Improvement of the slotless tubular permanent magnet linear motor with lighter mover mass for punching applications

Fundamental Research Funds for the Central Universities, Grant/Award Number: 2019CDXYDQ0010; National undergraduate innovation and entrepreneurship training program, Grant/Award Number: S201910611325 Abstract The slotless tubular permanent magnet linear motor (STPMLM) offers obvious advantages when it is applied to the punching system, such as space saving and controllable impact strength. Herein, the mechanical and electromagnetic coupled differential equations were established to present a comprehensive analysis of the relationship between mover mass and motor performance in the punching system. The travel time of the punching movement was expressed analytically under the condition of the mover obtaining the same impact kinetic energy, and the relationship between the mover mass and the resistive energy consumption in one punching process was also deduced under the same condition. Then the analytical results were validated by a numerical model. The studies show that a lighter mover mass can contribute to less resistive energy consumption and faster dynamic response in the punching applications. Based on that, a STPMLM with special‐shaped yoke was designed for lighter mover mass. Moreover, prototype experiments including static thrust characteristic and dynamic performance were conducted. The experimental results proved that the proposed permanent magnet linear motor with special‐shaped yoke can improve the dynamic performance and reduce resistive energy consumption while ensuring that the electromagnetic performance is not affected.


| INTRODUCTION
The punching system is increasingly demanded in a wide range of applications such as packaging and leather manufacturing. However, the conventional-driven mechanisms are subject to many problems like low efficiency and complex structure. Linear motors performing a direct linear motion are an attractive candidate for those applications. Such devices offer numerous advantages over their rotary-to-linear counterparts, notably the absence of mechanical gears and transmission systems, which results in a higher dynamic performance and improved reliability [1].
Slotless tubular permanent magnet linear motors (STPMLMs) are an important branch of linear electric machines. They are characterized by compact structure, high thrust capability, zero detent force and linearity of force-current relationship [2][3][4][5]. Therefore, many endeavors have been devoted to STPMLMs in recent years for further improvement of its electromagnetic performance. Analytical models were proposed in Refs. [6,7] for magnetic field calculation, and other techniques, such as lumped equivalent magnetic circuit [8], finite-element analysis [9] and magnetic charge method [10], were also developed to provide alternative ways to achieve the purpose. To enhance the decreased air-gap flux density resulting from slotless structure, Halbach permanent magnet array [11] was commonly adopted in the motor design. Besides, Novel PM structures such as trapezoidal-shaped PMs and dual Halbach array have been proposed for further enhancement of the flux density [12][13][14][15]. What's more, the thrust characteristic of the slotless tubular permanent magnet linear motor (STPMLM) was also studied [16,17] to yield the maximum thrust density within the stroke. The aforementioned studies mainly focused on magnetic field modelling and thrust density enhancement of STPMLM, which has been of great prominence in increasing the thrust density of the motor and further facilitating its application in the punching system.
With the growing demand of high-frequency motion of the punching system for higher productivity, a better dynamic performance was pursued for motor development. The studies on dynamic performance of the linear oscillating motor [18][19][20][21] show that the most effective way to achieve satisfying dynamic performance is to decrease mover mass [12]. However, two problems should be fully considered when it comes to reducing mover mass. First, the mover mass is generally associated with the electromagnetic performance for a welldesigned motor, and any inappropriate change of the mover mass may damage the electromagnetic property. Therefore, it is challenging to reduce the mover mass while maintaining the electromagnetic performance. Second, the working mechanism in punching system is different from other reciprocation systems such as compressors or linear pumps. The mover accelerates in the stroke without load force, and finally, all accumulated kinetic energy was absorbed by the material during the perforation process [22]. For this reason, the kinetic energy of the mover can be the criterion of impact strength in terms of energy balance. To get the same impact strength, the final velocity must be increased by adjusting the drive current after reducing the mover mass. As a result, other parameters associated with the resistive energy consumption, such as the travel time, are also changed accordingly. These changes may result in greater resistive energy consumption and a poor thermal condition of the motor. Therefore, it is of great significance to reveal the inherent relationship between the resistive energy consumption and the mover mass for a comprehensive assessment of the effect caused by reducing mover mass.
Herein, the analytical relationship between the mover mass and the resistive energy consumption was deduced, and the travel time was expressed at the same time, and other factors which affect the resistive energy consumption were also discussed. Then, a special-shaped yoke structure was proposed to reduce the mover mass based on the comprehensive finite element analysis of the magnetic distribution. Then, two prototypes, that is special-shaped yoke structure and original structure, respectively, were manufactured and tested. Finally, experimental results of two structures are presented to illustrate the effectiveness of the proposed analysis.

| BASIC MODEL DESCRIPTION OF THE STPMLM
The STPMLM was selected for the punching application because of its simple structure, easy eddy-current confinement and zero detent force compared with the slotted structure. Figure 1 shows schematically the basic structure of a two-pole single-phase STPMLM in r-z plane [16,23]. The iron core of the motor is equipped with the surface-mounted Halbach PM array. The outer yoke and the iron core are fixed together and form a single-end open structure. Two cylindrical coils are wrapped in the aluminium bobbin and are connected in seriesopposing connection. Particularly, to restrain the eddy current loss in the aluminium bobbin, four slits were cut along the axis direction in the bobbin and each slit was filled with a slice of mica to prevent the deformation of the bobbin. When a DC current is switched across the winding, the thrust is generated according to the Ampere's Laws and further forces the motor to perform a linear motion.
According to the structure shown in Figure 1, a STPMLM can be fabricated into two types, that is, moving coil and moving magnet. The fundamental difference between the two types is the size of the mover mass. Despite the fact that the moving coil machines have a lighter mover mass and faster dynamic response, it suffers from a problem of moving cable bundle during the motion. As a result, the cable bundle is susceptible to wear, and the rotary motion required in the punching system is also limited. Therefore, the moving magnet structure was chosen in punching system for better reliability and multiple degree of freedom motion requirement.
The major parameters of the motor are given in Table 1, which have been optimized. The density values of each parts of the motor are listed in Table 2.  -311

| STUDY ON THE RELATIONSHIP BETWEEN THE MOVER MASS AND THE MOTOR PERFORMANCE OF STPMLM
It is known that the lighter the mover mass is, the better the dynamic performance becomes. However, the relationship between the mover mass and the resistive energy consumption is more implicit to be understood. Due to the long electromagnetic air-gap of the STPMLM, a larger winding current is needed to reach the same thrust force, resulting in an inevitable increase of copper loss [3] and a significant temperature rise [24], which may have a negative influence on the PMs and eventually to the failure of the motors. Therefore, it is unreasonable to improve the dynamic performance at the expense of increasing the resistive energy consumption. To analyse how the mover mass affects the motor performance in the punching system, the relationship between mover mass and the resistive energy consumption was investigated through analytical solution and numerical solution, respectively, in this section.

| Analytic solution
The motor is assembled vertically into the punching system. A spring is used to restore the mover to its initial position. Therefore, when a current is exerted into the winding, the mover accelerates in response to the resultant force of electromagnetic force, gravity and spring force. Considering that the impact motion is a non-linear and multi-parameter coupled physical process, the following assumptions are adopted to simplify the analysis to get the analytical solution.
1. Eddy current losses are ignored. First, the eddy current in the aluminium bobbin is well restrained by the slit-cutting operation. In addition, the PMs, iron core and yoke remain stationary, so no eddy-current is induced by the PM flux in the mover iron. What's more, the core loss generated by armature current is also negligible because of the long air gap, which is also validated by the experiment where the core loss is no more than 4% of the total loss when the motor is excited by 50 Hz AC current with rms value equals to 5 A and the mover is blocked. 2. The motor is driven by a constant current source, which means that the electric circuit equation is decoupled with magnetic field. 3. The thrust coefficient of the motor remains constant within the stroke and therefore replaced by the average thrust coefficient, which makes the equation a linear differential equation. 4. The mechanical losses are ignored.
Based on the above hypotheses, the equation set describing the impact motion can be written as follows [25]: where I is source current; v(t) is function of velocity; x(t) is function of displacement; M is mover mass; B m is the average thrust coefficient which is defined as the integral mean value of the thrust coefficient curve within the stroke and g is gravity constant. The spring stiffness k equals to 360 N/m, and initial spring deformation Δx 0 ¼ 50 mm. Then, the second-order differential equation of velocity can be deduced by taking a derivative with respect to formula (b) in Equation (1). Thus, the velocity function can be solved: According to the velocity function, the mover exhibits a simple harmonic motion on the premise of the hypotheses mentioned above. Integrating the velocity function, the displacement function is derived in (3) Then, the travel time T can be solved by bringing the stroke s into Equation (3), and the resistive energy consumption in one punching process and the final kinetic energy of the mover can be calculated and expressed in Equations (5) and (6).
Equation (6) displays the composition of the mover kinetic energy W k . To get the same impact strength for the punching system, the source current must be adjusted accordingly to keep the kinetic energy constant when mover mass is changed. The relationship between mass increment ΔM and current increment ΔI can be expressed as Equation (7). Therefore, the arc-cosine function part in Equations (4) and (5) will remain unchanged, and the travel time T is a monotonous increasing function with respect to the mover mass, which indicates that a lighter mover mass leads to a better dynamic performance.
By bringing Equation (7) into the expression (5), the resistive energy consumption can be expressed as a function of the mass increment, as shown in Equation (8). It should be noted that the mass increment ΔM is used for subsequent analyses and graphics, and the negative values of the ΔM denote a decrease of the mover mass compared with the original parameter while the positive values denote an increase.
By taking the derivative of Equation (8) with respect to ΔM and assigning zero to the derivative function, the extreme value points of resistive energy consumption were given by where ΔM 1 is the minimum value point in which the solution is meaningless due to the fact that all kinetic energy is converted from the gravitational potential energy; the ΔM 2 is the maximum value point. In other words, if the values of all parameters exactly meet Equation (10), the maximum resistive energy consumption will be generated.
The distribution of the two extreme points determines how the mover mass affects the resistive energy consumption in the real working condition. Therefore, by bringing the motor parameters designed in Section 2 into Formula (8) and assuming that the required kinetic energy is 2.5 J, the relationship between the resistive energy consumption and the mass increment can be depicted.
As shown in the subplot inset in Figure 2, to present a comprehensive observation of the resistive energy consumption changing with the mass increment, the mass increment was restricted to the range between -1.36 and 15 kg. The resistive energy consumption rises rapidly, peaking at around 5.1 J, and then drops steadily to zero before increasing again. Two extreme points of the curve are consistent with the calculation results of Equation (9). However, it should be noted that the mass increment has practical significance only in the vicinity of zero (labelled by a dashed circle in Figure 2). The resistive energy consumption value produced with original design parameters is 4.4 J, as shown at point A in Figure 2. It is worth noting that the point A is in the monotonic increasing interval of the curve, which means that the resistive energy consumption can be reduced by decreasing the mover mass.
In addition, the relationship between the resistive energy consumption and the mass increment with different kinetic energies is shown in Figure 3. It is always conducive to reduce resistive energy consumption by decreasing mover mass when kinetic energy range from 1 to 3 J (in which all working conditions are covered). Besides, it should be noted that this trend will be more remarkable in higher kinetic energy conditions.
Moreover, the effect of the average thrust coefficient on the resistive energy consumption was also analysed. It is assumed that the average thrust coefficient B m is another independent variate, thus the current increment ΔI should compensate for the changes in kinetic energy caused by the change of the B m .
By bringing (11) into Formula (5), the relationship surface (also the isosurface where the kinetic energy equals to 2.5 J) of the resistive energy consumption with mass increment and thrust coefficient increment can be depicted in Figure 4. For the same mover mass, a higher thrust coefficient leads to a lower resistive energy consumption. What's more, it is worth noting that the thrust coefficient even has a bigger impact on the resistive energy consumption than mass increment within certain areas, so special attention should be paid to the thrust coefficient when it comes to reducing the mover mass.

| Numerical solution
Several assumptions have been adopted to simplify the equation set (1) to get the analytical solution. But for more accurate results, numerical computation method is needed because it provides an efficient way to calculate any complex differential equations. Therefore, a non-linear and multi-parameter coupled model, which is more consistent with the reality, can be described in Equation (12) [26] in which the thrust coefficient is no longer a constant but a variable changing with the displacement and the motor is driven by a constant voltage source.
where B m (x(t)) is the function of the thrust coefficient with respect to the displacement, which can be obtained by fitting the static thrust experiment data. L is the winding inductance and assumed to be constant within the stroke. The average value of inductance measured by impedance analyser is 3 mH. U is the source voltage value. Then, Equation (12) can be solved through the structural block diagram established in MATLAB with the i(t), v(t) and x(t) as state variables. The mass increment ΔM varies between -1 and 1.4 kg when solving the numerical model, and the input voltage is adjusted to ensure that the kinetic energy remains constant at 2.5 J. Thus, the corresponding resistive energy consumption value is calculated and depicted in Figure 5, and the results of the analytical model are also displayed for comparative analysis. As can be seen, with the same motor parameters and kinetic energy requirement, the numerical model presents higher resistive energy consumption than the analytical model. This difference is mainly caused by the different current waveforms of two models. Figure 6 shows the current and the motor thrust force waveform of the analytical model (constant current source supplied) and the numerical model (constant voltage source supplied). Both models are calculated under the condition of 2.5 J kinetic energy and 1.36 kg mover mass. As can be seen, the current in numerical model increases rapidly at the beginning of the calculation, peaking at about 16 A which is much bigger than the constant current 10.1 A of the analytical model, and then reduce to below the constant current. The variation of the current in numerical model leads to the higher resistive energy consumption than analytical model, but the variation of the resistive energy consumption with mass F I G U R E 4 Resistive energy consumption with mass increment and thrust coefficient increment F I G U R E 5 Resistive energy consumption versus mass increment of two models 314increment of two models show a similar tendency. Therefore, the relationship between the mover mass and the resistive energy consumption which was predicted by analytical model can be verified. In addition, the travel time in different mover masses can be calculated by the analytical model and the numerical model, respectively, and the results are listed in Figure 7. The results of two models achieve a good agreement, and the curves show that a considerable improvement in dynamic performance will be achieved by reducing the mover mass. For example, if the mass reduces by 0.4 kg, the travel time will reduce by about 3.2 ms, which is account for over 15% of the original travel time.
In general, the studies on the differential equation of punching movement reveals that a lighter mover mass not only improve the dynamic performance but also reduce the resistive energy consumption. Therefore, it is absolutely reasonable to pursue a lighter mover mass in the punching system while ensuring that the electromagnetic property is not affected in the process.

| SPECIAL-SHAPED YOKE DESIGN OF THE TPMLM
According to the analytical conclusion derived in Section 3, the mover mass should be reduced for a better dynamic performance and a lower resistive energy consumption. But generally, reducing the mover mass is achieved by cutting the ferromagnetic material of the magnetic circuit, which may raise a problem of decreased average thrust coefficient of the motor and further result in increasing the resistive energy consumption. Therefore, it becomes necessarily a challenge to reduce the mover mass while maintaining the electromagnetic performance. Figure 8a shows the flux distribution of the original motor. Halbach permanent magnet array generates two radial magnetic poles in the air gap, and hence three magnetic flux paths are formed through the iron core and the yoke. Three magnetic circuits are independent to each other, and the two areas which are sandwiched between three magnetic circuits in the yoke feature lower flux density (labelled by two dotted circles in Figure 8a. The same result can also be obtained from the cloud map shown in Figure 8b. In addition, the flux density of left circuit also decreases significantly due to the open structure on the left side. It is worth noting that these low flux density areas in the yoke are nearly constant due to the fixed structure between the yoke and the PMs. Therefore, those foregoing areas with low flux density in the yoke can be removed to reduce the mover mass, which will be a valuable structural improvement if the electromagnetic performance is not damaged. The STPMLM features a radial magnetic field in the air gap and an axial magnetic field in the yoke. To take an example of a sufficiently small section of the yoke, the flux lines pass through the cross-sectional area along the axial direction. Therefore, the magnetic flux density can be adjusted by changing the thickness of the yoke. Moreover, because the flux density is evenly and symmetrically distributed along the circumference of the iron yoke, the outer radius r 0 o (z) of the shaped yoke is calculated by Equation (13), where B(z) is the flux density of the original yoke, which was calculated by finite element model (FEM); r i is the inner radius of the yoke. This formula can only be applied to the cutting area in which the flux density in the yoke is lower than 1.5 T.
Generally, the mover iron can be seen to be saturated at the flux density of about 1.8 T. Therefore, the flux density of the cutting areas is expected to increase to about 1.5 T, at which the permeability of the iron is still much greater than that of the surrounding air. Consequently, the reluctance of the yoke is negligible so that the electromagnetic performance will not be damaged at all. What's more, considering the mechanical restriction for the thickness in reduced region, the thinnest yoke thickness is 0.5 mm. The yoke has sufficient mechanical F I G U R E 7 Travel time versus mass increment comparing with analytical solution F I G U R E 6 Current and motor thrust waveform of two models LUO ET AL. strength since it will not be subject to the lateral force during the punching movement.
The magnetic distribution of the motor with specialshaped yoke was visualized in Figure 9. As can be seen, the flux density was significantly increased in the cutting areas, while keeping substantially unchanged in the remaining areas of the yoke. For quantitative analysis, the values of the flux density were exported from the FEM and depicted in Figure 10. The flux density values of two yokes shown in Figure 10 are as a function of the spatial position along the zaxis, and the right end of the yoke is indicated by zero point of the abscissa. As can be seen, the magnetic flux density of original yoke is subject to an approximate waveform distribution, and the area with flux density below 1.5 T accounts for about 73% in the yoke. For the new-shaped yoke, the flux density in the cutting areas increased to about 1.5 T, whereas the flux density in originally saturated areas remains exactly constant. Therefore, the electromagnetic performance of the motor will not be affected by the special-shaped yoke structure. Figure 11 shows the thrust characteristics of the two motors which are calculated by the FEM. The average thrust force for a current of 10 A changed from 132.3 to 131.2 N after cutting operation, in other words, the average thrust coefficient decreased by 0.11 N/A, which further indicates that the effect of structural changes of the iron yoke on thrust characteristics can be ignored. By processing the iron yoke and other components which are not related to the electromagnetic performance such as the shaft, the mover mass was reduced by 0.28 kg, 95 g of which was decreased by cutting the iron yoke. Then the travel time and the resistive energy consumption under 2.5 J kinetic energy requirement can be calculated by the dynamic FEM and compared with the numerical results.
As shown in Table 3, the results of the numerical solution and the FEM show a same tendency on the travel time and the resistive energy consumption when reducing the mover mass. Therefore, the conclusion of the analytical model can be validated.

| EXPERIMENTAL VERIFICATION
In this section, two prototype motors with different yokes were manufactured and tested. The static thrust experiment and dynamic experiment were conducted to validate the previous analyses.
The mover prototypes and the stator winding are shown in Figure 12a and the static thrust test platform is shown in Figure 12b. The pressure sensor (JHBM-H1, 0.04 N precision) was fixed on the z-axis positioning platform (HGVM0160, 0.1 mm precision), and hence, the mover position can be regulated by the z-axis positioning platform with an accuracy of 0.1 mm. The reading on the displayer is 50 times the real thrust force. The supporting spring is removed to measure the net thrust. The motors were energized by a DC current source, and the drive current was set to 4.5 A to ensure the operability and the safety of the experiment. The static thrust of both prototypes was measured every 1 mm within the working stroke. Besides, the static thrust characteristic under 4.5 A drive current was also calculated by the finite element method. Both test results and FEM results are presented in Figure 13. As can be seen, four curves show a good agreement, and therefore, it is feasible to reduce the mover mass by means of a special-shaped yoke.
The dynamic performance testing platform is shown in Figure 14. Based on the analytical expression, the resistive energy consumption should be measured on the premise of keeping kinetic energy of two prototypes exactly the same. Therefore, a precise measurement for the kinetic energy and the resistive energy consumption is essential for dynamic experiments. As is known, the kinetic energy can be calculated on the basis of velocity. Therefore, an accelerometer (CT1005LC, 48.9 mV/g charge sensitivity) was mounted on the top of the mover for acceleration measurement, and its output signal is magnified 10 times by -317 adapter (CT5201). Finally, this signal was integrated into velocity signal by an analog integrating circuit (configured by dual precision, rail-to-rail amplifier AD823ARZ-R7). Besides, due to the fact that the current data exported by the oscilloscope was discrete, the composite trapezoidal formula shown in Equation (14) was adopted for numerical integration to get the resistive energy consumption.
where T S is the sampling period; N is the sampling data length and I n is nth current sampling value. Besides, considering that the large impact acceleration may exceed the measurement range of the sensor, the kinetic energy was controlled at about 2.5 J, that is the final velocity of two movers was about 1.90 and 2.12 m/s, respectively. The waveforms recorded by the oscilloscope for two prototypes are shown Figure 15.
It should be noted that the acceleration signal changes abruptly at the moment of impact, which provides an obvious indicator of determining the travel time. Besides, the acceleration waveforms, which are consistent with the mover resultant force, can also reflect the dynamic motor thrust force because the resultant force of the gravity and the spring force is much smaller than the electromagnetic force. The final speed signal is used to indicate the kinetic energy of two movers and adjust the input voltage during the experiment. The peak current value of the optimized motor is higher than the original one, because a higher input voltage is required in the optimized motor to get the same kinetic energy as its original counterpart. In addition, the current and acceleration waveforms show a similar tendency at the first half of the stroke because of the approximate linearity of force-current relationship. However, at the rest of the stroke, the current increases slightly, but the acceleration shows a continuous decline as a result of increased spring force and reduced thrust coefficient. Overall, the acceleration of the optimized motor is higher than that of the original one. Thus, the travel time of the motor with specialshaped yoke is significantly shorter, which not only improves its dynamic response, but also shortens the heating time of its winding. Furthermore, the experimental data are detailed in Table 4, and compared with the numerical and FEM results in Table 5.
As shown in Table 4, when the kinetic energy is set to be 2.5 J for both prototype motors, the mover mass of the shaped structure was reduced by 0.28 kg compared with the original one, and thus its travel time reduced by 1.9 ms (10.5% of the original value), and resistive energy consumption reduced by 0.23 J (3.36% of the original value). Table 5 compared the experimental results to the numerical and the FEM results with same kinetic energy requirement and mass change. The travel time could shorten by 2.2 ms (11.58% of the original value), and the resistive energy consumption could reduce by 0.31 J (5.85% of the original value) according to the numerical calculations. At the same time, the shortening of the travel time is 2.2 ms (11.28% of the original value), and the resistive energy consumption reduction is 0.16 J (2.96% of the original value) according to the FEM results. What's more, for further verification of the consistency of the three methods, the current, total thrust and speed waveforms with the special-shaped yoke structure are presented in Figure 16. Generally, the waveforms of three methods show a good agreement. The difference between test current and the theoretically calculated currents after 12 ms is probably caused by the recharge of the DC-bus capacitor after the voltage drop. Besides, because the FEM calculated inductance values are greater than those measured by the prototype, the FEM has a larger time constant than the prototype. As a result, the current in the FEM changes more slowly than the experiment and numerical results, which further causes the FEM calculated thrust and speed waveforms lag behind the test and numerical curves. In addition, other factors such as the friction or manufacturing error may also cause the difference of three methods. Nevertheless, the results of three methods indicate that a lighter mover mass in the punching system features better dynamic performance and lower resistive energy consumption, and the specialshaped yoke design is an effective way to achieve that goal.

| CONCLUSION
Herein, an analytical model for predicting the travel time and the resistive energy consumption of a STPMLM which is applied to punching system has been developed, and the results were verified by numerical computations. The studies indicate that a lighter mover mass is conducive to improving the dynamic performance and reducing the resistive energy consumption. Based on that, a novel special-shaped yoke structure was proposed to reduce the mover mass for better motor performance. Two prototypes with different yokes were built and tested, and the experimental results in static thrust characteristic and dynamic performance proved the effectiveness of this novel structure.