CHAPTER 5 PREDICTIVE MODELING STUDIES TO DETERMINE THE CONVEYING VELOCITY OF PARTS ON VIBRATORY FEEDER

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1 93 CHAPTER 5 PREDICTIVE MODELING STUDIES TO DETERMINE THE CONVEYING VELOCITY OF PARTS ON VIBRATORY FEEDER 5.1 INTRODUCTION The development of an active trap based feeder for handling brakeliners was discussed in Chapter 4. The appropriate gate and feeder input parameters such as track angle, trap angle, excitation frequency (f) and amplitude of vibration (A) for maximum smooth conveying velocity were determined experimentally. In this chapter, predictive models to determine the conveying velocity of parts based on the gate and trap parameters was developed. In industries, there is a need for such models to set the input parameters based on the desired conveying velocity to maintain a continuous flow of parts. A theoretical model, regression model and Artificial Neural Network (ANN) model were developed to predict the conveying velocity of the part on the trap. The limitations of theoretical model and comparison of regression and ANN models with experimental results are discussed. The development of theoretical model is explained in the next section. 5.2 DEVELOPMENT OF THEORETICAL MODEL The analysis of part motion on a vibratory feeder was done by Lim (1997), Keraita (2008) and Ramalingam and Samuel (2009) and discussed in section 2.4. Lim (1997), in his analysis, assumed that the forcing element in case of a vibratory feeder would provide Simple Harmonic Motion (SHM). The same assumption was made here in developing the theoretical model as well. An introduction to representation of the SHM for mathematical modeling is found in book by Rao (2000). The SHM of the trough used in a vibratory feeder was

2 94 discussed by Ramalingam and Samuel (2009). Similarly, a typical harmonic motion of the trap on the feeder was developed and is shown in the Figure 5.1. The vibration force was provided to the horizontal feeder at an angle e the line of action of vibrational force was along PQ in Figure 5.1. f Figure 5.1 SHM horizontal to the Since the motion was SHM, the length of OP, OQ and OR represent the amplitude of vibration. The amplitude of vibration is denoted as A and frequency of vibration as f (as discussed in Chapter 4). As the plane of vibration is along the line PQ, the line of motion of the feeder also lies along the line PQ. Considering the initial position of the feeder to be at the point Q and after time period, it was displaced to point S along the line of motion PQ. This displacement of the feeder is given by SQ, which is represented as d f (Figure 5.1), inclined at an angle with the horizontal. The horizontal and the

3 95 vertical components of the displacement d f are denoted as X df and Y df respectively as shown in Figure 5.1. The harmonic motion could be represented by vector of magnitude A rotating at constant angular with the plane of vibration PQ (Rao 2000), as shown in Figure 5.1. The angular velocity, = 2 f and hence, where, f excitation frequency, Hz t time, s Considering ORS as highlighted in Figure 5.2, From Equation (5.1), (5.1) (5.2) (5.3) f Figure 5.2 ORS highlighted Considering the encircled section in Figure 5.3, SQ = d f = OQ-OS Since, OQ=A and OS= A cos2 ft (5.4)

4 96 Figure 5.3 Line OSQ highlighted The horizontal and the vertical components of feeder displacement (encircled in Figure 5.4) are derived as follows: (5.5) From Equation (5.4), Figure 5.4 STQ highlighted (5.6) (5.7) (5.8) The horizontal velocity component ( of the feeder (i.e conveying velocity) is determined by differentiating the Equation (5.6) (5.9) The maximum velocity of feeder in horizontal direction is obtained when sin 2 ft = 1, in Equation (5.9)

5 97 (5.10) The maximum velocity of the feeder obtained matches with the equation derived by Ramalingam and Samuel (2009). It is found that the direction of vibration force ( ) does not affect the conveying velocity of the feeder. The acceleration of the feeder ( obtained by double differentiating Equation (5.4), ) in the direction of oscillation is (5.11) The acceleration obtained matches with the equation derived by Ramalingam and Samuel (2009). It is found that the acceleration is not affected by the direction of vibration force ( ). From Equation (5.9), it could be observed that the horizontal velocity and trap angle (as discussed in section 4.4) could not be included in the 2D representation of SHM of feeder. So, regression analysis was done to have a model to predict the conveying velocity based on f and A, which is discussed in the next section. 5.3 DEVELOPMENT OF REGRESSION MODEL Regression is a procedure, which selects, from a class of functions, the one which best fits a given set of empirical data. The application of regression analysis is elaborated in section 2.4. The popularity of regression is due to its applicability to different types of problems, easiness in interpretation, robustness to violations of the underlying assumptions and widespread availability (Mason and Perreault 1991). When two or more quantitative variables are used to predict the quantitative response variable, then it is termed as multiple regression. The conveying velocity of the part

6 98 on the trap was the dependent variable whereas the track angle, trap angle excitation frequency (f) and amplitude of vibration (A) were the input variables. The velocity of orientations 3, 4, 7 and 8 were only considered because of reasons discussed in section The conveying velocity of orientation 3, 4, 7 and 8 on the trap were denoted as V3, V4, V7 and V8 respectively. The experimental results as discussed in section were used to develop the regression model. The model using Minitab software is as follows: R 2 = 82.4% (5.12) R 2 = 83.1% (5.13) R 2 = 81.3% (5.14) R 2 = 84.2% (5.15) where, R 2 - percent of total variation that could be explained by the regression equation. is track angle (degree) is trap angle (degree) f is excitation frequency (Hz) A is amplitude of vibration (% of input voltage).

7 99 The development of ANN model is discussed in the next section and the comparison of regression and ANN model results with the experimental values is discussed in section DEVELOPMENT OF ANN MODEL The significance of ANN and its application in predicting the response variables is discussed in section 2.4. Neural networks donot get stuck in local minima and could be trained faster to converge (Eskandari et al 2004). The parameters of ANN proposed by Yusoff and Aziz (2004) were used for developing the ANN model to predict the conveying velocity of the part in orientations 3, 4, 7 and 8 and are listed in Table 5.1. The learning rate of 0.01 and momentum 0.95 were used to enable faster simulation. The weights and bias were initialized randomly. The simulations were stopped after it reached 1000 maximum epoch or when square error reached a value of between the actual and predicted value. The program was developed using Matlab software. The network was trained using the experimental results discussed in the section Table 5.1 Parameters for developing ANN model Structure Feed Forward Algorithm Back propagation Type of Training Trainlm Network structure 15,h and l Transferfunction TANSIG Number of iterations 1000(max epoch) Performance function MSE = Data division Random Learning rate 0.01 Momentum 0.95

8 100 The input combinations of track angle, trap angle, excitation frequency (f) and amplitude of vibration (A) were fed as input data in the Matlab software. The corresponding output value (conveying velocity of orientations 3, 4, 7 and 8) were fed as target to develop the network model using the parameters listed in Table 5.1 and the ANN architecture is shown in Figure 5.5. Input layer Hidden layer Output layer Trap angle Track angle Conveying velocity Excitation frequency Amplitude of vibration Figure 5.5 ANN architecture To determine the number of neurons in the hidden layer (Figure 5.5), trial and error method was followed (Ricca et al 2012). ed the correlation between outputs and targets. f 1 meant a closer relationship (Ricca et al 2012). The number of neurons had to be selected such 1.The training, validation and test data samples for determining the number of neurons in the hidden layer is listed in Table 5.2.

9 101 Table 5.2 Test and validation data S.No Description % of samples No.of samples 1 Training 70 % Validation 15 % 38 3 Testing 15 % 38 The training samples were provided to the developed network during training phase and based on the error, the network was adjusted. The validation samples were provided to measure network generalization and to stop the training when the generalization fails to improve. The testing samples were provided to have an independent measure of performance of network during and after training. The hidden neurons were initially varied from 10 in increments of 10 (i.e, 10, 20, The by varying the hidden neurons is shown in Figure overall R value No.of neurons Figure 5.6

10 102 From Figure 5.6, it could be inferred that hidden neuron size of 50 had (shown in dashed line) and hence 50 was chosen as the appropriate number of hidden neuron. On increasing the neurons in hidden layer and hence the test was stopped at neuron size of 70. Using this, the ANN network was again created and trained. An ANN model capable of predicting the conveying velocity based on input parameters was thus ready. The comparison of conveying velocity values predicted by regression and ANN models were compared with the experimental results in the next section. 5.5 COMPARISON OF EXPERIMENTAL RESULTS WITH REGRESSION AND ANN MODEL RESULTS To validate the developed regression and ANN models, twelve random sets of input values as listed in Table 5.3 were considered. Table 5.3 Set of input values considered for validation Experiment set no. Track angle, (degree) Trap angle, (degree) Excitation frequency, f (Hz) Amplitude, A (% of input voltage)

11 103 Table 5.4 to Table 5.7 list the conveying velocity of parts in orientations 3, 4, 7 and 8 predicted by regression and ANN models along with the percentage deviation (%) with the experimental values. The average deviation of results predicted by regression and ANN models for orientation 3 is 16.56% and 2.5% respectively (Table 5.4). Similarly, the average deviation of results predicted by regression and ANN models for orientation 4 is 14.01% and 2.8 % respectively (Table 5.5). 24.8% and 3.72% were the average deviation of results predicted by regression and ANN models respectively for orientation 7 (Table 5.6). Similarly, 10.42% and 3.34% were the average deviation of results predicted by regression and ANN models respectively for orientation 8 (Table 5.7). These models will be of great use in industries to match the conveying velocity of parts with the subsequent processes to maintain an uninterrupted flow. Exp Set. No Table 5.4 Comparison of conveying velocity of orientation 3 Conveying Velocity, V3 (mm/s) Deviation % Experiment Regression ANN Regression ANN Average

12 104 Orientation 3 60 Conveying velocity, mm/s CONVEYING VELOCITY V3 EXPERIMENT CONVEYING VELOCITY V3 REGRESSION CONVEYING VELOCITY V3 ANN Experimental set no. Figure 5.7 Comparison of experimental conveying velocity of orientation 3 with results predicted by regression and ANN models Exp Set. No Table 5.5 Comparison of conveying velocity of orientation 4 Conveying Velocity, V4 (mm/s) Deviation % Experiment Regression ANN Regression ANN Average

13 105 Orientation 4 60 Conveying velocity, mm/s CONVEYING VELOCITY V4 EXPERIMENT CONVEYING VELOCITY V4 REGRESSION CONVEYING VELOCITY V4 ANN Experimental set no Figure 5.8 Comparison of experimental conveying velocity of orientation 4 with results predicted by regression and ANN models Exp Set. No Table 5.6 Comparison of conveying velocity of orientation 7 Conveying Velocity, V7 (mm/s) Deviation % Experiment Regression ANN Regression ANN Average

14 106 Orientation 7 70 Conveying velocity, mm/s Experimental set no CONVEYING VELOCITY V7 EXPERIMENT CONVEYING VELOCITY V7 REGRESSION CONVEYING VELOCITY V7 ANN Figure 5.9 Comparison of experimental conveying velocity of orientation 7 with results predicted by regression and ANN models Exp Set. No Table 5.7 Comparison of conveying velocity of orientation 8 Conveying Velocity, V8 (mm/s) Deviation % Experiment Regression ANN Regression ANN Average

15 107 Orientation 8 60 Conveying velocity, mm/s Experimental set no CONVEYING VELOCITY V8 EXPERIMENT CONVEYING VELOCITY V8 REGRESSION CONVEYING VELOCITY V8 ANN Figure 5.10 Comparison of experimental conveying velocity of orientation 8 with results predicted by regression and ANN models From the results, it could be inferred that ANN model was able to predict the results much closer to the experimental values than the regression model. This is in agreement with Abounoori and Bagherpour (2007), who also concluded that ANN could predict results much better than regression models. Since, ANN could make rules without any implicit formula, they were able to predict the results much accurately than the regression equations (Ahangar et al 2010). Further, ANN are robust to noise in the training data and execute faster than regression. Though the regression models could not predict the results much accurately as that of ANN models, they follow the same pattern as that of the experimental results which is evident from Figure 5.7 to Figure 5.10.However, the limitation is that, a model in the form of an equation (as that of regression model) is not available in ANN model.

16 CONCLUDING REMARKS In this chapter, a theoretical model, regression model and ANN model were developed to predict the conveying velocity of part. The theoretical model could not accommodate the effect of track angle, trap angle and orientation of the part, since it was developed based on 2D free body diagram of the feeder. The predictive models to determine the conveying velocity of part orientations 3,4,7 and 8 amplitude (A) were developed using regression and ANN. The results predicted by the models were compared with the experimental results. The average deviation of regression results with the experimental results for the part orientations 3, 4, 7 and 8 were found to be 16.56%, 14.01%, 24.8% and 10.42% respectively. The average deviation of ANN results with the experimental results for the part orientations 3, 4, 7 and 8 were found to be 2.5%, 2.8%, 3.72% and 3.34% respectively. From the results, it could be inferred that ANN model was able to predict the results much closer to the experimental values than the regression model. ANN could make rules without any implicit formula and hence were able to predict the results much accurately than the regression equations. These models will be of great use in industries to match the conveying velocity of parts with the subsequent processes to maintain a continuous flow. Though the regression models could not predict the results much accurately as that of ANN models, they follow the same pattern as that of the experimental results.

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