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衣架注塑模具设计与制造,衣架,注塑,模具设计,制造

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M. Kamel et al. (Eds.): AIS 2011, LNAI 6752, pp. 102111, 2011. Springer-Verlag Berlin Heidelberg 2011 Thermal Dynamic Modeling and Control of Injection Moulding Process Jaho Seo, Amir Khajepour, and Jan P. Huissoon Department of Mechanical and Mechatronics Engineering University of Waterloo j7seoengmail.uwaterloo.ca Abstract. Thermal control of a mould is the key in the development of high ef-ficiency injection moulds. For an effective thermal management system, this re-search provides a strategy to identify the thermal dynamic model for the design of a controller. Using neural networks and finite element analysis, system iden-tification is carried out to deal with various cycle-times for moulding process and uncertain dynamics of the mould. Based on the system identification, a self-adaptive PID controller with radial basis function (RBF) is designed to tune controller parameters. The controllers performance is studied in terms of track-ing accuracy under different moulding processes. Keywords: Plastic injection moulding; thermal dynamic modeling; cycle-times, neural networks; finite element analysis; RBF based self-adaptive PID control. 1 Introduction Injection moulding is a primary manufacturing process to produce parts by injecting molten plastic materials into a mould. Thermal control is a key issue in this process since uniform temperature in the moulds contributes to production quality by reducing problems such as shrink porosity, poor fill and prolonged cycle-times for part solidification 1, 2. Many approaches have been proposed to deal with the thermal control in mould (or die) systems. A PI 3 and PID algorithms 1 were applied to manage the cavity temperature on a plastic injection moulding and high-pressure die-casting, respectively. To improve limitation of PID control in presence of uncertain or nonlinear dynamics, a Dahlin controller 4 and the Model Predictive Control (MPC) have been utilized in diverse range of mould systems for thermal control 5, 6, 7, 8. Despite of improved performance compared to PID control, the addressed controllers are not robust in some circumstances with more complex nonlinearities Specifically, because these controllers are based on a linear “best-fit” approximation (e.g., ARX and ARMAX), the performance of the controllers is affected largely by modeling errors arisen from uncertain dynamics. Although accurate modeling of the thermal dynamics of moulds is a prerequisite to successful thermal control, it is a difficult task in practice due to mould uncertainties. For example, a mould is a complex continuous system with cooling and heating Thermal Dynamic Modeling and Control of Injection Moulding Process 103 channels causing the modeling and control to be quite complicated. Unmodeled thermal dynamics of moulds (such as convection and radiation) also provide further modeling challenges. To deal with inherent challenges of modeling in a mould system, this paper considers a neural network (NN) approach. By applying NN techniques, the thermal dynamics with uncertainties in a plastic injection mould is modeled. In addition, our modeling covers various cycle-times for plastic moulding process that has not been studied (i.e., most documented approaches for thermal management of mould systems have only considered a fixed cycle-time). In this study, the system identification is conducted using the temperature distribution obtained through a finite element analysis (FEA). Based on this system identification, a controller is designed using the RBF based self-adaptive PID control. Section 2 describes the mould system. Section 3 presents a methodology for modeling thermal dynamics using FE simulation and NN techniques. In Section 4, a controller is designed and its performance is discussed. Finally, Section 5 provides concluding remarks. 2 Injection Mould and Various Cycle-Times Figure 1 shows a plastic injection mould used for the analysis in this study. Hot polymer injected into the mould cavity is cooled to the demoulding temperature by heat transfer to coolant through the cooling channel in close proximity to the cavity. Since the dominant heat for the injection moulding process is transferred by means of the conduction and convection by the coolant 3, 9, the flow rate and temperature of the coolant are chosen as the control parameters. Finite element simulations for ther-mal dynamic analysis of the mould are carried out based on this input-output model. In the plastic injection moulding process, the cycle-time is allocated for the main phases of injection, packing, holding, cooling and ejection. Since the cooling phase among these phases takes a large portion of the cycle-time to cool the polymer down to solidification temperature 10, the cooling time plays an important role in the cycle. The previous studies dealing with the system identification and thermal control in the injection moulding process have used a predetermined cooling time (thus cycle-time). However, the system identification using a fixed cycle-time cannot cope with the wide range of thermal dynamics variation when the cycle-time is not fixed. The control strategy based on this limited identification cannot make full use of the benefits of the cost-effective manufacturing which can be achieved by reducing the cooling time (or cycle-times). In this study, various cycle-times of the moulding process are considered in the process of thermal dynamics modeling. 3 Thermal Dynamics Modeling 3.1 Finite Element Analysis The objective of the FEA is to identify the thermal dynamics (i.e., temperature distribution) during the moulding process in the injection mould and to obtain the input-output data set required for the system identification. 104 J. Seo, A. Khajepour, and J.P. Huissoon Fig. 1. Injection mould with cooling channel The injection moulding process of the mould can be briefly described as follows: Polymer is injected into the cavity at a temperature of 205 oC before the injection moulding cycle starts. During the cycle, the heat removal from the melt and mould is achieved through heat conduction with the coolant and natural convection with air. After the cooling phase, the polymer part is ejected from the cavity of opened mould. For the FEA, ANSYS CFX 11 software package was used with 3D CAD files from SolidWorks. For the polymer and mould, material properties of Santoprene 8211-45 11 and Cast Iron GG-15 12 were applied, respectively. The properties of water were used for the coolant. Flow of the coolant in the cooling channel was assumed to be turbulent. The initial temperature of the mould was assumed as the room tempera-ture (25 oC). For natural convection with the stagnant air, the value of 6 W/m2 oC was chosen as the convective heat transfer coefficient. A transient state analysis in the FEA was carried out to observe the temperature distribution over the cycle-times of injec-tion moulding process at some nodes near the cavity. Four nodes (Mo1, Mo2, Mo3 and Mo4) located inside the mould in Fig. 1 were chosen to monitor the temperature distribution. For the FEA meshing, the convergence study for mesh refinement was conducted until a convergence in the temperature distribution was achieved. 3.2 System Identification Using NARX Model and FEA To capture a dynamical model to describe the temperature distribution in the mould using the input-output data from the FEA, NARX (nonlinear autoregressive with exogenous inputs) model as a neural network approach is provided. NARX is a pow-erful class of nonlinear dynamical model which enables to deal with uncertain dynam-ics of the mould and in which embedded memory is included as temporal changes to reflect the dynamic response of the system. Due to this feature, the NARX has the following structure: .( )( )( ) ( (1), (2), ., (), ( ), (1), (2), ., ()( )Nyuy tyte tf y ty ty tnu tu tu tu tne t=+=+ (1) Thermal Dynamic Modeling and Control of Injection Moulding Process 105 where y(t), yN (t), u(t) and e(t) stand for the system output, neural network output, system input and error between y(t) and yN(t) at time t, respectively. The inputs are composed of past values of the system inputs and outputs. ny and nu are the number of past outputs and past inputs, respectively and are referred to the model orders. When applying the NARX model to our modeling, the outputs in the NARX model are the temperatures at 4 nodes. However, the rapid change of the temperature during the cycle makes it difficult to use this temperature profile as a set-point and thus alternative variable is required as an output. Instead, cycle average temperature 4 defined in Eq. (2) serves as an output. 00aT.PMnMTdtTP= (2)where t is time, T is the instantaneous temperature at a specific location, P is the cycle-time, M is the total number of samples during one cycle, and Tn is the temperature at the nth sample time. The use of Ta has an advantage of its insensitiveness to process noise. The past states of temperatures at each node (i.e., past outputs) and the flow rate (input) of the coolant are used as inputs to the NARX model. A real plastic moulding process uses the coolant with constant temperature (15.5 oC) and thus, the coolant temperature was utilized as a boundary condition instead of an input in the NARX model. As mentioned before, our modeling is extended to deal with various cycle-times rather than a predetermined cycle-time that previous studies have considered. Therefore, the cycle-time for the moulding process is additionally included as input variable in the model. The NARX model to cover all above inputs (flow rate, cycle-time, past values of outputs at 4 nodes) and outputs (each Ta at 4 nodes) is a multi-input multi-output (MIMO) system. The following table shows inputs conditions for flow rate and cycle-time to generate the steady states of outputs used for training and validation of the NARX modeling. A part of training data sets is demonstrated in Fig. 2, which is generated for node Mo2 using all flow rates in Table 1 and one cycle-time (91 sec) among 4 types of cycle-times. From the figure, it can be seen that for data gathering, each flow rate (input) with a given cycle-time keeps constant until a corresponding steady state of temperature at the node (output) is reached over many cycles 3. Additional data sets for modeling at the same node (Mo 2) were generated by using the same flow rate ranges (0, 1, 3, 4, 5, 6, 8 gpm) with each different cycle-time (i.e., 81, 71, 61 sec). This method was also applied for data generation at the remaining three nodes (Mo1, Mo3, Mo4). Table 1. Input conditions of FEA for NARX modeling Data type Input variables Range Flow rate (gpm) 0, 1, 3, 4, 5, 6, 8 Training Cycle-times (sec) 61, 71, 81, 91 Flow rate (gpm) 0, 2, 7.13 Validation Cycle-times (sec) 66, 86 106 J. Seo, A. Khajepour, and J.P. Huissoon Fig. 2. Training data using all flow rate ranges (0, 1, 3, 4, 5, 6, 8 gpm) and one cycle-time (91 sec) at node Mo2 Next step was to determine the model orders in the NARX model of Eq. (1) by ap-plying the Lipschitz criterion 13 and using the training data sets. The obtained model orders are presented in the schematic of NARX model of Fig.3. For better results and reduction of the calculation time, data was normalized by Eq. (3) before they were used for training and validation process. max,maxminmax( )( )( )( ); .mNorNor mmiiiiitutytutuyyyy= (3)where yNor,i, yimax, yimin are the normalized value, maximum, minimum of the output (temperature), y at ith node, respectively. uNor,m, and ummax are the normalized value and maximum of the mth input, um (cycle-time or flow rate), respectively. ? Temperature at Mo1, Mo2, Mo3, Mo4: Y(t)= yMo1(t); yMo2(t); yMo3(t); yMo4(t)? Neural network output: YN(t)? System inputs:u1(t): cycle-time,u2(t): water flow rate,YN(t-1): past values of YN(t) with one cycle-time delay. Fig. 3. Schematic of NARX model Thermal Dynamic Modeling and Control of Injection Moulding Process 107 The Levenberg-Marquardt (LM) algorithm was adopted as a training algorithm, and the number of hidden layer nodes was determined by trial and error, based on the smallest mean square error (MSE). After training with the training data with 200 epochs, the NARX models performance (i.e., accuracy) was tested with the valida-tion data sets. Using the validation input conditions (from Table 1), the NARX model is validated (see Fig.4). From Fig. 4, it can be noted that our NARX model has a good performance of temperature estimation at all nodes for new input conditions that are not utilized for training data. 4 Controller Design 4.1 Control of Cavity-Wall Temperature The cavity-wall temperature after part ejection can indicate the quality of the final product. However, because the cavity wall is an area where locating sensors is chal-lenging, the temperature measured at 4 nodes can be used instead of the cavity-wall temperature by utilizing the relation between them. From the training data sets used for NARX modeling, a strong linear relation (R=0.787) between an average of four Ta at 4 nodes and cavity-wall temperature right after ejection is found. 02040608010012014016017417410203040 020406080100120140160174174203040020406080100120140160174203040020406080100120140160174174203040YYN Temp.at Mo1 (oC) Temp.at Mo2 (oC) Temp.at Mo3 (oC) Temp.at Mo4 (oC) No. of cycles Fig. 4. Validation results (comparison between model (YN) and actual outputs (Y) Since an average of four Ta is relatively insensitive to noise or unique response which can be generated at specific node, this value is more suitable to use for analyzing the relation with cavity-wall temperature rather than each Ta. When the part reaches the desired demoulding temperature for this material (Santoprene 8211-45: 85 oC), the cavity-wall temperature is around 25 oC, which corresponds to 22 oC for an 108 J. Seo, A. Khajepour, and J.P. Huissoon average of four Ta from the linear relationship. Therefore, 22 oC will be used as an objective value to control the quality of moulding process (or product quality). 4.2 Self-tuning PID with RBF Neural Network As an effective control strategy, neural networks using the radial basis function (RBF) is applied for the design of a self-tuning PID controller. The advantages of RBF neu-ral network include simple structure, faster learning algorithms and effective mapping between a controlling systems input and output 14. This PID controller overcomes a limitation of conventional PID controller that the determination of control parame-ters is not easy in nonlinear or uncertain systems, by the self-learning ability of tuning PID parameters online. Fig. 5. Self-adaptive PID control with RBF neural network Figure 5 describes the structure of self-adaptive PID controller using the RBF neural network. The main function of this control technique is to first identify the plant dynamics and secondly adjust the PID parameters adaptively based on RBF neural network identification. The input vector, output vector in hidden layer and weight vector between hidden layer and output layers in RBF network are respectively, 121212, , ., , , ., , , ., ; ; .TTTnmmXxxxHhhhWwww= (4)where X, H and W are the input vector, output vector in hidden layer and weight vector between hidden and output layers. Each element (hj) of vector H has a format of the Gaussian kernel function given by ()2212exp/ 2 1, . , ; , , ., , ., ,.TjjjjijnjjjXCbjmccccCh= (5)where hj is the Gaussian kernel function, Cj and bj are the center and width of hj. The output of RBF network, my is expressed as: Thermal Dynamic Modeling and Control of Injection Moulding Process 109 1 122.mmmywhw hw h=+? (6)The incremental PID control algorithm is given as follows: 2( )(1)( ) (1)( )( )( ).PIDu ku ku ku kKe kK e kKe k=+ =+ (7)where u(k) is the control variable at the time instant k. Kp, KI and KD (PID parame-ters) are proportional, integral and derivative gains, respectively. e(k)=r(k)-y(k) is the control error; e(k)=e(k)-e(k-1); 2e(k)=e(k)-2e(k-1)+e(k-2). The W, Cj, bj and PID parameters are updated by iterative algorithms using the gradient-descent technique addressed in 15 in which the Jacobian information (i.e., sensitivity of output to controlled input, /( )myu k) identified by RBF network is used for adjusting the PID parameters online as seen in Fig. 5.4.3 Controller Performance By adopting the aforementioned control technique, the performance of designed controller was evaluated using MATLAB/Simulink. The simulation results in Fig. 6 show the variation of average of four Ta (at 4 nodes) in NARX model with the Fig. 6. RBF based self-adaptive PID Controllers performance for multi-setpoints with various cycle-times controller. For the first range (0-6000 sec), the cycle-time of 91 sec (thus 66 no. of cycles) is considered with multi-setpoints of 25 oC and 22 oC (which corresponds to the demoulding temperature). Then, set-point is increased to 24 oC and kept constant during 6000 sec 12000 sec with a different cycle-time of 71 sec (total 85 cycles). 110 J. Seo, A. Khajepour, and J.P. Huissoon Then multi-setpoints of 22 oC and 25 oC are used during 12000 sec 18000 sec with the cycle-time of 61 sec (total 99 cycles). The self-tuning PID controller with RBF shows a good tracking performance for all cycle-time ranges. 5 Conclusion This study proposed an effective thermal control strategy for the plastic injection moulding process. The thermal dynamics was modeled using FEA and NN techniques allowing to tackle the problem of various cycle-times and uncertain dynamics of an injection mould system. Based on the model, a self adaptive PID controller with RBF neural network was introduced. By utilizing on-line learning algorithms to tune con-trol parameters, the self-adaptive PID controller showed accurate control performance for the moulding process with diverse cycle-times. Acknowledgements The author would like to acknowledge the financial support of Natural Sciences and Engineering Research Council of Canada (NSERC) and Ontario Graduate Scholarship (OGS). References 1. Bishenden, W., Bhola, R.: Die temperature control. In: 20th International Die Casting Congress and Exposition, North American Die Casting Association, pp. 161164 (1999) 2. Kong, L., She, F., Gao, W., Nahavandi, S., Hodgson, P.: Integrated optimization system for high pressure die casting processes. Journal of Materials Processing Technology 201 (1-3), 629634 (2008) 3. Dubay, R., Pramujati, B., Hernandez, J.: Cavity temperature control in injection molding. In: