A multi-objective optimization design method for the spatial layout of a gear transmission system

Abstract

The present invention provides a multi-objective optimization design method for the spatial layout of a gear transmission system, which belongs to the field of gear transmission, and more specifically, relates to a multi-objective optimization design method for the spatial layout of a gear transmission system. The method first constructs the basic structural units of the gear transmission system, then describes the spatial posture based on the connection relationship of the basic structural units, then establishes a shaft system mechanical model, then establishes a gear transmission system spatial layout model and its optimization model, and finally performs multi-objective optimization on the gear transmission system spatial layout based on the NSGA-Ⅱ genetic algorithm. This method can effectively solve the problem of multi-objective optimization design of spatial layout schemes of various types of multi-stage gear transmission systems.

Description

Technical Field

[0001] The invention belongs to the field of gear transmission and relates to a multi-objective optimization design method for a spatial layout scheme of a gear transmission system. Background Art

[0002] Gear transmission systems, with their advantages of high transmission efficiency, compact structure, strong reliability, and long service life, are widely used in modern industrial equipment. With the wave of digital transformation sweeping across industries worldwide, gears and gear products are moving towards digitalization, intelligence, and miniaturization. However, the design process of a gear transmission system is highly complex. The parameter design and spatial layout of the gear pairs have a significant impact on the spatial structure, operating performance, and mechanical properties of the entire transmission system. Traditional design methods rely heavily on the designer's experience, supplemented by specialized software for verification. This results in long gear transmission system design cycles, a degree of subjectivity, and blindness, and the designed product fails to comprehensively consider the impact of various factors on the system's structure and performance. Therefore, to meet the diverse design requirements of gear transmission systems across various industries, this paper utilizes the principle of discretization followed by assembly for the selected gear transmission scheme to obtain a spatial layout scheme for the gear transmission system with multiple scales, structures, and performance.

[0003] Currently, researchers at home and abroad have simplified various types of gear transmission systems and abstracted various layout models. This has optimized key design parameters and spatial layouts, reduced system weight, and improved system compactness. However, most methods are limited in scope and lack versatility and scalability. Furthermore, existing layout modeling methods lack research on the mechanical properties of the gears and shafts in the system.

[0004] Chinese patent application CN113987903A discloses a method for coupled optimization design of the dimensions and layout of a multi-stage cylindrical gear transmission system. This method can solve the coupled optimization problem of the dimensions and layout of cylindrical gear transmission systems with any number of transmission stages. However, this method is limited in its applicability and cannot solve the layout problems of other types of gear transmission systems, such as bevel gear transmissions and worm gear transmissions. Summary of the Invention

[0005] This invention provides a method for multi-objective optimization design of the spatial layout of a gear transmission system. This method first constructs the basic structural units of the gear transmission system, then describes the spatial position based on the connection relationships of the basic structural units. A shafting mechanical model is then established, followed by a spatial layout model and its optimization model for the gear transmission system. Finally, the multi-objective optimization of the spatial layout of the gear transmission system is performed using the NSGA-II genetic algorithm. This method can effectively solve the multi-objective optimization design problem of the spatial layout of various types of multi-stage gear transmission systems.

[0006] In order to solve the above problems, this application proposes a multi-objective optimization design method for the spatial layout of a gear transmission system, comprising the following steps:

[0007] A multi-objective optimization design method for the spatial layout of a gear transmission system, the specific steps of the method are as follows:

[0008] Step 1: construct the basic structural unit of the gear transmission system;

[0009] The physical structures of various gears and shafts in the gear transmission system are mapped into cylindrical models. According to the connection relationship, the mapped cylindrical models of the gears and shafts in the first-level gear transmission are assembled into the mapping structure of the gear transmission. The mapping structure is the basic structural unit of the gear transmission system; wherein the size parameters and connection parameters of the cylindrical model are the basic properties of the basic structural unit; different types of basic structural units of the gear transmission system are constructed according to different types of gear pairs.

[0010] Step 2: Describe the spatial pose based on the connection relationship of the basic structural units;

[0011] There are three types of connection relationships among basic structural units. The first is a series connection relationship, that is, the output component of the previous level basic structural unit is connected in series to the input component of the next level basic structural unit; the second is a parallel connection relationship, that is, the previous level basic structural unit is connected to multiple next level basic structural units at the same time, or multiple previous level basic structural units are connected to the next level basic structural unit at the same time; the third is a closed-loop connection relationship, which can be considered as a combination of two parallel connection relationships; according to different basic structural unit connection relationships, homogeneous coordinate transformation is used to describe the spatial posture of the basic structural unit.

[0012] Step 3, establish the shaft system mechanical model;

[0013] Step 3.1, solve the projection cuboid Cub of the basic unit structure: Project the cylindrical model in the basic structure unit onto the x, y, and z coordinate axes of the global coordinate system to obtain the minimum and maximum values of the cylindrical projection; then calculate the minimum and maximum values of the projections of all cylinders on the x, y, and z axes. Finally, reduce and increase the safety distance ΔS at the minimum and maximum values of the x, y, and z axis projections, respectively, to obtain the cuboid Cub;

[0014] Step 3.2, extend the connecting axis to the plane of the cuboid Cub and solve the axis system support: select two points P1 and P2 on any axis, find the plane of the cuboid Cub in the same direction as P2P1 and P1P2 respectively, extend the axis to the plane in the directions of P2P1 and P1P2 respectively, and get the intersection points P 11 、P 22 Coordinates, that is, the fulcrum coordinates of the selected axis;

[0015] Step 3.3, perform mechanical analysis of basic structural units;

[0016] Step 3.4: Establish a shafting mechanical model: First, establish a local rectangular coordinate system o-xyz with the geometric center point of the gear connected to the shaft in the previous basic structural unit of the shafting as the origin, with x along the axis. Second, decompose the forces acting on the shaft support into radial and axial directions, and establish radial force mechanical models and axial force mechanical models for the shafting support, respectively. The axial force mechanical model for the shafting support is divided into three cases: the first case is a support method with two ends floating; the second case is a support method with a single support point and two directions fixed; and the third case is a support method with two supports and one direction fixed. Finally, obtain the support forces in the x, y, and z directions for each shafting support to complete the construction of the shafting mechanical model.

[0017] Step 4, establishing a gear transmission system spatial layout model and its optimization model;

[0018] First, according to the transmission scheme, the corresponding basic structural units are assembled into a complete system. The spatial structure of the gear transmission system is described by using the internal design parameters of the basic structural units and the connection parameters between the basic structural units, thereby forming a spatial layout model of the gear transmission system.

[0019] Secondly, the optimization variables, optimization objectives, and constraints in the spatial layout model of the gear transmission system are defined;

[0020] Step 5, multi-objective optimization of the spatial layout of the gear transmission system is performed based on the NSGA-II genetic algorithm;

[0021] Substitute the optimization variables, optimization objectives, and constraints established in step 4 into the NSGA-II genetic algorithm for solution. The algorithm's solution process is as follows:

[0022] Step 5.1, generate the initial population P0;

[0023] Step 5.2, perform fast non-dominated sorting and crowding calculation on population P0;

[0024] Step 5.3, through the evolution operation: selection > recombination > mutation, obtain the offspring population Q t , and construct a new population R t =P t +Q t ;

[0025] Step 5.4, for population R t Perform fast non-dominated sorting and congestion calculation;

[0026] Step 5.5, in the population R tA certain number of individuals are selected to calculate their fitness, and the individuals with the highest fitness are selected to enter the next generation population;

[0027] In step 5.6, determine whether the population size has reached N. If not, jump to step 5.5 and continue iterating until the population size reaches N.

[0028] In step 5.7, determine whether the termination condition is met. If not, jump to step 5.3 and continue iterating until the termination condition is met. If so, output the optimal solution.

[0029] Furthermore, the step 1 constructs the basic structural unit of the gear transmission system: the local right-handed input coordinate system O is established with the geometric center of the cylinder mapped by the small gear and the large gear as the origin. ik -x ik y ik z ik and the output coordinate system O ok -x ok y ok z ok , the z-axis direction of the coordinate system is along the axis direction of the gear; the radius and height of the two cylinders are r ik 、h ik and r ok 、h ok ; Establish connecting semi-axes along the positive direction of z axis from the origin of the two local coordinate systems, and the connection points are J ik 、J ok , the axial directions are W ik 、W ok The connecting semi-axle diameters are d ik d ok ; Use formula (1) to output coordinate system {O ok P in} ok Point coordinates are transformed to the input coordinate system {O ik P under} ik Point coordinates:

[0030]

[0031] In the formula, [X ok Y ok Z ok 1] T and [X ik Y ik Z ik 1] T They are respectively P point in the output coordinate system {O ok} and input coordinate system {O ik} Homogeneous coordinates; in RPY angle α k To rotate around the fixed input coordinate system {Oik The rotation angle of the z axis, β k To rotate around the fixed input coordinate system {O ik}y-axis rotation angle, γ k To rotate around the fixed input coordinate system {O ik}Rotation angle of x-axis; a k 、b k 、c k For the output coordinate system {O ok}The origin is in the input coordinate system {O ik}Next x, y, z coordinate values;

[0032] Determine the size parameter r for different types of gear transmission ik 、h ik 、r ok 、h ok and position and attitude parameters α k , β k , γ k 、a k 、b k 、c k Construct different types of basic structural units.

[0033] Furthermore, in step 2, the spatial position of the basic structural unit is described: the input semi-axis of the kth basic structural unit is connected to the output semi-axis of the k-1th basic structural unit to establish a connection relationship, and the two semi-axis are merged into one connection axis through connection; the input coordinate system of the basic structural unit k is described using formula (2) { ik} Output coordinate system relative to basic structural unit k-1 {O ok-1} position and posture;

[0034]

[0035] Where m is the total number of basic structural units in the gear transmission system; θ k When the basic structure unit k is connected to the previous level basic structure unit k-1, the basic structure unit k rotates around the output semi-axis of the previous level basic structure unit k-1; d k The signed distance along the z-axis between the origin of the input coordinate system of the basic structural unit k and the origin of the output coordinate system of the basic structural unit k-1.

[0036] Furthermore, in step 3.1, the cylindrical model in the basic structural unit is projected onto the x, y, and z coordinate axes of the global coordinate system using formula (3) to obtain the minimum and maximum values of the cylindrical projection;

[0037]

[0038] Where λ eminis the projection minimum, λ emax is the maximum value of the projection, D is the unit vector of the projection direction, C e 、r e 、w e 、h e They are respectively the center point coordinates, radius value, axial unit vector, and height value of the e-th cylinder.

[0039] Furthermore, in step 3.2, the method for finding the plane of the cuboid Cub in the same direction as P2P1 and P1P2 is: and i p ∈{1,2,···,6} simultaneously holds true, then P2P1 and the plane Normal vectors are in the same direction; when and i p ∈{1,2,···,6} simultaneously holds true, then P1P2 and the plane Normal vectors are in the same direction.

[0040] Furthermore, in step 3.2, the intersection point P 11 、P 22 The coordinate acquisition method is shown in formula (4):

[0041]

[0042] Where, P 11 、P 22 are the coordinates of the intersection of the axis and the two planes, The axis intersection planes The coefficients of the plane equation, P1 and P2 are the coordinate points.

[0043] Furthermore, in step 3.4, the mechanical model of radial force at the shaft support is shown in formula (5);

[0044]

[0045] Where, F xa 、F ya 、F za , a=1,2 is the force acting on the shaft at the support; F b、 P b ,b∈{1,2,···,n F} is the force acting on the gear meshing of the shaft system and its point of action, n F is the total number of forces; M k , k∈{1,2,···,n M} is the torque acting on the shaft system, n M is the total number of couple moments; x, y, z are unit vectors in three directions.

[0046] Furthermore, step 4 defines the optimization variables, optimization objectives, and constraints in the spatial layout model of the gear transmission system, including the following specific steps:

[0047] Step 4.1, define the optimization variables. There are two types of optimization design variables in the gear transmission system spatial layout optimization model. One type is used to characterize the internal structural dimensions of the basic structural unit, and the other type is used to characterize the connection relationship of the basic structural unit. The optimization variables are shown in formula (6):

[0048] X=[X U ,X C ] (6)

[0049]

[0050] Where, X is the overall optimization design variable of the gear transmission system spatial layout optimization model, X U is the sum of the optimization variables of the basic structural unit size, x r is the design variable inside the basic structural unit r, r∈{1,2,···,n type} is the type of basic structural unit, n type is the total number of basic structural unit types; X C is the optimization design variable of the connection pair, θ is the rotation angle of the input semi-axis of the next basic structural unit around the output semi-axis of the previous basic structural unit, d is the signed distance along the z-axis between the origin of the input coordinate system of the next basic structural unit and the origin of the output coordinate system of the previous basic structural unit, and c∈{2,···,m} is the number of transmission stages;

[0051] Step 4.2, define the first optimization goal, which is a compact system structure, as shown in formula (7):

[0052] V=V gear +V shaft +V box (7)

[0053] Where V is the volume of the main parts in the gear transmission system. The smaller the volume, the more compact the structure. gear 、V shaft 、V box , are the volume of gear and shaft, and the volume of housing respectively; where V gear 、V shaft 、V box , is calculated as shown in formula (8):

[0054]

[0055] Where h i1 、hi2 are the equivalent widths of the small gear and large gear of each transmission stage, r i1 、r i2 are the equivalent radius of the small gear and large gear of each stage, r i1,s 、r i2,s are the radii of the shafts where the pinion and gear of each stage are located; r j,s is the radius of the axis, h j,s is the length of the axis; L ou 、W ou 、H ou are the length, width and height of the gearbox outer contour in the three coordinate directions respectively; L in 、W in 、H in are the length, width and height of the inner boundary of the smallest cuboid formed by all basic structural units and axes; Δl, Δw and Δh are the safety distances defined between the inner wall of the box and the inner frame in the three coordinate directions; in 、W in 、H in The calculation of is shown in formula (9):

[0056]

[0057] Where x min , x max ,y min ,y max , z min , z max are the minimum and maximum values of the projections of all cylinders in the x-axis, y-axis, and z-axis directions respectively;

[0058] Step 4.3: Define the second optimization goal. The second optimization goal is to make the transmission strength of each level equal, as shown in formula (10):

[0059]

[0060] Where S Hi1 、S Hi2 、S Fi1 、S Fi2 The safety factors for contact and bending fatigue calculations of the i-th level transmission are M H 、M F Calculate the safety factor for average contact and bending fatigue at each level, where m is the number of basic structural units;

[0061] Step 4.4, the third optimization goal is to reduce the magnitude of the support reaction force at the shaft support point, as shown in formula (11):

[0062]

[0063] Where, F xa 、F ya 、F za , a=1,2 is the load at the two supporting points of the shaft system;

[0064] Step 4.5: Define the constraints, which include upper and lower limit constraints of parameters, coprime constraints of the number of teeth, transmission ratio constraints of each level, total transmission ratio constraints, strength constraints, and interference constraints, as shown in formula (12):

[0065]

[0066] Where x jnmin 、x jnmax is the design variable x jn The minimum and maximum values are given by x jn Variable type determination; n X is the dimension of the overall design variable X of the spatial layout model; Pn is a positive number; i kmin 、i kmax is the minimum and maximum transmission ratio allowed for the k-th transmission; i k is the kth transmission ratio; m is the number of transmission stages; i represents the total transmission ratio of the gear transmission system; δ is the allowable error of the transmission ratio; Gapd s ≤0 is the sth interference constraint, and q is the total number of interference constraints.

[0067] The beneficial effect of the present invention is that a spatial layout model of a gear transmission system based on the assembly of basic structural units is proposed. By analyzing the basic composition of the gear transmission system, the concept of the basic structural unit of the gear transmission system is introduced, and a method for establishing the basic structural unit is proposed. The spatial position of the basic structural unit is described based on the connection relationship of the basic structural unit, and a mechanical model of the gear transmission system shaft system is established. By adopting the idea of first discretizing and then assembling, a spatial layout model of the gear transmission system with certain versatility is constructed. At the same time, a multi-objective optimization design method for the spatial layout of the gear transmission system based on the NSGA-II genetic algorithm is proposed. A spatial layout optimization model for the gear transmission system is constructed, and a multi-objective optimization of the spatial layout of the gear transmission system is carried out based on the NSGA-II genetic algorithm. Taking the design of a three-stage bevel-cylindrical gear reducer as an example, the optimization goals are to improve the structural compactness and equal strength design. Compared with the traditional design, the total volume of the gear transmission system is reduced by 30.65%, while ensuring that the transmission strength of each level is basically the same, improving the system structural compactness and load-bearing capacity, and verifying the effectiveness of the multi-objective optimization design method for the spatial layout of the gear transmission. BRIEF DESCRIPTION OF THE DRAWINGS

[0068] Figure 1 It is a schematic diagram of the process of the present invention.

[0069] Figure 2-(a), (b), (c), and (d) are the basic structural units of external meshing cylindrical gear pairs, the basic structural unit of straight bevel gear pairs, the basic structural unit of ordinary cylindrical worm pairs, and the basic structural unit of internal meshing cylindrical gear pairs, respectively.

[0070] Figure 3 This is a schematic diagram of the connection of basic structural units.

[0071] Figure 4 Schematic diagram of the projected cuboid of the basic structural unit.

[0072] Figure 5-(a) and (b) are schematic diagrams before and after the projection of the connecting axis, respectively.

[0073] Figure 6-(a), (b), (c), and (d) are the force analysis diagrams of the external meshing cylindrical gear pair, the force analysis diagram of the spur bevel gear pair, the force analysis diagram of the ordinary cylindrical worm pair, and the force analysis diagram of the internal meshing cylindrical gear pair, respectively.

[0074] Figure 7 Schematic diagram of the axis system spatial mechanics model.

[0075] Figure 8 This is a cross-sectional view of the gearbox model. DETAILED DESCRIPTION

[0076] The following is combined with Figure 1-8 The specific embodiment of the present invention is described in detail, and the method includes the following steps:

[0077] Step 1: Construct the basic structural unit of the gear transmission system

[0078] As shown in Figure 2-(a) and (b), taking a certain conical-cylindrical gear three-stage reducer as an example, the basic structural unit of the bevel gear and external meshing cylindrical gear pair is established, where the size parameter α of the bevel gear is k =0,β k =90°,γ k =0, b k =0, The design variables are Dimensional parameter α of external cylindrical gear pair k =0,β k =0,γ k =0, a k =m n (z1+z2) / (2cosβ), b k =0, c k =0, the extracted design variables are

[0079] Step 2: Describe the spatial posture based on the connection relationship of the basic structural units

[0080] like Figure 3 As shown in the figure, the input half-shaft of the kth basic structural unit is connected to the output half-shaft of the k-1th basic structural unit, and the two half-shafts are merged into one connecting shaft through connection.

[0081] The gear transmission system is formed in series connection. The input shaft of the k-th basic structural unit is connected to the output shaft of the k-1-th basic structural unit. There is θ k d k , k=1,2,···,m, there are 2m connection parameters in total.

[0082] Assume the homogeneous coordinates of the origin of the local coordinate system are [0 0 0 1] T , then the coordinates of the origin of the input and output coordinate systems of the k-th level basic structural unit in space [X O,ik Y O,ik Z O,ik ] T 、[X O,ok Y O,ok Z O,ok ] T They are:

[0083]

[0084]

[0085] At the same time, the axial vector of the basic structural unit gear mapping cylindrical model can also be obtained.

[0086] Step 3: Establish the shaft system mechanical model

[0087] First, solve the projection cuboid Cub and its plane equations that contain all the basic structural units in the transmission scheme. Use the following formula to project the cylindrical model in the basic structural unit onto the x, y, and z coordinate axes of the global coordinate system to obtain the minimum and maximum values of the cylindrical projection; then calculate the minimum and maximum values of the projections of all cylinders on the x, y, and z axes; finally, reduce and increase the safety distance ΔS at the minimum and maximum values of the x, y, and z axis projections, respectively, to obtain the cuboid Cub, as shown in the following example: Figure 4 As shown in the figure i t ∈{1,2,···,8} are the 8 vertices of the cuboid, i p ∈{1,2,···,6} are the six planes of the cuboid, i p ∈{1,2,···,6} are the normal vectors of the six planes of the cuboid, and the positive direction of the normal vector points from the inside of the cuboid to the outside.

[0088]

[0089] Where λ emin is the projection minimum, λ emax is the maximum value of the projection, D is the unit vector of the projection direction, C e 、r e 、w e 、h e They are respectively the center point coordinates, radius value, axial unit vector, and height value of the e-th cylinder.

[0090] Secondly, select three different points P on each face of the cuboid Cub em =(x m ,y m ,z m ), P en =(x n ,y n ,z n ), P eq =(x q ,y q ,z q ), m, n, q∈{1,2,···,8}, the plane equation of each face of the cuboid Cub can be obtained using the following formula i p ∈{1,2,···,6}.

[0091]

[0092] The normal vectors of the six planes of Cub can be obtained by using the eight vertices of Cub i p ∈{1,2,···,6}.

[0093] Then, extend the connecting shaft to the plane of the rectangular parallelepiped Cub and solve for the fulcrum. Take the two-stage cylindrical gear transmission as an example to illustrate the fulcrum solution process, as shown in Figure 5-(a). The figure shows a cross-sectional view of the box. The thick solid line frame is the projection of the partial plane of the rectangular parallelepiped Cub in step 1, and ΔS is the safe distance between the basic structural unit and the inner wall of the gear box. Extend the connecting shaft in the basic structural unit along the axial direction to the plane on the rectangular parallelepiped Cub to obtain the fulcrum, as shown in Figure 5-(b). The detailed solution process is as follows:

[0094] (1) Determine the intersection plane of the connecting axis and the cuboid Cub: Taking the middle axis as an example, select two points P1 and P2 on the axis, and find the plane of the cuboid Cub in the same direction as P2P1 and P1P2 respectively. The judgment method is: if and i p ∈{1,2,···,6} simultaneously holds true, then P2P1 and the plane The normal vectors are in the same direction, and P1 and P2 can be determined in the same way.

[0095] (2) Solve the intersection point to obtain the support coordinates: After obtaining the plane with the same direction as P2P1 and P1P2, extend the axis to the plane in the directions of P2P1 and P1P2 respectively, and use the following formula to obtain the intersection point P 11 、P 22 Coordinates, thus obtaining the fulcrum coordinates, and similarly the coordinates of other fulcrums can be obtained.

[0096]

[0097] Among them, a4, b4, c4, d4 and a2, b2, c2, d2 are coefficients of the plane equations of planes M4 and M2 respectively, and P1 and P2 are coordinate points.

[0098] Then, a mechanical analysis is performed on the basic structural unit. Taking the internal force analysis of the basic structural unit of the external meshing cylindrical gear pair as an example, as shown in Figure 6, the calculation formulas for the tangential force, radial force, and axial force acting on the gear are as follows.

[0099]

[0100] Where, T1 is the torque transmitted by the driving wheel (N·m); β is the pitch circle pressure angle; α n is the normal pressure angle, standard helical gear α n =20°.

[0101] Finally, the mechanical model of the shaft system is established, and the geometric center point of the gear connected to the shaft in the basic structural unit of the shaft system is used as the origin to establish the local rectangular coordinate system o-xyz, with x along the axis. For general cases, it is assumed that the bearing support is the equivalent support point P S1 、P S2 , the force F acting on the shaft at the support xa 、F ya 、F za , a=1,2; the force F acting on the shaft at the gear meshing point b Its action point is P b ,b∈{1,2,···,n F}, the force can be obtained by the force analysis in the basic structural unit; the moment of force acting on the shaft system is M k , k∈{1,2,···,n M}, n M is the total number of couple moments; then the mechanical model of the shaft system in space is as follows Figure 7 shown.

[0102] Step 4: Establish the spatial layout model of the gear transmission system and its optimization model

[0103] A three-stage conical-cylindrical gear transmission scheme was used, with the optimization goals being a compact gear system with equal transmission strength at all levels. The design parameters for this example were P = 20kW, i = 40, n = 1500r / min, L = 24000h, IT = 7, gear hardness of 628HBS, shaft material of 45 steel, prime mover operating characteristics of uniformity and stability, operating characteristics of slight impact, and general reliability. Other parameters in this example were: m = 3, r = 1, and c = 3.

[0104] (4.1) The optimization variables are as follows:

[0105] X=[X U ,X C ]

[0106]

[0107] Where, X is the overall optimization design variable of the gear transmission system spatial layout optimization model, X U is the sum of the optimization variables of the basic structural unit size, x r is the design variable inside the basic structural unit r, r∈{1,2,···,n type} is the type of basic structural unit, n type is the total number of basic structural unit types. C is the optimization design variable of the connection pair, θ is the rotation angle of the input semi-axis of the subsequent basic structure unit around the output semi-axis of the previous basic structure unit, d is the signed distance along the z-axis between the origin of the input coordinate system of the subsequent basic structure unit and the origin of the output coordinate system of the previous basic structure unit, and c∈{2,···,m} is the number of transmission stages.

[0108] (4.2) The optimization goal is to achieve a compact system structure and equal transmission strength at all levels. The formula is as follows:

[0109] V=V gear +V shaft +V box

[0110] Where V is the volume of the main parts in the gear transmission system. The smaller the volume, the more compact the structure. gear 、V shaft 、V box , respectively the volume of the gear and the volume of the shaft and the volume of the box. Among them, V gear 、V shaft 、V box The calculation formula is as follows.

[0111]

[0112] Where h i1 、h i2 are the equivalent widths of the small gear and large gear of each transmission stage, r i1 、r i2 are the equivalent radius of the small gear and large gear of each stage, r i1,s 、r i2,s are the radii of the shafts where the pinion and gear of each stage are located. j,s is the radius of the axis, h j,s is the length of the axis. ou 、W ou 、H ou L are the length, width and height of the gearbox outer contour in the three coordinate directions respectively. in 、W in 、H in are the length, width, and height of the inner boundary of the smallest cuboid formed by all basic structural units and axes. Δl, Δw, and Δh are the safety distances defined between the inner wall of the box and the inner frame in the three coordinate directions. in 、W in 、H in The calculation formula is as follows.

[0113]

[0114] Where x min , x max ,y min ,y max , z min , z max are the minimum and maximum values of the projections of all cylinders in the x-axis, y-axis, and z-axis directions, respectively.

[0115]

[0116] Where S Hi1 、S Hi2 、S Fi1 、S Fi2 The safety factors for contact and bending fatigue calculations of the i-th level transmission are M H 、M F Safety factors are calculated for average contact and bending fatigue at each level, where k is the number of basic structural units.

[0117] (4.3) In this embodiment, the constraints include upper and lower limit constraints of parameters, coprime constraints of the number of teeth, constraints on the transmission ratios of each level, constraints on the total transmission ratio, strength constraints, and interference constraints. The formula is as follows:

[0118]

[0119] Where x jmin 、xjmax is the design variable x j The minimum and maximum values are given by x j Variable type determination; n X is the dimension of the overall design variable X of the spatial layout model; Pn is a positive number; i kmin 、i kmax is the minimum and maximum transmission ratio allowed for the k-th transmission; i k is the kth transmission ratio; m is the number of transmission stages; i represents the total transmission ratio of the gear transmission system; δ is the allowable error of the transmission ratio; Gapd s ≤0 is the sth interference constraint, and q is the total number of interference constraints.

[0120] Step 5: Multi-objective optimization of the spatial layout of the gear transmission system based on the NSGA-II genetic algorithm

[0121] Substitute the optimization variables, optimization objectives, and constraints established in step 4 into the NSGA-II genetic algorithm for solution. The algorithm's solution process is as follows:

[0122] (5.1) Determine the number of individuals P0 in the initial population based on the number of basic structural units in the gear transmission system.

[0123] (5.2) Perform fast non-dominated sorting and crowding calculation on the population P0. The fast non-dominated sorting mainly calculates the parameter n of each individual i in the population. i and s i , where n i represents the number of individuals that dominate individual i in the population, s i represents the number of individuals in the population that are dominated by individual i. The fast non-dominated sorting process of individuals in the population is as follows:

[0124] ① Find n in the population i = 0, save the individual to the set F, which is the first-level individual.

[0125] ② The individual geometry dominated by each individual i in the first layer is s i , traverse s i For each individual L in , nL = nL-1, if nL = 0, save L into the set S, which is the second-level individual.

[0126] ③ With S as the current geometry, repeat step ② until all individuals in the population are hierarchically sorted.

[0127] The calculation process of congestion is as follows:

[0128] ① Assume that the crowding degree P of N individuals in the population is id is 0;

[0129] ② For each target, sort the individuals and set the crowding degree of the two individuals at the boundary to ∞, that is: P 1d =P Nd =0

[0130] ③ The calculation formulas for the remaining individual crowding degrees are as follows:

[0131]

[0132] in, Represent the function values of individuals i+1 and i-1 under the j-th target respectively.

[0133] (5.3) Through the evolutionary operations: selection > recombination > mutation, the offspring population Qt is obtained, and a new population Rt = Pt + Qt is constructed.

[0134] (5.4) Repeat step (5.2)

[0135] (5.5) Select a certain number of individuals from the population Rt and calculate their fitness, and select the individuals with the highest fitness to enter the next generation population.

[0136] (5.6) Determine whether the population size has reached N. If not, jump to step (5.5) and continue iterating until the population size reaches N.

[0137] (5.7) Determine whether the termination condition is met. If not, jump to 5.3 and continue iterating until the termination condition is met. If so, output the optimal solution.

[0138] Table 1 is a comparison of the main geometric design parameters of a reducer with the traditional design and the optimized design of the present invention.

[0139]

[0140] Table 2 shows the calculated safety factors and objective function values for each level of transmission of a certain reducer with a traditional design and the optimized design of this embodiment.

[0141]

[0142] As shown in Table 2, the total volume of the gear transmission system after optimization is reduced by 30.65% compared with the traditional design. At the same time, the calculated fatigue safety factors of each level of transmission are closer, ensuring that the strength of each level of transmission is basically equal, verifying the feasibility and effectiveness of the multi-objective optimization design method for spatial layout.

Claims

1. A multi-objective optimization design method for the spatial layout of a gear transmission system, characterized in that: The specific steps of this method are as follows: Step 1: construct the basic structural unit of the gear transmission system; The physical structures of various gears and shafts in the gear transmission system are mapped into cylindrical models. According to the connection relationship, the mapped cylindrical models of the gears and shafts in the first-level gear transmission are assembled into the mapping structure of the gear transmission. The mapping structure is the basic structural unit of the gear transmission system. The size parameters and connection parameters of the cylindrical model are the basic properties of the basic structural unit. Different types of basic structural units of the gear transmission system are constructed according to different types of gear pairs. Step 2: Describe the spatial posture based on the connection relationship of the basic structural units; There are three types of basic structural unit connection relationships. The first is a series connection relationship, that is, the output component of the previous basic structural unit is connected in series to the input component of the next basic structural unit; the second is a parallel connection relationship, that is, the previous basic structural unit is connected to multiple next basic structural units at the same time, or multiple previous basic structural units are connected to the next basic structural unit at the same time; the third is a closed-loop connection relationship, which can be considered as a combination of the two parallel connection relationships. According to different basic structural unit connection relationships, the spatial position of the basic structural unit is described using homogeneous coordinate transformation. Step 3, establish the shaft system mechanical model; Step 3.1, solve the projection cuboid Cub of the basic unit structure: Project the cylindrical model in the basic structure unit onto the x, y, and z coordinate axes of the global coordinate system to obtain the minimum and maximum values of the cylindrical projection; then calculate the minimum and maximum values of the projections of all cylinders on the x, y, and z axes. Finally, reduce and increase the safety distance ΔS at the minimum and maximum values of the x, y, and z axis projections, respectively, to obtain the cuboid Cub; Step 3.2, extend the connecting axis to the plane of the cuboid Cub and solve the axis system support: select two points P1 and P2 on any axis, find the plane of the cuboid Cub in the same direction as P2P1 and P1P2 respectively, extend the axis to the plane in the directions of P2P1 and P1P2 respectively, and get the intersection points P 11 、P 22 Coordinates, that is, the fulcrum coordinates of the selected axis; Step 3.3, perform mechanical analysis of basic structural units; Step 3.4: Establish a shafting mechanical model: First, establish a local rectangular coordinate system o-xyz with the geometric center point of the gear connected to the shaft in the previous basic structural unit of the shafting as the origin, with x along the axis. Second, decompose the forces acting on the shaft support into radial and axial directions, and establish radial force mechanical models and axial force mechanical models for the shafting support, respectively. The axial force mechanical model for the shafting support is divided into three cases: the first case is a support method with two ends floating; the second case is a support method with a single support point and two directions fixed; and the third case is a support method with two supports and one direction fixed. Finally, obtain the support forces in the x, y, and z directions for each shafting support to complete the construction of the shafting mechanical model. Step 4, establishing a gear transmission system spatial layout model and its optimization model; First, according to the transmission scheme, the corresponding basic structural units are assembled into a complete system. The spatial structure of the gear transmission system is described by using the internal design parameters of the basic structural units and the connection parameters between the basic structural units, thereby forming a spatial layout model of the gear transmission system. Secondly, the optimization variables, optimization objectives, and constraints in the spatial layout model of the gear transmission system are defined; Step 5, multi-objective optimization of the spatial layout of the gear transmission system is performed based on the NSGA-II genetic algorithm; Substitute the optimization variables, optimization objectives, and constraints established in step 4 into the NSGA-II genetic algorithm for solution. The algorithm's solution process is as follows: Step 5.1, generate the initial population P0; Step 5.2, perform fast non-dominated sorting and crowding calculation on population P0; Step 5.3, through the evolution operation: selection > recombination > mutation, obtain the offspring population Q t , and construct a new population R t =P t +Q t ; Step 5.4, for population R t Perform fast non-dominated sorting and congestion calculation; Step 5.5, in the population R t A certain number of individuals are selected to calculate their fitness, and the individuals with the highest fitness are selected to enter the next generation population; Step 5.6: Determine whether the population size has reached N. If not, jump to step 5.5 and continue iterating until the population size reaches N. In step 5.7, determine whether the termination condition is met. If not, jump to step 5.3 and continue iterating until the termination condition is met. If so, output the optimal solution.

2. The multi-objective optimization design method for the spatial layout of a gear transmission system according to claim 1, characterized in that: Step 1: Construct the basic structural unit of the gear transmission system: establish the local right-handed input coordinate system O with the geometric center of the cylinder mapped by the small gear and the large gear as the origin. ik -x ik y ik z ik and the output coordinate system O ok -x ok y ok z ok , the z-axis direction of the coordinate system is along the axis direction of the gear; the radius and height of the two cylinders are r ik 、h ik and r ok 、h ok ; Establish connecting semi-axes along the positive direction of z axis from the origin of the two local coordinate systems, and the connection points are J ik 、J ok , the axial directions are W ik 、W ok The connecting semi-axle diameters are d ik d ok ; Use formula (1) to output coordinate system {O ok P in} ok Point coordinates are transformed to the input coordinate system {O ik P under} ik Point coordinates: In the formula, [X ok Y ok Z ok 1] T and [X ik Y ik Z ik 1] T They are respectively P point in the output coordinate system {O ok } and input coordinate system {O ik } Homogeneous coordinates; in RPY angle α k To rotate around the fixed input coordinate system {O ik The rotation angle of the z axis, β k To rotate around the fixed input coordinate system {O ik }y-axis rotation angle, γ k To rotate around the fixed input coordinate system {O ik }Rotation angle of x-axis; a k 、b k 、c k For the output coordinate system {O ok }The origin is in the input coordinate system {O ik }Next x, y, z coordinate values; Determine the size parameter r for different types of gear transmission ik 、h ik 、r ok 、h ok and position and attitude parameters α k , β k , γ k 、a k 、b k 、c k Construct different types of basic structural units.

3. The multi-objective optimization design method for the spatial layout of a gear transmission system according to claim 1, characterized in that: Step 2: Describe the spatial position of the basic structural unit: connect the input semi-axis of the kth basic structural unit to the output semi-axis of the k-1th basic structural unit to establish a connection relationship. The two semi-axis are merged into one connection axis through connection. Formula (2) is used to describe the input coordinate system of the basic structural unit k {O ik } Output coordinate system relative to basic structural unit k-1 {O ok-1 } position and posture; Where m is the total number of basic structural units in the gear transmission system; θ k When the basic structure unit k is connected to the previous level basic structure unit k-1, the basic structure unit k rotates around the output semi-axis of the previous level basic structure unit k-1; d k The signed distance along the z-axis between the origin of the input coordinate system of the basic structural unit k and the origin of the output coordinate system of the basic structural unit k-1.

4. The multi-objective optimization design method for the spatial layout of a gear transmission system according to claim 1, characterized in that: In step 3.1, use formula (3) to project the cylindrical model in the basic structural unit onto the x, y, and z coordinate axes of the global coordinate system to obtain the minimum and maximum values of the cylindrical projection; Where λ emin is the projection minimum, λ emax is the maximum value of the projection, D is the unit vector of the projection direction, C e 、r e 、w e 、h e They are respectively the center point coordinates, radius value, axial unit vector, and height value of the e-th cylinder.

5. The multi-objective optimization design method for the spatial layout of a gear transmission system according to claim 1, characterized in that: In step 3.2, the method for finding the plane of the cuboid Cub in the same direction as P2P1 and P1P2 is: and If both are true, then P2P1 and plane Normal vectors are in the same direction; when and If both are true, then P1P2 and the plane Normal vectors are in the same direction.

6. The multi-objective optimization design method for the spatial layout of a gear transmission system according to claim 1, characterized in that: In step 3.2, the intersection point P 11 、P 22 The coordinate acquisition method is shown in formula (4): Where, P 11 、P 22 are the coordinates of the intersection of the axis and the two planes, The axis intersection planes The coefficients of the plane equation, P1 and P2 are the coordinate points.

7. The multi-objective optimization design method for the spatial layout of a gear transmission system according to claim 1, characterized in that: In step 3.4, the mechanical model of radial force at the shaft support is shown in formula (5); Where, F xa 、F ya 、F za , a=1,2 is the force acting on the shaft at the support; F b 、P b ,b∈{1,2,···,n F } is the force acting on the gear meshing of the shaft system and its point of action, n F is the total number of forces; M k , k∈{1,2,···,n M } is the torque acting on the shaft system, n M is the total number of couple moments; x, y, z are unit vectors in three directions.

8. The multi-objective optimization design method for the spatial layout of a gear transmission system according to claim 1, characterized in that: Step 4 defines the optimization variables, optimization objectives, and constraints in the gear transmission system spatial layout model; The specific steps include: Step 4.1, define the optimization variables, which are shown in formula (6): X=[X U ,X C ] (6) Where, X is the overall optimization design variable of the gear transmission system spatial layout optimization model, X U is the sum of the optimization variables of the basic structural unit size, x r is the design variable inside the basic structural unit r, r∈{1,2,···,n type } is the type of basic structural unit, n type is the total number of basic structural unit types; X C is the optimization design variable of the connection pair, θ is the rotation angle of the input semi-axis of the next basic structural unit around the output semi-axis of the previous basic structural unit, d is the signed distance along the z-axis between the origin of the input coordinate system of the next basic structural unit and the origin of the output coordinate system of the previous basic structural unit, and c∈{2,···,m} is the number of transmission stages; Step 4.2, define the first optimization goal, which is a compact system structure, as shown in formula (7): V=V gear +V shaft +V box (7) Where V is the volume of the main parts in the gear transmission system, V gear 、V shaft 、V box , are the volume of gear and shaft, and the volume of housing respectively; where V gear 、V shaft 、V box , is calculated as shown in formula (8): Where h i1 、h i2 are the equivalent widths of the small gear and large gear of each transmission stage, r i1 、r i2 are the equivalent radius of the small gear and large gear of each stage, r i1,s 、r i2,s are the radii of the shafts where the pinion and gear of each stage are located; r j,s is the radius of the axis, h j,s is the length of the axis; L ou 、W ou 、H ou are the length, width and height of the gearbox outer contour in the three coordinate directions respectively; L in 、W in 、H in are the length, width and height of the inner boundary of the smallest cuboid formed by all basic structural units and axes; Δl, Δw and Δh are the safety distances defined between the inner wall of the box and the inner frame in the three coordinate directions; in 、W in 、H in The calculation of is shown in formula (9): Where x min , x max ,y min ,y max , z min , z max are the minimum and maximum values of the projections of all cylinders in the x-axis, y-axis, and z-axis directions respectively; Step 4.3: Define the second optimization goal. The second optimization goal is to make the transmission strength of each level equal, as shown in formula (10): Where S Hi1 、S Hi2 、S Fi1 、S Fi2 The safety factors for contact and bending fatigue calculations of the i-th level transmission are M H 、M F Calculate the safety factor for average contact and bending fatigue at each level, where m is the number of basic structural units; Step 4.4, the third optimization goal is to reduce the magnitude of the support reaction force at the shaft support point, as shown in formula (11): Where, F xa 、F ya 、F za , a=1,2 is the load at the two supporting points of the shaft system; Step 4.5: Define the constraints, which include upper and lower limit constraints of parameters, coprime constraints of the number of teeth, transmission ratio constraints of each level, total transmission ratio constraints, strength constraints, and interference constraints, as shown in formula (12): Where x jnmin 、x jnmax is the design variable x jn The minimum and maximum values are given by x jn Variable type determination; n X is the dimension of the overall design variable X of the spatial layout model; Pn is a positive number; i kmin 、i kmax is the minimum and maximum transmission ratio allowed for the k-th transmission; i k is the kth transmission ratio; m is the number of transmission stages; i represents the total transmission ratio of the gear transmission system; δ is the allowable error of the transmission ratio; Gapd s ≤0 is the sth interference constraint, and q is the total number of interference constraints.

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