Definition of Optimization Problem for Electromagnetic Linear Actuator

Transcription

1 Excerpt fro the Proceedings of the COMSOL Conference 2009 Milan Definition of Optiiation Proble for Electroagnetic Linear Actuator P. Piskur*, W. Tarnowski, K. Just Kosalin University of Technology, Poland *Corresponding author: Departent of Mechatronics, Nanotechnology and Vacuu Technology; Kosalin University of Technology; , Kosalin, Poland; eail address: Abstract: In this paper a poly-optiiation of the design of the electroechanical actuator is presented [5]. The shape of the actuator is defined by the decision. The nuber of decision under consideration is up to ten but in the next step while the ulti-s syste will be analyed the nuber of decision will increase up to hundred, so the genetic algorith has been used. The genetic algorith progra has been ipleented in the Matlab progra which operates together with the Cosol Multiphysics package. For each decision variable vector generated in Matlab progra the electroagnetic force has been calculated in Cosol Multiphysics by using FEM (finite eleent ethod). The ai of the optiiation process [1] is to find the axial electroagnetic force and the inial ass of all the device. The shape of the and the current density in the are taken to be constant. Three kind of the construction ferroagnetic case have been taken under consideration. The final result is a set of the Pareto optial solutions, which akes possible to draw out soe ore general conclusions on a design of the actuator. Keywords: electroagnetic linear actuator, optiiation, Pareto-optial solution, genetic algorith. 1. Introduction Electroagnetic actuators are coonly used for various purposes, the ain drawback however is the low energy efficiency and the large diensions and ass of the device. In the case of an electroagnetic linear launcher, there is a set of s, displayed in series. However, in this paper for a preliinary analysis there is only one syste under consideration (Figure 1). The syste consists of one cylindrical and a ferroagnetic plunger. The is supplied by a constant voltage ipulse of a finite period of tie. The electroagnetic field is pulling the plunger inside the. v v ferroagnetic Figure 1. The cross section view of the electroagnetic linear actuator A forer analysis of the electroagnetic linear actuators without ferroagnetic case [6] shows that the electroagnetic force is ten ties lower than for the electroagnetic linear actuators with ferroagnetic case, and is to sall to proposed application. In Figure 2 the electroagnetic force is depicted as a function of the plunger displaceent for both solutions. The axial value of the ferroagnetic force is about 1 Newton for surrounded by air (brown color line) and about 24 Newtons for covered by the ferroagnetic case (green color line). Electragnetic force [N] Fe*10-1 Fe position [] Figure 2. Coparison between the electroagnetic force for with and without ferroagnetic case (see text)

2 In the optiiation process three kinds of the device shape have been taken under consideration. The first one is shown in Figure 3 and consists of three ferroagnetic rings which thickness is defined as decision p(3). The length of the ferroagnetic plunger is defined as. Figure 4. The length of the ferroagnetic plunger is defined as p(6). p(5) p(3) p(3 ) r p(8) p(7) p(6) Figure 5. Cross sectional view of the half of the ferroagnetic actuator with nine decision Figure 3. Cross sectional view of the half of the ferroagnetic actuator with four decision In Figure 5 another exaple of electroagnetic linear actuator with seven decision is shown. The eight decision variable defines the length of ferroagnetic plunger. There is uch ore cobinations of shape of the ferroagnetic linear actuator, but the ain ones have been chosen to deterine the relationship between the shape and the ferroagnetic force. Another question was the device anufacturing process difficulties. 2. Governing Equations p(6) p(3) p(5) Figure 4. Cross sectional view of the half of the ferroagnetic actuator with six decision The second version of geoetry is defined by the decision p(5) presented in r The proble of electroagnetic analysis is to solve the Maxwell s equations subjected to certain boundary conditions. Maxwell s equations are written in a differential or integral for, defining the relationship between the fundaental electroagnetic quantities. The differential for are presented here, because it leads to differentials equations that the finite eleent ethod can handle [2]. Here only the second and the fourth Maxwell s equation have been used: D H = J + (1) B = 0 (2) Where:

3 B is the agnetic flux density [T]; H is the agnetic field intensity [A/]; D is the electric flux density [C/ 2 ]; J is the current density in the [A/ 2 ]. The quasi-static analysis is ade under the assuption that: D = 0 (3) This iplies that the equation (1) can be rewritten in the following anner: H = J (4) If we assue the electroagnetic field is stationary then: B = 0 (5) That is, if the field is varying so slowly that we can neglect the contribution fro induced currents. We also assue that the odeled object = is not oving: v 0, so that there is no contribution fro Lorent forces. This assuption is appropriate also, if the ferroagnetic part of the device is ade in that way, that the eddy currents are low. It can be helpful to forulate the agnetic field intensity generated by the current in the in ters of the agnetic vector potential A [3]. It is given by the equation: B = A (6) where the relation between the agnetic field intensity and the agnetic flux density is given by the equation: µ = µ 0 ; µ r B = µ H (7) 7 µ = 4 π 10 [ H / ] is the pereability 0 of vacuu. B µ f ( ) is the relative pereability of the H r = aterial. Assuing static currents and fields, the agnetic vector potential A ust satisfy the following equation: 1 ( µ A ) = J (8) In Figures 6 9 agnetic potential A is depicted as a result of the finite eleent ethod coputation process. The force is coputed on the electroagnetic energy W analysis of the syste with respect to the sall displaceent. The ethod of a virtual work utilies the fact that under constant agnetic flux condition, the total agnetic force on a syste is coputed as: Fe = W 3. Theory The independent variable (an operand): in the optiiation process it is the displaceent of the ferroagnetic plunger. Values to be searched: diensions p(n). Given data: The diensions of (length and width = 2 x 2 [c]) and the inner radius equal to 1 [c]. The outer radius of a plunger equals to 0.9 [c]. The difference between both radiuses is the air gap. The is supplied with 3 [A] current. The ain optiiation criteria: the first is a kinetic energy of the plunger to be axial and the other is a total ass of the device to be inial, with an assuption of the constant electric energy delivered to the device. The kinetic energy of the plunger is: 1 2 E = k pv 2 p, v is the ass and the velocity of the plunger, respectively. The velocity depends on the electroagnetic force: Fe v = dt p The ass of the plunger varies depending on the decision variable defining the length of the plunger. The ass of the device is to be directly proportional to the volue and density (ρ = const.) of the aterial. = ρv ρ is the density of the aterial; V is the device volue. Global criterion: P = (1 - w)fe +s(1-w) p + w ; w is a weigh coefficient s is a scaling factor. Liitation range: The ass of the device: < ax ;

4 The electroagnetic force: Fe > Fe in 4. Nuerical odel = 0.90 [kg] The electroagnetic actuator odel has been ipleented in the Cosol Multiphysics package. Because of the odel syetry, only a half of the device has been taken under consideration. In Figures 6 9 the lines of the agnetic potential and arrows of the agnetic flux density have been presented for the various solution of the device shape. Figure 7. Magnetic field for the surrounded by ferroagnetic rings and for the ferroagnetic plunger (4 decision ) Figure 6. Magnetic field for surrounded by air and for the ferroagnetic plunger. One of the Pareto-optial solution is presented in the Figure 7, for the diension (see Figure 3): = [] = [] p(3) = [] = [] Electroagnetic force achieves the value: Fe = 39 [N] = 0.89 [kg] Another Pareto optia solution is presented in Figure 8, for the following diensions (see Figure 4): = [] = [] = [] p(5) = [] p(3) = [] p(6) = [] Electroagnetic force achieves the value: Fe = [N] Figure 8. Magnetic field for surrounded by ferroagnetic rings and for the ferroagnetic plunger (6 decision ) The last one of the Pareto-optial solutions is presented in the Figure 9, for the following diensions (see Figure 5): = [] P(5) = [ = [] p(6) = [] p(3) = [] p(7) = [] = [ p(8) = [

5 Electroagnetic force achieves the value: Fe = [N] = 0.75 [kg] In Figure 10 all the Pareto-optial solutions are presented for the three kinds of construction shapes. It can be found that the siplest kind of ferroagnetic case (with 3 decision ) would be enough for the sall value of the electroagnetic force (data 1). Then the ass of the device reaches the lowest value. Another advantage is an easy construction. If the electroagnetic force is the ost iportant optiiation criterion (a coefficient weigh w < 0.5) then the geoetry specified by nine decision should be applied (Figures 5 and 9, and data 2 in Figure 10). The design presented in Figure 4 and 8 has the poorest paraeters (data 3 in Figure 10). 5. Conclusions Figure 9. Magnetic field for surrounded by ferroagnetic rings and for the ferroagnetic plunger (9 decision ) 4. Siulation results The poly-optiiation of electroagnetic devices requires a very accurate siulation odel, which has been built in the Cosol Multiphysics progra and presented in this paper. A cobination of the two progras (Cosol and Matlab) gives the powerful opportunity to analye the object without building the prototype. Although the Genetic Algorith needs the high coputing power of a coputer, it is able to find the global inia. 6. References 1. W. Tarnowski Syulacja i optyaliacja w MATLAB ie page nubers: , WSM, Gdynia Cosol Multiphysics Electroagnetic Module, User s Guide, page nubers: COMSOL AB., Cosol Multiphysics Scripting Guide, page nubers: , COMSOL AB S. Wincenciak, Metody i algoryty optyaliacji kstałtu obiektów w polu elektroagnetycny, Oficyna Wydawnica Politechniki Warsawskiej, page nubers 23-33, Warsawa P. Piskur, W. Tarnowski, K Just Polyoptiiation of in electroagnetic linear actuator, Recent Advances in Machatronics, Springer 2007, page nubers: , Figure 10. Pareto optial solution for three different kind of the ferroagnetic casing