© J. Wang and H. Zhou, Published by EDP Sciences 2025

1 Introduction

Gear transmission system, which is used for transmitting motion and power, is the most important and widely used transmission mechanism [1]. Complex gear transmission, which usually consists of both parallel shaft and planetary gear transmission, is widely used in many machines such as aircraft engine, helicopters, automobiles, mining machinery, wind turbines, robots, etc. [2–6], as shown in Figure 1. Some complex gear transmissions have numerous transmission parts for reasons such as enabling actuator to reach the designated position like arms (Fig. 1a), more transmission stages (Figs. 1b, 1c). More transmission parts means larger dynamic model in studying dynamic characteristics of complex gear transmission, and results in higher demand on computational resources. In some cases, we need to focus on the dynamics of only some parts of gear transmission, not the whole. For example, when studying the dynamics of coal mining machine gearbox, planetary gear transmission gear transmission is the dynamic reliability weak part since it is closer to shock loads caused by cutting coal than parallel shaft gear transmission. Hence, it is essential to simplify the dynamic model of parallel shaft gear transmission for paying more attention on dynamics of planetary gear transmission with less computing accuracy loss. Therefore, it is essential to propose a new method for simplifying the dynamic model of complex PSGS which can solve the problem of numerical calculation of large dynamic model while meeting certain accuracy requirements.

Two main research methods are commonly used in establishing dynamic model of gear transmission- finite element model (FEM) and lumped mass model (LMM). A large number of papers focused on FEM [7–10] in researching dynamics of gear transmission system, which can reflect the flexibility of gear components and obtain more accurate dynamic results than LMM. However, FEM cannot be used in establishing dynamic model of complex gear transmission system which has both multiple parallel shaft and planetary gear transmission due to the large amount of computing cost. Published paper mostly established only one gear meshing pair with FEM. LMM [11–20] is commonly used in establishing dynamic model of complex gear transmission with numerous gear components, and numerical analysis methods are applied in solving differential equations of LMM. However, LMM cannot be used in establishing dynamic model when the complex gear transmission has too many gear components, which means there are numerous degrees of freedom in dynamic model due to the following reasons: (1) Dimension disaster. Numerous degrees of freedom will lead to computational explosion and large demand of storage. (2) Challenges for numerical methods. Taking Runge-Kutta method as an example, the time step dt must be very small to maintain numerical stability, and this stability step should be much smaller than the resolution step required to describe the physical evolution of the system. This will result in the need to calculate millions or even billions of time steps, which is unacceptable in terms of computation time. Moreover, complex gear transmission is non-linear system for the dynamic model point of view, and it apparently increased the difficulties in solving differential equations of LMM. For the best knowledge of the authors, maximum degree of freedom in published papers for establishing LMM of complex gear transmission is 72. The case study objective is MG500-1180 coal mining machine gearbox, there are 22 components (including gears, carriers, motor and cutting drum) in the gearbox, which means 22 × 6 = 132 degrees of freedom when considering all of the six moving directions of one component. Therefore, it is essential to establish a simplified dynamic model for directly reducing the number of degrees of freedom of complex gear transmission.

The simplified modeling method for PSGS have been a challenging issue. Only a few efforts have been spent to solve this problem. Some studies [21,22] treat gear pairs that need to be simplified as rigid connections and provide calculation methods for equivalent stiffness and equivalent moment of inertia, thereby reducing the number of degrees of freedom in dynamic model. Furthermore, gear backlash, bearing clearance and gear flexibility are neglected in these studies. Hence, the number of degrees of freedom was reduced to 1. However, key factors affecting the dynamic characteristics of PSGS, such as nonlinear vibration caused by backlash, time-varying meshing stiffness excitation, meshing force coupling between different transmission stages, are neglected along with the vanish of gear meshing pairs, and then inevitably causing significant errors of results of dynamic characteristics. Other papers [23,24] equated PSGS to motor shaft or connecting shaft, and angular displacements are recalculated depending on gear transmission ratios. However, the deficiencies exist in references [21,22] are still unresolved. The deficiencies of existing studies are summarized as follows:

• PSGS is excessively simplified in existing methodologies. All gear meshing pairs are neglected along with the vanish of key factors affecting the dynamic characteristics of PSGS, such as nonlinear vibration caused by backlash, time-varying meshing stiffness excitation, meshing force coupling between different transmission stages. Therefore, significant errors of results of dynamic characteristics are inevitable in these existing studies. How to describe these nonlinear dynamic behaviors of PSGS through equivalent parameters more accurately?

• The stiffness and deformation of gearbox housing are not found in previous papers. Stiffness and deformation of gearbox housing can obviously affect dynamic characteristics of PSGS. How to simplified this continuum? How to calculate the supported stiffness provided by both bearings and gearbox housing? How to reposition the equivalent discrete point of gearbox housing and establish deformation coordination relationships since the number of bearing holes are reduced in simplified dynamic model of PSGS?

• No experiment is conducted to validate these proposed theoretical methodologies.

With the best knowledge of previous studies, this paper proposes a more accurate simplified modeling methodology of PSGS by the following contributions:

• PSGS is equivalent to one gear meshing pair which can reserve key factors affecting the dynamic characteristics of PSGS, such as nonlinear vibration caused by backlash, time-varying meshing stiffness excitation, meshing force coupling between different transmission stages. The values of these nonlinear dynamic factors are calculated based on mechanical system vibration transmission theory.

• Based on agglutination method, gearbox housing was simplified into discrete nodes to make the continuum gearbox housing available for centralized parameter modelling. Series stiffness relationship is proposed in this study for calculate the supported stiffness provided by both bearings and gearbox housing. Deformation coordination relationships is discussed in this paper, since the number of bearing holes is reduced and the positions of the discrete point of gearbox housing are changed.

• In case study, the simplified dynamic model of PSGS in coal mining machine gearbox is proposed. Dynamic experiment under real operating conditions is conducted to validate the methodology of this study.

Fig. 1 Parallel shaft gear transmission system.

2 Simplified dynamic modeling methodology

A parallel shaft gear transmission usually contains gears, bearings, shafts, and gearbox housing. Idler gears are sometimes used for an extended actuator. When discussing the simplified methodology of dynamic model, there are two key factors: mass/moment of inertia and vibration energy. Hence, the principles of simplified methodology are: (1) The mass/moment of inertia in before/after simplification must be equal. (2) The vibration energy-no matter linear or nonlinear-in before/after simplification must be equal. According to these principles, the simplified dynamic modeling methodology of a n-stages parallel shaft gear transmission is shown as follows.

2.1 Equivalent of mass/moment of inertia

Figure 2 shows the overall simplified model of a general n-stages PSGS which is simplified as one gear pair. Previous papers eliminated all gear pairs when simplifying dynamic model of gear transmission system. However, when all gear pairs are eliminated, the dynamic characteristics caused by gear meshing will not be shown, and definitely lead to unacceptable errors. Hence, one gear pair is reserved in this paper, and the mass/moment of inertia and vibration energy of other gear pairs can be equivalent to this gear pair. Gear pair in the middle position is chosen for the balance of mass/moment. The equivalent moment of inertia on gear is [25–28]:

$$J_{se}={\displaystyle {{\displaystyle \sum }}_{s=1}^n}{J}_s/i_s^2+{\displaystyle {{\displaystyle \sum }}_{k=1}^m}{J}_k.$$(1)

where J_s is the moment of inertia of the sth gear; i_s is transmission ratio of the sth parallel transmission stage; J_k is the moment of inertia of idler gears. “sth” is the sth gear in a paralleled stage.

Equivalent mass m_se is the sum of all gears which are equivalent to chosen gear:

$$m_{se}={\displaystyle {{\displaystyle \sum }}_{k=1}^m}{m}_k.$$(2)

Fig. 2 Simplified model of a n-stages parallel shaft gear transmission.

2.2 Equivalent vibration energy of gear pairs

When a gear pair runs, vibration will be caused because of stiffness and damping of gear tooth. If the stiffness is unchanged, or there is no clearance between gear tooth, vibration will be linear, otherwise will be nonlinear. These vibrations make up the excitation source of gear transmission system, whether they are linear or nonlinear. When the vibrations caused by gear meshing occur, they will transfer in gear system along gears, bearings, shafts, and gearbox housing, along with the vibration energy attenuation caused by contact interfaces. Yimin Shao et al. [29] conducted deep and detailed research on vibration energy loss in gear transmission system. In references [15], according to the characteristics of the interfaces between gears, shafts, bearings, bearing seats, and metal plates, the interfaces for transmitting vibration energy can be classified into four categories: impact interface, interference fit interface, bearing multi-interface, continuous medium interface. The relationship between the peak energy loss and RMS value on the transmission interface can be summarized in the following form:

$$E_o=\left[\text{a},\text{b},\text{c}\right]{\left[E_i^2,E_i,1\right]}^T.$$(3)

Here, [a, b, c] is coefficient matrix related to interface structure and coordination; $\left[E_i^2,E_i,1\right]$ is interface input energy matrix.

In references [29], a table of energy relationship coefficients on multiple interfaces was obtained through experiments (Tab. 1). The coefficient matrix of the energy relationship coefficient matrix for the corresponding type of interface in practical applications can be found in Table 1.

The vibration of a gear pair is not only composed of itself, but also includes the vibration of other gear pairs after transmission loss. Hence, when a gear transmission system is simplified (that means less gears, shafts and bearings), the vibration of removed gear pairs must be considered in simplified model. As mentioned above, one gear pair which can indicate gear vibration is reserved in this paper. Through changing the stiffness and damping of this gear pair, vibration energy can be changed in simplified model. Therefore, according to the simplify principle (2), the rule is: the vibration energy of the meshing pair after equivalence (using equivalent stiffness and damping) within one cycle should be equal to the sum of vibration energy when using the original meshing pair's stiffness and damping and the vibration energy transmitted by other meshing pairs to this meshing pair after attenuation. Formally, this rule can be expressed as follows:

$$E_e=E+\mathrm{\Delta }E=\frac{1}{2}{m}_{se}\left(x_{se}^2+y_{se}^2\right)+\frac{1}{2}{m}_{pe}\left(x_{pe}^2+y_{pe}^2\right)+\frac{1}{2}{J}_{se}\left({\theta }_{se}^2\right)+\frac{1}{2}{J}_{pe}\left({\theta }_{pe}^2\right)+\frac{1}{2}{k}_e\left(L_{sp}^2\right)$$(4)

where E is the origin energy of simplified gear pair; se is the equivalent parameter of parallel shaft gears; pe is the equivalent parameter of planetary gears; ΔE is the energy transmitted from other gear pairs to simplified gear pair; k_e is the equivalent stiffness parameter; L_sp is the relative displacement on direction of gear meshing.

It should be pointed out that, the equivalent energy of other gear pairs on the reserved gear pair is vibration energy, not all kinds of mechanical energy such as rotation of gears/shafts. From the perspective of energy conversion, the mechanical energy in gear transmission system is provided by electric. Stable operation without vibration will not be considered because this paper focuses on vibration.

2.3 Equivalent supporting stiffness, and deformation in bearing hole

2.3.1 Equivalent supporting stiffness

Supporting stiffness in gear transmission system is provided by bearings and gearbox housing. Previous papers mainly focused on bearings, but supporting stiffness provided by gearbox housing is less studied. In this paper, supporting stiffness of gearbox housing is considered, and the stiffness value can be calculated by finite element method (calculation process can be found in part 3-case study). According to the assembly relationship between bearings and gearbox housing, total stiffness is the series connection of bearing and gearbox housing, as shown in Figure 3. Total supporting stiffness of one bearing-gearbox housing connection is:

$$\frac{1}{k_{total}}=\frac{1}{k_b}+\frac{1}{k_g}$$(5)

where k_total is total supporting stiffness of one bearing-gearbox housing connection; k_b is stiffness of bearing; k_g is stiffness of gearbox housing.

It should be pointed out that the direction of supporting stiffness is not just vertical. Direction depends on degree of freedom of gear transmission system. Hence, supporting stiffness also exists in the horizonal direction.

Fig. 3 Dynamic model of supporting part in gear transmission system.

2.3.2 Equivalent deformation in bearing hole

When a gear transmission system with n gear pairs is equivalented to one gear pair, the number of bearing holes decreases with the reduction of gear pair, as shown in Figure 2. As a result, the number/position of support point (center of bearing hole) of the gearbox housing to gear transmission system reduces, and the deformation of supporting point changes accordingly. For example, Figure 4 shows the mechanical structure of gearbox housing on one side after simplification of dynamic model. It can be seen that the number of support point decreases from 2 to 1. Supporting stiffness on equivalent bearing hole position is the sum of supporting stiffness on each original bearing hole position, as shown in equation (6). As mentioned in Section 2.3.1, stiffness on each original bearing hole position can be calculated by equation (5). Therefore,

$$k_{bpe}=k_{bp1}+k_{bp2}$$(6)

where k_bpe is supporting stiffness on equivalent bearing hole position; k_bp1 and k_bp2 are supporting stiffness on each original bearing hole position.

Similarly, deformation of gearbox housing at the original 2 bearing holes in the parallel shaft stage also needs to be equivalent to 1 bearing hole. The principle of equivalence is the deformation coordination condition (as shown in Fig. 4), which assumes that the parallel axis part of the gearbox housing is a rigid body (in order to achieve equivalence, this will lose some accuracy), and the deformation of 2 bearing holes and the deformation of 1 equivalent bearing hole satisfy a certain quantity relationship:

$$y_{bpe}=\left(y_{bp1}\times l_2+y_{bp2}\times l_1\right)/\left(l_1+l_2\right).$$(7)

Here, y_bpe is the deformation of equivalent bearing hole; y_bp1 and y_bp2 are deformation of original bearing hole.

It should be mentioned that equations (6) and (7) are applicable for all situations, not just the example in this paper. When the number of original bearing holes is n (n > 2), two adjacent bearing holes of equivalent bearing holes will be chosen for calculating y_bpe. The value of y_bp1 and y_bp2 can be calculated by finite element model of gearbox housing.

Fig. 4 Deformation coordination relationship between bearing holes on one side of gearbox housing.

3 Case study

3.1 Theoretical dynamic model

Case study is conducted for validating proposing simplified dynamic modeling methodology for parallel shaft gear transmission system. In this paper, cutting unit of MG500-1180/WD coal mining machine is chosen as the research object since its gear transmission system contains multiple parallel shaft transmission stages and idle gears, as shown in Figure 5. Gear transmission system consists of one electric motor, three-stage parallel shaft gear transmission (totally 7 gears), two idle gears, two-stage planetary gear transmission(totally 11 gears), and a central gear. The electric motor is connected to the parallel shaft gear system through a torque shaft; The parallel shaft gear system is connected to the planetary gear system through a transmission shaft. The planetary gear system is connected to the drum through a transmission shaft. Parameters of gears, shaft, bearings in dynamic model are shown in references [30].

Three theoretical dynamic models of gear transmission system in cutting unit of MG500-1180/WD coal mining machine are established for validating the accuracy and computational efficiency of the proposed methodology in this paper: (1) Complete dynamic model, as shown in Figure 6. In this model, all the transmission components are reserved for accurate theoretical results. (2) Simplified dynamic model using methodology in references [10], as shown in Figure 7. Liu et al. [10] equated PSGS to motor shaft or connecting shaft, and angular displacements are recalculated depending on gear transmission ratios. (3) Dynamic model using the methodology in this paper, as shown in Figure 8.

Due to limitations on space, only some representative equations are proposed in this paper. Translation in direction x and y, and rotation around the axis of gears are chosen as the degrees of freedom in dynamic model.

(1) Planetary gear transmission

Relative displacements in sun-planet and planet-ring gears are:

$$L_{sp}={\theta }_s{R}_s+{\theta }_p{R}_p+\left(x_s-x_p\right)\sin \left(\phi +\alpha \right)+\left(y_s-y_p\right)\cos \left(\phi +\alpha \right)-e_{sp}\left(t\right)$$(8)

$$L_{rp}={\theta }_r{R}_r-{\theta }_p{R}_p+\left(-x_r-x_p\right)\sin \left(\phi -\alpha \right)+\left(y_{ri}-y_p\right)\cos \left(\phi -\alpha \right)+e_{rp}\left(t\right).$$(9)

Here, θ, φ and α are torsional vibration degrees of freedom, planetary gear position angle, and pressure angle, respectively. R, x and y are the base radius of the gear and the forced vibration response in the x and y directions. Here, s p r represents the sun gear, planetary gear, and ring gear. sp and rp respectively represent the meshing of the sun-planet and planet-ring gear pairs.

The meshing force in sun-planet gear pairs and planet-ring gear pairs can be expressed as:

$$F_{spij}=k_{spij}{L}_{spij}+c_{spij}{{\displaystyle \dot{L}}}_{spij}$$(10)

$$F_{rpij}=k_{rpij}{L}_{spij}+c_{rpij}{{\displaystyle \dot{L}}}_{rpij}.$$(11)

Lateral (x and y direction) force and torque on connecting shafts are:

$$T_{cs}=k_{cs}\left({\theta }_c-{\theta }_s\right)+c_{cs}\left({\displaystyle \dot{{\theta }_c}}-{\displaystyle \dot{{\theta }_s}}\right)$$(12)

$$F_{xcs}=k_{xcs}\left(x_c-x_s\right)+c_{xcs}\left({\displaystyle \dot{x_c}}-{\displaystyle \dot{x_s}}\right)$$(13)

$$F_{ycs}=k_{ycs}\left(y_c-y_s\right)+c_{ycs}\left({\displaystyle \dot{y_c}}-{\displaystyle \dot{y_s}}\right).$$(14)

Here, cs is the connecting shaft; c, s are the connected components by shafts. It should be mentioned that the carrier rotates around the center of the sun gear, so the lateral force of the connecting axis between the carrier and the planetary gear needs to be projected in the x and y directions.

(2) Parallel shaft gear transmission

Dynamic model of parallel shaft gear transmission is only used in complete dynamic model since the dynamic model proposed by references [10] have eliminated parallel shaft gear transmission, and the dynamic model proposed by this paper has proposed in Section 2.

Relative displacements in parallel shaft gear transmission are:

$$L_{i\left(i+1\right)}={\theta }_i{R}_i+{\theta }_{\left(i+1\right)}{R}_{\left(i+1\right)}+\left(x_i-x_{\left(i+1\right)}\right)\sin \left(\alpha \right)+\left(y_i-y_{\left(i+1\right)}\right)\cos \left(\alpha \right)-e_{i\left(i+1\right)}\left(t\right)$$(15)

where i represents the ith parallel shaft gear;

Supporting forces are:

$$F_{xs}=k_{xs}\left(x-x_s\right)+c_{xcs}\left({\displaystyle \dot{x}}-{{\displaystyle \dot{x}}}_s\right)$$(16)

$$F_{ys}=k_{ys}\left(y-y_s\right)+c_{ycs}\left({\displaystyle \dot{y}}-{{\displaystyle \dot{y}}}_s\right).$$(17)

Lateral (x and y direction) force and torque on connecting shafts between motor and gear transmission system are:

$$T_{mg}=k_{mg}\left({\theta }_m-{\theta }_g\right)+c_{mg}\left({{\displaystyle \dot{\theta }}}_m-{{\displaystyle \dot{\theta }}}_g\right)$$(18)

$$F_{mg}=k_{mg}\left(x_m-x_g\right)+c_{xmg}\left({{\displaystyle \dot{x}}}_m-{{\displaystyle \dot{x}}}_g\right).$$(19)

$$F_{mg}=k_{mg}\left(y_m-y_g\right)+c_{ymg}\left({{\displaystyle \dot{y}}}_m-{{\displaystyle \dot{y}}}_g\right).$$(20)

The meshing force in gear pairs can be expressed as:

$$F_{i\left(i+1\right)}=k_{i\left(i+1\right)}{L}_{i\left(i+1\right)}+c_{i\left(i+1\right)}{{\displaystyle \dot{L}}}_{i\left(i+1\right)}.$$(21)

Here, i, (i + 1) means the number of gears, and the nearby gear; subscript “mg” means motor-gear transmission system; Subscripts “ys”, “xs” mean supporting force in y and x directions of the sth parallel gear transmission.

Bearing stiffness, which is a part of supporting stiffness to gear transmission system, is calculated based on the method in references [11].

For obtaining the support stiffness provided by gearbox housing, finite element model for the gearbox housing of MG500-1180/WD is established, as shown in Figure 9. There are totally 18293 Solid 185 units and 329,800 nodes in this model. Grid adopts ANSYS software automatic partitioning technology. According to the actual stress state of the box during operation, fixed constraints are set at the hinge ears, and the load is applied to the bearing holes connecting the box and the drum. Young's modulus of elasticity, yield limit, Poisson's ratio, and density are 200 GPa, 300 MPa, 0.3, and 7800 kg/m³, respectively.

By applying a load to the finite element model of the cutting gear box, the lateral deformation (i.e. x and y directions) of the bearing housing hole was obtained. According to the stiffness calculation formula of the bearing housing hole based on the finite element model provided in references [13], the radial stiffness of the bearing housing hole for each transmission stage is shown in Table 2.

Fig. 5 Gear transmission system in cutting unit of MG500-1180/WD coal mining machine.

Fig. 6 Complete dynamic model of gear transmission system in cutting unit of MG500-1180/WD coal mining machine.

Fig. 7 Simplified dynamic model using methodology in references [10].

Fig. 8 Simplified dynamic model using methodology in this paper.

Fig. 9 Finite element model for the gearbox housing of MG500-1180/WD.

3.2 Experimental validation

For a more accurate validation to theoretical model, MG500-1180/WD coal mining machine which is identical to the object in case study is chosen as the study object in dynamics experiment. This experiment relies on the comprehensive mining work interview and testing platform of the National Energy Mining Equipment Research and Development Test Center, completely utilizing the existing testing instruments and equipment of Northeastern University to measure the inherent characteristics of the coal mining machine, as well as the dynamic response under no-load and heavy-duty conditions. The National Energy Mining Equipment Research and Development Testing Center (as shown in Fig. 10) is located in Zhangjiakou City, Hebei Province. It is the world's first coal mining machinery testing center independently developed and invested in by China.

This test platform is composed of comprehensive mining equipment and simulated coal walls. Comprehensive mining equipment includes: coal mining machine, scraper conveyor, hydraulic support, transfer machine, etc. Among them, the coal mining machine is the MG500-1180/WD double mining machine produced by Xi'an Coal Mining Machinery Co., Ltd., which is also the research object of the theoretical dynamic model in this article. Its parameters are shown in Table 3; The scraper conveyor is SGZ1000/1050 of middling coal Group Zhangjiakou Coal Mining Machinery Co., Ltd. The scraper machine on the test bench has a length of 73 m, a scraper chain movement speed of 1.25 m/s, and a rated voltage of 3300 V; Hydraulic support is a ZY9000/15/28D support shield hydraulic support produced by Beijing Coal Mining Machinery Company. Its support height is 1.5–2.8 m, with a center distance of 1.75 m between frames and a maximum displacement distance of 800 mm. Table 3 shows some main parameters of MG500-1180/WD.

Based on actual working conditions of the coal mining machine in coal mines, the working conditions which are chosen for the comparison between theoretical and experimental results are as follows: no-load operating and straight line cutting. The torque on drum of coal mining machine when cutting coal can be calculated as follows:

$$h_z=h_{max}\left(1-\text{cos}\left({\phi }_b\right)\right)/{\phi }_b$$(22)

$$h_{max}=100v_q/S_p\omega$$(23)

$$Z_i=A{h}_z\left(0.3+350b_p\right)$$(24)

$$A=150f$$(25)

$$T_d={\displaystyle {{\displaystyle \sum }}_{i=1}^N}{Z}_i{D}_i/2.$$(26)

Here, h_z, f, A, D_i, T_d, N, S_p, ω, φ_b are average cutting thickness, coal rock hardness, cutting impedance, drum diameter, drum load torque, number of cutting teeth simultaneously participating in cutting, number of cutting teeth on each cutting line, drum speed and surrounding angle, respectively.

Four three-direction acceleration sensors are used for obtaining vibration data of the gearbox, and they are put on gearbox housing nearby parallel gear transmission stages, as shown in Figure 11. Signal processing system can collect the data obtained by sensors and transmits data to computer, as shown in Figure 12.

The testing process under no-load conditions is relatively simple. Start the traction motor and cutting motor, and the drum will idle without cutting coal. The coal mining machine relies on the scraper to move back and forth along the coal mining face. Figure 13 shows the operating sketch map of coal mining machine in experiment in straight line cutting. The coal mining machine moves from left to right with the cutting depth from 0 to 600 mm for the objective of experimental research.

Fig. 10 National Energy Mining Equipment Research and Development Test Center.

Fig. 11 Locations of the sensors.

Fig. 12 Signal processing system.

Fig. 13 Operating sketch map of coal mining machine in experiment.

4 Results and discussion

4.1 Results of theoretical models and experiments and comparison

Acceleration results obtained from vibration experimentation are shown in Figure 14. Based on the models in Section 3.1, theoretical results of PSGS computed by complete dynamic model, dynamic model proposed by references [10] and model proposed by this paper are obtained for comparison. Vertical and horizonal vibration deformation results of 1st sun gear (Fig. 6) obtained by both theoretical dynamic models and experimentation are selected for comparison because the origin gears in parallel shaft gear transmission system are eliminated by simplified models. Here, the vibration results on the rotational direction will not be shown in comparison analysis since the sensors put on gearbox housing can't measure these vibrations. Cutting depth of coal mining machine here ranges from 0 to 600 mm based on actual working conditions. Cutting depth can determine loads of PSGS, and the loads can be computed from cutting depth with the help of equations in Section 3.2. In the following discussion, the direction of horizontal, vertical and axial are referred to as x, y and z directions, respectively.

Figure 15 shows vibration displacement of sun gear 1 and sun gear 2 on the directions of rotational vertical, and horizontal computed by dynamic model in this paper with 600 mm cutting depth. Through further analyzing the vibration data, it can be found that the maximum vibration displacement of sun gear 1 on x and y directions are 0.75 mm and 0.45 mm, respectively. The maximum vibration displacement of sun gear 2 on x and y directions are 0.70 mm and 0.38 mm, respectively. Vibration displacement values of sun gear 1 are larger than that of sun gear 2 in identical direction. This is because mass and moment of inertia of gears in the second planetary transmission are larger than that of the first planetary transmission, and the rotational speed of gears decreases from the first to the second planetary transmission. It can also be seen from the pictures that X vibration displacement is larger than Y in the same sun gear. Because coal mining machine is moving on scraper conveyor in x direction. Hence, this moving direction will cause additional cutting loads on PSGS in x direction. Vertical load of PSGS is caused by the coal cutting of cutting picks installed on the rotating drum.

Figure 16 shows the vibration displacement of sun gear 1 on x and y directions computed by theoretical models and experimentation, including complete dynamic model, simplified dynamic model proposed by references [10], simplified dynamic model proposed by this paper, and experiment. Here, choosing only one gear as the comparison object in this paper is because when the PSGS is simplified, the structure of parallel shaft gear transmission system is changed. All of the gears are eliminated by only one mass in references [10] simplified dynamic model. Therefore, vibration results obtained by sensors nearby parallel shaft gears can't be used in the comparison. In addition, sensors are fixed on the gearbox housing, not on gears, vibration displacements on rotational direction can't be obtained by these sensors. Therefore, rotational vibration displacements are not concluded in the discussion here. It can be obviously seen in the pictures that complete dynamic model has the closest results to experimentation with the maximum error 0.05 mm in x direction and 0.05 mm in y direction. references [10] simplified dynamic model has the largest error compared with experiments with the maximum error 0.3 mm in x direction and 0.19 mm in y direction. The error of dynamic model proposed by this paper is between complete dynamic model and references [10] dynamic model with the maximum error 0.1 mm in x direction and 0.09 mm in y direction. Complete dynamic model is the most accurate since it considers the all the transmission components without any simplification in establishing mathematical model. Complete dynamic model is the least accurate since it eliminates all the transmission components and these components are just simplified as one mass when establishing mathematical model. However, Complete dynamic model which has 66 degrees of freedom is so time-consuming with the average time cost more than 25 hours in author's computer which has Intel Core i7 CPU. In contrast, dynamic model proposed by this paper has only 42 degrees of freedom with the average time cost less than 8 hours in the same computer. Therefore, considering both accuracy and computing efficiency, dynamic model proposed by this paper is the best model based on the results.

Fig. 14 Acceleration results obtained from experimentation.

Fig. 15 Vibration displacement computed by the model proposed by this paper.

Fig. 16 Comparison of theoretical and experimental deformation results of sun gear 1.

4.2 Further analysis

In further analysis, two key factors-cutting depth and wall thickness which can affect meshing forces in gear pairs are discussed. Cutting depth determines external loads of PSGS, and wall thickness determines the stiffness of gearbox housing and then affect supporting stiffness and displacement to PSGS. Meshing forces calculated by simplified dynamic model proposed by this paper of the first and second sun-planet gear pairs are analyzed in this part. Figure 17 shows the maximum meshing forces of the first and second sun-planet gear pairs with cutting depth range from 100 mm to 600 mm. It can be easily seen that meshing forces in both gear pairs increase with cutting depth, but the trends are a little different. Meshing force of the first sun-planet gear pair in 600 mm cutting depth increases by 15.8% compared with 100 mm cutting depth, and the value is 28.3% in the second sun-planet gear pair. This is because cutting loads cause the vibration deformation of gearbox housing, but the deformation values are different with the locations. According to the results calculated by finite element (FE) model, vibration deformation nearby second sun-planet gear pair is larger than that nearby first sun-planet gear pair. Hence, it will cause larger supporting vibration displacement to PSGS and then cause larger meshing forces when meshing stiffness of gear pairs are invariable. However, the relative increase value of second sun-planet gear pair is not too larger than the first due to the structure characteristics, as shown in Figure 1a. Different from gearbox housing nearby parallel shaft gear transmission with cantilever structure, external cutting loads can't cause that large deformation of gearbox housing nearby planetary gear transmission compared with gearbox housing nearby parallel shaft gear transmission. Therefore, supporting deformation caused by gearbox housing vibration to two planetary gear transmission are close to each other compared with parallel shaft gear transmission.

Figure 18 shows the maximum meshing forces of the first and second sun-planet gear pairs with wall thickness range from 10 mm to 50 mm. It can be seen that meshing forces in both gear pairs decrease with wall thickness, but the trends are a different. Meshing force of the first sun-planet gear pair in 50 mm wall thickness increases by 5% compared with 10 mm wall thickness, and the value is 6.5% in the second sun-planet gear pair. This is because when the wall thickness becomes thinner, stiffness of gearbox housing becomes smaller. Supporting stiffness of PSGS decreases since it is provided by gearbox housing. Vibration displacement of gearbox housing will increase due to the decrease of stiffness with invariable external cutting loads, and then cause the increase of vibration displacement on location of bearing holes. Therefore, this increase of vibration displacement will cause larger meshing forces with invariable meshing stiffness.

Fig. 17 Effect of cutting depth on gear meshing force.

Fig. 18 Effect of gearbox wall thickness on gear meshing force.

5 Conclusion

• A simplified dynamic modelling methodology for PSGS is proposed to find an equivalent simplified dynamic model. The model proposed in this paper is compared with experimental results, complete dynamic model, and references [10] dynamic model in case study of coal mining machine gear transmission system. Results show both the accuracy and computing efficiency of the methodology in this paper.

• Further discussions about the coupling dynamic characteristics between gearbox housing and gear transmission system are conducted in this paper. Results show that the second planetary gears are greater influenced by cutting loads and wall thickness due to larger deformation vibrations of gearbox housing which connected to gear transmission system with bearing holes.

Funding

This study was funded by Basic Research Project of Liaoning Provincial Department of Education for Universities (LJ212412899002).

Conflicts of interest

The authors declare that they have no conflict of interest.

Data availability statement

No dataset was generated or analyzed during this study.

Author contribution statement

Jue Wang is responsible for writing the main paper and establishing theoretical model and experiments. Hang Zhou is responsible for making images.

References

All Tables

All Figures

	Fig. 1 Parallel shaft gear transmission system.
In the text

	Fig. 2 Simplified model of a n-stages parallel shaft gear transmission.
In the text

	Fig. 3 Dynamic model of supporting part in gear transmission system.
In the text

	Fig. 4 Deformation coordination relationship between bearing holes on one side of gearbox housing.
In the text

	Fig. 5 Gear transmission system in cutting unit of MG500-1180/WD coal mining machine.
In the text

	Fig. 6 Complete dynamic model of gear transmission system in cutting unit of MG500-1180/WD coal mining machine.
In the text

	Fig. 7 Simplified dynamic model using methodology in references [10].
In the text

	Fig. 8 Simplified dynamic model using methodology in this paper.
In the text

	Fig. 9 Finite element model for the gearbox housing of MG500-1180/WD.
In the text

	Fig. 10 National Energy Mining Equipment Research and Development Test Center.
In the text

	Fig. 11 Locations of the sensors.
In the text

	Fig. 12 Signal processing system.
In the text

	Fig. 13 Operating sketch map of coal mining machine in experiment.
In the text

	Fig. 14 Acceleration results obtained from experimentation.
In the text

	Fig. 15 Vibration displacement computed by the model proposed by this paper.
In the text

	Fig. 16 Comparison of theoretical and experimental deformation results of sun gear 1.
In the text

	Fig. 17 Effect of cutting depth on gear meshing force.
In the text

	Fig. 18 Effect of gearbox wall thickness on gear meshing force.
In the text