# Dynamic and Static Multiobjective Topology Optimization for Gears of Directional Drill Transmission System

Authors: Xiaohong Zhao; Xianguo Yan; Zhi Chen; Linbo Li; Ruibin Ren
Journal: Advances in Civil Engineering
 (2023)

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## Abstract

Aiming at the multiobjective topology optimization design of the structure, this paper proposes a method to establish the comprehensive objective function based on the normalized subobjective of the compromise programming method and to determine the weight coefficient of the subobjective of the comprehensive objective function by the analytic hierarchy process (AHP). By using statistical analysis and modal analysis, the static and dynamic characteristics of the gear can be obtained. The single objective topology optimization is used to reduce the compliance of the gear and elevate the low-order natural frequency, and the frequency weighting method is used to suppress the oscillation phenomenon in the frequency-single-objective optimization. AHP was used to determine the weight coefficient of each subtarget. The compromise programming method is used for multiobjective topology optimization, and the gear design is improved according to the optimization results. By analyzing the improved gear structure, the weight of the optimized gear is reduced by 25.3%. The overall stiffness performance and strength performance are enhanced, and the natural frequencies of each order are improved to different degrees.


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## Body

## 1. Introduction

Down hole, directional drilling technology is the key technology to control coal mine gas and ensure efficient extraction, and it is also one of the basic measures to ensure coal mine production safety [1]. A directional drill is equipment used for gas extraction, it drives the drill pipe by slewing device, and the drill pipe is connected with a bending joint to realize the drilling process. A slewing device is the core part of a drilling rig, which drives the drilling tool to turn and provides torque. It is primarily used to clamp the drill pipe, convert the speed and torque output from the hydraulic motor into those compatible with the drilling process requirements, transfer them to the the drill pipe to drive the slewing, and realize the omni–directional operation of the roadway [2–4]. In the process of drilling, the drill bit will produce a resistance moment when breaking rock, and the disturbance of the drill pipe and hole wall will also produce a resistance moment. All these external dynamic loads will act on the rotary, making it bear circumferential torsion and axial impact, which will affect the performance of the drill. At the same time, due to the narrow working environment characteristics of underground roadways in the coal mines, the drill must have a lighter weight and smaller volume to facilitate the installation and disassembly of the drill underground. Therefore, this paper optimizes the large gear at the output end of the gyrator to improve its static and dynamic characteristics and achieve the purpose of lightweighting and miniaturization.Topology optimization is the most promising and innovative technique in structure optimization, which is to obtain the lightest design by finding the best material distribution or force transmission path in the design space. It is widely used in structural conceptual design [5, 6]. In the process of drilling, the external dynamic load will directly act on the helical gear at the output end of the rotary device. As the output end of drilling power, the performance of the helical gear has a great influence on the rotary device to make the optimized gear not only meet certain mechanical characteristics but also have the ability to inhibit external excitation and at the same time achieve the purpose of being lightweight [7, 8]. In this paper, the multiobjective topology optimization method is adopted. The big gear at the output end of the gyrator is selected as the research object, the dynamic frequency characteristics and static stiffness characteristics of the gear are respectively taken as the optimization objectives for single objective optimization, and the compromise programming method is used to normalize each subobjective. The weight coefficients of each subobjective were determined by AHP, and the multiobjective topology optimization of gear was carried out [9, 10]. According to the optimization results, the web design of the gear is improved. The analysis results show that the improved gear strength performance and modal natural frequency are both improved, and the quality reduction effect is obvious. The topology optimization process is shown in Figure 1.Figure 1
Schematic of the topology optimization process.


## 2. Topology Optimization Model

### 2.1. Material Interpolation Method

The variable density method is an efficient method for structural topology optimization, the continuum structure is discretized by the finite element method [11]. The corresponding relation between the relative density of the element and the elastic modulus of the material is expressed by the density interpolation function of the continuous variable. The relative density of each cell is taken as the optimization variable. In the optimization process, each element corresponds to an optimization variable. By changing the value of the optimization variable, the elastic modulus of the element in the structure changes, thus adjusting the change of the overall stiffness matrix of the structure and making the material layout in the structure tends to be optimal. There are two kinds of material interpolation models of the variable density method [12, 13]: solid isotropic material with penalization and rational approximation of material properties. Since the topology optimization problem based on density description is a discrete programming problem [14]. To solve the topology optimization problem efficiently, The discrete element density variable can be continuous, that is, ρi can take any value between 0 and 1. To avoid intermediate density elements in the optimization process, a penalty factor P is introduced, to avoid intermediate density elements in the optimization process, a penalty factor P is introduced to make the density value of the optimization result close to the original discrete variable and get a clear boundary, thus improving the quality of topology optimization [15, 16].(1)
Solid isotropic material with penalization:(1)Ei=ρiP·E0i=1,2,⋯,n.(2)
Rational approximation of material properties:(2)Eρe=ρe1+q1−ρeE0,where Ei is the elastic modulus of the element i; P is the density penalty factor, generally taken as P=3; ρi is the density of the element i; to avoid singular phenomena in the solution process, its value range is as follows: 0≤ρmin≤ρe≤1,ρmin is the density value of the void part; and E0 is the elastic modulus of the part ρi=1.From a lot of engineering practices, the SIMP interpolation model is better than the RAMP interpolation model, so this paper chooses the SIMP interpolation model for optimization design.


### 2.2. The Topology Optimization Algorithm

Optistruct uses three methods to build approximate models: Paired method, Feasible direction method and the optimization criterion method. The optimization criterion method is applied to solve the classical topology optimization problem. The optimization steps of the optimization criterion method are as follows: Start with an initial designxk, k is the number of iterations, this method does not need to consider the constraint conditions and the state of the objective function, as long as the optimal computing conditions for each iteration are achieved, then an improved design xk+1 can be calculated, its iterative formula can be expressed as follows:(3)xk+1=Ckxk.The advantage of the optimization criterion method is that it is simple, has small computation, has clear physical meaning, fast iteration convergence speed, and structure reanalysis fewer times.


### 2.3. Finite Element Model of Gear

The reducer transmission system uses a double planetary deceleration and helical gear deceleration combined with a two-stage deceleration type. The gear transmission structure of the rotary reducer is shown in Figure2. The transmission system is a two-stage deceleration. The first stage of deceleration has two NGW-type planetary gear trains with the same structure. NGW is shorthand for a planetary array structure (N for internal engagement, G for a common planetary wheel, and W for a external engagement). Each planetary gear train comprises a sun gear A, a planetary wheel C, an inner ring B, and a planetary frame X. The inner gear ring is fixed, the central wheel is the input end, the planetary rack is the output end, and the number of planetary wheels is four. The second stage of deceleration is a helical gear set driven by a planetary frame. The helical gear set comprises a small helical gear D and a large helical gear E.Figure 2
Schematic of the rotary transmission system. (a) Schematic of the transmission system. (b) Structure schematic of the large helical gear.
(a)(b)The large helical gear E is taken as the research object, and the gear material is 20CrNi4. The material properties and geometric parameters are shown in Table1.Table 1
The gear parameter.
PropertyNumerical valueYoung’s modulus (Pa)2.07E + 11Poisson’s ratio0.29Density (kg/m³)7800Helix angle8°Tooth widthb (mm)60Number of teethz65Normal modulusmn (mm)5Reference diameter (mm)328.28The 3D solid model of the gear is imported into Hypermesh, and the hexahedral mesh of the gear is divided. Gears are divided into design area and nondesign area, and the gears are given material property andPsolid property. The gear holes are connected with the main shaft by keys, create a node in the center of the gear, and connect all the nodes of the inner ring with the central node by RBE2 rigid unit to create coupling constraints. The boundary condition is to constrain the five degrees of freedom, only releasing the them around the axis of rotation. In the statics analysis, a fixed constraint is added near the pitch line of two gears 12 apart to simulate the force of gear meshing. According to the transmission diagram and force situation of the gear, the helical gear E is subjected to a circular force of 503.28 N applied by two small helical gears at the meshing place, and the output torque of the gear center is 7200 Nm. The finite element model of gear is shown in Figure 3.Figure 3
Schematic of a gear finite element model.


## 2.1. Material Interpolation Method

The variable density method is an efficient method for structural topology optimization, the continuum structure is discretized by the finite element method [11]. The corresponding relation between the relative density of the element and the elastic modulus of the material is expressed by the density interpolation function of the continuous variable. The relative density of each cell is taken as the optimization variable. In the optimization process, each element corresponds to an optimization variable. By changing the value of the optimization variable, the elastic modulus of the element in the structure changes, thus adjusting the change of the overall stiffness matrix of the structure and making the material layout in the structure tends to be optimal. There are two kinds of material interpolation models of the variable density method [12, 13]: solid isotropic material with penalization and rational approximation of material properties. Since the topology optimization problem based on density description is a discrete programming problem [14]. To solve the topology optimization problem efficiently, The discrete element density variable can be continuous, that is, ρi can take any value between 0 and 1. To avoid intermediate density elements in the optimization process, a penalty factor P is introduced, to avoid intermediate density elements in the optimization process, a penalty factor P is introduced to make the density value of the optimization result close to the original discrete variable and get a clear boundary, thus improving the quality of topology optimization [15, 16].(1)
Solid isotropic material with penalization:(1)Ei=ρiP·E0i=1,2,⋯,n.(2)
Rational approximation of material properties:(2)Eρe=ρe1+q1−ρeE0,where Ei is the elastic modulus of the element i; P is the density penalty factor, generally taken as P=3; ρi is the density of the element i; to avoid singular phenomena in the solution process, its value range is as follows: 0≤ρmin≤ρe≤1,ρmin is the density value of the void part; and E0 is the elastic modulus of the part ρi=1.From a lot of engineering practices, the SIMP interpolation model is better than the RAMP interpolation model, so this paper chooses the SIMP interpolation model for optimization design.


## 2.2. The Topology Optimization Algorithm

Optistruct uses three methods to build approximate models: Paired method, Feasible direction method and the optimization criterion method. The optimization criterion method is applied to solve the classical topology optimization problem. The optimization steps of the optimization criterion method are as follows: Start with an initial designxk, k is the number of iterations, this method does not need to consider the constraint conditions and the state of the objective function, as long as the optimal computing conditions for each iteration are achieved, then an improved design xk+1 can be calculated, its iterative formula can be expressed as follows:(3)xk+1=Ckxk.The advantage of the optimization criterion method is that it is simple, has small computation, has clear physical meaning, fast iteration convergence speed, and structure reanalysis fewer times.


## 2.3. Finite Element Model of Gear

The reducer transmission system uses a double planetary deceleration and helical gear deceleration combined with a two-stage deceleration type. The gear transmission structure of the rotary reducer is shown in Figure2. The transmission system is a two-stage deceleration. The first stage of deceleration has two NGW-type planetary gear trains with the same structure. NGW is shorthand for a planetary array structure (N for internal engagement, G for a common planetary wheel, and W for a external engagement). Each planetary gear train comprises a sun gear A, a planetary wheel C, an inner ring B, and a planetary frame X. The inner gear ring is fixed, the central wheel is the input end, the planetary rack is the output end, and the number of planetary wheels is four. The second stage of deceleration is a helical gear set driven by a planetary frame. The helical gear set comprises a small helical gear D and a large helical gear E.Figure 2
Schematic of the rotary transmission system. (a) Schematic of the transmission system. (b) Structure schematic of the large helical gear.
(a)(b)The large helical gear E is taken as the research object, and the gear material is 20CrNi4. The material properties and geometric parameters are shown in Table1.Table 1
The gear parameter.
PropertyNumerical valueYoung’s modulus (Pa)2.07E + 11Poisson’s ratio0.29Density (kg/m³)7800Helix angle8°Tooth widthb (mm)60Number of teethz65Normal modulusmn (mm)5Reference diameter (mm)328.28The 3D solid model of the gear is imported into Hypermesh, and the hexahedral mesh of the gear is divided. Gears are divided into design area and nondesign area, and the gears are given material property andPsolid property. The gear holes are connected with the main shaft by keys, create a node in the center of the gear, and connect all the nodes of the inner ring with the central node by RBE2 rigid unit to create coupling constraints. The boundary condition is to constrain the five degrees of freedom, only releasing the them around the axis of rotation. In the statics analysis, a fixed constraint is added near the pitch line of two gears 12 apart to simulate the force of gear meshing. According to the transmission diagram and force situation of the gear, the helical gear E is subjected to a circular force of 503.28 N applied by two small helical gears at the meshing place, and the output torque of the gear center is 7200 Nm. The finite element model of gear is shown in Figure 3.Figure 3
Schematic of a gear finite element model.


## 3. Single Objective Topology Optimization of Gears

### 3.1. Topology Optimization Based on Static Stiffness Characteristics

Structural compliance reflects the strain capacity of the structure, which can be expressed by the strain energy stored in the structure or the work done by external forces in the process of structural deformation. Under the action of external force, the structural flexibility is equal to the reciprocal of the stiffness; that is, the larger the structural stiffness, the smaller the structural flexibility [17]. To control the maximum deformation of gear, overall structural compliance is adopted as the optimization objective [18]. Taking unit material density value ρi as the design variable, the minimum compliance is taken as the optimization objective, and the volume ratio before and after optimization is taken as the constraint condition. The optimization model of the solid isotropic material penalty method for gear is expressed as follows [19]:(4)findx=x1,x2,x3,⋯,xnT∈R,minCx=UTKU,s.t.∑j=1nvjxj−v=0,0<xmin≤xi≤xmax≤1,i=1,2,⋯,l,j=1,2,⋯,n,where x is the unit variable; n is the total number of units; C is structural compliance; K represents the system stiffness matrix; U represents the displacement vector of the structure; v is the volume constraint; xmin is the lower limit of design variables; and xmax is the upper limit of design variables.Import the 3D model of gear into Hypermesh, the gear was meshed by hexahedron, and the material attribute andPsolid attribute were given to the gear. The gear is divided into a design area and a nondesign area and created a node in the center of the gear. All the nodes of the inner ring are connected with the central node by RBE2 rigid unit to create coupling constraints. A fixed constraint is added to the contact surface of two gears with 12 adjacent teeth to simulate the force of gears during meshing [20]. The load condition is to add T=7200N·m torque at the center node. The boundary constraint is to release only the degree of freedom about the rotation axis.Firstly, the gear is analyzed statically. The Von Mises stress and displacement cloud diagram of gear can be obtained, as shown in Figure4. According to the figure, the maximum stress of the gear appears on the meshing line of the gear, and the stress value is 263.9 MPa. The maximum displacement is 0.09428 mm. Based on the statics analysis, the material element density value is created as the topology optimization variable. To control the occurrence of the checkerboard phenomenon during the optimization process, the minimum cell size is controlled to be 2–3 times the average grid size, and axial symmetric manufacturing constraints are added [19]. The constraint condition is that the volume ratio before and after optimization is ≤50%. Take compliance minimization as the design objective to conduct topology optimization, and obtain topology optimization results, as shown in Figure 5. Figure 6 shows the compliance iteration process of static topology optimization. It can be seen from the figure that the maximum value of gear compliance is 1.377 mm/N and the minimum value is 1.210 mm/N.Figure 4
Schematic of gear statics analysis results. (a) Schematic of gear displacement cloud. (b) Schematic of gear Von Mises stress cloud.
(a)(b)Figure 5
Schematic of static topology optimization results.Figure 6
Schematic of the compliance iteration process.


### 3.2. Topology Optimization Based on Dynamic Frequency Characteristics

Because the low-order natural frequency of the structure is close to the external excitation, it is easy to cause the gear resonance phenomenon. The topology optimization of dynamic frequency characteristics is aimed at improving the low-order frequency. In modal single objective optimization, the optimization process of each order frequency is not a positive correlation, which may occur when the frequency of one order is maximum, other times frequency will have a larger lower phenomenon, which can appear even the frequency switching between different orders, thus cause the iterative oscillation of the objective function, so that the optimization process is difficult to converge [10]. To solve this problem, make the lower order frequency relatively best on the whole. The average frequency method is introduced. The more the average frequency participates in the order, the more obvious the effect of reducing the iterative oscillation is [21]. However, the optimized frequency value may be less than that of direct single-objective optimization. The optimization objective is to maximize the average frequency of the first six orders. To obtain the maximum value of the average frequency and improve the low-order frequency at the same time, the weight values of each order frequency are 0.3, 0.3, 0.2, 0.2, 0.1, and 0.1. Taking the unit material density value ρi as the design variable, the volume ratio before and after optimization is taken as the constraint condition. The gear dynamic frequency topology optimization model is expressed as follows:(5)findx=x1,x2,x3,⋯,xnT∈R,maxΛx=λ0+α∑k=1mwkλk−λ0−1,s.t.∑j=1nvjxj−v=0,0<xmin≤xi≤xmax≤1,k=1,2,⋯,m,j=1,2,⋯,n,where Λx is the average eigenvalue; λ0, α is the given parameter, used to adjust the objective function; m is the order of eigenvalues to be optimized; wk is the weight coefficient of k-order eigenvalues; and λk is an eigenvalue of order k.Preprocessing is similar to static topology optimization, the gear was meshed by a hexahedron, and the material attribute andPsolid attribute were given to the gear, creating a node in the center of the gear. All the nodes of the inner ring are connected with the central node by the RBE2 rigid unit to create coupling constraints. Firstly, the modal analysis of the gear was carried out to obtain the first six-order frequency values of the gear, as shown in Table 2. Based on the modal analysis, the material element density value is created as the topology optimization variable. To control the occurrence of the checkerboard phenomenon during the optimization process, the minimum cell size is controlled to be 2–3 times the average grid size, and axial symmetric manufacturing constraints are added. The constraint condition is that the volume ratio before and after optimization is ≤50%. The optimization objective is the maximum value of the average frequency of the first six orders. After solving, the dynamic topology optimization result of gear structure can be obtained, as shown in Figure 7. The iteration process is shown in Figure 8. It can be seen from the figure that the objective function converges after 28 iterations. The first six frequency iterations are shown in Figure 9, After optimization, the first and second-order natural frequencies are increased to 2309.015 Hz, the third-order frequency is reduced to 2399.557 Hz, the fourth-order frequency is increased to 2684.151 Hz, and the fifth and sixth-order frequencies are increased to 2804.088 Hz. The frequency of each order oscillates slightly in the iteration, but the changing trend is stable in the late iteration, and there is no alternating oscillation phenomenon in the first six order frequencies. Because the gear has a symmetric structure, the first-order frequency value is equal to the second-order frequency value, and the fifth-order frequency value is equal to the sixth-order frequency value.Table 2
Gear modal analysis results.
OrderNatural frequency (Hz)12268.6522268.6532422.3842611.6652611.6662635.52Figure 7
Schematic of gear dynamic topology optimization results. (a) Schematic of topology optimization results. (b) Schematic of design area optimization results.
(a)(b)Figure 8
Schematic of gear dynamic topology optimization iteration process.Figure 9
Schematic of the first six-order frequency iteration process of gear dynamic topology optimization.


## 3.1. Topology Optimization Based on Static Stiffness Characteristics

Structural compliance reflects the strain capacity of the structure, which can be expressed by the strain energy stored in the structure or the work done by external forces in the process of structural deformation. Under the action of external force, the structural flexibility is equal to the reciprocal of the stiffness; that is, the larger the structural stiffness, the smaller the structural flexibility [17]. To control the maximum deformation of gear, overall structural compliance is adopted as the optimization objective [18]. Taking unit material density value ρi as the design variable, the minimum compliance is taken as the optimization objective, and the volume ratio before and after optimization is taken as the constraint condition. The optimization model of the solid isotropic material penalty method for gear is expressed as follows [19]:(4)findx=x1,x2,x3,⋯,xnT∈R,minCx=UTKU,s.t.∑j=1nvjxj−v=0,0<xmin≤xi≤xmax≤1,i=1,2,⋯,l,j=1,2,⋯,n,where x is the unit variable; n is the total number of units; C is structural compliance; K represents the system stiffness matrix; U represents the displacement vector of the structure; v is the volume constraint; xmin is the lower limit of design variables; and xmax is the upper limit of design variables.Import the 3D model of gear into Hypermesh, the gear was meshed by hexahedron, and the material attribute andPsolid attribute were given to the gear. The gear is divided into a design area and a nondesign area and created a node in the center of the gear. All the nodes of the inner ring are connected with the central node by RBE2 rigid unit to create coupling constraints. A fixed constraint is added to the contact surface of two gears with 12 adjacent teeth to simulate the force of gears during meshing [20]. The load condition is to add T=7200N·m torque at the center node. The boundary constraint is to release only the degree of freedom about the rotation axis.Firstly, the gear is analyzed statically. The Von Mises stress and displacement cloud diagram of gear can be obtained, as shown in Figure4. According to the figure, the maximum stress of the gear appears on the meshing line of the gear, and the stress value is 263.9 MPa. The maximum displacement is 0.09428 mm. Based on the statics analysis, the material element density value is created as the topology optimization variable. To control the occurrence of the checkerboard phenomenon during the optimization process, the minimum cell size is controlled to be 2–3 times the average grid size, and axial symmetric manufacturing constraints are added [19]. The constraint condition is that the volume ratio before and after optimization is ≤50%. Take compliance minimization as the design objective to conduct topology optimization, and obtain topology optimization results, as shown in Figure 5. Figure 6 shows the compliance iteration process of static topology optimization. It can be seen from the figure that the maximum value of gear compliance is 1.377 mm/N and the minimum value is 1.210 mm/N.Figure 4
Schematic of gear statics analysis results. (a) Schematic of gear displacement cloud. (b) Schematic of gear Von Mises stress cloud.
(a)(b)Figure 5
Schematic of static topology optimization results.Figure 6
Schematic of the compliance iteration process.


## 3.2. Topology Optimization Based on Dynamic Frequency Characteristics

Because the low-order natural frequency of the structure is close to the external excitation, it is easy to cause the gear resonance phenomenon. The topology optimization of dynamic frequency characteristics is aimed at improving the low-order frequency. In modal single objective optimization, the optimization process of each order frequency is not a positive correlation, which may occur when the frequency of one order is maximum, other times frequency will have a larger lower phenomenon, which can appear even the frequency switching between different orders, thus cause the iterative oscillation of the objective function, so that the optimization process is difficult to converge [10]. To solve this problem, make the lower order frequency relatively best on the whole. The average frequency method is introduced. The more the average frequency participates in the order, the more obvious the effect of reducing the iterative oscillation is [21]. However, the optimized frequency value may be less than that of direct single-objective optimization. The optimization objective is to maximize the average frequency of the first six orders. To obtain the maximum value of the average frequency and improve the low-order frequency at the same time, the weight values of each order frequency are 0.3, 0.3, 0.2, 0.2, 0.1, and 0.1. Taking the unit material density value ρi as the design variable, the volume ratio before and after optimization is taken as the constraint condition. The gear dynamic frequency topology optimization model is expressed as follows:(5)findx=x1,x2,x3,⋯,xnT∈R,maxΛx=λ0+α∑k=1mwkλk−λ0−1,s.t.∑j=1nvjxj−v=0,0<xmin≤xi≤xmax≤1,k=1,2,⋯,m,j=1,2,⋯,n,where Λx is the average eigenvalue; λ0, α is the given parameter, used to adjust the objective function; m is the order of eigenvalues to be optimized; wk is the weight coefficient of k-order eigenvalues; and λk is an eigenvalue of order k.Preprocessing is similar to static topology optimization, the gear was meshed by a hexahedron, and the material attribute andPsolid attribute were given to the gear, creating a node in the center of the gear. All the nodes of the inner ring are connected with the central node by the RBE2 rigid unit to create coupling constraints. Firstly, the modal analysis of the gear was carried out to obtain the first six-order frequency values of the gear, as shown in Table 2. Based on the modal analysis, the material element density value is created as the topology optimization variable. To control the occurrence of the checkerboard phenomenon during the optimization process, the minimum cell size is controlled to be 2–3 times the average grid size, and axial symmetric manufacturing constraints are added. The constraint condition is that the volume ratio before and after optimization is ≤50%. The optimization objective is the maximum value of the average frequency of the first six orders. After solving, the dynamic topology optimization result of gear structure can be obtained, as shown in Figure 7. The iteration process is shown in Figure 8. It can be seen from the figure that the objective function converges after 28 iterations. The first six frequency iterations are shown in Figure 9, After optimization, the first and second-order natural frequencies are increased to 2309.015 Hz, the third-order frequency is reduced to 2399.557 Hz, the fourth-order frequency is increased to 2684.151 Hz, and the fifth and sixth-order frequencies are increased to 2804.088 Hz. The frequency of each order oscillates slightly in the iteration, but the changing trend is stable in the late iteration, and there is no alternating oscillation phenomenon in the first six order frequencies. Because the gear has a symmetric structure, the first-order frequency value is equal to the second-order frequency value, and the fifth-order frequency value is equal to the sixth-order frequency value.Table 2
Gear modal analysis results.
OrderNatural frequency (Hz)12268.6522268.6532422.3842611.6652611.6662635.52Figure 7
Schematic of gear dynamic topology optimization results. (a) Schematic of topology optimization results. (b) Schematic of design area optimization results.
(a)(b)Figure 8
Schematic of gear dynamic topology optimization iteration process.Figure 9
Schematic of the first six-order frequency iteration process of gear dynamic topology optimization.


## 4. Multiobjective Topology Optimization Based on AHP

### 4.1. Topology Optimization Model and Weight Coefficient

Because the single-objective topology optimization of gear considers few factors. It is difficult to meet the actual engineering needs to make the gear structure have good mechanical properties, the static stiffness characteristics and dynamic frequency characteristics should be considered in the topology optimization stage. Therefore, multiobjective topology optimization of gears is carried out in this paper. Due to the different properties of stiffness and frequency, the numerical difference is obvious. The stiffness values and deformation modes of different working conditions are very different, and the frequency values of different orders are also obviously different, so it is difficult to achieve the optimal solution at the same time. Therefore, when the multiobjective optimization problem is transformed into a single objective optimization problem, it is necessary to normalize each subobjective [10, 22, 23]. In this paper, the compromise programming method with weight coefficients is used to deal with multiobjective optimization problems, and the comprehensive objective function is expressed as follows:(6)minFx=w2∑i=1lwiCimax−CixCimax−Cimin2+1−w2Λimax−ΛixΛimax−Λimin21/2,s.t.∑j=1nvjxj−v=0,βy≥β∗,0<xmin≤xi≤xmax≤1,i=1,2,⋯,n,where l is the load condition; wi is the weight coefficient of i-order eigenvalues; xj is the density of the j the element; Cix is the compliance value of working condition k; Cimax and Cimin are the maximum and minimum compliance values of working condition k; Λimax and Λimin are the maximum and minimum values of the frequency objective function, and wk is the weight value of each subtarget.When the compromise programming method is used to solve the multiobjective topology optimization problem, the corresponding weight of each optimization objective should be found [24]. According to the paper [25], the scheme of AHP to determine the weight coefficient has higher computational efficiency in topology optimization design, and the generated topology structure is more detailed, the force transmission path is more reasonable, and the stability is better. So this paper uses the AHP to determine the weight of each target. The basic idea of AHP is to decompose decision-making-related elements into goals, criteria, schemes, and other levels and then conduct quantitative analysis [26]. The AHP builds the decision layer of the problem to be evaluated. Then, according to the paired comparison method, the importance degree of each subobjective is obtained by pairwise comparison. Constructing decisions matrix W=Wijn×n,n represents the number of subtargets, Wij is the importance of the paired comparison between decision level i and decision level j [27]. The decision matrix is expressed as follows:(7)W=W11W12⋯W1nW21W22⋯W2n⋮⋮⋮Wn1Wn2⋯Wnn.The eigenvector of the decision matrix is the weight ratio of each working condition in the integrated objective function. To transform the decision level comparison into a numerical comparison, the initial importance reference definition is given, as shown in Table3.Table 3
The definition of the importance scale of the judgment matrix.
Importance DegreeRatio of importanceWiji is equally important with j1i is slightly more important than j3i is more important than j5i is far more important than j7i is significantly more important than j9j is slightly more important than i1/3j is more important than i1/5j is far more important than i1/7j is significantly more important than i1/9In practice, the pairing decision matrix given by the designer is generally inconsistent. If the number of nonzero eigenvectors of the matrix is greater than 1, it is necessary to determine which eigenvector represents the desired weight ratio vector. Therefore, the prerequisite is to ensure the consistency of the pairing matrix. If the consistency requirements are not met, the pairing matrix needs to be reestablished [28]. The consistency ratio (CR) can be used to judge the consistency degree of each pairing matrix [29]. If the consistency ratio is satisfied,(8)CR=CIRI<0.1,then the consistency is acceptable.WhereCI—coincident indicator, CI=λmax−n/n−1,λmax—The largest eigenvalue of the matrix; RI—randomly generated matrix consistency index.Therefore, the decision matrix of gear multi-objective optimization is expressed as follows:(9)W=W11W12W21W22,where W11, W12—the relevant elements of the objective function of static compliance; W21, W22—the relevant elements of the objective function of modal natural frequency; Define diagonal elements according to analytic hierarchy process W11=W22=1Because the gear needs a certain stiffness to determine the stability of the transmission, the dynamic characteristics are more important. According to the AHP, the decision matrix is parameterized as follows:(10)W=11331.By solving the decision matrix, it can be known that the maximum eigenvalue isλWmax=2, the eigenvector is P=13. By verification, the matrix meets the consistency requirements. Therefore, the weight coefficients of each optimization objective of multi-objective topology optimization are 0.25 and 0.75.


### 4.2. Topology Optimization Process and Results

Due to the limitation of OptiStruct, only a single objective function optimization design can be carried out, especially in nonconvex optimization problems that cannot get all Pareto solutions. Use the equation function definition panel in Optistruct, according to the maximum and minimum values of compliance and average frequency in single objective topology optimization, based on the multiobjective optimization formula, and using the sum of residual squares, the objective function is defined as follows:(11)Fx1,x2=rss0.25x1−1.203950.17256,0.75284864000−x271297000,where x1 is the compliance value of static topology optimization, and x2 is the frequency value of dynamic topology optimization.The preprocessing settings of multiobjective optimization are consistent with those of single-objective extension optimization. The material element density value is created as the topology optimization variable. To control the occurrence of the checkerboard phenomenon during the optimization process, the minimum cell size is controlled to be 2–3 times the average grid size, and axial symmetric manufacturing constraints are added. The constraint condition is that the volume ratio before and after optimization is≤50%. The optimization objective is to find the minimum value of the objective function. By solving, the multiobjective topology optimization results of gear structure can be obtained, as shown in Figures 10–14.Figure 10
Schematic of the results of multiobjective topology optimization.Figure 11
Schematic of the iterative process of multiobjective optimization.Figure 12
Schematic of the first six-order frequency iteration process of multiobjective optimization.Figure 13
Statics analysis of a multiobjective optimization model. (a) Schematic of gear displacement cloud. (b) Schematic of gear Von Mises stress cloud.
(a)(b)Figure 14
Schematic of the compliance iterative course of multiobjective optimization.Figure11 shows that the objective function converges after 10 steps of iteration. It can be seen from Figures 12–14, after multiobjective topology optimization, the low-order natural frequency of gear increases, and the compliance value decreases from 1.348 N·mm to 1.225 N·mm. The statistical analysis of the optimized gear is carried out. The cloud image of the gear is obtained, as shown in Figure 13, the maximum stress of the gear appears on the meshing line of the gear, and the stress value is 264 MPa, and the maximum displacement is 0.07763 mm. The compliance and frequency are mutually restricted. From the fourth iteration, the frequency value first decreases and then suddenly increases, and the flexibility value first increases and then suddenly decreases. This is because Optistruct uses the SIMP material interpolation model in its optimization process to obtain a more discrete structure, the minimum member size is set in the optimization. In the initial stage of optimization, the penalty coefficient ρ=2, with the progress of optimization, the penalty coefficient increases to ρ=3. Because there are intermediate density elements in the optimization process, a jump will occur for the objective function when ρ is increased by 1, which can be observed in Figures 12 and 14 [30]. There is a slight oscillation phenomenon in each order frequency during the iteration process, but the changing trend is stable in the late iteration, and there is no alternating oscillation phenomenon in the first six order frequencies, which proves that the optimization is feasible. Because the gear has a symmetric structure, the first-order frequency value is equal to the second-order frequency value, and the fifth-order frequency value is equal to the sixth-order frequency value.Therefore, a jump will occur for the objective function whenq is increased by 1, which can be observed in Figures 8 and 9.


### 4.3. Model Reconstruction

The material density distribution cloud image after multi-objective topology optimization is used as the basis for improving the design. In Hypermesh, the geometry is extracted through OSSmooth in the postpanel and the gears are remodeled taking into account the processing conditions of the structure, as shown in Figure15. Static analysis and modal analysis of the improved gear are carried out, and the analysis results are shown in Table 4.Figure 15
Schematic diagram of the improved gear structure.Table 4
Comparison of gear performance before and after multiobjective optimization.
Subunit objectivesBefore optimizationAfter optimizationVariable quantity (%)Maximum deflection (mm)0.0940.078−17Maximum stress (MPa)263.92640.04The first-order natural frequencies (Hz)2268.652314.392.02The second-order natural frequencies (Hz)2268.652314.392.02The third-order natural frequencies (Hz)2422.382407.74−0.6The fourth-order natural frequencies (Hz)2611.662519.51−3.53The fifth-order natural frequencies (Hz)2611.662642.021.06The sixth-order natural frequencies (Hz)2635.522642.020.25Mass (kg)36.06926.9428−25.3The weight of the improved gear is 26.9482 kg, which is 25.3% less than that of the original structure. Static and modal analysis of the improved gear is carried out, and the results are shown in Table4. The deformation of the modified gear structure is reduced from 0.09428 mm to 0.07763 mm, the stiffness performance is improved, the first six natural frequencies are slightly reduced in the third and fourth orders, and the other four orders are increased to varying degrees, and the dynamic characteristics are significantly improved.


## 4.1. Topology Optimization Model and Weight Coefficient

Because the single-objective topology optimization of gear considers few factors. It is difficult to meet the actual engineering needs to make the gear structure have good mechanical properties, the static stiffness characteristics and dynamic frequency characteristics should be considered in the topology optimization stage. Therefore, multiobjective topology optimization of gears is carried out in this paper. Due to the different properties of stiffness and frequency, the numerical difference is obvious. The stiffness values and deformation modes of different working conditions are very different, and the frequency values of different orders are also obviously different, so it is difficult to achieve the optimal solution at the same time. Therefore, when the multiobjective optimization problem is transformed into a single objective optimization problem, it is necessary to normalize each subobjective [10, 22, 23]. In this paper, the compromise programming method with weight coefficients is used to deal with multiobjective optimization problems, and the comprehensive objective function is expressed as follows:(6)minFx=w2∑i=1lwiCimax−CixCimax−Cimin2+1−w2Λimax−ΛixΛimax−Λimin21/2,s.t.∑j=1nvjxj−v=0,βy≥β∗,0<xmin≤xi≤xmax≤1,i=1,2,⋯,n,where l is the load condition; wi is the weight coefficient of i-order eigenvalues; xj is the density of the j the element; Cix is the compliance value of working condition k; Cimax and Cimin are the maximum and minimum compliance values of working condition k; Λimax and Λimin are the maximum and minimum values of the frequency objective function, and wk is the weight value of each subtarget.When the compromise programming method is used to solve the multiobjective topology optimization problem, the corresponding weight of each optimization objective should be found [24]. According to the paper [25], the scheme of AHP to determine the weight coefficient has higher computational efficiency in topology optimization design, and the generated topology structure is more detailed, the force transmission path is more reasonable, and the stability is better. So this paper uses the AHP to determine the weight of each target. The basic idea of AHP is to decompose decision-making-related elements into goals, criteria, schemes, and other levels and then conduct quantitative analysis [26]. The AHP builds the decision layer of the problem to be evaluated. Then, according to the paired comparison method, the importance degree of each subobjective is obtained by pairwise comparison. Constructing decisions matrix W=Wijn×n,n represents the number of subtargets, Wij is the importance of the paired comparison between decision level i and decision level j [27]. The decision matrix is expressed as follows:(7)W=W11W12⋯W1nW21W22⋯W2n⋮⋮⋮Wn1Wn2⋯Wnn.The eigenvector of the decision matrix is the weight ratio of each working condition in the integrated objective function. To transform the decision level comparison into a numerical comparison, the initial importance reference definition is given, as shown in Table3.Table 3
The definition of the importance scale of the judgment matrix.
Importance DegreeRatio of importanceWiji is equally important with j1i is slightly more important than j3i is more important than j5i is far more important than j7i is significantly more important than j9j is slightly more important than i1/3j is more important than i1/5j is far more important than i1/7j is significantly more important than i1/9In practice, the pairing decision matrix given by the designer is generally inconsistent. If the number of nonzero eigenvectors of the matrix is greater than 1, it is necessary to determine which eigenvector represents the desired weight ratio vector. Therefore, the prerequisite is to ensure the consistency of the pairing matrix. If the consistency requirements are not met, the pairing matrix needs to be reestablished [28]. The consistency ratio (CR) can be used to judge the consistency degree of each pairing matrix [29]. If the consistency ratio is satisfied,(8)CR=CIRI<0.1,then the consistency is acceptable.WhereCI—coincident indicator, CI=λmax−n/n−1,λmax—The largest eigenvalue of the matrix; RI—randomly generated matrix consistency index.Therefore, the decision matrix of gear multi-objective optimization is expressed as follows:(9)W=W11W12W21W22,where W11, W12—the relevant elements of the objective function of static compliance; W21, W22—the relevant elements of the objective function of modal natural frequency; Define diagonal elements according to analytic hierarchy process W11=W22=1Because the gear needs a certain stiffness to determine the stability of the transmission, the dynamic characteristics are more important. According to the AHP, the decision matrix is parameterized as follows:(10)W=11331.By solving the decision matrix, it can be known that the maximum eigenvalue isλWmax=2, the eigenvector is P=13. By verification, the matrix meets the consistency requirements. Therefore, the weight coefficients of each optimization objective of multi-objective topology optimization are 0.25 and 0.75.


## 4.2. Topology Optimization Process and Results

Due to the limitation of OptiStruct, only a single objective function optimization design can be carried out, especially in nonconvex optimization problems that cannot get all Pareto solutions. Use the equation function definition panel in Optistruct, according to the maximum and minimum values of compliance and average frequency in single objective topology optimization, based on the multiobjective optimization formula, and using the sum of residual squares, the objective function is defined as follows:(11)Fx1,x2=rss0.25x1−1.203950.17256,0.75284864000−x271297000,where x1 is the compliance value of static topology optimization, and x2 is the frequency value of dynamic topology optimization.The preprocessing settings of multiobjective optimization are consistent with those of single-objective extension optimization. The material element density value is created as the topology optimization variable. To control the occurrence of the checkerboard phenomenon during the optimization process, the minimum cell size is controlled to be 2–3 times the average grid size, and axial symmetric manufacturing constraints are added. The constraint condition is that the volume ratio before and after optimization is≤50%. The optimization objective is to find the minimum value of the objective function. By solving, the multiobjective topology optimization results of gear structure can be obtained, as shown in Figures 10–14.Figure 10
Schematic of the results of multiobjective topology optimization.Figure 11
Schematic of the iterative process of multiobjective optimization.Figure 12
Schematic of the first six-order frequency iteration process of multiobjective optimization.Figure 13
Statics analysis of a multiobjective optimization model. (a) Schematic of gear displacement cloud. (b) Schematic of gear Von Mises stress cloud.
(a)(b)Figure 14
Schematic of the compliance iterative course of multiobjective optimization.Figure11 shows that the objective function converges after 10 steps of iteration. It can be seen from Figures 12–14, after multiobjective topology optimization, the low-order natural frequency of gear increases, and the compliance value decreases from 1.348 N·mm to 1.225 N·mm. The statistical analysis of the optimized gear is carried out. The cloud image of the gear is obtained, as shown in Figure 13, the maximum stress of the gear appears on the meshing line of the gear, and the stress value is 264 MPa, and the maximum displacement is 0.07763 mm. The compliance and frequency are mutually restricted. From the fourth iteration, the frequency value first decreases and then suddenly increases, and the flexibility value first increases and then suddenly decreases. This is because Optistruct uses the SIMP material interpolation model in its optimization process to obtain a more discrete structure, the minimum member size is set in the optimization. In the initial stage of optimization, the penalty coefficient ρ=2, with the progress of optimization, the penalty coefficient increases to ρ=3. Because there are intermediate density elements in the optimization process, a jump will occur for the objective function when ρ is increased by 1, which can be observed in Figures 12 and 14 [30]. There is a slight oscillation phenomenon in each order frequency during the iteration process, but the changing trend is stable in the late iteration, and there is no alternating oscillation phenomenon in the first six order frequencies, which proves that the optimization is feasible. Because the gear has a symmetric structure, the first-order frequency value is equal to the second-order frequency value, and the fifth-order frequency value is equal to the sixth-order frequency value.Therefore, a jump will occur for the objective function whenq is increased by 1, which can be observed in Figures 8 and 9.


## 4.3. Model Reconstruction

The material density distribution cloud image after multi-objective topology optimization is used as the basis for improving the design. In Hypermesh, the geometry is extracted through OSSmooth in the postpanel and the gears are remodeled taking into account the processing conditions of the structure, as shown in Figure15. Static analysis and modal analysis of the improved gear are carried out, and the analysis results are shown in Table 4.Figure 15
Schematic diagram of the improved gear structure.Table 4
Comparison of gear performance before and after multiobjective optimization.
Subunit objectivesBefore optimizationAfter optimizationVariable quantity (%)Maximum deflection (mm)0.0940.078−17Maximum stress (MPa)263.92640.04The first-order natural frequencies (Hz)2268.652314.392.02The second-order natural frequencies (Hz)2268.652314.392.02The third-order natural frequencies (Hz)2422.382407.74−0.6The fourth-order natural frequencies (Hz)2611.662519.51−3.53The fifth-order natural frequencies (Hz)2611.662642.021.06The sixth-order natural frequencies (Hz)2635.522642.020.25Mass (kg)36.06926.9428−25.3The weight of the improved gear is 26.9482 kg, which is 25.3% less than that of the original structure. Static and modal analysis of the improved gear is carried out, and the results are shown in Table4. The deformation of the modified gear structure is reduced from 0.09428 mm to 0.07763 mm, the stiffness performance is improved, the first six natural frequencies are slightly reduced in the third and fourth orders, and the other four orders are increased to varying degrees, and the dynamic characteristics are significantly improved.


## 5. Conclusion

(1)
Topology and multiobjective theory are adopted to optimize the structure design of the large gear in the transmission system of the directional drilling rig, considering the dynamic and static characteristics of the gear, By combining the compromise programming method with the analytic hierarchy process (AHP), an optimization method was proposed, which used the compromise programming method to normalize the subobjectives and determined the weight coefficients of subobjectives through the hierarchical analysis. OptiStruct was used to optimize the multiobjective topology of the gear.(2)
To reduce the frequency altercation and oscillation phenomenon IN the process of natural frequency subobjective optimization of gear, a new objective function is obtained by using the weighted average frequency method, which makes the iteration of natural frequency subobjective optimization tends to be stable, the convergence speed is faster, the optimization effect is better. The optimized gear model is reconstructed, and its static and dynamic analysis is carried out. The analysis results show that the stiffness performance of the gear is significantly enhanced, the first six natural frequencies are improved to varying degrees, and the weight reduction effect is obvious, which also provides a theoretical basis for the lightweight design of other structures.



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*Source: 2977092-2023-01-10.xml*