CN106528991A - Method for performing optimization design on gearbox based on Taylor random finite element - Google Patents

Abstract

The invention provides a method for performing optimization design on a gearbox based on a Taylor random finite element. The gearbox is subjected to optimization design. According to the core scheme, an optimized mathematic model is built; a target function is a sum of actual center distances of overall gear transmission; constraint conditions of bending fatigue strength of gears are calculated; the Taylor random finite element needs to be used for constraint conditions of shaft strength; meshes used by the gears adopt 20-node hexahedral parameter units; meshes used by shafts adopt axial symmetric quadrilateral annular units; an optimization problem is solved by adopting a random direction method; a computer operation program is written; and finally the computer operation program runs to obtain an optimal solution. According to the method, the optimization effect is remarkable, the gearbox weight is reduced, the volume is reduced, the raw material expense is reduced, the product quality is improved, and a product is more competitive.

Description

Method for optimally designing gear box based on Taylor random finite element

Technical Field

The invention provides a method for optimally designing a gear box based on Taylor random finite elements, and belongs to the field of mechanical design, mechanical optimal design and mechanical modern design methods.

Background

The gear box is widely applied to the fields of machine tools, engineering machinery, metallurgical machinery, mining machinery, petroleum machinery, agricultural machinery, vehicles and the like. With the development of computer technology, mechanical optimization design methods are generated. The combination of the mechanical reliability design and the optimization design forms a reliability optimization design, which can not only quantitatively predict the reliability of the product, but also obtain an optimized solution for the design parameters of the product. Mechanical reliability designs can only be designed for simple parts. Many modern structural systems have a high structural complexity. Under random loading and operating conditions, advanced numerical techniques, well known finite element methods, are used to analyze the structure. Most applications are limited to certain loads and operating environments, despite random and uncertain factors to a considerable extent. The influence of random factors on the structure is more and more emphasized by scholars at home and abroad. With the advance of human understanding, it is not practical to ignore random finite elements. In order to improve the calculation accuracy, the finite element analysis must consider the influence of random factors. Finite elements that take into account random factors are called random finite elements. The calculation method of the random finite element mainly comprises a direct Monte Carlo method, a Taylor expansion method, a perturbation method, a Neumann expansion method, a Neumann-PCG method and the like.

At present, no scheme for optimally designing the gearbox based on Taylor random finite elements exists.

Disclosure of Invention

The invention provides a method for optimally designing a gear box based on Neumann random finite elements, which is used for optimally designing the gear box, so that the weight of the gear box is reduced, and the product quality is improved.

Therefore, the technical scheme of the invention is as follows: the method for optimally designing the gearbox based on the Taylor random finite element comprises the following steps:

(1) calculating a constraint condition of the bending fatigue strength of the gear, wherein Taylor random finite elements are required to be used for the constraint condition of the shaft strength, a grid used by the gear adopts twenty-node hexahedron parameter units, a grid used by the shaft adopts axisymmetric quadrilateral annular units, and a random direction method is adopted to solve an optimization problem to generate a finite element model; solving the mean value and the variance of the bending fatigue stress of the gear, the allowable mean value and the allowable variance of the bending fatigue strength of the gear, the mean value and the variance of the dangerous section stress of the shaft and the allowable mean value and the allowable variance of the shaft strength;

(2) establishing an optimized mathematical model of a gearbox

The design variables are: gear module, gear tooth number, shaft diameter and shaft length;

the constraint conditions are as follows: mean and variance of gear bending fatigue stress, allowable mean and allowable variance of gear bending fatigue strength, mean and variance of shaft dangerous section stress, allowable mean and allowable variance of shaft strength;

the objective function is: the sum of actual center distances of transmission of all levels of gears;

establishing an optimized mathematical model of the gearbox;

(3) and compiling a computer operation program according to the optimized mathematical model of the gear box, and finally running the computer operation program to obtain an optimal solution.

Has the advantages that: the optimization design optimization method is based on the Taylor random finite element to optimize the gearbox, the optimization effect is obvious, the quality of the optimized gearbox is reduced, the volume is reduced, the cost of raw materials is reduced, the product quality is improved, and the product has higher competitiveness.

Drawings

FIG. 1 is a block diagram of a gearbox that requires an optimized design.

FIG. 2 is a block diagram of Taylor random finite element calculation of mean and variance of gear bending stresses.

Detailed Description

The present invention is described in further detail by the following.

FIG. 1 is a gear box structure with 12 gears and 4 shafts, wherein the reference numbers 1-12 represent the gears, and the reference numbers I, II, III and IV represent the shafts.

The invention is described in detail with reference to fig. 1, and the method for optimally designing the gearbox based on the Taylor random finite element comprises the following steps:

(1) constructing a three-dimensional entity model of the gearbox by using three-dimensional modeling software according to parameters of an original design drawing of the gearbox;

(2) importing the three-dimensional solid model of the gear box obtained in the step (1) into finite element software, calculating constraint conditions of gear bending fatigue strength, wherein Taylor random finite elements are required to be used for the constraint conditions of shaft strength, twenty-node hexahedron parameter units are adopted for meshes used by the gear, axisymmetric quadrilateral annular units are adopted for the meshes used by the shaft, and a random direction method is adopted to solve an optimization problem to generate a finite element model; solving the mean value and the variance of the bending fatigue stress of the gear, the allowable mean value and the allowable variance of the bending fatigue strength of the gear, the mean value and the variance of the dangerous section stress of the shaft and the allowable mean value and the allowable variance of the shaft strength;

the detailed process of obtaining the mean and variance functions is as follows:

taylor random finite element

The finite element governing equation under dead load can be written as

[K]{}＝{F}

[K] For the global stiffness matrix, { } is each node-displaced array, { F } is each node-loaded array.

The mechanical part material performance parameters, the geometric dimension and the load are observed as n random variables a of normal random variables₁,a₂,…,a_i,…,a_n.

Upper pair a_iCalculating partial derivative to obtain

The above formula is to a_jCalculating partial derivative to obtain

Shift at mean point of random variableThe region is spread and averaged at both sides simultaneously to obtain

(E { } denotes the mean of { }, Cov (a)_i,a_j) Is a_iAnd a_jThe covariance of (a). The variance of { } can be calculated by the following equation

{σ}＝[D][B]{}

[D] Is an elastic matrix, [ B ] is a strain matrix;

the above formula is to a_iCalculating partial derivative to obtain

The above formula is to a_jCalculating partial derivative to obtain

Stress is at mean point of random variableThe region is spread and averaged at both sides simultaneously to obtain

E { σ } represents the mean value of { σ }, Cov (a)_i,a_j) Is a_iAnd a_jThe covariance of (a). The variance of { σ } can be calculated by the following equation

(3) Establishing an optimized mathematical model of a gearbox

The gearbox in fig. 1 consists of 12 gears and 4 shafts;

for the sake of clarity, the reference numerals I, II, III, IV in FIG. 1 indicate that the axes are replaced by 1, 2, 3, 4;

the design variables are: x ═ m₁，z₁，z₂，m₂，z₃，z₄，m₃，z₅，z₆，m₄，z₇，z₈，m₅，z₉，z₁₀，m₆，z₁₁，z₁₂，b₁，b₂，b₃，b₄，d₁，l₁，d₂，l₂，d₃，l₃，d₄，l₄)^TWherein m is the gear module, Z is the number of gear teeth, d is the diameter of the shaft, and l is the length of the shaft;

the objective function is: the sum of actual center distances of transmission of all stages of gears is specifically as follows:

the constraint condition is

Wherein,mean and variance of gear bending stress.The allowable mean value and the allowable variance of the bending fatigue strength of the gear.

Wherein,the mean and variance of the axial critical section stress,the allowable mean and allowable variance of the axis intensities.

m_kl≤m_k≤m_ks(k＝1，2，…，6)

z_kl≤z_k≤z_ks(k＝1，2，…，12)

b_kl≤b_k≤b_ks(k＝1，2，3，4)

d_kl≤d_k≤d_ks(k＝1，2，3，4)

l_kl≤l_k≤l_ks(k＝1，2，3，4)

Wherein m is_kl,z_kl,b_kl,d_kl,l_klTo design the lower bound value of the variable. m is_ks,z_ks,b_ks,d_ks,l_ksDesigning an upper bound value of a variable;

(4) and compiling a computer operation program according to the optimized mathematical model of the gear box, and finally running the computer operation program to obtain an optimal solution.

Table 1 below compares the original design to the optimized design parameters for the gearbox of FIG. 1;

TABLE 1 comparison of design parameters

	m1	m2	m3	m4	m5	m6	z1	z2	z3	z4	z5	z6	z7	z8	z9
																Original design	4	4	4	4	4	4	18	44	27	43	35	35	31	39	25
Optimized design	3	3.5	3.5	3.5	4	4	20	40	28	44	36	36	32	40	27
																	z10	z11	z12	b1	b2	b3	b4	d1	l1	d2	l2	d3	l3	d4	l4
Original design	41	19	47	25	25	25	25	50	350	50	280	50	340	65	290
																Optimized design	40	19	42	18	22	25	25	45	285	46	210	48	286	62	235

As can be seen from Table 1, the optimization effect is very remarkable, the quality of the gear box is reduced, the volume is reduced, the cost of raw materials is reduced, and the product quality is improved.

Claims (1)

1. The method for optimally designing the gearbox based on the Taylor random finite element comprises the following steps:

(1) constructing a three-dimensional entity model of the gearbox by using three-dimensional modeling software according to parameters of an original design drawing of the gearbox;

(2) importing the three-dimensional solid model of the gear box obtained in the step (1) into finite element software, calculating constraint conditions of gear bending fatigue strength, wherein Taylor random finite elements are required to be used for the constraint conditions of shaft strength, twenty-node hexahedron parameter units are adopted for meshes used by the gear, axisymmetric quadrilateral annular units are adopted for the meshes used by the shaft, and a random direction method is adopted to solve an optimization problem to generate a finite element model; solving the mean value and the variance of the bending fatigue stress of the gear, the allowable mean value and the allowable variance of the bending fatigue strength of the gear, the mean value and the variance of the dangerous section stress of the shaft and the allowable mean value and the allowable variance of the shaft strength;

(3) establishing an optimized mathematical model of a gearbox

The design variables are: gear module, gear tooth number, shaft diameter and shaft length;

the constraint conditions are as follows: mean and variance of gear bending fatigue stress, allowable mean and allowable variance of gear bending fatigue strength, mean and variance of shaft dangerous section stress, allowable mean and allowable variance of shaft strength;

the objective function is: the sum of actual center distances of transmission of all levels of gears;

establishing an optimized mathematical model of the gearbox;

(4) and compiling a computer operation program according to the optimized mathematical model of the gear box, and finally running the computer operation program to obtain an optimal solution.