three-dimensional schematic diagram of a multi-stage fixed-axis helical gear transmission system provided by an embodiment of the present application.

[0022] Figure 3 It is a side schematic diagram of a multi-stage fixed-axis helical gear transmission system provided by an embodiment of the present application.

[0023] Figure 4 It is a schematic diagram of the displacement-deformation relationship of bearing elements provided by an embodiment of the present application.

[0024] Figure 5 It is a schematic diagram of the relative sliding speed relationship between the rolling element and the outer raceway of a bearing element provided by an embodiment of the present application.

[0025] Figure 6 It is a schematic diagram of the relative sliding speed relationship between the rolling element and the inner raceway of a bearing element provided by an embodiment of the present application.

[0026] Figure 7 It is a schematic diagram of the force analysis of the rolling element of a bearing element in the xz direction provided by an embodiment of the present application.

[0027] Figure 8 It is a schematic diagram of the force analysis of the rolling element of a bearing element in the yz direction provided by an embodiment of the present application.

[0028] Figure 9 It is a schematic diagram of the force analysis of the rolling element of a bearing element in the xy direction provided by an embodiment of the present application.

[0029] Figure 10 It is a schematic diagram of a single tooth slice of a helical gear provided by an embodiment of the present application.

[0030] Figure 11 It is a schematic diagram of the maximum contact line length of a helical gear in the first case provided by an embodiment of the present application.

[0031] Figure 12 It is a schematic diagram of the maximum contact line length of a helical gear in the second case provided by an embodiment of the present application.

[0032] Figure 13It is a schematic diagram of the time-varying contact line length of the helical gear in the first case provided by the embodiment of the present application.

[0033] Figure 14 It is a schematic diagram of the time-varying contact line length of the helical gear in the second case provided by the embodiment of the present application.

[0034] Figure 15 It is a block diagram of the multi-body dynamics model of the multi-stage fixed-axis helical gear transmission system in the Simpack simulation platform provided by the embodiment of the present application.

[0035] Figure 16 It is the stiffness of the support bearing of the gear shaft Z4-1 in the multi-stage fixed-axis helical gear transmission system provided by the embodiment of the present application K xx The variation law with bearing clearance and bearing radial deformation.

[0036] Figure 17 It is the stiffness of the support bearing of the gear shaft Z4-1 in the multi-stage fixed-axis helical gear transmission system provided by the embodiment of the present application K θxθx The variation law with bearing clearance and bearing radial deformation.

[0037] Figure 18 It is the variation law of the tooth meshing stiffness between the gear shaft Z3 and the gear shaft Z4-1 with the gear rotation angle in the multi-stage fixed-axis helical gear transmission system provided by the embodiment of the present application.

[0038] Figure 19 It is the change of the root mean square value of the vibration acceleration of the gear shafts Z2 and Z3 with the change of the bearing support positions of the gear shafts Z4-1, Z4-2, and Z4-3 in the multi-stage fixed-axis helical gear transmission system provided by the embodiment of the present application.

[0039] Figure 20 It is the change of the root mean square value of the vibration acceleration of the gear shaft Z4-1 with the change of the bearing support positions of the gear shafts Z4-1, Z4-2, and Z4-3 in the multi-stage fixed-axis helical gear transmission system provided by the embodiment of the present application.

[0040] Figure 21 It is the change of the root mean square value of the vibration acceleration of the gear shaft Z4-2 with the change of the bearing support positions of the gear shafts Z4-1, Z4-2, and Z4-3 in the multi-stage fixed-axis helical gear transmission system provided by the embodiment of the present application.

[0041] Figure 22 It is the change of the root mean square value of the vibration acceleration of the gear shaft Z4-3 with the change of the bearing support positions of the gear shafts Z4-1, Z4-2, and Z4-3 in the multi-stage fixed-axis helical gear transmission system provided by the embodiment of the present application.

[0042] Figure 23It is the variation of the root mean square value of the vibration acceleration of gear shaft Z5-1 in the multi-stage fixed-axis helical gear transmission system provided by the embodiments of the present application with the bearing support positions of gear shafts Z4-1, Z4-2, and Z4-3.

[0043] Figure 24 It is the variation of the root mean square value of the vibration acceleration of gear shaft Z5-2 in the multi-stage fixed-axis helical gear transmission system provided by the embodiments of the present application with the bearing support positions of gear shafts Z4-1, Z4-2, and Z4-3.

[0044] Figure 25 It is the variation of the root mean square value of the vibration acceleration of gear shaft Z5-3 in the multi-stage fixed-axis helical gear transmission system provided by the embodiments of the present application with the bearing support positions of gear shafts Z4-1, Z4-2, and Z4-3.

[0045] Figure 26 It is the variation of the root mean square value of the vibration acceleration of gear shaft Z6 in the multi-stage fixed-axis helical gear transmission system provided by the embodiments of the present application with the bearing support positions of gear shafts Z4-1, Z4-2, and Z4-3. Detailed implementation manners

[0046] Next, the technical solutions in the embodiments of the present application will be clearly and completely described in conjunction with the accompanying drawings in the embodiments of the present application. Obviously, the described embodiments are only a part of the embodiments of the present application, rather than all the embodiments. Based on the embodiments in the present application, all other embodiments obtained by those of ordinary skill in the art without creative efforts shall fall within the protection scope of the present application.

[0047] Figure 1 It is a flowchart of a design method for a multi-stage fixed-axis helical gear transmission system provided by the embodiments of the present application. The embodiments of the present application provide a design method for a multi-stage fixed-axis helical gear transmission system, including:

[0048] S100, establishing an internal deformation coordination equation of the bearing and a force balance equation of the rolling elements.

[0049] Exemplarily, as Figure 2 and 3 shown is a schematic diagram of a multi-stage fixed-axis helical gear transmission system. The system consists of a total of 10 gear shafts, namely Z1, Z2, Z3, Z4-1, Z4-2, Z4-3, Z5-1, Z5-2, Z5-3, and Z6. Among them, Z1 is the input shaft and Z6 is the output shaft. Each gear shaft is supported by 2 angular contact ball bearings.

[0050] Furthermore, when establishing the internal deformation coordination equation and the force balance equation of the rolling elements of the bearing, first obtain the geometric dimension parameters of the bearing and the force analysis of the rolling elements, and then establish the internal deformation coordination equation and the force balance equation of the rolling elements based on the geometric dimension parameters and the force analysis respectively.

[0051] Specifically, the internal deformation coordination equation of the bearing is established based on the geometric dimension parameters of the bearing, and the force balance equation of the rolling elements is established based on the force analysis of the rolling elements. The bearing mentioned here is an angular contact bearing that supports the gear shaft.

[0052] When establishing the internal deformation coordination equation of the bearing, it is necessary to introduce the radial clearance dimension of the bearing into the equation. Specifically, as Figure 4 shown, the established internal deformation coordination equation of the bearing is shown in Equations (1) and (2):

[0053] (1)

[0054] (2)

[0055] In the equations, δ ik and δ ok are the contact deformation amounts of the inner raceway and the outer raceway of the bearing respectively, r i and r o are the curvature radii of the inner raceway and the outer raceway of the bearing respectively, D is the diameter of the steel ball. X 1k and X 2k are the distances between the geometric center of the deformed rolling element and the curvature center of the inner raceway of the bearing and the curvature center of the outer raceway of the bearing respectively, A 1k and A 2k are the horizontal and vertical distances between the curvature centers of the deformed inner raceway and the outer raceway of the bearing respectively, as shown in Equations (3) and (4):

[0056] (3)

[0057] (4)

[0058] In the equations, δ x 、 δ y and δ z are respectively the inner ring of the bearing at x 、y and z displacements in three directions θ y and θ z are respectively the angular displacements of the inner ring of the bearing about the y axis and the z axis, α 0 is the initial contact angle, φ k is the k th position angle of the rolling element, c 0 is the radial clearance of the bearing, ℜ i is the radial position of the contact point between the inner ring raceway of the bearing and the steel ball, which can be expressed by Equation (5):

[0059] (5)

[0060] In the formula, d m is the pitch diameter of the bearing.

[0061] The method of establishing the force balance equation of the rolling element successively includes determining the contact force, frictional force and frictional torque of the rolling element. Specifically, the contact forces between the rolling element and the inner ring raceway and the outer ring raceway can be expressed by Equation (6) and Equation (7):

[0062] (6)

[0063] (7)

[0064] In the formula, Q ik and Q ok are respectively the contact forces between the rolling element and the inner ring raceway and the outer ring raceway, K i and K o are respectively the Hertz contact stiffnesses of the inner ring raceway and the outer ring raceway.

[0065] The frictional forces and frictional torques exerted on the rolling element by the inner ring raceway and the outer ring raceway of the bearing can be obtained by Coulomb's law. Specifically, the elliptical contact area between the rolling element and the raceway is divided into v × w square contact elements, where v is the row label of the square contact element, w is the column label of the square contact element. By accumulating the frictional forces and frictional torques of each square contact element respectively, the frictional forces and frictional torques exerted on the rolling element by the inner ring raceway and the outer ring raceway can be obtained, which are expressed by Equation (8) and (9):

[0066] (8)

[0067] (9)

[0068] In the formula, f ix / iy and f ox / oy are the frictional forces exerted on the rolling elements by the inner raceway and the outer raceway, respectively; S i and S o are the Hertz contact ellipse areas between the rolling elements and the inner raceway and the outer raceway, respectively; τ ix / iy ( x c , y c ) and τ ox / oy ( x c , y c ) are the frictional stresses between the rolling elements and the inner raceway and the outer raceway, respectively; sign () is the sign function; u ix / iy ( x v , y w ) and u ox / oy ( x v , y w ) are the relative sliding velocities between the rolling elements and the inner raceway and the outer raceway, respectively; P i ( x v , y w ) and P o ( x v , y w ) are the Hertz contact pressures of the square contact elements above the rolling elements and the inner raceway and the outer raceway ([[]] x v , y w ); A i ( x v , y w ) and A o (x v , y w are the areas of each square contact unit on the inner raceway and the outer raceway, respectively; m i and m o are the frictional torques exerted on the rolling elements by the inner raceway and the outer raceway, respectively; x v and y w are the positions of each square contact unit along the coordinate axes x c and y c respectively; u ix ( x v , y w ) and u iy ( x v , y w ) are the relative sliding velocities of the rolling element and the inner raceway along the coordinate axes x c and y c respectively, u ox ( x v , y w ) and u oy ( x v , y w ) are the relative sliding velocities of the rolling element and the outer raceway along the coordinate axes x c and y c respectively; μ ic and μ oc are the Coulomb friction coefficients of the inner raceway and the outer raceway respectively, which can be expressed by Equation (10):

[0069] (10)

[0070] wherein, μ is μ ic and μ oc collectively; AL , B L , C L and D L are the fitting coefficients of the friction coefficient varying with load, temperature, and relative sliding speed; SR is the rolling - sliding ratio of the rolling element. Solve the corresponding fitting coefficients according to the load, temperature, and relative sliding speed of the rolling element with the inner - raceway and outer - raceway A L , B L , C L and D L , and then the parameters can be obtained according to Equation (10) μ ic and μ oc .

[0071] As Figure 5 , Figure 6 shown, the relative sliding speed of the rolling element with the inner - raceway and outer - raceway at the contact position ( x c , y c ) is shown by Equation (11):

[0072] (11)

[0073] In the formula, is u ix ( x c , y c ) and u ox ( x c , y c ) in general, is u iy ( x c , y c ) and u oy ( x c , y c ) in general, u ix ( x c , yc ) u iy ( x c , y c ) u ox ( x c , y c ) and u oy ( x c , y c ) are respectively the relative sliding speeds at the contact positions of the rolling elements and the inner raceway x c and y c and the relative sliding speeds at the contact positions of the rolling elements and the outer raceway x c and y c ; is the collective term for two parameters and , and are respectively the linear velocity vectors of the surfaces of the inner raceway and the outer raceway along the coordinate axis x c direction; is the collective term for two parameters and , and are respectively the linear velocity vectors of the contact surfaces of the rolling elements with the inner raceway and the outer raceway along the coordinate axis x c direction; is the collective term for two parameters and , and are respectively the linear velocity vectors of the surfaces of the inner raceway and the outer raceway along the coordinate axis y c direction; is the collective term for two parameters and , and are respectively the linear velocity vectors of the contact surfaces of the rolling elements with the inner raceway and the outer raceway along the coordinate axis y c direction; is the collective term for two parameters and , and are the angular velocity vectors of the inner raceway and the outer raceway surfaces around the coordinate axes z c respectively; are the general terms for two parameters and respectively; and are the angular velocity vectors of the contact surfaces between the rolling elements and the inner and outer raceways around the coordinate axes z c respectively. The linear velocity vectors of the rolling elements and the raceway surfaces are expressed by Equations (12) and (13) respectively:

[0074] (12)

[0075] (13)

[0076] In the equations, ω xj , ω yj and ω zj are the self-rotation angular velocity components of the rolling elements in the x , y and z directions respectively; α j is the contact angle between the j th rolling element and the raceway; ω i / o are the general terms for two parameters ω i and ω o respectively; ω i and ω o are the rotational angular velocities of the inner and outer rings of the bearing respectively; ω cj is the common angular velocity of the rolling elements; d bi / bo are the general terms for two parameters d bi and d bo respectively; d bi and d bo are the distances between the geometric center of the rolling element and the contact points of the inner and outer raceways( x c , y c ) respectively; d zi / zo are the general terms for two parameters dzi and d zo collectively referred to as d zi and d zo are respectively the distances between the rolling element contact points and the rotation axes of the inner and outer raceways of the bearing:

[0077] (14)

[0078] In the formula, r ie / oe are two parameters r ie and r oe collectively referred to as r ie and r oe are respectively the equivalent curvature radii between the rolling elements and the inner and outer raceways, r ie / oe =2 r i / o d b / (2 r i / o + d b ); r i / o are two parameters r i and r o collectively referred to as d b is the rolling element diameter; a i / o are two parameters a i and a o collectively referred to as a i and a o are respectively the major semi-axis lengths of the elliptical contact areas between the rolling elements and the inner and outer raceways, d is the inner diameter of the bearing.

[0079] The force conditions of the rolling elements are as shown in Figures 7 - 9 The established force balance equations of the rolling elements are expressed by Equations (15) to (19):

[0080] (15)

[0081] (16)

[0082] (17)

[0083] (18)

[0084] (19)

[0085] In the formula, α ik and α ok are the contact angles between the rolling elements and the inner and outer raceways respectively; f ix and f ox are the frictional forces between the rolling elements and the inner and outer raceways along the coordinate axis x c direction respectively; f iy and f oy are the frictional forces between the rolling elements and the inner and outer raceways along the coordinate axis y c direction respectively; F c is the centrifugal force of the rolling element; F d is the fluid resistance suffered by the rolling element; m ix and m ox are the components of the frictional torque of the rolling element and the inner and outer raceways around the coordinate axis x c direction respectively; m iy and m oy are the components of the frictional torque of the rolling element and the inner and outer raceways around the coordinate axis y c direction respectively; M gy’ is y the gyroscopic torque in the Figure 9 axis direction, M gz’ in z is the gyroscopic torque in the

[0086] S110, solve the internal deformation coordination equation of the bearing and the force balance equation of the rolling element, and obtain the contact force between the bearing and the raceway, as well as the contact force, frictional force and frictional torque of the rolling element respectively.

[0087] Exemplarily, when solving the internal deformation compatibility equation and the force balance equation of the rolling elements of the bearing, based on the given external load conditions and the bearing displacements updated iteratively, the Newton-Raphson method is used to iteratively solve the internal deformation compatibility equation and the force balance equation of the rolling elements of the bearing.

[0088] Specifically, when updating the bearing displacements iteratively, an iterative equation is defined according to the force balance equation of the inner ring of the bearing. If the iterative equation does not satisfy the convergence constraint conditions, the increment of the bearing displacements is calculated, and the initial values of the bearing displacements are updated based on the increment. The initial values are substituted into the force balance equation of the rolling elements, and the updating process is repeated until the iterative equation satisfies the convergence constraint conditions to obtain the final bearing displacements.

[0089] The iterative equation is expressed by Equations (20) to (24):

[0090] (20)

[0091] (21)

[0092] (22)

[0093] (23)

[0094] (24)

[0095] In the equations, H 1, H 2, H 3, H 4 and H 5 constitute the iterative equation H g , F x , F y and F z are the external loads along the coordinate axes x , y and z respectively; N b is the number of rolling elements; M x and M y are the external torques about the coordinate axes x , y respectively.

[0096] If the iterative equation H g does not satisfy the convergence constraint conditions, the increment of the bearing displacements {Δ δ} is calculated, which is expressed by Equation (25):

[0097] (25)

[0098] In the formula, the superscript T represents the transpose, θ x is the angular displacement of the bearing inner ring around the x axis. Based on the increment of the bearing displacement {Δ δ}, update the initial value of the bearing displacement, substitute the updated initial value into equations (15) to (19), and repeat the above solution process until the iterative equation H g satisfies the convergence constraint condition.

[0099] S120, determine the bearing stiffness considering the influence of clearance and friction based on the contact force between the bearing and the raceway, the contact force of the rolling elements, the friction force, and the friction torque.

[0100] Exemplarily, according to the relationship between the bearing inner ring force and displacement, solve to obtain the bearing stiffness considering the influence of clearance and friction, as shown in equation (26):

[0101] (26)

[0102] In the formula, K xx 、K yy 、K zz 、 and are the main stiffness of the bearing, and the remaining matrix elements are the cross stiffness.

[0103] S130, combine the radial deformation and clearance in different cases, fit the bearing stiffness, and obtain the bearing stiffness curve.

[0104] S140, determine the meshing stiffness curve of the helical gear pair.

[0105] Exemplarily, when determining the meshing stiffness curve of the helical gear pair, first divide the tooth along the tooth thickness direction into multiple spur gear micro-elements, calculate the meshing stiffness of each spur gear micro-element, and then parallelly connect the meshing stiffness of each spur gear micro-element to solve and obtain the meshing stiffness curve of the helical gear pair.

[0106] As Figure 10 shown, the helical gear can be equivalent to countless spur gear micro-elements with a width of db . Since db the width is infinitesimal, the meshing stiffness of the spur gear micro-element with a width of db can be expressed by equation (27):

[0107] (27)

[0108] In the formula, K d is the meshing stiffness of the spur gear element, K i is the time-varying meshing stiffness of the spur gear element with a tooth width of db . The time-varying meshing stiffness K i can be simply obtained by conventional calculation methods such as the energy method or the material mechanics method. The meshing stiffness of the helical gear pair can be obtained by the parallel connection of the meshing stiffnesses of all spur gear elements participating in meshing. To determine the spur gear elements participating in meshing, it is necessary to calculate the time-varying contact line length of the helical gear. According to the geometric relationship shown in Figures 11 - 12 , the maximum value of the contact line length can be expressed by Equation (28):

[0109] (28)

[0110] In the formula, L tmax is the maximum value of the contact line length, L cd is the transverse length of the meshing plane, that is, the involute length from the tooth root to the tooth tip, and its expression is L cd = ε a · pbt , where ε a is the transverse contact ratio of the helical gear, pbt is the base pitch of the driving wheel end face, β is the helix angle of the helical gear, b is the tooth width of the helical gear.

[0111] As shown in Figures 13 - 14 , the contact line length of the helical gear changes continuously with the gear rotation angle. Corresponding to the two cases of the maximum contact line length shown in Equation (28), the time-varying contact line length of the helical gear is expressed by Equation (29):

[0112] (29)

[0113] In the formula, L t1 and L t2 are the time-varying contact line lengths of the helical gear in the first case and the second case respectively, θ is the gear rotation angle, θ 1 is the gear rotation angle corresponding to the maximum value of the time-varying contact line length of the helical gear in the first case, θ 2 is the gear rotation angle corresponding to the maximum value of the time-varying contact line length of the helical gear in the second case.

[0114] Taking the case where the rotation angle is in the range of 0 ≤ θ ≤ θ 2 as an example, the meshing stiffness expression of the helical gear pair is expressed by Equation (30): n

[0115] (30) (30)

[0116] In the formula, is the meshing stiffness of the n th helical gear pair, θ is is the gear rolling angle corresponding to the starting point of the contact line, θ if is the rolling angle corresponding to the ending point of the contact line.

[0117] Thus, the meshing stiffness of the helical gear pair is expressed by Equation (31):

[0118] (31)

[0119] In the formula, K h is the meshing stiffness of the helical gear pair, m is the maximum number of simultaneously meshing teeth, and its expression is m = ceil( ε a + ε b ), where ceil() is the ceiling function, ε a and ε b are the face contact ratios of the two meshing helical gears.

[0120] S150. According to the geometric information of the multi-stage fixed-axis helical gear transmission system, a multi-body model of each component in the system is established.

[0121] Exemplarily, after establishing the multi-body model of each component in the system, the motion states of each component in the system are also analyzed, and the corresponding degrees of freedom of motion of the corresponding components are restricted. Specifically, kinematic pairs can be used to restrict the degrees of freedom of motion of each component.

[0122] For example Figure 15As shown, based on the Simpack multi-body dynamics simulation platform, geometric information such as the mass and moment of inertia of each gear shaft in the multi-stage fixed-axis helical gear transmission system is defined, and multi-body models of each gear shaft component are established. According to the motion states of the components in the multi-stage fixed-axis helical gear transmission system, corresponding kinematic pairs are used to restrict the degrees of freedom of the components. The usage of kinematic pairs in the multi-stage fixed-axis helical gear transmission system provided in this embodiment is shown in Table 1. In Table 1, the component "ground" is the fixed coordinate system of the Simpack simulation platform. To facilitate modeling, a fictional component "box" is added, which is fixedly connected to the ground through the kinematic pair Joint #0, and its degree of freedom of motion is 0.

[0123] Table 1 Usage of kinematic pairs in the multi-stage fixed-axis helical gear transmission system

[0124]

[0125] S160. According to the force transfer relationship between multi-body models, gear force elements and bearing force elements are defined.

[0126] Exemplarily, as Figure 15 shown, each gear shaft is supported by two bearings. The bearing force element in the Simpack multi-body dynamics simulation platform is 88: Rolling bearing. The gear force element 225: Gear pair is used to represent the tooth meshing force between two meshing helical gears.

[0127] S170. The bearing stiffness curve and the meshing stiffness curve are introduced into the gear force element and the bearing force element to establish a multi-body dynamics model of the multi-stage fixed-axis helical gear transmission system.

[0128] S180. Solve the multi-body dynamics model to obtain the dynamic characteristics of the multi-stage fixed-axis helical gear transmission system.

[0129] Exemplarily, the solver of the Simpack simulation platform can be used to simulate and solve the multi-body dynamics model. The simulation step size and the simulation time can be adjusted according to actual needs. The smaller the simulation step size, the better the convergence of the results, but the simulation time increases accordingly. It is recommended that the simulation solution time step size be Δ t =1×10 -5 s. The kinematic and dynamic states of each component of the gear transmission system are obtained by solving the multi-body dynamics model.

[0130] S190. Adjust the structural parameters of the multi-stage fixed-axis helical gear transmission system, compare the dynamic characteristics of the multi-stage fixed-axis helical gear transmission system after adjusting the structural parameters, and obtain the optimized structural design parameters.

[0131] Exemplarily, the distance between the support bearings of the gear shaft can be adjusted in batches through the DoE (Design of Experiments) module of the Simpack simulation platform to obtain the influence of the bearing support position on the system vibration level, so as to obtain the optimal support position parameters of the gear shaft. Taking the multi-stage fixed-axis helical gear transmission system provided in this embodiment as an example, the bearing movement distance conditions are shown in Table 2. Table 2 shows that there are a total of 4096 calculation conditions for the multi-stage fixed-axis helical gear transmission system provided in this embodiment. When using the finite element method or the combination method of the finite element method and the numerical method for calculation, there are problems of huge workload. The method adopted in this application can use the DoE module to automatically adjust parameters and calculate these 4096 calculation conditions.

[0132] Table 2 Bearing movement distance conditions of the gear shaft of the multi-stage fixed-axis helical gear transmission system

[0133]

[0134] Stiffness of the support bearing of gear shaft Z4-1 K xx and The variation laws with bearing clearance and bearing radial deformation are as shown in Figure 16 、 Figure 17 The results show that the bearing stiffness K xx and decrease with the increase of bearing clearance, and with the continuous increase of bearing radial deformation, the influence of clearance on bearing stiffness gradually decreases.

[0135] The variation law of the tooth meshing stiffness between gear shaft Z3 and gear shaft Z4-1 with the gear rotation angle is as shown in Figure 18 The results show that the meshing stiffness of helical gears fluctuates less than that of spur gears, which is beneficial to improving the vibration level of the multi-stage fixed-axis helical gear transmission system.

[0136] Figures 19 to 26 Respectively show the changes in the root mean square values of the vibration acceleration of gear shafts Z2 and Z3, gear shafts Z4-1 to Z4-3, gear shafts Z5-1 to Z5-3, and gear shaft Z6 with the change of the bearing support positions of gear shafts Z4-1, Z4-2, and Z4-3. The results show that increasing the bearing support position of the shunt gear shaft Z4 has an obvious inhibitory effect on the vibration levels of other gear shaft components. When the rear bearing movement distance is -15 mm and the front bearing movement distance is 15 mm, it is the optimal bearing support position for the shunt gear shaft Z4.

[0137] Although the preferred embodiments of the present application have been described, additional changes and modifications can be made to these embodiments by those skilled in the art once they learn of the basic creative concept. Therefore, the appended claims are intended to be construed to include the preferred embodiments as well as all changes and modifications that fall within the scope of the present application.

[0138] Obviously, those skilled in the art can make various changes and modifications to the present application without departing from the spirit and scope of the present application. Thus, if these modifications and variations of the present application fall within the scope of the claims of the present application and their equivalent technologies, the present application is also intended to include these modifications and variations.

Claims

1. A design method for a multi-stage fixed-axis helical gear transmission system, characterized in that, Including: Establish the internal deformation coordination equation of the bearing and the force balance equation of the rolling elements; Solve the internal deformation coordination equation of the bearing and the force balance equation of the rolling elements to obtain the contact force between the bearing and the raceway, as well as the contact force, friction force and friction torque of the rolling elements respectively; Determine the bearing stiffness considering the influence of clearance and friction based on the contact force between the bearing and the raceway, the contact force of the rolling elements, the friction force and the friction torque; Fit the bearing stiffness by combining the radial deformation and clearance under different conditions to obtain the bearing stiffness curve; Determine the meshing stiffness curve of the helical gear pair; Establish the multi-body models of each component in the system according to the geometric information of the multi-stage fixed-axis helical gear transmission system; Define the gear force element and the bearing force element according to the force transfer relationship between the multi-body models; Introduce the bearing stiffness curve and the meshing stiffness curve into the gear force element and the bearing force element to establish the multi-body dynamics model of the multi-stage fixed-axis helical gear transmission system; Solve the multi-body dynamics model to obtain the dynamic characteristics of the multi-stage fixed-axis helical gear transmission system; Adjust the structural parameters of the multi-stage fixed-axis helical gear transmission system, compare and adjust the dynamic characteristics of the multi-stage fixed-axis helical gear transmission system after adjusting the structural parameters, and obtain the optimized structural design parameters; When establishing the internal deformation coordination equation of the bearing and the force balance equation of the rolling elements, first obtain the geometric dimension parameters of the bearing and the force analysis of the rolling elements, and then establish the internal deformation coordination equation of the bearing and the force balance equation of the rolling elements respectively based on the geometric dimension parameters and the force analysis; When establishing the force balance equation of the rolling elements, divide the elliptical contact area between the rolling elements and the raceway into multiple square contact units, accumulate the friction force and the friction torque of each square contact unit respectively, and obtain the friction force and the friction torque applied to the rolling elements by the inner raceway and the outer raceway; When determining the meshing stiffness curve of the helical gear pair, first divide the tooth along the tooth thickness direction into multiple spur gear micro-elements, calculate the meshing stiffness of each spur gear micro-element, and then parallelly connect the meshing stiffness of each spur gear micro-element to solve and obtain the meshing stiffness curve of the helical gear pair; 2. The design method of a multi-stage fixed-axis helical gear transmission system according to claim 1, characterized in that, When solving the internal deformation coordination equation of the bearing and the force balance equation of the rolling elements, based on the given external load conditions and the bearing displacement updated by iteration, use the Newton-Raphson method to iteratively solve the internal deformation coordination equation of the bearing and the force balance equation of the rolling elements; 3. The design method of a multi-stage fixed-axis helical gear transmission system according to claim 2, characterized in that, When iteratively updating the bearing displacement, define the iteration equation according to the force balance equation of the bearing inner ring. If the iteration equation does not meet the convergence constraint conditions, calculate the increment of the bearing displacement, update the initial value of the bearing displacement based on the increment, substitute the initial value into the force balance equation of the rolling elements, and repeat the update process until the iteration equation meets the convergence constraint conditions to obtain the final bearing displacement; 4. A design method for a multi-stage fixed-axis helical gear transmission system according to claim 1, characterized in that After establishing the multi-body models of each component in the system, also analyze the motion states of each component in the system and restrict the corresponding motion degrees of freedom of the corresponding components; 5. The design method of a multi-stage fixed-axis helical gear transmission system according to claim 4, characterized in that, Use kinematic pairs to restrict the motion degrees of freedom of each component.

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