A Robust Optimization Method on the Transmission System of the Cutting Unit of Shearer


 The transmission system of the cutting unit of shearer is divided into three basic components: planetary reduction form, one gear on one shaft form and a double gears on one shaft. The dynamic differential equations of three basic components are established respectively, and the volume functions of each structure are obtained. The characteristics of the internal excitation of the transmission system are analyzed, and the solution methods of the motion parameters of each component are obtained based on the harmonic balance method. Taking the parameters such as tooth number, modulus and tooth width as optimized variables, and a robust optimization method with the minimum value of multi-parameter evaluation function weighted linearly by dimensionless volume and vibration for the transmission system of the cutting unit of shearer is presented. Taking a certain type of shearer as an example, the transmission system of the cutting unit is optimized by using the presented method. After the design, the size is reduced by 5.4%, the maximum torsional acceleration of the drum is reduced by 17.8%, and the maximum torsional acceleration of the first gear is reduced by 9.6%. The results show that the design method can reduce the manufacturing cost of shearer and reduce the failure rate of the cutting unit.

at the end of the drive side bearing installation position and around the triangular column, the . 52 excitation, the gear transmission system will produce greater vibration, which accelerates the 66 transmission mechanism to be damaged and affects the normal operation of the whole shearer 67 [17]. From the above research results, it can be seen that taking volume as the objective function 68 is an effective method to reduce the size of the cutting unit, and the high frequency vibration of 69 the cutting unit is closely related to the structure of the transmission system itself. It is of great 70 significance to reduce the volume of the cutting unit and the fault caused by high frequency 71 vibration for the stable operation of shearer under the mine. Therefore, it is worth studying to 72 optimize the design of the transmission of the cutting unit system to reduce its volume and high 73 frequency vibration. 74 The method of robust design was put forward by Japanese engineering expert Taguchi G. in 75 1970s [17]. The basic idea of robust design is to minimize the influence of some factors on 76 product quality without eliminating and reducing the factors related to the product. After the idea 77 of robust design was put forward, many scholars soon applied the concept of robust design to the 78 design of machinery and equipment [2, 3, 6, 18, 21,]. In the above researches, the analysis and 79 optimal design of the transmission system of the cutting unit are generally carried out under a 80 certain load, and most designers adopt conservative load as the design criterion. Not only is the 81 result likely to be a large horse pulling a small car, but it may also be uncertain whether the result 82 will meet demand as the load changes. Robust optimization can solve these problems. 83 84 At present, the objective function of the optimal design of transmission system of is 85 relatively single: the single mass is the smallest or the similar volume is the smallest as the 86 objective function. The vibration of shearer has a direct and important influence on the failure of 87 transmission system and even the whole machine. It is also even more important to reduce the 88 faults caused by vibration when the shearer works. Since the vibration intensity is directly 89 proportional to the acceleration in the case of high-frequency vibration (over 1000HZ), 90 acceleration is adopted in this study to represent the vibration quantity. At the same time, in 91 order to reduce the manufacturing and transportation cost of shearer, the volume of shearer 92 should be reduced as much as possible. Therefore, the contribution of this paper is presenting an 93 optimal design method for the transmission system of the cutting unit of shearer by taking the 94 optimal linearly weighted minimum value of dimensionless vibration and volume as the 95 objective function and the changing load as the external excitation. 8 118 Figure 1. Relationship diagram between product quality and design factors 119 In figure 1, the set of controllable factors is all the design variables, that is,

Scope and contribution of this study
The value used in product design is nominal, and its collection is compared to design 122 variables as follows: There is a deviation between the nominal value and the actual variable in machining and 125 manufacturing, so the maximum allowable value-tolerance T x ∆ of the deviation can be used to 126 limit the range of variation.
It can be seen that in order to design the required functional quality products, it is not only 129 necessary to determine the nominal value n i x R ∈ , but also to determine the range of 130 tolerance n i x R ∆ ∈ , the type of probability distribution and the distribution parameters.
Generally, T z is a random variable that obeys a certain probability distribution, and its 134 fluctuation range may also change with time and space. 135 As can be seen from Fig.1 The relationship between product design and functional factors shown in Fig. 1 According to the results of the above equations, it can be seen that the variation of the 149 design results is as follows: It can be seen from Eq. (8) that the change of the design results of the nonlinear system is 152 not only the result of the change of a certain design parameter, but also related to the change of 153 other parameters caused by the change of the parameter. The mathematical expectation and 154 variance of the statistical parameters of the design results are as follows: means, respectively, that the smaller the hope is, the better (but not negative); the larger the hope 175 is, the better (infinity is the best); and the smaller the hope is, the better. When the design variable is taken x 2 , the calculated value of the objective function is larger than 187 that when the design variable is taken x 1 . However, in the same tolerance range A of design 188 variables, the variation of the corresponding objective function value is smaller the design 189 variable is taken x 2 . That is, the objective function is more robust at The mathematical model used for general optimization problems is as follows [5]: 198 ( ) 0 ( 1, 2, , ) In summary, robust optimization is the intersection of robust design and optimal design, so the 201 mathematical model of robust optimization can be expressed as follows: It can be seen that the mathematical method of solving design variables by robust 213 optimization described above belongs to a two-level, multi-objective optimization process.  215 Taking some shearer as an example, the structure diagram of the transmission system of the 216 cutting unit is shown in figure 3. In this transmission system, each shafting is a high-speed motor 217 spindle rotor-bearing system supported by rolling bearings. The function of each shafting in the 218 shafting is to transfer the energy from the motor to the drum through the torque of rotation, and 219 then to complete the coal cutting work. Therefore, the dynamic equation of the forced rotor-220 bearing system can be used to describe the transmission process of the rotor-bearing system as 221 follows:  Dynamic differential equations of solar wheels s:

DYNAMIC MODEL OF TRANSMISSION SYSTEM
Dynamic differential equations of planet wheel p: Dynamic differential equations for planetary framework c: where, R bi is the radius of the gear base circle, R bc is the radius of rotation of the planet frame, F spi 252 is the meshing force between the solar wheel and the planet wheel, F rpi is the meshing force 253 between the planet gear and the inner gear ring, wheel, α is the pressure angle. The following symbolic meaning is the same as this kind. 257 The volume of the single-stage planet reducer shown in Fig.4 (a) can be expressed as 258 follows: where, xx m is the modulus of meshing gears in gear train, Z s is the number of teeth of the sun 261 wheel, b xx is tooth width, i is the transmission ratio of Planet reducer. The following symbolic 262 meaning is the same as this kind. In Fig. 4 (b), if gear 1 represents the input end and its input torque is T in , gear 3 represents 265 the output end and its the output torque is T out , the dynamic differential equation of the 266 transmission system in this transmission form is as follows: The volume of one gear on one shaft reducer shown in Fig. 4 where, Z 1 and Z 3 are not independent variables, they are related to transmission ratio i, i= Z 3 /Z 1 .
The volume of a double gears on one shaft reducer shown in Fig. 4 where, Z 4 , Z 5 ,Z 6 and Z 7 are related to transmission ratio i, where, m K is the average meshing stiffness, a K is the amplitude of gear meshing fluctuation, e ω 294 is gear meshing frequency, r θ is the initial phase angle. 295 The relationship between the meshing force between gear pairs and gear stiffness in each

DESIGN ANALYSIS AND SOLUTION METHOD OF ROBUST OPTIMIZATION
where, E spi and E rpi are the amplitudes of the comprehensive tooth frequency error of the gear 299 pair respectively, spi φ and rpi φ are the initial phases of the two, respectively. 300 The damping parameters in each dynamic differential equation can be obtained from Eq.

301
(26), as follows: 302 , , where, K is the average meshing stiffness of gear pairs, g m is the total mass of the gear on the 304 sun wheel mounting shaft, eq m is the equivalent mass on the radius of the base circle, damping 305 ratio ζ can be found in the manual.

318
According to the basic principle of harmonic balance method [11], it is assumed that the 319 dynamic harmonic response of each basic unit in section 4 is as follows:     3  11  11  4  5  6  7  12  12   11  12  13  14  15  16  17  18  21  22  23  24  25  31  32  33  34  35  36   , , , ,  ,  , , , , , , , , , , , , , ,  Equations to the corresponding value at the average cutting resistance moment can be used as the 391 optimized term, that is, The value of the evaluation function obtained by the linearly reweighting of the above 399 Equations is taken as the final objective function, that is,   It can be seen from Table 1 493 We all declare that we have no conflict of interest in this paper. 494