Robust Design and Control of Linear Actuator Dedicated to Stamping Press Application

In this paper, we present a robust design and control approach of all converter-linear actuator dedicated to Stamping Press Application. The linear actuator is designed by a systemic design approach taking in account the constraints of the application such as displacement limit and interactions between the design and the control. The model developed is implanted under the simulation environment Matlab / Simulink. The obtained results encourage the industrialization of the studied structure of the Stamping Press Machine.


Introduction
Several research works address the problem of design and variable speed driving of linear motors for industrial applications requiring an accuracy of the position and speed of the movable shaft [1][2][3]. These studies have shown several drawbacks of linear motors stepping and variable reluctance those which may be mentioned [1][2][3]: Vibration. Average precision of the position control. Complication control algorithms of reluctance linear motors since the inductance is variable depending on the position the movable shaft. Our application is to automate a Stamping Press to make repetitive cycles to transform pieces of sheet metal in average thickness in parts to be used in industrial applications such manufacturing of the yokes of the electric machines. Our choice was directed on an innovated linear motor cylindrical type structure with permanent magnets. This type of motor is with concentrated winding to reduce congestion and inductance of winding heads. The procedure for manufacturing of this type of actuator is easy to automate. The control law chosen is a scalar type control to monitor the speed and moving force of the movable shaft, it is following robust in front of view simplicity and precision of the speed and position control. The movable shaft carries the sheet metal punch tool.
In this context, this paper presents a systemic design and modeling methodology of the actuator resting on works carried out in [4][5][6][7].

Analytical Sizing of the Actuator
The linear motor structure with permanent magnets is illustrated by figure 1: a. Motor cutting plane b. Front view of the motor Several research studies show that the analytical method is recommended for the design of electrotechnical devices, since it leads to sizing models highly parameterized and compatible thereafter for large optimization problems. In addition, certain factors need to be adjusted by the finite element method which can be cited as an example the coefficient of leakage flux. This method is also based on well-justified simplifying assumptions. For this, we set to establish an analytical model adjusted by the finite element method. The method chosen is then an hybrid Analytic / Finite Element method. This method combines the advantages of the analytical method as compatibility with large optimization programs and advantages of the finite element method such that accuracy of the results [5][6][7][8][9][10][11][12][13][14][15].

Modeling of the Actuator
The motor is controlled by a DC-AC converter with two voltage levels controlled in current by pulse width modulation. The converter feed voltage is derived from a three-phase rectifier connected to a three-phase 220 V / 380 V-50 Hz sector.
The average rectified voltage is estimated by the following equation: Where V max is maximal phase voltage of the sector. Le actuator electric constant is expressed by the flowing relation: Where L ra is the width of the width of the radial opening of the magnets in meter, N sp is the phase number of turns and B e is the flux density in the air-gap.
The maximal value of the back electromotive force is expressed by the following relation: Where V is the velocity of the movable stem. We deduce the expression of the electromagnetic force of attraction of the moving stem: Where I is the maximal phase current. The average length of a coil turn is expressed by the following relation: Where D cm is the movable stem diameter, e is the air-gap thickness, H a is the magnet thickness, H d is the tooth height, α is the tooth opening coefficient in the radial direction, L enc is the slot width and A dentm is the angular teeth opening in the axial direction.
We deduce the expression of the phase resistance in function of the temperature of copper: Where r cu is the copper resistivity, I dim is the dimensioning current and δ admissible current density in the copper.
An average value of resistance is considered for a copper temperature maintained constant equal to 80°C, assuming that the motor is cooled by a cooling system controlled in temperature (the temperature of the copper equal to 80°C) by acting on the thermal convection coefficient of the refrigerated fluid.
The phase inductance is expressed by the following relation: Where N d is the total number of teeth, S d is the teeth section, C td is a coefficient taking into account the three dimensional effects, L m is the width of the radial opening of the tooth and L d is the tooth width in the axial direction.
The phase mutual inductance is expressed by the following relation: The C td coefficient is identified by a two-dimensional finite element model of the motor after a cutting along an horizontal plane passing through the axis of the actuator. The model is obtained by spreading the two portions of the actuator.

Finite Element Model of the Actuator
The actuator is studied in two dimensions basing on the following assumptions: Three-dimensional effects are neglected since the radial opening of the teeth and the magnets is important. The motor is axially symmetrical, only half of the actuator is investigated. Mesh refinement in the air gap to increase the accuracy of calculations. The problem is solved by varying the static position of the movable axis of a step size equal to 26.66 mm by Maxell-2 D software.
With all these assumptions, the two-dimensional model of actuator with refined mesh is shown in Figure 2. The distribution of field lines at load in the planar portion of the actuator is shown in Figure 3:

Simulation Results
The three phase fluxes a load captured by one coil calculated by finite element method are illustrated by Figure 4: The fluxes are close in form and value of those estimated analytically which validates the analytical design procedure.
The back electromotive forces a load for movable stem speed equal to 2 m/s calculated by finite element method are illustrated by Figure 5: The back electromotive forces are close in form and value of those estimated analytically which validates the analytical design procedure.
The electromagnetic strength calculated by finite element method related to a current value equal to 200 A is illustrated by Figure 6: The electromagnetic strength is close in form and value of that estimated analytically which validates the analytical design procedure entirely.

Motion Equation
The vehicle motion equation is derived from the fundamental relationship of dynamics: M m is the movable stem mass, V is the movable stem velocity, F em is electromagnetic attraction strength, F p is the gravity strength, F f is the strength due to iron losses, F mec is the strength due to mechanical losses, F pr is the Stamping Press resistance force and d is the movable stem displacement.
The movable stem displacement is deducted from the following relation: The gravity strength is expressed by the following relation.
Where g is considered equal to 9.8 N.kg -1 .
The strength due to iron losses is expressed by the following relation : Where f is the frequency of the stator currents, M ds is the mass of the stator teeth, M cs is the mass of the stator yoke, B d is the flux density in the teeth and B cs is the flux density in the stator yoke.
The strength due to mechanical losses is expressed by : Where s is the dry friction coefficient, nu is the viscous friction coefficient, xsi is the fluid friction coefficient and Re is the movable stem bore radius.
The motion equation is implanted under the simulation environment Matlab / Simulink according to Figure 7.

Speed Control
The speed comparator outputs the amplitude of the reference currents minimizing the error between the reference speed and the response speed. Indeed, the reference speed is compared to the response speed. The comparator output drives a proportional / integral controller type (PI) to provide the amplitude of reference currents minimizing the speed error. The Simulink model of the speed controller is shown in Figure  8

Current Regulation
Regulators currents allow the imposition of currents having the same shape and in phase with the back electromotive forces. Indeed, the reference currents are compared to the phase currents of the actuator. The outputs of the three comparators attack three proportional / integral (PI) regulators to provide three reference voltages necessary to impose ideal currents in phase with the back electromotive forces and to minimize the error between the reference speed and response speed of the movable stem .
Simulink model of the current regulator is illustrated in Figure 9:

Model of the Back Electromotive Forces
The back electromotive forces are expressed by the three following equations : Where K e is the electric constant of the motor and V is the linear velocity of the motor and p is the number of pole pairs. These equations are implanted under Matlab-Simulink environment according to Figure 10:

Generator of Control Signals
The control signal generator compares the three reference voltages to a triangular signal having a frequency much greater than the voltages provided by the regulators of the currents. The output of each comparator drives an hysteresis variant between 0 and 1 to reproduce the control signals of the IGBT 1, 3 and 5. The speed controller and current controller adjusts the pulse width of the control signals so as to impose currents in phase with the back electromotive forces and to minimize the error between the reference speed and the speed of response. The control signals of the IGBT 2, 4 and 6 are respectively complementary to the control signals of IGBT 1, 3 and 5 to avoid short circuit, the control pulses of S2, S4 and S6 are shortened to avoid duplication between two control signals of an arm. The Simulink model of the control signals generator is illustrated by Figure 11:

Model of the Motor-Converter
The motor is powered by a two-level voltage inverter with IGBTs. Each phase of the motor is equivalent to a resistor in series with an inductance and a back electromotive force.
The three phases models of the motor are described by the following equations: Where R, L, M and K e are respectively the resistance, inductance, mutual inductance and the motor electric constant, ii and Vi are the current and the voltage of the phase i.
The electromagnetic torque is given by the following relation:

Global Model of the Power Chain
The coupling of different models of the power system leads to the global model implanted under the Matlab / Simulink simulation environment according to Figure 13:

Description of the Simulation Results
The simulation parameters (Table 1) are extracted from the developed design and modeling program of the studied power system. The response speed of a stamping cycle is shown in Figure  14:  Figure 14 shows that the response speed follow with good accuracy the stamping cycle, which shows the performance of the selected technique of control. Figure 15 illustrates the displacement of the movable stem: Figure 15 shows that the displacement of the movable stem is accurate, since it joined to its return the initial position exactly (d = 0 m).   Figure 16 show that the phase currents present their strong values during the phase of gone up since the gravity strength is important. Figure 17 shows the change of the back electromotive force compared to the phase current: The phase shift between the current and the electromotive force is very low leading to a significant reduction in the energy consumed. This characteristic shows the effectiveness of the developed control technique.
The figure 18 shows the evolution of the electromagnetic strength and the resistance strength: During the attack phase of the sheet, the electromagnetic strength becomes positive and important to overcome the stamping strength resistance. Figure 19 illustrates the evolution of the electromagnetic power: Figure 19. Evolution of the electromagnetic power. Figure 19 show that the electromagnetic power present their strong values during the phase of gone up since the gravity strength is important. During the attack phase of the sheet, the electromagnetic power becomes positive and important to overcome the stamping resistance strength. Figure 20 illustrates the evolution of the losses strengths: Figure 20 shows that the strengths due to iron losses and mechanical losses are negligible, which shows the performance of the developed design approach.

Conclusion
In this paper is shown a comprehensive approach of robust design and control of a linear motor with permanent magnets dedicated to Stamping Press Application. A design model of the linear motor based on joint method analytic / finite element is developed. This model takes into account the constraints of the application and the interactions between the control and the sizing of the actuator. The model developed can be applied to several industrial applications other than the present application. This model is also highly parameterized, and eventually leads to problems of optimization of large dimensions. One proposed solution to the automation of Stamping Press is presented.
This study is completed by the development of a robust control law. Simulation results are with good scientific level, leading to the full validation of the design and control approach.
As prospects, it will be very interesting to industrialize this innovative product.