Multi-objective optimization design method for gear stress relief holes based on interval analysis

Abstract

The present invention discloses a multi-objective optimization design method for gear stress release holes based on interval analysis, the method comprising the following steps: the first step is to establish finite element models of gears with holes and without holes, and perform strength analysis; the second step is to define design variables and perform parametric modeling; the third step is to establish a Kriging approximation model based on the sample points obtained in the second step; the fourth step is to establish an optimization model for gear stress release holes: setting the reduction of the maximum bending stress at the root of the gear and the maximum contact stress on the tooth surface as the optimization target, and forming constraint conditions for the positioning and size parameters of the gear stress release holes and the values ​​of the driving wheel torque; the fifth step is to formulate a solution strategy for the optimization model of gear stress release holes: using a genetic particle swarm hybrid algorithm to optimize the inner and outer layers. The present invention reduces the bending stress at the root of the gear and the contact stress on the tooth surface by taking into account the uncertainty of the driving wheel torque and taking the reduction of the maximum bending stress at the root of the gear and the maximum contact stress on the tooth surface as the optimization target.

Description

Technical Field

[0001] The present invention relates to the technical field of gear structure optimization, and in particular to a multi-objective optimization design method for gear stress relief holes based on interval analysis. Background Art

[0002] Gears are extremely critical components in mechanical transmission systems. Gear failure directly affects the normal operation of the transmission system. The main forms of gear failure include tooth surface pitting, tooth surface wear and gear tooth breakage. Reducing the bending stress at the root of the gear tooth and the contact stress on the tooth surface can effectively avoid or delay the occurrence of these failures.

[0003] Opening stress relief holes in the gear teeth and tooth root areas is a method to reduce the maximum bending stress at the gear tooth root and the maximum contact stress on the tooth surface, and the quality of stress release performance is closely related to the position and size of the hole. If the position and size of the hole are not selected appropriately, it is very likely that the stress release effect will not be achieved, and it may even increase the maximum bending stress at the gear tooth root and the maximum contact stress on the tooth surface. During the movement of the gear, due to factors such as transmission loss, the actual load on the gear is often different from the set value, that is, it is uncertain, and the bending stress at the gear tooth root and the contact stress on the tooth surface are very sensitive to the load. Therefore, when optimizing the design of the gear stress release hole, it is necessary to take the uncertainty of the load into account in order to obtain a more scientific gear structure with stress release holes.

[0004] Uncertain factors can be described mathematically as probability models, fuzzy models, and convex models. The establishment of probability models and fuzzy models depends on mastering enough statistical data, which is often not available in reality. Therefore, these two models are ideal uncertainty evaluation models with great limitations. In engineering practice, although the probability distribution of uncertain parameters is difficult to obtain, its value range is relatively easy to determine. Summary of the invention

[0005] In order to overcome the defects and shortcomings of the prior art, the present invention provides a multi-objective optimization design method for gear stress release holes based on interval analysis. The method combines the uncertainty of the driving wheel torque and takes reducing the maximum bending stress of the gear tooth root and the maximum contact stress of the tooth surface as the optimization goal, thereby obtaining a more scientific and reasonable hole position and hole size.

[0006] In order to achieve the above object, the present invention adopts the following technical solutions:

[0007] A multi-objective optimization design method for gear stress relief holes based on interval analysis includes the following steps:

[0008] The first step is to establish the finite element model of the gear with and without holes and conduct strength analysis on it;

[0009] The second step is to define design variables and perform parametric modeling:

[0010] 2.1) There are three holes in total in the gear teeth and tooth root area, and the three holes are determined according to the positioning and size parameters of the stress relief holes;

[0011] 2.2) The positioning and size parameters of the gear stress relief hole are defined as the deterministic design vector X, and the driving wheel torque is defined as the uncertain design variable T;

[0012] 2.3) Using the Latin hypercube experimental design method, uniform random sampling is performed within the value range of each variable to obtain a number of initial sample points including X and T;

[0013] The third step is to establish a Kriging approximation model based on the sample points obtained in the second step;

[0014] The fourth step is to establish the optimization model of gear stress relief holes:

[0015] 4.1) Set the reduction of the maximum bending stress of the gear tooth root and the maximum contact stress of the tooth surface as the optimization goal, and form constraints on the positioning and size parameters of the gear stress relief hole and the value of the driving wheel torque so that the constraints have upper and lower limits. The expression of the gear stress relief hole optimization model is:

[0016] min{σ H (X,T),σ F (X,T)}

[0017] xT L ≤X≤X R

[0018] T∈[T L ,T R ]

[0019] Where T L , T R They represent the lower and upper limits of the uncertain variable T, respectively. L , X R They represent the lower and upper limits of the deterministic design vector X, σ F represents the maximum bending stress at the tooth root, σ H Indicates the maximum contact stress on the tooth surface;

[0020] The fifth step is to formulate a solution strategy for the gear stress relief hole optimization model and obtain the optimization results:

[0021] The genetic-particle swarm hybrid algorithm is used to optimize the inner and outer layers. The deterministic design vector X is found in the outer layer optimization, and the value of the uncertain design variable T is found in the inner layer optimization.

[0022] As a preferred technical solution, the first step specifically includes:

[0023] 1.1) Taking spur gears as the research object, circular, elliptical and straight notch-shaped stress relief holes are opened in the gear teeth and tooth root areas in turn;

[0024] 1.2) Establish the geometric model and finite element model of the gear with and without holes, perform finite element simulation on them, and calculate the maximum bending stress at the root of the gear and the maximum contact stress on the tooth surface under specific loads.

[0025] As a preferred technical solution, in step 2.1, determining three holes according to the positioning and size parameters of the stress relief holes specifically includes:

[0026] Three circular holes are determined according to the positioning parameters θ1, θ2, θ3, R1, R2, R3 and the diameter parameters d1, d2, d3 of the centers of the three circles;

[0027] Three elliptical holes are determined according to the positioning parameters θ1', θ2', θ3', R1', R2', R3' of the three ellipse center points, the three major axis parameters a1, a2, a3, the three minor axis parameters b1, b2, b3 and the angle parameters β1', β2', β3' between the three major axes and the gear center line;

[0028] The three straight slot-shaped holes are determined according to the positioning parameters θ1”, θ2”, θ3”, R1”, R2”, R3” of the center points of the three straight slot-shaped holes, the radius parameters r1, r2, r3 of the three straight slot-shaped holes, the distance parameters l1, l2, l3 between the two centers of the three straight slot-shaped holes, and the angle parameters β1”, β2”, β3” between the straight lines passing through the two centers of the three straight slot-shaped holes and the center line of the gear;

[0029] Among the positioning parameters, θ1, θ2, θ3, θ1', θ2', θ3', θ1", θ2", and θ3" belong to angle positioning parameters, and R1, R2, R3, R1', R2', R3', R1", R2", and R3" belong to radius positioning parameters. The subscripts 1, 2, and 3 in each parameter represent the serial numbers marked in sequence in the direction from the central axis to one side.

[0030] As a preferred technical solution, in step 2.2, specifically including:

[0031] For a circular hole, X = [θ1 θ2 θ3 R1 R2 R3 d1 d2 d3];

[0032] For an elliptical hole, X = [θ1' θ2' θ3' R1' R2' R3' a1 a2 a3 b1 b2 b3 β1' β2' β3'];

[0033] For a straight slot shape hole, X = [θ1” θ2” θ3” R1”R2” R3” r1 r2 r3 l1 l2 l3 β1”β2” β3”];

[0034] The driving wheel torque is defined as the uncertain design variable T, and the value range of each variable is determined according to the positioning and size parameters of the gear stress relief hole and the value of the driving wheel torque;

[0035] 2.3) Using the Latin hypercube experimental design method, uniform random sampling is performed within the value range of each variable to obtain several initial sample points including X and T.

[0036] As a preferred technical solution, in step 2.3, specifically including:

[0037] The finite element model corresponding to the initial sample point is established, and simulation calculation is performed to obtain the corresponding maximum bending stress σ of the tooth root. F , maximum contact stress on tooth surface σ H Finally, we get the deterministic design vector X, the uncertain variable T, and the maximum bending stress σ of the tooth root. F , maximum contact stress on tooth surface σ H of sample points.

[0038] As a preferred technical solution, in the third step, the following steps are specifically included:

[0039] 3.1) According to the sample points obtained in the second step, the maximum bending stress σ of the tooth root is established F , Maximum contact stress on tooth surface σ H Kriging approximation model with deterministic design vector X and uncertain variable T;

[0040] 3.2) For each hole shape, the Latin hypercube method is used to randomly generate multiple groups of samples, and the maximum bending stress σ of the tooth root corresponding to the multiple groups of sample points is calculated. F , Maximum contact stress on tooth surface σ H The finite element simulation value is used to calculate the maximum bending stress σ of the tooth root corresponding to multiple groups of sample points. F , Maximum contact stress on tooth surface σ H The Kriging approximation model response value;

[0041] Divide the difference between the simulation value and the approximate model response value by the simulation value, and the calculation result is the relative error between the simulation value and the approximate model response value;

[0042] When each relative error does not exceed the preset error threshold, the approximate model is successfully constructed.

[0043] As a preferred technical solution, in step 3.2, if each relative error does not exceed 5%, the approximate model is successfully constructed, otherwise resampling is performed to establish a new Kriging approximate model.

[0044] As a preferred technical solution, in the fourth step, the following steps are also included:

[0045] 4.2) Minimizing the maximum contact stress on the tooth surface and the maximum bending stress on the tooth root are taken as the optimization objectives of the gear stress relief hole optimization model;

[0046] The weighted coefficient method is used to transform the optimization model of gear stress relief holes:

[0047]

[0048] xT L ≤X≤X R

[0049] T∈[T L ,T R ]

[0050] Where β represents the weighting coefficient, 0≤β≤1.

[0051] As a preferred technical solution, in the fourth step, the following steps are also included:

[0052] 4.3) When the deterministic design vector X is determined, the maximum bending stress σ at the tooth root is F , Maximum contact stress on tooth surface σ H The value of is: The upper and lower bounds of the two intervals are obtained by the following formula:

[0053]

[0054] in are the maximum bending stress at the tooth root σ F The lower and upper bounds of are the maximum contact stress σ on the tooth surface H The lower and upper bounds of

[0055] The linear weighted method is used to transform the multi-objective optimization problem into a single-objective optimization problem:

[0056] Using interval order relation ≤ CW To achieve the deterministic transformation of the uncertain objective function, the maximum bending stress at the tooth root σ F , Maximum contact stress on tooth surface σ H Midpoint of the interval and radius for:

[0057]

[0058] Among them, the interval order relation ≤ CW It is a calculation method to measure the superiority of two intervals;

[0059] The gear stress relief hole optimization model is further transformed into the following form:

[0060]

[0061] xT L ≤X≤X R

[0062] T∈[T L ,T R ]

[0063] Where 0≤β i ≤1(i=1,2,3,4) and β1+β2+β3+β4=1,β i represents the i-th optimized weighted coefficient.

[0064] As a preferred technical solution, the genetic particle swarm hybrid algorithm is used to optimize the inner and outer layers, specifically including:

[0065] In the outer optimization, the population size is m, and in the inner optimization, the population size is n. For each deterministic design vector X i (i=1,2,…,m), taking T as the variable, randomly generate n different individuals T1,T2,…,T n , T1, T2, …, T n Respectively with X i Combined together, we get n different Kriging approximation model inputs: [X i ,T1],[X i ,T2],…,[X i ,T n ];

[0066] Calculate the σ corresponding to the n inputs F , σ H The value of σ F (X i ,T1),σ F (X i ,T2),…,σ F (X i ,T n ) and σ H (X i ,T1),σH (X i ,T2),…,σ H (X i ,T n );

[0067] According to the size of the value, the σ corresponding to the n inputs is F , σ H Sort the values ​​and find at this time:

[0068]

[0069] The first iteration is completed by using crossover, mutation, and selection operations. After several iterations, the deterministic design vector X is output. i The corresponding That is, the inner layer optimization is finished;

[0070] Based on these numbers, the deterministic design vector X is calculated. i The corresponding σ d , according to σ d The size of each deterministic design vector is sorted, and then the crossover, mutation, and selection operations are used to complete the iteration. After multiple iterations, the outer optimization is completed and the σ in the last iteration step is output. d The minimum deterministic design vector is the optimal solution.

[0071] Compared with the prior art, the present invention has the following advantages and beneficial effects:

[0072] (1) The multi-objective optimization design method for gear stress relief holes based on interval analysis proposed in the present invention establishes Kriging approximate models of the maximum bending stress at the gear tooth root, the maximum contact stress on the tooth surface, the positioning and size parameters of the gear stress relief holes, and the driving wheel torque. These approximate models can better predict the maximum bending stress at the gear tooth root and the maximum contact stress on the tooth surface under given parameters, and can replace finite element calculations.

[0073] (2) In the multi-objective optimization design method for gear stress release holes based on interval analysis proposed in the present invention, the interval model is a type of convex model. The present invention uses interval numbers to represent uncertain parameters, and optimizes the positioning and size parameters of the gear stress release holes based on the interval optimization method. The uncertainty of the driving wheel torque is also considered, and the reduction of the maximum bending stress at the root of the gear and the maximum contact stress on the tooth surface is taken as the optimization goal. The gear stress release holes are optimized by multi-objective based on the interval analysis method, and a more scientific and reasonable hole position and hole size are obtained, thereby reducing the bending stress at the root of the gear and the contact stress on the tooth surface, thereby effectively avoiding or delaying the occurrence of gear failure. BRIEF DESCRIPTION OF THE DRAWINGS

[0074] Figure 1 It is a flow chart of a multi-objective optimization design method for gear stress relief holes based on interval analysis in Example 1 of the present invention;

[0075] Figure 2 is an assembly diagram of a gear pair in Example 2 of the present invention;

[0076] Figure 3 This is a schematic diagram of the positioning and size parameters of the circular stress relief hole in Example 2 of the present invention;

[0077] Figure 4 This is a schematic diagram of the positioning and size parameters of the elliptical stress relief hole in Example 2 of the present invention;

[0078] Figure 5 This is a schematic diagram of the positioning and size parameters of the straight slot-shaped stress relief hole in Example 2 of the present invention;

[0079] Figure 6 This is the finite element simulation result of the gear without a hole in Example 2 of the present invention;

[0080] Figure 7 This is the finite element simulation result of the gear with a circular stress relief hole in Example 2 of the present invention;

[0081] Figure 8 This is the finite element simulation result of the gear with an elliptical stress relief hole in Example 2 of the present invention;

[0082] Fig. 9 The finite element simulation result of the gear with a straight notch-shaped stress relief hole in Example 2 of the present invention;

[0083] Fig.10 A flow chart for constructing a Kriging approximation model in Example 2 of the present invention;

[0084] Fig.11 This is a flowchart of the multi-objective optimization of gear stress relief holes based on interval analysis in Example 2 of the present invention. DETAILED DESCRIPTION

[0085] In the description of the present disclosure, it should be noted that the terms "center", "up", "down", "left", "right", "vertical", "horizontal", "inside", "outside", etc. indicate orientations or positional relationships based on the orientations or positional relationships shown in the accompanying drawings, and are only for the convenience of describing the present disclosure and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation, and therefore cannot be understood as a limitation on the present disclosure.

[0086] In addition, the terms "first", "second", and "third" are used for descriptive purposes only and are not to be understood as indicating or implying relative importance. Similarly, words such as "one", "an", or "the" do not indicate a quantitative limitation, but rather indicate the presence of at least one. Words such as "include" or "comprise" and the like mean that the elements or objects appearing before the word include the elements or objects listed after the word and their equivalents, without excluding other elements or objects. Words such as "connected" or "connected" and the like are not limited to physical or mechanical connections, but may include electrical connections, whether direct or indirect.

[0087] In the description of the present disclosure, it should be noted that, unless otherwise clearly specified and limited, the terms "installed", "connected", and "connected" should be understood in a broad sense. For example, it can be a fixed connection, a detachable connection, or an integral connection; it can be a mechanical connection or an electrical connection; it can be a direct connection, or it can be indirectly connected through an intermediate medium, or it can be internal communication between two elements. For ordinary technicians in this field, the specific meanings of the above terms in the present disclosure can be understood according to specific circumstances. In addition, the technical features involved in the different embodiments of the present disclosure described below can be combined with each other as long as they do not conflict with each other.

[0088] In order to make the purpose, technical solution and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention.

[0089] Example

[0090] Example 1

[0091] like Figure 1 As shown, this embodiment provides a multi-objective optimization design method for gear stress relief holes based on interval analysis, and the method includes the following steps:

[0092] The first step is to establish the finite element model of the gear with and without holes and perform strength analysis on it:

[0093] 1.1) Taking spur gears as the research object, circular, elliptical and straight slot-shaped stress relief holes are opened in the gear teeth and tooth root areas in turn.

[0094] 1.2) Establish the geometric model and finite element model of the gear with and without holes, perform finite element simulation on them, and calculate the maximum bending stress at the root of the gear and the maximum contact stress on the tooth surface under specific loads.

[0095] The second step is to define design variables and perform parametric modeling:

[0096] 2.1) There are three holes in the gear teeth and tooth root area. Nine parameters are needed to determine the three circular holes, including the positioning parameters of the centers of the three circles θ1, θ2, θ3, R1, R2, R3, and the diameter parameters of the three circles d1, d2, d3; 15 parameters are needed to determine the three elliptical holes, including the positioning parameters of the center points of the three ellipses θ1', θ2', θ3', R1', R2', R3', the three major axis parameters a1, a2, a3, the three minor axis parameters b1, b2, b3, and the angle parameters between the three major axes and the center line of the gear β1', β2', β3'; 15 parameters are needed to determine the three straight slot-shaped holes, including the positioning parameters of the center points of the three straight slot-shaped holes θ1", θ2", θ3", R1", R2", R3", radius parameters r1, r2, r3 of the three straight slot holes, distance parameters l1, l2, l3 between the two centers of the three straight slot holes, angle parameters β1", β2", β3". Among them, θ1, θ2, θ3, θ1', θ2', θ3', θ1", θ2", θ3" of the positioning parameters belong to angle positioning parameters, R1, R2, R3, R1', R2', R3', R1", R2", R3" belong to radius positioning parameters, and the subscripts 1, 2, 3 in each parameter represent the serial numbers marked in sequence in the direction from the central axis to one side. The present invention collectively refers to these parameters as positioning and size parameters of stress release holes.

[0097] 2.2) Define the positioning and size parameters of the gear stress relief hole as the deterministic design vector X. For a circular hole, X = [θ1 θ2 θ3 R1 R2 R3 d1 d2 d3]; for an elliptical hole, X = [θ1' θ2' θ3' R1' R2' R3' a1 a2 a3 b1b2 b3 β1' β2' β3']; for a straight slot hole, X = [θ1' θ2' θ3' R1” R2” R3” r1 r2 r3 l1 l2 l3β1” β2” β3”]. Define the driving wheel torque as an uncertain design variable, and use T to represent the driving wheel torque. According to the positioning and size parameters of the gear stress relief hole and the value of the driving wheel torque established in the first step, determine the value range of each variable.

[0098] 2.3) Using the Latin hypercube experimental design method, uniform random sampling is performed within the value range of each variable to obtain a number of initial sample points including X and T. Referring to the content of the first step, a finite element model corresponding to the initial sample point is established, and simulation calculation is performed to obtain the corresponding maximum bending stress σ of the tooth root F , Maximum contact stress on tooth surface σ H Finally, we get the deterministic design vector X, the uncertain variable T, and the maximum bending stress σ of the tooth root. F, Maximum contact stress on tooth surface σ H of sample points.

[0099] The third step is to establish the Kriging approximation model:

[0100] 3.1) According to the sample points obtained in the second step, the maximum bending stress σ of the tooth root is established F , Maximum contact stress on tooth surface σ H Kriging approximation model with deterministic design vector X and uncertain variables T.

[0101] 3.2) For each hole shape, 10 groups of samples are randomly generated using the Latin hypercube method, and the maximum bending stress σ of the tooth root corresponding to these sample points is calculated. F , Maximum contact stress on tooth surface σ H The finite element simulation values ​​are calculated, and the maximum bending stress σ of the tooth root corresponding to these sample points is calculated. F , Maximum contact stress on tooth surface σ H The Kriging approximate model response value. The difference between the simulation value and the approximate model response value is divided by the simulation value. The calculation result is the relative error between the simulation value and the approximate model response value. If the relative errors do not exceed 5%, the approximate model is successfully constructed. Otherwise, resample and establish a new Kriging approximate model.

[0102] The fourth step is to establish the optimization model of gear stress relief holes:

[0103] 4.1) The optimization goal is to reduce the maximum bending stress of the gear tooth root and the maximum contact stress of the tooth surface. The constraints are: the positioning and size parameters of the gear stress relief hole and the value of the driving wheel torque cannot exceed their upper and lower limits. The expression of the gear stress relief hole optimization model is:

[0104] min{σ H (X,T),σ F (X,T)}

[0105] xT L ≤X≤X R

[0106] T∈[T L ,T R ]

[0107] Where T L , T R They represent the lower and upper limits of the uncertain variable T, respectively. L , X R They represent the lower and upper limits of the deterministic design vector X, σ F represents the maximum bending stress at the tooth root, σ HIndicates the maximum contact stress on the tooth surface;

[0108] 4.2) The optimization model of gear stress relief holes contains two optimization objectives, namely, minimizing the maximum contact stress on the tooth surface and minimizing the maximum bending stress at the tooth root. For optimization problems with multiple optimization objectives, it is often not possible to obtain the optimal solution for each optimization objective. The weighted coefficient method is used to deal with this problem, and the optimization model of gear stress relief holes can be transformed as follows:

[0109]

[0110] xT L ≤X≤X R

[0111] T∈[T L ,T R ]

[0112] Where 0≤β≤1. The value of the weighting coefficient β should be selected according to actual needs.

[0113] 4.3) Since the uncertain design variable driving wheel torque is an interval T, when the deterministic design vector X is determined, the maximum bending stress σ at the tooth root is F , Maximum contact stress on tooth surface σ H The values ​​of are all intervals, that is, The upper and lower bounds of the two intervals can be obtained by the following formula:

[0114]

[0115] in are the maximum bending stress at the tooth root σ F The lower and upper bounds of are the maximum contact stress σ on the tooth surface H The lower and upper bounds of

[0116] Using interval order relation ≤ CW To achieve the deterministic transformation of the uncertain objective function, the maximum bending stress at the tooth root σ F , Maximum contact stress on tooth surface σ H Midpoint of the interval and radius for:

[0117]

[0118] Among them, the interval order relation ≤ CW It is a calculation method to measure the superiority of two intervals;

[0119] The optimization model of gear stress relief holes can be further transformed into the following form:

[0120]

[0121] xT L ≤X≤X R

[0122] T∈[T L ,T R ]

[0123] Where 0≤β i ≤1(i=1,2,3,4) and β1+β2+β3+β4=1. The weighting coefficient β should be selected according to the actual needs of the project. i The value of (i=1,2,3,4).

[0124] The fifth step is to formulate a solution strategy for the gear stress relief hole optimization model and obtain the optimization results:

[0125] The optimization problem shown in the above equation belongs to a two-layer nested optimization problem, in which the outer optimization is used to find a suitable deterministic design vector X so that the calculated σ d Close enough to its minimum value; the inner optimization is used to find the appropriate value of the uncertain design variable T to ensure that the Kriging approximation model calculates is close enough to its true value.

[0126] Both the inner and outer optimizations use a genetic-particle swarm hybrid algorithm. In the outer optimization, the population size is m. In the inner optimization, the population size is n. For each deterministic design vector X i (i=1,2,…,m), taking T as the variable, randomly generate n different individuals T1,T2,…,T n , T1, T2, …, T n Respectively with X i Combined together, we get n different Kriging approximation model inputs: [X i ,T1],[X i ,T2],…,[X i ,T n ] Calculate the σ corresponding to these n inputs F , σ H The value of σ F (X i ,T1),σ F (X i ,T2),…,σ F (X i ,T n ) and σ H (X i ,T1),σ H (X i ,T2),…,σ H (X i,T n ). According to the value of F (X i ,T1),σ F (X i ,T2),…,σ F (X i ,T n ) and σ H (X i ,T1),σ H (X i ,T2),…,σ H (X i ,T n ) to sort and find at this time

[0127]

[0128] After the above content is completed, crossover, mutation, selection and other operations are performed to complete the first iteration. After several iterations, the deterministic design vector X is output. i The corresponding The inner optimization is finished. Based on these numbers, the deterministic design vector X is calculated. i The corresponding σ d , according to σ d The size of each deterministic design vector is sorted, and then crossover, mutation, selection and other operations are performed to complete the iteration. After multiple iterations, the outer optimization is completed and the σ in the last iteration step is output. d The minimum deterministic design vector is the optimal solution.

[0129] Example 2

[0130] This embodiment 2 is further explained by taking the application scenario of a gear as an example.

[0131] like Figure 2 As shown in the figure, it is the assembly drawing of the gear pair. The basic parameter information of the gear pair used is: the number of teeth of the small gear (driving gear) is 18, the number of teeth of the large gear (driven gear) is 42, the module is 5, the center distance between the two gears is 150mm, the pressure angle is 20°, and the tooth width is 20mm. In order to reduce the amount of calculation, only the three gear teeth of the large gear and the two gear teeth of the small gear are selected for finite element simulation calculation.

[0132] Three circular stress relief holes are opened on the gear teeth and tooth roots. Figure 3 As shown, nine parameters are required to determine the three circular stress relief holes:

[0133] θ1, θ2, θ3, R1, R2, R3, d1, d2, d3;

[0134] Three elliptical stress relief holes are opened on the gear teeth and tooth roots. Figure 4 As shown, fifteen parameters are required to determine these three elliptical stress relief holes:

[0135] θ1', θ2', θ3', R1', R2', R3', a1, a2, a3, b1, b2, b3, β1', β2', β3';

[0136] Three straight slot-shaped stress relief holes are opened on the gear teeth and tooth roots, as shown in the figure. Figure 5 As shown, fifteen parameters are required to determine the three straight slot shape holes:

[0137] θ1”, θ2”, θ3”, R1”, R2”, R3”, r1, r2, r3, l1, l2, l3, β1”, β2”, β3”;

[0138] Specific values ​​of the positioning and size parameters of these stress relief holes are assigned, and the corresponding finite element models of gears with stress relief holes are established. The values ​​of the maximum bending stress at the tooth root and the maximum contact stress on the tooth surface of these finite element models under the same load (250000N·mm) are calculated. At the same time, the values ​​of the maximum bending stress at the tooth root and the maximum contact stress on the tooth surface of gears without holes under the same load are calculated.

[0139] The circular hole positioning and size parameter assignment are shown in Table 1, the elliptical hole positioning and size parameter assignment are shown in Table 2, and the straight slot shape hole positioning and size parameter assignment are shown in Table 3. The finite element simulation results of the gear without holes are shown in Figure 6 , the finite element simulation results of gears with circular holes are shown in Figure 7 , the finite element simulation results of gears with elliptical holes are shown in Figure 8 , the finite element simulation results of gears with straight slot shape holes are shown in Fig. 9 .

[0140] Table 1 Positioning and size parameter assignment of circular stress relief holes

[0141]

[0142] Table 2 Positioning and size parameter assignment of elliptical stress relief holes

[0143]

[0144] Table 3 Positioning and dimensional parameter assignment of straight slot shape stress relief holes

[0145]

[0146] The simulation results show that the opening of holes in the gear teeth and tooth roots can play a role in stress release, and the maximum bending stress at the tooth root and the maximum contact stress on the tooth surface are reduced. Compared with the gear without holes, the maximum bending stress at the tooth root of the gear with circular holes is reduced by 2.82%, and the maximum contact stress at the tooth surface is reduced by 4.85%; the maximum bending stress at the tooth root of the gear with elliptical holes is reduced by 3.64%, and the maximum contact stress at the tooth surface is reduced by 6.28%; the maximum bending stress at the tooth root of the gear with straight slot holes is reduced by 2.84%, and the maximum contact stress at the tooth surface is reduced by 5.41%. It should be pointed out that the stress release effects of the circular, elliptical and straight slot stress release holes opened in this chapter are not much different. In order to further study the influence of the hole position and size on the stress release performance, it is necessary to improve the relevant parameters with the help of optimization design methods.

[0147] In order to better study the stress release characteristics of circular holes, elliptical holes and straight slot holes, the present invention establishes the maximum bending stress σ of the gear tooth root through the Kriging approximation model. F , Maximum contact stress on tooth surface σ H The relationship between the positioning and dimensional parameters X of the gear stress relief hole and the driving wheel torque T.

[0148] Matlab has a DACE toolbox specifically for establishing the Kriging approximation model. Based on this toolbox, the present invention establishes the maximum bending stress σ of the gear tooth root with circular holes, elliptical holes, and straight slot holes. F , Maximum contact stress on tooth surface σ H Kriging approximation model between X and T.

[0149] like Fig.10 As shown, the construction process of the Kriging approximation model is as follows:

[0150] (1) Latin hypercube sampling was implemented in Matlab software using the lhsdesign function. For gears with circular stress relief holes, 10 design variables were sampled 60 times in the design space; for gears with elliptical stress relief holes, 16 design variables were sampled 80 times in the design space; for gears with straight slot-shaped stress relief holes, 16 design variables were sampled 80 times in the design space.

[0151] (2) Based on the sampling values ​​of step (1), a finite element model of the open hole gear is established, and a strength analysis is performed on it to calculate the maximum bending stress value σ of its tooth root. F , Maximum contact stress on tooth surface σ H .

[0152] (3) Set the parameters of the Kriging approximation model, use the zero-order polynomial for the regression part, and use the Gaussian correlation equation for the co-correlation equation. Adjust the directional parameter θ i The Kriging approximate model is constructed by combining the sampling values ​​with the simulation values ​​of the maximum stress at the tooth root and the maximum contact stress at the tooth surface.

[0153] (4) Verify the accuracy of the Kriging approximate model. For each approximate model, the Latin hypercube method is used to obtain the other 10 sample points. The maximum bending stress at the tooth root and the maximum contact stress on the tooth surface corresponding to the sample data are calculated by the approximate model and finite element simulation, and the relative error between the two response values ​​is calculated. If the relative error is less than 5%, the approximate model is successfully constructed. Otherwise, re-sample and calculate the simulation values ​​of the maximum bending stress at the tooth root and the maximum contact stress on the tooth surface corresponding to the new sample data to construct a new approximate model. Repeat this process until an approximate model that meets the accuracy requirements is obtained.

[0154] For gears with circular stress relief holes, the value ranges of the ten variables involved in the construction of the approximate model are shown in Table 4; for gears with elliptical stress relief holes, the value ranges of the sixteen variables involved in the construction of the approximate model are shown in Table 5; for gears with straight slot-shaped stress relief holes, the value ranges of the sixteen variables involved in the construction of the approximate model are shown in Table 5.

[0155] Table 4 Design variable value range (circular hole)

[0156]

[0157] Table 5 Design variable value range (elliptical hole)

[0158]

[0159]

[0160] Table 6 Design variable value range (straight slot shape hole)

[0161]

[0162] Furthermore, the optimization problem involved in this embodiment is to reduce the maximum bending stress of the gear tooth root and the maximum contact stress of the tooth surface, which is a multi-objective optimization problem. This embodiment uses a linear weighted method to transform the multi-objective optimization problem into a single-objective optimization problem.

[0163] Using interval order relation ≤ CW To achieve the deterministic transformation of the uncertain objective function, the gear stress relief hole optimization model is finally transformed into the following form:

[0164]

[0165] xT L ≤X≤X R

[0166] T∈[T L ,T R ]

[0167] Where 0≤β i ≤1(i=1,2,3,4) and β1+β2+β3+β4=1. The weighting coefficient β should be selected according to the actual needs of the project. i The value of (i=1,2,3,4), this example stipulates that β1=β2=β3=β4=0.25.

[0168] The optimization problem shown in the above equation belongs to a two-layer nested optimization problem, in which the outer optimization is used to find a suitable deterministic design vector X so that the calculated σ d Close enough to its minimum value; the inner optimization is used to find the appropriate value of the uncertain design variable T to ensure that the Kriging approximation model calculates is close enough to its true value.

[0169] This embodiment uses a genetic particle swarm hybrid algorithm in both inner and outer layer optimization. In the outer layer optimization, the population size is 200. In the inner layer optimization, the population size is 50. The selection rate of the inner and outer layer optimization is 0.6, the crossover rate is 0.2, and the mutation rate is 0.1.

[0170] Combination Fig.11 As shown, for each deterministic design vector X i (i=1,2,…,200), taking T as the variable, randomly generate 50 different individuals T1,T2,…,T 50 , T1, T2, …, T 50 Respectively with X i Combined together, we get 50 different Kriging approximation model inputs: [X i ,T1],[X i ,T2],…,[X i ,T 50 ] Calculate the σ corresponding to these n inputs F , σ H The value of σ F (X i ,T1),σ F (X i ,T2),…,σ F (X i ,T 50 ) and σ H (X i ,T1),σ H(X i ,T2),…,σ H (X i ,T 50 ). According to the value of F (X i ,T1),σ F (X i ,T2),…,σ F (X i ,T 50 ) and σ H (X i ,T1),σ H (X i ,T2),…,σ H (X i ,T 50 ) to sort and find at this time

[0171]

[0172] After the above content is completed, crossover, mutation, selection and other operations are performed to complete the first iteration. After several iterations, the deterministic design vector X is output. i The corresponding The inner layer optimization is finished.

[0173] Based on these numbers, the deterministic design vector X is calculated. i The corresponding σ d , according to σ d The size of each deterministic design vector is sorted, and then crossover, mutation, selection and other operations are performed to complete the iteration. After multiple iterations, the outer optimization is completed and the σ in the last iteration step is output. d The minimum deterministic design vector is the optimal solution.

[0174] The above embodiments are preferred implementation modes of the present invention, but the implementation modes of the present invention are not limited to the above embodiments. Any other changes, modifications, substitutions, combinations, and simplifications that do not deviate from the spirit and principles of the present invention should be equivalent replacement methods and are included in the protection scope of the present invention.

Claims

1. A multi-objective optimization design method for gear stress relief holes based on interval analysis, characterized in that: The steps include: The first step is to establish the finite element model of the gear with and without holes and conduct strength analysis on it; The second step is to define design variables and perform parametric modeling: 2.1) There are three holes in total in the gear teeth and tooth root area, and the three holes are determined according to the positioning and size parameters of the stress relief holes; 2.2) The positioning and size parameters of the gear stress relief hole are defined as the deterministic design vector X, and the driving wheel torque is defined as the uncertain design variable T; 2.3) Using the Latin hypercube experimental design method, uniform random sampling is performed within the value range of each variable to obtain a number of initial sample points including X and T; The third step is to establish a Kriging approximation model based on the sample points obtained in the second step; The fourth step is to establish the optimization model of gear stress relief holes: 4.1) Set the reduction of the maximum bending stress of the gear tooth root and the maximum contact stress of the tooth surface as the optimization goal, and form constraints on the positioning and size parameters of the gear stress relief hole and the value of the driving wheel torque so that the constraints have upper and lower limits. The expression of the gear stress relief hole optimization model is: min{s H (X,T),σ F (X,T)} s.t.X L ≤X≤X R T∈[T L ,T R ] Where T L , T R They represent the lower and upper limits of the uncertain variable T, respectively. L , X R They represent the lower and upper limits of the deterministic design vector X, σ F represents the maximum bending stress at the tooth root, σ H Indicates the maximum contact stress on the tooth surface; The fifth step is to formulate a solution strategy for the gear stress relief hole optimization model and obtain the optimization results: The genetic-particle swarm hybrid algorithm is used to optimize the inner and outer layers. The deterministic design vector X is found in the outer layer optimization, and the value of the uncertain design variable T is found in the inner layer optimization.

2. The multi-objective optimization design method for gear stress relief holes based on interval analysis according to claim 1 is characterized in that: The first step specifically includes: 1.1) Taking spur gears as the research object, circular, elliptical and straight slot-shaped stress relief holes are opened in the gear teeth and tooth root areas in turn; 1.2) Establish the geometric model and finite element model of the gear with and without holes, perform finite element simulation on them, and calculate the maximum bending stress at the root of the gear and the maximum contact stress on the tooth surface under specific loads.

3. The multi-objective optimization design method for gear stress relief holes based on interval analysis according to claim 1 is characterized in that: In step 2.1, determining three holes according to the positioning and size parameters of the stress relief holes specifically includes: Three circular holes are determined according to the positioning parameters θ1, θ2, θ3, R1, R2, R3 and the diameter parameters d1, d2, d3 of the centers of the three circles; Three elliptical holes are determined according to the positioning parameters θ1', θ2', θ3', R1', R2', R3' of the three ellipse center points, the three major axis parameters a1, a2, a3, the three minor axis parameters b1, b2, b3 and the angle parameters β1', β2', β3' between the three major axes and the gear center line; The three straight slot-shaped holes are determined according to the positioning parameters θ1”, θ2”, θ3”, R1”, R2”, R3” of the center points of the three straight slot-shaped holes, the radius parameters r1, r2, r3 of the three straight slot-shaped holes, the distance parameters l1, l2, l3 between the two centers of the three straight slot-shaped holes, and the angle parameters β1”, β2”, β3” between the straight lines passing through the two centers of the three straight slot-shaped holes and the center line of the gear; Among the positioning parameters, θ1, θ2, θ3, θ1', θ2', θ3', θ1", θ2", and θ3" belong to angle positioning parameters, and R1, R2, R3, R1', R2', R3', R1", R2", and R3" belong to radius positioning parameters. The subscripts 1, 2, and 3 in each parameter represent the serial numbers marked in sequence in the direction from the central axis to one side.

4. The multi-objective optimization design method for gear stress relief holes based on interval analysis according to claim 3 is characterized in that: In step 2.2, specifically include: For a circular hole, X = [θ1 θ2 θ3 R1 R2 R3 d1 d2 d3]; For an elliptical hole, X = [θ1' θ2' θ3' R1' R2' R3' a1 a2 a3 b1 b2 b3 β1' β2' β3']; For a straight slot shape hole, X = [θ1” θ2” θ3” R1” R2” R3” r1 r2 r3 l1 l2 l3 β1” β2” β3”]; The driving wheel torque is defined as the uncertain design variable T, and the value range of each variable is determined according to the positioning and size parameters of the gear stress relief hole and the value of the driving wheel torque; 2.3) Using the Latin hypercube experimental design method, uniform random sampling is performed within the value range of each variable to obtain several initial sample points including X and T.

5. The multi-objective optimization design method for gear stress relief holes based on interval analysis according to claim 4 is characterized in that: In step 2.3, specifically include: The finite element model corresponding to the initial sample point is established, and simulation calculation is performed to obtain the corresponding maximum bending stress σ of the tooth root. F , maximum contact stress on tooth surface σ H Finally, we get the deterministic design vector X, the uncertain variable T, and the maximum bending stress σ of the tooth root. F , maximum contact stress on tooth surface σ H of sample points.

6. The multi-objective optimization design method for gear stress relief holes based on interval analysis according to claim 1 is characterized in that: In the third step, the following steps are specifically included: 3.1) According to the sample points obtained in the second step, the maximum bending stress σ of the tooth root is established F , maximum contact stress on tooth surface σ H Kriging approximation model with deterministic design vector X and uncertain variable T; 3.2) For each hole shape, the Latin hypercube method is used to randomly generate multiple groups of samples, and the maximum bending stress σ of the tooth root corresponding to the multiple groups of sample points is calculated. F , maximum contact stress on tooth surface σ H The finite element simulation value is used to calculate the maximum bending stress σ of the tooth root corresponding to multiple groups of sample points. F , maximum contact stress on tooth surface σ H The Kriging approximation model response value; Divide the difference between the simulation value and the approximate model response value by the simulation value, and the calculation result is the relative error between the simulation value and the approximate model response value; When each relative error does not exceed the preset error threshold, the approximate model is successfully constructed.

7. The multi-objective optimization design method for gear stress relief holes based on interval analysis according to claim 6 is characterized in that: In step 3.2, if the relative errors do not exceed 5%, the approximate model is successfully constructed; otherwise, resampling is performed to establish a new Kriging approximate model.

8. The multi-objective optimization design method for gear stress relief holes based on interval analysis according to claim 1 is characterized in that: In the fourth step, the following steps are also included: 4.2) Minimizing the maximum contact stress on the tooth surface and the maximum bending stress on the tooth root are taken as the optimization objectives of the gear stress relief hole optimization model; The weighted coefficient method is used to transform the optimization model of gear stress relief holes: s.t.X L ≤X≤X R T∈[T L ,T R ] Where β represents the weighting coefficient, 0≤β≤1.

9. The multi-objective optimization design method for gear stress relief holes based on interval analysis according to claim 8 is characterized in that: In the fourth step, the following steps are also included: 4.3) When the deterministic design vector X is determined, the maximum bending stress σ at the tooth root is F , maximum contact stress on tooth surface σ H The value of is: The upper and lower bounds of the two intervals are obtained by the following formula: in are the maximum bending stress at the tooth root σ F The lower and upper bounds of are the maximum contact stress σ on the tooth surface H The lower and upper bounds of The linear weighted method is used to transform the multi-objective optimization problem into a single-objective optimization problem: Using interval order relation ≤ CW To achieve the deterministic transformation of the uncertain objective function, the maximum bending stress at the tooth root σ F , maximum contact stress on tooth surface σ H Midpoint of the interval and radius for: Among them, the interval order relation ≤ CW It is a calculation method to measure the superiority of two intervals; The gear stress relief hole optimization model is further transformed into the following form: s.t.X L ≤X≤X R T∈[T L ,T R ] Where 0≤β i ≤1(i=1,2,3,4) and β1+β2+β3+β4=1,β i represents the i-th optimized weighted coefficient.

10. The multi-objective optimization design method for gear stress relief holes based on interval analysis according to claim 9 is characterized in that: The optimization of the inner and outer layers by using the genetic particle swarm hybrid algorithm specifically includes: In the outer optimization, the population size is m, and in the inner optimization, the population size is n. For each deterministic design vector X i (i=1,2,…,m), taking T as the variable, randomly generate n different individuals T1,T2,…,T n , T1, T2, …, T n Respectively with X i Combined together, we get n different Kriging approximation model inputs: [X i ,T1],[X i ,T2],…,[X i ,T n ]; Calculate the σ corresponding to the n inputs F , σ H The value of σ F (X i ,T1),σ F (X i ,T2),…,σ F (X i ,T n ) and σ H (X i ,T1),σ H (X i ,T2),…,σ H (X i ,T n ); According to the size of the value, the σ corresponding to the n inputs is F , σ H Sort the values ​​and find at this time: The first iteration is completed by using crossover, mutation, and selection operations. After several iterations, the deterministic design vector X is output. i The corresponding That is, the inner layer optimization is finished; Based on these numbers, the deterministic design vector X is calculated. i The corresponding σ d , according to σ d The size of each deterministic design vector is sorted, and then the crossover, mutation, and selection operations are used to complete the iteration. After multiple iterations, the outer optimization is completed and the σ in the last iteration step is output. d The minimum deterministic design vector is the optimal solution.

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