Multi-objective optimization model of transmission error of nonlinear dynamic load of double helical gears

Xingling Yao

doi:10.1515/nleng-2022-0323

Article Open Access

Multi-objective optimization model of transmission error of nonlinear dynamic load of double helical gears

Xingling Yao

Published/Copyright: October 28, 2023

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information

From the journal Nonlinear Engineering Volume 12 Issue 1

Abstract

In order to address the impact of reduced transmission stability and reliability caused by volume reduction on the quality of gear transmission, this article proposes a multi-objective optimization model for nonlinear dynamic load transmission errors of double helical gears. This study aims to introduce a multi-objective design method for gear transmission, using the volume and smooth reliability of helical gears as objective functions, and establish a multi-objective optimization design mathematical model for helical cylindrical gear transmission. In order to solve this multi-objective optimization problem, we utilized the optimization toolbox in the scientific calculation software MATLAB with examples. The results show that after the joint optimization design of volume and coincidence degree, it is calculated that the volume after the joint optimization design is still 2.2624 × 10⁷ mm³, and the coincidence degree is 5.9908. After rounding, the design result is m n = 3 , Z 1 = 31 , β = 20 ∘ , Ψ d = 1.2 . The optimization design results show that the joint optimization design with the minimum volume and the maximum coincidence as the objective function can reduce the volume and improve the output stability of the helical gear.

Keywords: helical gear transmission; multi-objective optimization design; MATLAB

1 Introduction

Gear transmission is a widely used form of mechanical transmission, which has been widely used in multiple industrial sectors, such as automobiles, aviation, instrumentation, shipbuilding, power, mining, and agricultural machinery. This transmission method has many important advantages, making it the preferred transmission choice. First, gear transmission has the characteristic of high efficiency. Due to the high precision of meshing and matching between gears, they can effectively transmit power and torque, resulting in high transmission efficiency. Compared to other transmission forms, gear transmission has less energy loss and can provide higher transmission efficiency, making the energy utilization of the system more effective. Second, gear transmission has an accurate transmission ratio. By selecting the module, number of teeth, and meshing method of the gear reasonably, the transmission ratio can be accurately controlled to meet the design requirements. This accurate transmission ratio gives gear transmission important advantages in applications that require precise speed matching and torque transformation. In addition, gear transmission also has the characteristics of compact structure and reliable operation. The gear transmission system has a simple and compact structure, occupying relatively small space, and is suitable for limited space application scenarios. Meanwhile, due to the relatively simple working principle of gear transmission and the absence of complex electronic control systems, it has high reliability and stability and can work stably for a long time. Finally, gear transmission has the characteristic of long service life. The design and manufacturing process of gear transmission undergo strict technology and quality control. The selection of gear materials and heat treatment results in high hardness and wear resistance, thereby extending the service life of the transmission system. With the rapid development of science and technology, the gear system has developed in terms of high-speed, heavy-duty, light-weight, high-pressure, and automation, increasing the need for its operation, especially its good operation [1].

With the development of high-speed and high-precision mechanical equipment, and the continuous development of technology, the need for people to work well, the quality of technology is increasing, and the reliability of the delivery system is increasing higher. As the main transmission system in industry and production, the transmission has the advantages of large bearing capacity, stable transmission, compact structure, etc., and is widely used in civil and military equipment [2]. Traditional mechanical models typically do not consider the need for good performance. Glass machine design is related to technology, standard work installation, and technology. The analysis of various properties of glass is the main basis for interpreting glass distribution (noise, vibration, strength, etc.). Under the influence of external loads, the working gear causes large deformations and stress gradients in the transmission gear area, thereby reducing the fatigue life of the glasses. Vibration and noise from gear mechanisms are often the source of system vibration and noise [3]. Therefore, the performance characteristics are studied, and the performance characteristics of transmission gears have an important impact on improving the performance characteristics of the whole system. Particularly with the widespread use of high-speed and heavy-duty vehicles, glass strength analysis has become an important topic of future research [4].

The main objective of this article is (i) to study the characteristics and effects of nonlinear dynamic load transmission errors in double helical gear transmission. The goal is to gain a deeper understanding of nonlinear dynamic phenomena in double helical gear transmission systems and to analyze the sources, mechanisms, and influencing factors of load transmission errors. (ii) Construct a multi-objective optimization model for nonlinear dynamic load transmission error of double helical gears. The goal is to establish an optimization model that comprehensively considers multiple design objectives, with helical gear volume and transmission smoothness reliability as the main objective functions. (iii) Develop effective optimization algorithms and calculation methods. The goal is to research and develop optimization algorithms and calculation methods suitable for the optimization design of double helical gear transmission, in order to solve multi-objective optimization problems and obtain a set of optimal solutions. By achieving the above objectives, this article aims to deeply study the multi-objective optimization problem of nonlinear dynamic load transmission error of double helical gears, provide a theoretical basis and practical methods for the optimization design of double helical gear transmission, and further improve the performance and quality of the transmission system.

With the development of gear transmission systems toward high speed and heavy load, the vibration and noise of the gear transmission system are intensified under the combined action of internal and external excitation and even affect the service life and reliability of the gear transmission system. Tooth profile modification can effectively alleviate the sharp changes in meshing stiffness caused by changes in the number of meshing teeth, thereby reducing the impact of meshing in and out and reducing the vibration and noise of the gear transmission system. The main task of tooth profile modification is to determine the three elements of tooth profile modification: modification amount, modification length, and modification type. The determination of these three factors is a complex problem, and in order to achieve the best tooth profile modification effect, optimization design is generally used to determine the optimal modification parameters. Domestic and foreign scholars have conducted extensive and in-depth research on the optimization design of tooth profile modification, focusing on optimization based on static models and optimization based on dynamic models. In the optimization design of tooth profile modification based on the static model of gear transmission, analytical methods or finite-element software are generally used. Calculate the static transmission error of the gear and take reducing the fluctuation of the static transmission error as the optimization objective. The optimization results obtained the optimal amount and length of modification for static performance. However, it is difficult to achieve the optimal dynamic performance of the system through the modification parameters when the static performance is optimal. Some scholars have carried out tooth profile modification optimization design based on the dynamic model of the gear transmission system, and combined with numerical simulation results to reduce the dynamic transmission error or root mean square of vibration acceleration as the optimization goal, obtained the modification parameters when the dynamic performance is optimal.

According to research literature, it can be seen that in the past, in the optimization design of tooth profile modification based on gear dynamics models, the gear dynamics model was generally simplified as a two degrees of freedom pure torsional nonlinear dynamics model, ignoring the influence of lateral and swing direction degrees of freedom; neglecting the coupling relationship between the actual motion state of the gear and the tooth profile modification parameters on the influence of meshing stiffness; the meshing stiffness of tooth profile modification gears is often calculated using finite-element methods, which reduces computational efficiency. Based on the integration of Isight optimization software with nonlinear dynamics programs, an optimization platform is built to optimize the tooth profile modification design of a heavy-duty vehicle gear transmission system and the effectiveness of the optimization model is verified.

In gearbox fault diagnosis, literature focuses on the vibration response signal of the gearbox as the research object and achieves fault diagnosis based on different signal processing methods. The signal processing technology based on vibration response basically follows the following process: information acquisition, signal processing, fault feature extraction, and fault pattern recognition. From this process, it can be seen that when it is difficult to establish a mathematical model of the object, the signal processing method based on vibration response avoids the difficulty of establishing a mathematical model, which is the advantage of these theoretical methods. However, from another perspective, the inability to analyze the vibration mechanism, properties, characteristics, and influencing factors generated by the gearbox is also its weakness. The use of nonlinear dynamics for fault diagnosis of gearboxes is one of the important research topics in dynamics, and it is also an effective method for fault diagnosis of gearboxes. This method not only reveals the mechanism, properties, characteristics, and influencing factors of gearbox vibration, but also has important theoretical significance in analyzing its spectral mechanism.

2 Literature review

As an important transmission device in the mechanical system, with its continuous development toward high speed and importance, the dynamic analysis and vibration response analysis of the gear system has become an inevitable requirement, and the dynamic design and parametric design of the gear has become an inevitable trend. The research results of gear system dynamics and vibration response have an important impact on improving the bearing capacity of gear transmission devices, reducing vibration and noise, and reducing failure and damage [5]. Wang et al. proposed a method for three-segment modification of the tooth profile of small gears. By changing the shape of the cutting edge and replacing the straight tooth profile with a three-segment parabolic curve, the equation of the cutting surface was derived. A coordinate system for the meshing of double helical gears has been established, and tooth surface contact analysis and load contact analysis methods have been proposed. Taking the minimum transmission error as the optimization objective, the complex method is used to obtain the optimization parameters. Case studies have shown that in the presence of axis error, compared with optimization without considering axis error, optimization results without considering axis deviation in tooth profile and axial correction can still significantly reduce load transmission error. Finally, a calculation method for correcting the tooth profile of the small gear is provided [6].

Wang and Zhu proposed an improved meshing stiffness calculation model of helical gear pair, which fully considered the influence of transverse gear tooth stiffness, transverse gear base stiffness, axial gear tooth stiffness, and surface roughness under the condition of elastohydrodynamic lubrication [7]. Liang et al. proposed an internal meshing gear transmission based on the spatial involute helix. First, according to the theory of gear geometry, the generation principle of spatial involute helix is proposed. The mathematical design model of spatial conjugate involute helix is established by establishing the meshing relationship solved by the relative velocity and the normal vector of the contact point [8]. Yan et al. proposed a new meshing stiffness analysis model for helical gears, which can well predict the dimensionless time-varying meshing stiffness. On this basis, the influence of gear basic parameters on the meshing stiffness fluctuation coefficient is studied [9].

Based on the above references, it can be seen that the main aspects that have had a significant impact on the study of nonlinear dynamics of gear transmission in the past are the problem of parameter vibration of the system caused by elastic deformation and the problem of nonlinear vibration of the system caused by gaps between teeth. The key areas of research include the determination of vibration analysis models, excitation forms, selection of solution methods, and the impact of vibration parameters on dynamic characteristics.

The significance of studying gear system dynamics lies in improving the dynamic performance of its transmission, such as smoothness and reducing noise and vibration. In order to continuously improve the dynamic characteristics of products, the dynamic design of structures in transmission machinery has received widespread attention. In recent years, the focus of research has shifted from static design to dynamic optimization design with the goal of reducing product weight and structural volume in order to better adapt to the working needs of modern equipment. At the same time, as a key component of mechanism transmission, the theory of gear wheel system dynamics has also been continuously developed.

Based on the aforementioned research, this article proposes an innovative multi-objective optimization model for nonlinear dynamic load transmission errors of double helical gears. In order to achieve optimization design, we used the optimization toolbox in MATLAB language to calculate and solve multi-objective optimization design. This method has the following obvious advantages. The multi-objective optimization model for nonlinear dynamic load transmission error of double helical gears proposed in this article adopts the optimization toolbox of MATLAB language for optimization design, which has obvious advantages such as simple initial parameter input, small programming workload, and obvious advantages. This method can provide effective tools and frameworks for optimizing double helical gear transmission, helping designers quickly obtain the optimal design solution. The results show that after the joint optimization design of volume and coincidence degree, it is calculated that the volume after the joint optimization design is still 2.2624 × 10⁷ mm³, and the coincidence degree is 5.9908. After rounding, the design result is m n = 3 , Z 1 = 31 , β = 20 ∘ , Ψ d = 1.2 . From the comparison of design results, the multi-objective optimization design of helical gears is an effective design method. It is better than a single target optimization design and has certain practical value.

3 Research methods

3.1 Establishment of a coupled nonlinear dynamic model of a helical gear transmission system

The dynamic characteristics of the gear system depend on the external and internal excitation of the gear system. External excitation includes meshing force excitation in gear transmission. Internal excitation includes static transmission error excitation and stiffness excitation caused by elastic deformation, and stiffness excitation is one of the main factors that cause the nonlinearity of the system; at the same time, the backlash also causes the nonlinearity of the system [10]. The gear system will produce a dynamic response of the system under the action of external excitation and internal excitation; this periodic response causes structural vibration of the system and fatigue damage to the system components. Therefore, the dynamic model of the system is established to analyze the dynamic characteristics of the system, and it is one of the basic tasks to study the dynamics of the gear system to make it meet the better working characteristics.

Helical gear transmission is different from spur gear transmission, because the normal meshing force generated by the existence of a spiral angle is not along the radial direction but into a certain angle value, and this force is decomposed into radial force and axial force. Axial force produces axial vibration and radial vibration, and radial force produces bending vibration of the shaft [11]. Based on this, the vibration characteristics of helical gears are different from that of spur gears, and the performance is often more complex than that of spur gears, which is reflected in the coupling vibration of bending, torsion, and swing. Based on considering the influence of torsional stiffness and axial stiffness of the intermediate shaft of the high-speed and low-speed stages, the author establishes a complete coupling model of the two-stage helical gear reducer.

In general, the basic modeling methods for gear system dynamics can be roughly divided into two categories, as shown in Figure 1.

Figure 1

Basic method of system modeling.

A gear system is a continuous system composed of multiple elastomers, which is one of the more complex branches of rotor system dynamics. In general research, the box structure is often analyzed and treated as a rigid body component; although this treatment method is reasonable for the analysis of general gear transmission systems, it is often not applicable for some special applications, such as the gearbox used in aircraft and spacecraft, because its design principle is to reduce the weight of the overall structure as much as possible under the condition of meeting the work needs, and this will greatly reduce the rigidity of the box, in this case, the box must be treated as a flexible body, so the deformation of the box must be considered [12].

Assuming that the friction force is very small compared to the actual mesh force, the effects of inter-tooth friction force, gear misalignment, and transmission error are neglected. Only the effects of static errors in support stiffness, bearing damping, frame self-deformation, web damping, blade inversion, and mesh connection transmission are considered to be depressing.

The gear has four degrees of freedom in space for axial bending vibration (y-direction), torsional vibration, torsional vibration (horizontal direction), and axial vibration (z-direction). The author uses a large clustering method to construct a three-dimensional model of a secondary helical gear analysis model. As shown in formula (1), the total degree of freedom of the entire system is 16

(1) { δ } = { y 1 , z 1 , θ 1 y , θ 1 z , y 2 , z 2 , θ 2 y , θ 2 z , y 3 , z 3 , θ 3 y , θ 3 z , y 4 , z 4 , θ 4 y , θ 4 z } T .

In the above formula, y i , z i , θ iy , θ iz ( i = 1, 2, 3, 4) take the center point of the secondary helical gear as the origin, respectively, as the vibration generalized degree of freedom. The expression of meshing force at the transmission end of the high-speed stage is derived as follows [13].

The helix angle of the helical gear, the height angle of the end of the pair of gears is β , and the normal height angle is a n . The relationship between the movement of the mesh point in the y-direction and the movement in the z-direction is shown as z = y tan β . Using the bending torsional vibration model, the general motion of high-speed gears 1 and 2 is as follows:

(2) { δ } = { y 1 , z 1 , θ 1 y , θ 1 z , y 2 , z 2 , θ 2 y , θ 2 z } T ,

y i , z i , θ iy , θ iz ( i = 1, 2). The generalized degrees of freedom with the center point of the secondary helical gear as the origin are expressed as translation displacement and angular displacement.

The relationship between the vibration displacement of the engagement point p and the generalized displacement of the high-speed gear 1 is as follows:

(3) y p = y 1 + θ 1 z R 1 ,

(4) z p = z 1 − y p tan ⁡ β = z 1 − ( y 1 + θ 1 z R 1 ) tan ⁡ β .

The relationship between the vibration displacement of the engagement point G and the generalized displacement of the high-speed gear 2 is as follows:

(5) y g = y 2 − θ 2 z R 2 ,

(6) z g = z 2 − y g tan ⁡ β = z 2 − ( y 2 − θ 2 z R 2 ) tan ⁡ β .

The normal stiffness, normal damping, and normal meshing error of gear mesh are K Am , C Am , and e A , respectively, and the parameters along the transverse and axial directions are as follows:

(7) K Any = K Am cos ⁡ β , C Amy = C Am cos ⁡ β , e Ay = e A cos ⁡ β ,

(8) K Amz = K Am sin ⁡ β , C Amz = C Am sin ⁡ β , e Az = e A sin ⁡ β .

The lateral (y-direction) dynamic transmission error of the gear is as follows:

(9) y d ( t ) = y p − y g = y 1 + θ 1 z R − ( y 2 + θ 2 z R 2 ) ,

where R 1 and R 2 are the base circle radius of gear 1 and gear 2, respectively.

Y is the difference between the transmission after switching to the fault, and the static transmission of the high-speed shaft end gear pair, and formula (10) is as follows:

(10) y A ( t ) = y d ( t ) − e Ay = y 1 + θ 1 z R − ( y 2 + θ 2 z R 2 ) − e Ay ,

where e Ay is the static transfer error in the y-direction of the high-speed stage, and the calculation of the static transfer error is detailed in the next section [14]. The following formula (11) can be obtained by taking the derivative of the above formula:

(11) y ̇ A ( t ) = y ̇ d ( t ) − e ̇ Ay = y ̇ 1 + θ ̇ 1 z R − ( y ̇ 2 + θ ̇ 2 z R 2 ) − e ̇ Ay .

The dynamic transmission error of gear axial (z-direction) is as follows:

(12) z d ( t ) = z p − z g = z 1 − ( y 1 + θ 1 z R 1 ) tan ⁡ β − z 2 + ( y 2 − θ 2 z R 2 ) tan ⁡ β ,

where R 1 and R 2 are the base circle radius of gear 1 and gear 2, respectively.

z A ( t ) is the difference between the lateral dynamic transmission error and the static transmission error of the gear pair at the high-speed shaft end, which is shown in the following formula:

(13) z A ( t ) = z d ( t ) − e Az = z 1 − ( y 1 + θ 1 z R 1 ) tan ⁡ β − z 2 + ( y 2 − θ 2 z R 2 ) tan ⁡ β − e A z ,

where e Az refers to the static transmission error in the z-direction of the high-speed stage, the calculation of e Az is introduced in the next section, its size is the comprehensive result of the static transmission error of two gears, the influence factors of the static transmission error of the meshing tooth pair are related to the load, tooth top modification, tooth width, and other geometric factors.

In the same way, the derivation of the above formula has the following formula:

(14) Z ̇ A ( t ) = Z ̇ d ( t ) − e ̇ Az = z ̇ 1 − ( y ̇ 1 + θ ̇ 1 z R 1 ) tan ⁡ β − z ̇ 2 + ( y ̇ 2 − θ ̇ 2 z R 2 ) tan ⁡ β − e ̇ Az .

The above is the meshing model of the high-speed stage of the secondary helical gear; the meshing model of the low-speed stage is similar, replace the corresponding subscript with the subscript of the low-speed stage and unify the subscript A as the dynamic parameter of the low-speed stage, the subscript B is used as the mechanical parameter of the high-speed level. It is sufficient to make a corresponding substitution for other dimension parameters (such as pressure angle, base circle radius, etc.). The formulas are as follows:

(15) m 1 y y ̈ 1 + c 1 y y ̇ 1 + k 1 y y 1 + [ k Amy f A ( y ) + c Amy y ̇ A ] = 0 ,

(16) m 1 z z ̈ 1 + c 1 z z ̇ 1 + k 1 z z 1 + [ k Amz f A ( z ) + c Amz z ̇ A ] = 0 ,

(17) J 1 y θ ̈ 1 + c 1 θ y θ ̇ 1 y + k 1 θ y θ 1 y + R 1 [ k Amz f A ( z ) + c Amz z ̇ A ] = 0 ,

(18) I 1 z θ ̈ 1 z − T 1 d + R 1 [ k Amy f A ( y ) + c Amy y ̇ A ] = 0 ,

(19) m 2 y y ̈ 2 + c 2 y y ̇ 2 + k 2 y y 2 − [ k Amy f A ( y ) + c Amy y ̇ A ] = 0 ,

(20) m 2 z z ̈ 2 + c 2 z z ̇ 1 + k 1 z z 1 − [ k Amz f A ( z ) + c Amz z ̇ A ] + k s ( z 2 − z 3 ) = 0 ,

(21) J 2 y θ ̈ 2 y + c 2 θ y θ ̇ 2 y + k 2 θ y θ 2 y + R 2 [ k Amz f A ( z ) + c Amz z ̇ A ] = 0 ,

(22) I 2 z θ ̈ 2 z + R 2 [ k Amy f A ( y ) + c Amy y ̇ A ] + k 1 ( θ 2 z − θ 3 z ) = 0 ,

(23) m 3 y y ̈ 3 + c 3 y y ̇ 3 + k 3 y y 3 + [ k Bmy f B ( y ) + c Bmy y ̇ B ] = 0 ,

(24) m 3 z z ̈ 3 + c 3 z z ̇ 3 + k 3 z z 3 − [ k Bmz f B ( z ) + c Bmz z ̇ B ] − k s ( z 2 − z 3 ) = 0 ,

(25) J 3 y θ ̈ 3 y + c 3 θ y θ ̇ 3 y + k 3 θ y θ 3 y + R 3 [ k Bmz f B ( z ) + c Bmz z ̇ B ] = 0 ,

(26) I 3 z θ ̈ 3 z + R 3 [ k Bmy f B ( y ) + c Bmy y ̇ B ] − k 1 ( θ 2 z − θ 3 z ) = 0 ,

(27) m 4 y y ̈ 4 + c 4 y y ̇ 4 + k 4 y y 4 − [ k Bmy f B ( y ) + c Bmy y ̇ B ] = 0 ,

(28) m 4 z z ̈ 4 + c 4 z z ̇ 4 + k 4 z z 4 + [ k Bmz f B ( z ) + c Bmz z ̇ B ] = 0 ,

(29) J 4 y θ ̈ 4 y + c 4 θ y θ ̇ 4 y + k 4 θ y θ 4 y + R 4 [ k Bmz f B ( z ) + c Bmz z ̇ B ] = 0 ,

(30) I 4 z θ ̈ 4 z + T 2 d + R 4 [ k Bmy f B ( y ) + c Bmy y ̇ B ] = 0 .

Subscripts A and B represent high-speed shaft and low-speed shaft end gear pair, respectively; the meaning of each parameter is the same as the physical meaning mentioned above.

3.2 Establishment of optimization design mathematical model

The author takes a pair of helical gear transmissions as the research object. The optimization design mathematical model can abstract design problems into a mathematical model, formalizing the design objectives, constraints, and variables. This simplifies the complexity of design problems and makes the design process more systematic and controllable.

3.2.1 Selection of design variables

Many parameters affect the quality of helical gear transmission, and the author only selects the following key parameters as design variables. That is, the normal modulus m _n, the tooth number Z ₁, the helix angle β , and the tooth width coefficient Ψ d of the pinion; m _n directly affects the size and strength of the gear and is the standard series value specified in the national standard. The number of teeth can only be rounded. The helix angle has a direct impact on the shape, size, and stress state of the gear; the tooth width coefficient determines the tooth width of the gear and affects the stress distribution and transmission coincidence of the gear teeth. Therefore, the optimal design variables are as follows:

(31) X = ( x 1 , x 2 , x 3 , x 4 ) T = ( m n , z 1 , β , Ψ d ) T .

3.2.2 Establishment of the objective function

For high-speed transmission gears, we must reduce weight and save cost under the condition of meeting the requirements of use while improving the smoothness and reliability of transmission. Therefore, the first optimization objective is to minimize the sum of the indexing cylinder volumes of large and small helical gears f 1 ( X ) . Many parameters affect the stability and reliability of the transmission, and they are all related to the coincidence degree [15]. Therefore, the author chooses the maximum coincidence degree, that is, the minimum of its opposite number f 2 ( X ) , as the second optimization objective, namely, the following formulas:

(32) f 1 ( X ) = π 4 m n z 1 cos β 3 ( 1 + i 2 ) Ψ d ,

(33) f 2 ( X ) = − ε r = − ( 0.318 Ψ d z 1 tan ⁡ β ) − 1.88 − 3.2 1 z 1 + 1 i 1 ,

where i is the transmission ratio.

3.2.3 Determination of constraint conditions

Referring to the formula in the mechanical design manual, there are the following constraints.

3.2.3.1 Performance constraints

The contact stress σ H between the tooth surfaces is less than the allowable fatigue contact stress [ σ H ] , that is, the following formulas:
(34) σ H = Z E Z H Z ε cos β m n z 1 2,000 ( 1 + i ) T 1 cos ⁡ β K α K β K a K v i m n z 1 Ψ d ⩽ [ σ H ] ,

(35) G 1 ( X ) = σ H − [ σ H ] ⩽ 0 .
The bending stress σ F 1 of the pinion shall be less than the allowable bending fatigue stress [ σ F 1 ] , that is, the following formulas:
(36) σ F 1 = Y F 1 Y β Y ε K a K v K α K β 2,000 T 1 cos 2 ⁡ β m n 3 z 1 2 Ψ d ⩽ [ σ F 1 ] ,

(37) G 2 ( X ) = σ F 1 − [ σ F 1 ] ⩽ 0 .
The bending stress σ F 2 of the pinion shall be less than the allowable bending fatigue stress [ σ F 2 ] , that is, the following formulas:

(38) σ F 2 = Y F 2 Y F 1 σ F 1 ⩽ [ σ F 2 ] ,

(39) G 3 ( X ) = σ F 2 − [ σ F 2 ] ⩽ 0 .

3.2.3.2 Determination of geometric constraints

Modulus constraints

In order to prevent the gear tooth from breaking, the gear must have enough modulus; for power transmission gear, the modulus is generally not less than 2. The author takes the modulus greater than or equal to 3, that is, the following formula:
(40) G 4 ( X ) = 3 − x 1 ⩽ 0 .
Conditions for pinion not undercutting

Because the basic condition for helical gears not to undercut is that the number of teeth is not less than 16, that is, the following formula:
(41) G 4 ( X ) = 16 − x 2 ⩽ 0 .
Limitation of the helix angle

Generally, the value range of the helix angle is 8°–20°, i.e., the following formulas:
(42) G 6 ( X ) = x 3 − 0.3491 ⩽ 0 ,

(43) G 7 ( X ) = 0.1396 − x 3 ⩽ 0 .
Limitation of tooth width factor

The tooth width coefficient directly affects the volume and coincidence degree. The value shall be taken within a certain range according to the mechanical design manual, namely, the following formulas:
(44) G 8 ( X ) = x 4 − 1.2 ⩽ 0 ,

(45) G 9 ( X ) = 0.5 − x 4 ⩽ 0 .
Coincidence constraint limits

The coincidence e of helical gears is generally required to be no less than 2, that is, the following formula:

(46) G 9 ( X ) = 2 − ( 0.318 Ψ d z 1 tan β ) + 1.88 − 3.2 1 z 1 + 1 i z 1 ⩽ 0 .

3.3 Optimization method

There are many methods to solve multi-objective optimization, and the author adopts the unified objective method, also known as the comprehensive objective method. It is to transform the original multi-objective optimization problem into a unified objective function or a comprehensive objective function as the evaluation function of the multi-objective optimization problem (the function reconstructed by the multi-objective optimization problem) through a certain method and then solve it with the single-objective function optimization method [16].

Here, the linear weighting method is used to transform the multi-objective optimization problem into an evaluation function. That is, the following formula:

(47) min F ( X ) = min [ W 1 f 1 ( X ) + W 2 f 2 ( X ) ] .

Among them, W i ≥ 0 ( i = 1 , 2 ) , where W 1 is the volume weighting coefficient and W 2 is the weighting coefficient of coincidence degree.

The selection of the weighting coefficient reflects the different evaluations and compromises each sub-target, so it should be handled according to the specific situation.

4 Result analysis

4.1 Static transmission error of gear

4.1.1 Basic concept of transmission error

The meshing line increment method is adopted for the processing of gear error, that is, the change of the meshing angle is regarded as the change of meshing line increment, because the meshing line increment method involves the reading system of gear meshing error, the processing is relatively complex. Zhang Yimin and Yang Jian of Northeast University used the finite-element method to determine the static error of gear transmission, used the static analysis method to simulate the smooth transmission of gears, and obtained the contact state and static transmission error of meshing gears. In the past, in the overall modeling of gear systems, the static transmission error of gears was less handled, a simple harmonic function can be used to describe the static error of gear transmission, or the measured error value can be used in the modeling process. Although the measured error value can best reflect the real situation, it is difficult to achieve in actual production due to the limitation of test conditions [17]. Another general method is to take the frequency of gear meshing as the fundamental frequency and use the Fourier series to express the static transmission error of gear transmission. The error is expressed as follows:

(48) e ( t ) = e 0 + e r sin ⁡ ( ω m + φ ) .

In the above formula, e ( t ) is the function of the static transmission error of the gear with time t; e 0 , e r is the constant value and amplitude of gear error, usually e 0 = 0 ; ω m is the meshing frequency of gear, ω m = 2 π T z = 2 π zn 60 ; T z is the engagement period, T z = 60 nz ; n is the speed (rpm); z is the number of teeth of gear; and φ is the phase angle, usually φ = 0.

Although the above simple harmonic function can describe the static transmission error of the gear meshing process to a certain extent, the meshing error is regarded as a function related to the meshing period, but the influence of different load conditions and other geometric dimensions is not considered. The error processing method adopted by the author is a good supplement to the static error mentioned above and also accurately describes the error changes of helical gears in the meshing process.

4.1.2 Flake assumption

Due to the existence of the helical angle of helical gears, helical gear transmission is different from spur gear transmission. In the analysis of gear static transmission error, a compromise method is “slice model.” That is, the helical gear is regarded as a series of adjacent slices; each slice has no influence on each other, and the deformation only depends on the contact force between the slices [18].

4.1.3 Tooth shape assumption

The gear is processed and manufactured by relatively advanced processing and manufacturing methods, and the uniform tooth shape can be processed on the tooth surface to be processed. The errors in the manufacturing and installation of gears cause noise and vibration in the process of gear meshing, especially in high-power transmission devices such as megawatt wind power transmission devices; the problem of vibration and noise is more prominent. The gear modification technology is an effective method to solve the vibration and noise problem and has been more and more widely used in recent years. In gear continuous transmission, the base pitch of two gears must be equal in value. However, because the gear is a deformable body, the elastic deformation of the gear will occur under the action of meshing force, which changes the position of the base pitch of the driving gear and the passive gear on the meshing line and is no longer strictly equal. In addition, the non-uniform thermal deformation will also lead to the inequality of the base joint on the meshing line. In order to reduce the impact of meshing, the tooth profile must be modified. That is, the tooth profile is changed by removing part of the material along the tooth height direction.

The end thinning is to reduce the concentrated load at the end; this method is simple and easy to process and has been widely used in engineering practice. The thinning of the end is usually selected as a linearly variable dimension, the tooth profile modification parameter l c = 0.2 b (b is the tooth width), c = 0.03 mm, and the value of c is related to the load size and material selection. The author chooses 0.03 mm.

The modification of the drum tooth is applied to the full width of the tooth surface, and the tooth profile is parabolic. That is, the amount of trimming is proportional to the distance from the centerline [19].

Because the gear is an elastic body, some elastic deformation will occur under the action of external load, and some interference will be recorded as Cpp. Once the above interference is known, we can calculate the interference of other points on the meshing line. It is calculated by subtracting the trimming amount of the tooth top from the interference amount caused by the installation error and the modification. The meshing force between the two gears is calculated by multiplying the interference amount of each local slice by the corresponding stiffness. The meshing force obtained by this method may not be the correct meshing force for the first time, but we can correct the interference amount at the node according to the overall stiffness value, so as to obtain a more reasonable result. The iterative method is adopted until the error between the estimated meshing force and the actual meshing force is within an acceptable range, and the relative error adopted by the author is 0.05%. The degree of interference between two gears is only proportional to the distance from the center line of the tooth width. Its value is the angle between the two helices of β 1 x (usually very small), the deviation of β 1 manufacturing and installation, and the deformation under the elastic support leads to the overall movement of the gear. Drum trimming will also reduce interference, and its value is crrel (2x/b)², where cerel is the trimming amount of the end drum, and b is the tooth width of the gear. The thinning amount will also reduce the interference amount, and its value is endrel × x × bb 2 , in the above formula, endrel is the trimming amount of the gear end, and bb is the tooth width when there is no trimming amount at the end. When the value in the above formula is less than 0, the value is zero.

The size of the trimming amount of the sheet depends on the distance from the contact point to the pitch line, and the size of the specific trimming amount depends on the following formula:

(49) bprlf × ( | yppt | − p sf ) 0.5 p b − p sf ,

where bprlf is the trimming amount at 0.5 p b position, the pitch of p b base circle, and the rounding amount of all negative values in the above formula for p sf is set to 0. Each meshing cycle is divided into 16 sub-steps, and p b 16 is added to each meshing, at the same time, considering the influence of coincidence, there will still be one or two meshing lines in front of and behind the meshing line of the meshing point in Figure 2, only the distance from the base circle pitch.

Figure 2

Static error curve of gear pair.

4.1.4 Calculation of static transmission error

Figure 2 is the static transfer error curve drawn by MATLAB programming according to the calculation principle of static transfer error introduced above. According to the provisions of GB/T10095, the accuracy grade of gears is divided into 13 grades, 0–12 grades, and the accuracy grade is from high to low. Level 1–2 is the accuracy level to be developed, and level 3–5 is the high accuracy level; each accuracy level specifies the range of tooth deviation parameters. The author selects the accuracy class of the secondary helical gear transmission mechanism as 4. The installation deviation of high-speed stage and low-speed stage is 40 μm, the trimming amount (bprlf) at half of the base section is selected as 25 μm, the base circle pitch is 17.700 mm, the modification amount (crrel) of the end drum tooth is 8 μm, and the end thinning amount (endrel) is selected as 10 μm.

The above shows the error value curve of gear meshing in two cycles, and each meshing cycle is divided into 16 sub-steps to complete one meshing. Divide the length of each base node into 16 sub-steps. The change trend of the length of a base segment before the meshing point is similar to the change of the simple harmonic function, which has a similar trend to the previous treatment of static transmission error, indicating that this method is reasonable for the treatment of static transmission error. However, compared with the previous meshing line increment method, the error reading system is reduced, and the application range is expanded. Compared with the simple harmonic function, considering the change of static transmission error on the length of a base node rather than the error change on the whole cycle, this method can more accurately describe the change trend of static transmission error in the meshing process and has better practicability [20].

Because the static error transmission curve of gear meshing is a discrete point under the current step size, it is not convenient for subsequent differential and interpolation operations, so curve fitting is performed for it. The author uses a spline curve to fit discrete points, so that better results of polynomial meshing can be obtained. Use the spline function in the MATLAB curve fitting toolbox. The basic idea of solving spline function is to find a set of fitting polynomials when a set of data points is known. In the process of fitting, for each adjacent sample pair of this data set, the cubic polynomial is used to fit the curve between the sample points, in order to ensure the uniqueness of the fitting, the first and second derivatives of the cubic polynomial at the sample point are constrained. In this way, in addition to the endpoint of the interval to be studied, the first and second-order continuous derivatives are guaranteed at all internal sample points [21].

By comparing the original static transfer error curve and the spline fitting curve in Figures 3–6, it can be observed that the spline curve has a good fitting effect. Spline curves can accurately characterize the continuous and differentiable static transfer error distribution and exhibit the variation relationship of the error curve at the current step size. The derivatives of these error curves play an important role in the simulation process. A spline curve is a smooth interpolation curve obtained by interpolating between given data points. In Figures 3–6, a continuous and smooth curve is formed by connecting the data points of the original static transmission error curve using the spline fitting method. Compared with the original data, the spline fitting curve better approximates the shape of the actual error distribution and provides a more accurate and continuous error estimation. In the simulation process, the derivative of the error curve plays an important role in analyzing and controlling system performance. Derivatives can provide information on the rate of error change, which is crucial for determining the stability, response speed, and dynamic characteristics of a system. By utilizing the continuous differentiability of spline curves, the derivative of the error curve can be accurately calculated, providing necessary parameters for system simulation and performance evaluation. Therefore, by observing the fitting effect of the spline fitting curve in Figures 3–6 on the original static transfer error curve, we can conclude that the spline curve has better performance in characterizing error distribution, describing error variation relationships, and calculating error derivatives. This provides a more accurate and reliable foundation for subsequent simulation analysis and control strategy design [22].

Figure 3

Static error fitting curve of gear 1.

Figure 4

Derivative curve of gear 1.

Figure 5

Static error fitting curve of gear 2.

Figure 6

Static derivative curve of gear 2.

4.2 Application examples

Known T 1 = 440 Nm; transmission ratio i = 5; the material of the pinion is 40Cr (quenched), and the convenience is HRC45–55; big gear material 45 # steel (quenched and tempered), convenience HB230-280; Z E = 189.8 MPa； Z H = 2.43; Z ε = 0.93; K a = 1.40; K v = 1.07; K β = 1. 24; K a = 1.12; Y ε = 1; Y β = 1 − (1.268)/π; [ σ H ] = 604 MPa; [ σ F 1 ] = 302 MPa; [ σ F 2 ] = 232 MPa; Y F 1 = 3.78 − 0.045 z 1 /cos³ β ; Y F 2 = 2.23 − 0.0003 z 2 /cos³ β .

First, the parameters in the example are put into the objective function and constraint conditions for sorting and simplification and then the optimization calculation is carried out using MATLAB optimization toolbox programming.

First, set w 1 to 1 and w 2 to 0, that is, only the volume is optimized; the result is as follows:

(50) x = [ 3.9061 ; 25.1708 ; 0.3491 ; 0.9672 ] .

From this, the designed volume is 2.2624 × 10⁷ mm³, and the coincidence degree is 4.5456 by substituting the optimal value of design variables into formula (32).

Then, set w 1 to 0.8 and w 2 to 0.2, that is, only the volume and coincidence are jointly optimized, and the results are as follows:

(51) x = [ 3.0000 ; 30.5002 ; 0.3491 ; 1.200 ] .

From this, it can be calculated that the volume after the joint optimization design is still 2.2624 × 10⁷ mm³, and the coincidence degree is 5.9908.

After rounding, the design result is m n = 3 , Z 1 = 31 , β = 20 ∘ , Ψ d = 1.2 .

By comparing and analyzing the design results, it can be concluded that multi-objective optimization design of helical gears is an effective design method. Compared to single objective optimization design, it performs better in design effectiveness and has important practical value. In helical gear design, single objective optimization design only focuses on optimizing a single design indicator, ignoring other potentially related design elements. However, the performance of helical gear systems is often influenced by multiple factors, such as transmission efficiency, load distribution, noise, and vibration. Therefore, adopting multi-objective optimization design methods can comprehensively consider these factors and obtain more comprehensive and optimized design solutions. The advantage of multi-objective optimization design for helical gears lies in its ability to balance the relationships between different objectives during the design process and find a balance point to meet multiple design requirements. By using appropriate multi-objective optimization algorithms, such as genetic algorithm, particle swarm optimization, etc., multiple optimization solutions can be searched in the design space, forming a set of optimization solutions, known as Pareto Frontiers. This cutting-edge represents various possible design options, and designers can choose the most suitable design solution based on actual needs and trade-offs. The multi-objective optimization design method can also help designers better understand the behavior and interrelationships of the system during the design process, promoting innovation and improvement of design ideas. By exploring the trade-offs and compromises between different design schemes, the overall performance and reliability of the helical gear system can be further improved [23].

The basic concepts of modal analysis were introduced, the meshing stiffness matrix derived, the basic types of elements in ANSYS briefly introduced, a finite-element analysis model of the helical gear system established, and the corresponding coupling modal frequency and formation were obtained. Based on modal analysis, the displacement responses in the x- and y-directions at various coupling frequencies were analyzed in harmonic response analysis, and the impact of each response on the transmission system was analyzed. By analyzing the harmonic response and coupling modal analysis of the gear system, the displacement response of the meshing point at various resonant frequencies was determined. Based on the finite-element method, it provides a more effective method for further understanding the dynamic characteristics of the gear transmission system and also provides valuable information for further structural optimization of the gear system. This provides important reference value for the vibration reduction work of the transmission system and also provides a foundation and basis for the vibration characteristics of the system.

5 Conclusion

The author discusses the use of MATLAB optimization toolbox to carry out multi-objective optimization design of helical gear transmission and proves that the expected purpose has been achieved with an example. The optimization design results show that the joint optimization design with the minimum volume and the maximum coincidence as the objective function can reduce the volume and improve the output stability of the helical gear. However, many parameters affect the helical gear transmission, and the author only selects the main parameters as the design variables, which may have a certain impact on the accuracy of the design results. The author uses some simplified algorithms to select the objective function and constraint conditions, which may reduce the design accuracy to a certain extent. Therefore, the subsequent research goal will be how to improve the design accuracy. In the future, further research and application of multi-objective optimization models for nonlinear dynamic load transmission errors of double helical gears can be emphasized. Here are some possible research directions:

Model improvement and optimization algorithm: Further improve the accuracy and reliability of the multi-objective optimization model for nonlinear dynamic load transmission error of double helical gears. We can consider introducing more accurate dynamic models, further improving error evaluation methods, and exploring more efficient and optimized optimization algorithms.
Multi-objective optimization strategy: Study more objective functions and establish appropriate constraints to achieve a more comprehensive optimization design. For example, it is possible to consider minimizing transmission errors while maximizing transmission efficiency or minimizing noise and vibration.
Parameter sensitivity analysis: Study the sensitivity of different parameters to double helical gear transmission errors and analyze the impact of parameter changes on optimization results. This can provide a deeper understanding and guidance for optimizing design.

In summary, future research can further improve the model and algorithm, conduct experimental verification, and expand application fields based on multi-objective optimization models for nonlinear dynamic load transmission errors of double helical gears, in order to enhance the performance and application value of double helical gear transmission systems.

In the dynamic analysis of this article, only the influence of the elastic support stiffness and damping at the joint of the bearing and shaft is considered, while the influence of the joint of the upper and lower box bodies, as well as the joint of the box body and fixed bolts, is ignored. In future research work, more accurate analytical models can be established for factors not considered above. Considering that gear transmission is widely used in automotive gearboxes and aerospace applications, the requirements for shock absorption, noise reduction, and lightweight are becoming increasingly high. Finite element analysis technology can be used to optimize and analyze the structure, strength, and other indicators of gear transmission systems. (i) Minimize the vibration acceleration of each node of the box vibration (noise reduction) under the constraint of stress and small deformation of key parts. (ii) Minimize the weight of the gear transmission system (weight reduction) under the constraint of deformation.

Acknowledgments

The study was supported by Key Scientific Research Project of Colleges and Universities of Henan Province: Application Research of Beidou Satellite Positioning in Cutting Machine of Bicycle Sports Car (15B46D010).

Author contributions: Xingling Yao: writing and performing surgeries; data analysis and performing surgeries; article review and intellectual concept of the article.
Conflict of interest: The author declares that they have no competing interests.
Data availability statement: The data used to support the findings of this study are available from the corresponding author upon request.

References

[1] Dong H, Bi Y, Wen B, Liu ZB, Wang LB. Bending-torsional-axial-pendular nonlinear dynamic modeling and frequency response analysis of a marine double-helical gear drive system considering backlash. J Low Freq Noise Vib Act Control. 2022;41(2):519–39.10.1177/14613484211065742Search in Google Scholar

[2] He E, Yin S. Tooth profile design of a novel helical gear mechanism with improved geometry for a parallel shaft transmission. Mech Sci. 2022;13(2):1011–8.10.5194/ms-13-1011-2022Search in Google Scholar

[3] Liu Y, Li Y, Ma L, Chen B, Li R. Parametric design approach on high-order and multi-segment modified elliptical helical gears based on virtual gear shaping. Proc Inst Mech Eng C J Mech Eng Sci. 2022;236(9):4599–609.10.1177/09544062211058396Search in Google Scholar

[4] Liu Y, Zhu L. Research on non-circular helical gear NC hobbing linkage control scheme based on variable transmission ratio. Proc Inst Mech Eng B J Eng Manuf. 2022;(11):1433–42.10.1177/09544054221076240Search in Google Scholar

[5] Liu Y, Gong J. Study on TEHL flash temperature of helical gear pair considering profile modification. Ind Lubr Tribol. 2023;75(1):17–26.10.1108/ILT-06-2022-0199Search in Google Scholar

[6] Wang Y, Wang H, Liu X, Fu Y. Optimization of Topological Modification Parameters for Heavy-Duty Helical Gears. IEEE Access. 2022;10(6):41789–802.10.1109/ACCESS.2022.3168388Search in Google Scholar

[7] Wang S, Zhu R. An improved mesh stiffness model of helical gear pair considering axial mesh force and friction force influenced by surface roughness under EHL condition. Appl Math Model. 2022;102(8):453–71.10.1016/j.apm.2021.10.007Search in Google Scholar

[8] Liang D, Zhao W, Meng S, Peng S. Mathematical design and meshing analysis of a new internal gear transmission based on spatial involute-helix curve. Proc Inst Mech Eng C J Mech Eng Sci. 2023;237(1):237–49.10.1177/09544062221119079Search in Google Scholar

[9] Yan H, Wen L, Yin S, Cao H, Chang L. Research on gear mesh stiffness of helical gear based on combining contact line analysis method. Proc Inst Mech Eng C J Mech Eng Sci. 2022;236(16):9354–66.10.1177/09544062221092580Search in Google Scholar

[10] Patil PJ, Patil AJ. Whole field analysis of asymmetric helical gear using photo stress method and FEA method. Indian J Sci Technol. 2023;16(2):109–17.10.17485/IJST/v16i2.1817Search in Google Scholar

[11] Qin Y, Li D, Wang H, Liu Z, Wei X, Wang X. Multi-objective optimization design on high pressure side of a pump-turbine runner with high efficiency. Renew Energy. 2022;190(9):103–20.10.1016/j.renene.2022.03.085Search in Google Scholar

[12] Kouritem SA, Abouheaf MI, Nahas N, Hassan M. A multi-objective optimization design of industrial robot arms. Alex Eng J. 2022;61(12):12847–67.10.1016/j.aej.2022.06.052Search in Google Scholar

[13] Zhao Z, Tao Y, Wang J, Hu J. The multi-objective optimization design for the magnetic adsorption unit of wall-climbing robot. J Mech Sci Technol. 2022;36(1):305–16.10.1007/s12206-021-1228-2Search in Google Scholar

[14] Jiang M, Rui X, Yang F, Zhu W, Zhang Y. Multi-objective optimization design for a magnetorheological damper. J Intell Mater Syst Struct. 2022;33(1):33–45.10.1177/1045389X211006907Search in Google Scholar

[15] Chen L, Li P, Xia S, Kong R, Ge Y. Multi-objective optimization for membrane reactor for steam methane reforming heated by molten salt. Sci China Technol Sci. 2022;65(6):1396–414.10.1007/s11431-021-2003-0Search in Google Scholar

[16] Niu X, Xu F, Zou Z. Bionic inspired honeycomb structures and multi-objective optimization for variable graded layers. J Sandw Struct Mater. 2023;25(2):215–31.10.1177/10996362221127969Search in Google Scholar

[17] Bao N, Peng Y, Feng H, Yang C. Multi-objective aerodynamic optimization design of variable camber leading and trailing edge of airfoil. Proc Inst Mech Eng C J Mech Eng Sci. 2022;236(9):4748–65.10.1177/09544062211056012Search in Google Scholar

[18] Komiljonovna ML, Esonali o’g’li MY. Adjuster synthesizing for the heat process with Matlab. Tex J Eng Technol. 2022;7(4):63–6.Search in Google Scholar

[19] Shanmugasundaram N, Kumar SP, Ganesh EN. Modelling and analysis of space vector pulse width modulated inverter drives system using MatLab/Simulink. Int J Adv Intell Paradig. 2022;22(1–2):200–13.10.1504/IJAIP.2022.123023Search in Google Scholar

[20] Banik A, Shrivastava A, Potdar RM, Jain SK, Nagpure SG, Soni M. Design, modelling, and analysis of novel solar PV system using MATLAB. Mater Today Proc. 2022;51(2):756–63.10.1016/j.matpr.2021.06.226Search in Google Scholar

[21] Abdullah FN, Aziz GA, Shneen SW. Simulation model of servo motor by using matlab. J Robot Control (JRC). 2022;3(2):176–9.10.18196/jrc.v3i2.13959Search in Google Scholar

[22] Horri N, Pietraszko M. A tutorial and review on flight control co-simulation using matlab/simulink and flight simulators. Automation. 2022;3(3):486–510.10.3390/automation3030025Search in Google Scholar

[23] Papp D, Yıldız S. alfonso: Matlab package for nonsymmetric conic optimization. Informs J Comput. 2022;34(1):11–9.10.1287/ijoc.2021.1058Search in Google Scholar

Received: 2023-05-11

Revised: 2023-08-16

Accepted: 2023-08-30

Published Online: 2023-10-28

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/nleng-2022-0323

Keywords for this article

helical gear transmission; multi-objective optimization design; MATLAB

Creative Commons

BY 4.0