Optimal Design of a Novel Magnetic Twisting Device based on NSGA-II Algorithm

The magnetic twisting device is mainly composed of the outer rotor component, inner rotor component, spindle shaft, drive device, and yarn-guiding hook. As shown in Figure 1, the outer rotor component is mainly composed of the outer baseplate and outer spindle cover. The outer baseplate has a radical yarn-guiding hole. The inner rotor component is mainly composed of a magnetic drive mechanism and an inner spindle cover. The magnetic drive mechanism consists of active disc assembly and passive disc assembly. The outer rotor component is driven by a dragon belt, and the inner rotor component is driven by a dragon belt and magnetic drive mechanism.

Figure 1

The magnetic twisting device is a complex rotating system, and its schematic diagram is shown in Figure 2. The spindle shaft is a slender shaft with a hole in the center. The upper part of the spindle shaft has a yarn-guiding hole and the lower part has an air hole. The spindle shaft must be straight, tough, and elastic. The spindle is made of bearing steel. The outer baseplate has radial holes. The outer baseplate, spindle shaft, and belt wheel are fixedly connected and driven by the dragon belt. The active disc assembly is driven by another dragon belt through the belt wheel. The inner rotor component and outer rotor component run in opposite directions. The yarn-guiding hook is fixed on the frame.

Figure 2

Twisting is one of the essential processes in the spinning process. Its purpose is to make the fiber or yarn obtain a certain structure, increase its density and cohesion, and make the fiber or yarn have a certain strength, elasticity, and other physical-mechanical properties and appearance. When one end of the fiber or yarn is held and the other end rotates around its own axis, the relative angular displacement occurs between the sections of fiber or yarn. When the relative angular displacement is a circle, the fiber or yarn obtain one twist. The twist has a direction, as shown in Figure 3.

Figure 3

When the A end of the yarn AB is held, and the B end rotates clockwise or anticlockwise around its axis, the yarn AB will obtain Z-twist or S-twist. Twist is the relative angular displacement at unit length fiber or yarn. The larger the relative angular displacement is, the larger the twist is. On the contrary, the smaller the twist is. The twist direction is the rotation direction of the relative angular displacement.

Twisting mechanism of a magnetic twisting device is as follows:

Yarn path in the twisting device:

Yarn package→Inner spindle cover→Central hole of the spindle shaft→Radial hole of outer baseplate→Outer spindle cover→Yarn guiding-hook→Winding mechanism

As shown in Figure 2. The yarn package is placed on the passive disc assembly. The yarn unwinds from the point O of the package and enters the central hole of the spindle shaft along the inner spindle cover AB through point C at the end of the spindle. Then, it enters the radial hole DE of the outer baseplate and enters the winding mechanism along the outer spindle cover EF through the points F and G in the yarn-guiding hooks [15]. Specifying the rotation speed of the inner and outer rotor components as n (r/min), and the inner rotor component runs counterclockwise while the outer component runs clockwise. The output speed of the yarn is ν (m/min). The yarn AB runs anticlockwise at speed n with the inner rotor component, and the yarn DG runs clockwise at speed n with the outer rotor component. Obviously, point C is the first twisting point, point D is the second twisting point, and point G is the third twisting point [16]. The yarn obtains twist as follows by the twisting device [17]: (1) Twist of yarn AC: T1=nv {\rm{Twist}}\,{\rm{of}}\,{\rm{yarn}}\,{\rm{AC}}:\,{T_1} = {n \over v} (2) Twist of yarn CD: T2=2nv=2T1 {\rm{Twist}}\,{\rm{of}}\,{\rm{yarn}}\,{\rm{CD}}:\,{T_2} = {{2n} \over v} = 2{T_1} (3) Twist of yarn DG: T3=3nv=3T1 {\rm{Twist}}\,{\rm{of}}\,{\rm{yarn}}\,{\rm{DG}}:\,{T_3} = {{3n} \over v} = 3{T_1} Where: T1, T2, T3 – yarn twist, twist/m.

The final twist of the yarn is 3 n/v and the twist direction is Z-twist through the magnetic twisting device. The yarn can obtain S-twist by changing the rotation direction of the inner and outer rotor components.

The magnetic torque calculation is an important issue in the designs of the magnetic drive mechanism, which directly affects the performance of the magnetic drive mechanism. There are various methods of calculating magnetic torque, such as empirical method, numerical calculation method, and analytical method.

The analytical method of calculating magnetic torque has been discussed based on three main theories, including the equivalent magnetic charge theory [18,19,20], the equivalent current theory [21,22,23], and the magnetic energy theory [24, 25]. The mathematical formula of magnetic torque calculation based on the equivalent magnetic charge theory is a quadruple integral function on the structural parameters [26]. The mathematical formula of magnetic torque calculation based on the equivalent current theory is a double integral sum function on the structural parameters. The mathematical formulas of magnetic torque calculation are complex multi-integral functions on the structural parameters based on the equivalent current theory and the equivalent current theory. Furthermore, some parameters need correction by experiments before they can be applied to engineering practices. The finite element method was used in the analysis of the magnetic drive torque problems of 24-pole Halbach magnetic coupling [27]. The magnetic theory was used in magnetic torque calculation of the motor and permanent magnet coupling. A calculation formula for magnetic torque based on the magnetic energy theory was presented [28], in which the physical meaning of parameters is more intuitive and clearer, easy to understand, and facilitates the engineering practical application.

The mathematical model of magnetic torque calculation in this paper is based on the magnetic energy theory. The inner rotor of the magnetic twisting device bears the weight of the yarn package. One of the key performances of the design is that the magnetic torque of the magnetic drive meets the load torque. According to the magnetic energy theory, the calculation formula of magnetic torque of magnetic drive mechanism is as follows. The formula [25, 28] depends directly on the geometrical and physical parameters of the magnetic drive mechanism: (4) T=12gVm4μ0gBr2gKfKrgr2Lm sin θLg2+(r sinθ2)2 g(Lg2+(r sinθ2)2+KfKrgLm)2 T = {1 \over 2}g{{{V_m}} \over {4{\mu _0}}}g{B_r}^2g{{{K_f}} \over {{K_r}}}g{{{r^2}{L_m}\,\sin \,\theta } \over {\sqrt {{L_g}^2 + {{\left( {r\,\sin {\theta \over 2}} \right)}^2}\,g{{\left( {\sqrt {{L_g}^2 + {{\left( {r\,\sin {\theta \over 2}} \right)}^2}} + {{{K_f}} \over {{K_r}}}g{L_m}} \right)}^2}} }} (5) r=R1+R22 r = {{{R_1} + {R_2}} \over 2} Where T is magnetic torque, N·m, μ₀ is the permeability of vacuum, μ₀ = 4π × 10⁻⁷, H/m; V_m is the permanent magnets volume, m³; B_r is the remanence of the permanent magnet, T, K_f is the magnetic flux leakage coefficient, K_r is the magnetic-reluctance coefficient, L_m is the thickness of the permanent magnet, m; L_g is the gap length, m; R₁ is the inner radius of the magnets, m. R₂ is the outer radius of the magnets, m. r is the mean radius of the magnets, m. θ is the relative angle, °;

The main parameters that affect the magnetic torque are as follows: the relative angle, the working gap, and the magnetic flux leakage coefficient. There is a complex coupling relationship between them and magnetic torque. To analyze the effect of the parameters on magnetic torque, a preliminary model is used for simulation analysis. The parameters of the magnetic drive mechanism are shown in Table 1.

In the design parameters of the magnetic twisting device, L_g, K_f and K_r change monotonically, while θ, L_m and r change non monotonically. Therefore, when the system performance is optimized, θ, L_m and r are selected as the optimization parameters.

Taking the maximum transmission torque and the minimum magnets volume as the optimization objectives [29], the optimization model is as follows: (6) Vm=2π⋅(R22−R12)⋅Lm {V_m} = 2\pi \cdot \left( {{R_2}^2 - {R_1}^2} \right) \cdot {L_m} (7) T=12·Vm4μ0·Br2·Kf(C)Kr(C)·r2Lm sin θLg2+(r sinθ2)2·(Lg2+(r sinθ2)2+Kf(C)Kr(C)·Lm)2 T = {1 \over 2} \cdot {{{V_m}} \over {4{\mu _0}}} \cdot {B_r}^2 \cdot {{{K_f}^{(C)}} \over {{K_r}^{(C)}}} \cdot {{{r^2}{L_m}\,\sin \,\theta } \over {\sqrt {{L_g}^2 + {{\left( {r\,\sin {\theta \over 2}} \right)}^2}} \cdot {{\left( {\sqrt {{L_g}^2 + {{\left( {r\,\sin {\theta \over 2}} \right)}^2}} + {{{K_f}^{(C)}} \over {{K_r}^{(C)}}} \cdot {L_m}} \right)}^2}}} (8) min f(x)=min[f1(x), f2(x)]=min[Vm,−T] \min \,f\left( x \right) = \min \left[ {{f_1}\left( x \right),\,{f_2}\left( x \right)} \right] = \min \left[ {{V_m}, - T} \right]

The design variable of the model is: X=[Lm Lg R2 R1 θ]T X = {\left[ {{L_m}\,{L_g}\,{R_2}\,{R_1}\,\theta } \right]^{\rm{T}}} subject.to.{r=(R1+R2)/20.003≤Lm≤0.0150.012≤Lg≤0.020.025≤R1≤R2≤0.0480≤θ≤45Vm=2π×(R22−R12)×Lm {\rm{subject}}.{\rm{to}}.\left\{ {\matrix{ {r = \left( {{R_1} + {R_2}} \right)/2} \cr {0.003 \le {L_m} \le 0.015} \cr {0.012 \le {L_g} \le 0.02} \cr {0.025 \le {R_1} \le {R_2} \le 0.048} \cr {0 \le \theta \le 45} \cr {{V_{\rm{m}}} = 2\pi \times \left( {{R_2}^2 - {R_1}^2} \right) \times {L_{\rm{m}}}} \cr } } \right.

The NSGA-II algorithm is used to solve the double-objective optimization model of the magnetic twisting device. The optimization objectives, parameters, and constraints are determined to generate the initial population, and the optimal solution is found by continuous selection, crossover, variation, and non-dominated sorting. The algorithm can obtain multiple Pareto optimal solutions, and the designer can choose the final satisfactory solution according to the actual requirements of the system.

The result of NSGA-II optimization for the important parameters of a magnetic twisting device is a Pareto frontier diagram, as shown in Figure 5.

Figure 5

According to the requirement of magnetic torque, the design parameters can be selected from Figure 5, which lists the Pareto optimal solutions and the values of the corresponding parameters.

The key data of the main parts are as follows:

The spindle shaft is made of GCr15 (bearing steel), with a diameter of 10 mm and a length of 428 mm. The diameter of the center hole is 4 mm and the length is 300 mm, and the diameter of the air hole at the lower end is 5 mm and the length is 128 mm.

The outer baseplate material is made of Fe–Cr–Al Alloy with conductivity 6.25 x105 S/m. The diameter and thickness of the outer baseplate are 150 mm and 10 mm respectively.

The main material of the magnetic drive mechanism are listed below: Magnet material: N35SH; Magnet frame material: aluminum alloy; Yoke material: low carbon steel 20, μ_r = 1 x 10⁵.

The prototype of the magnetic twisting device is shown in Figure 6. The working gap is 0.012 m and the magnetic torque transmitted by the magnetic drive device is 3.28 N·m. According to the calculation model of magnetic torque, magnetic flux leakage coefficient K_f has an obvious influence on the magnetic torque. Figure 7 shows the trend of the calculated maximum torque and its relative error with the measured value under different magnetic flux leakage coefficient K_f.

Figure 6

Figure 7

Table 3 shows the calculated maximum torque and its relative error with the measured value under different magnetic flux leakage coefficient K_f.

It can be seen from Figure 7 and Table 3 that the relative error is the minimum when the magnetic flux leakage coefficient K_f is near 4. This result is consistent with the experimental experience: when the magnets arrangement form is the push-pull arrangement, the magnetic flux leakage coefficient is 4 [30].

The magnetic twisting device is a novel twisting device. Compared with the existing twisting device, the process performance is improved. The comparison index is shown in Table 4.