Studies on the prediction of springback in air vee bending of metallic sheets using an artificial neural network

Journal of Materials Processing Technology 108 (2000) 45±54

Studies on the prediction of springback in air vee bending of metallic sheets using an arti®cial neural network M.V. Inamdar*, P.P. Date, U.B. Desai1 Department of Mechanical Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400 076, India Accepted 13 June 2000

Abstract Springback in air vee bending process is large in the absence of bottoming. Inconsistency in springback might arise due to inconsistent sheet thickness and material properties. Among the various intelligent methods for controlling springback, an arti®cial neural network (ANN) may be used for real time control by virtue of their robustness and speed. The present work describes the development of an ANN based on backpropagation (BP) of error. The architecture, established using an analytical model for training consisted of 5 input, 10 hidden and two output nodes (punch displacement and springback angle). The ®ve inputs were angle of bend, punch radius/thickness ratio, die gap, die entry radius, yield strength to Young's modulus ratio and the strain hardening exponent, n. The effect of network parameters on the mean square error (MSE) of prediction was studied. The ANN was subsequently trained with experimental data generated from over 400 plane strain bending experiments using combinations of two punch radii, three die radii and three die gaps and ®ve different materials. Updating of the learning rate and the momentum term was found to be bene®cial. Testing of the ANN was carried out using experimental data not used during training. It was found that accuracy of predictions depended more on the number of training patterns used than on the ANN architecture. A comparison between batch and pattern modes of training showed that the pattern mode of learning was slower but more accurate. # 2000 Published by Elsevier Science B.V. Keywords: Springback; Metallic sheets; Arti®cial neural network

1. Introduction Arti®cial neural networks (ANNs) have found extensive applications in diverse ®elds like manufacturing, signal processing, bio-electric signal classi®cation, pattern recognition, speech recognition, image processing, communications, autonomous vehicle, navigation control of gantry crane [1,2] to name a few. Even in manufacturing, ANN applications to cold forging [3], for predicting the ¯ow stress in hot deformation [4], for tool wear monitoring [5,6], for prediction of machining behaviour [7], and for optimisation of manufacturing processes [8] among many others, are well documented and only a few illustrative references are cited here. The present work relates to the control of the sheet metal bending process in which an attempt is made to restrict springback and consequently the ®nal angle of bend to within a small tolerance. This would lead to a greater * Corresponding author. Present address: Machine Tool Division, Godrej and Boyce Mfg. Co. Ltd., Pirojshanagar, Vikhroli, Mumbai, India. 1 Department of Electrical Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400 076, India.

0924-0136/00/$ ± see front matter # 2000 Published by Elsevier Science B.V. PII: S 0 9 2 4 - 0 1 3 6 ( 0 0 ) 0 0 5 8 8 - 4

consistency in the angles of bend achieved and would be expected to substantially reduce ®tment problems during assembly of these components. For a given set of tooling, springback is in¯uenced by sheet thickness and material properties, inconsistency of which leads to inconsistent springback. Several methods are successfully employed to restrict springback like adaptive control in the machine tool or suitable tool design (bottoming, overbending, etc.). A number of analytical models based on the material properties and tool geometry are available to predict springback, only a few of which are cited here [9±15]. But they invariably involve simplifying assumptions and do not consider the machine tool parameters. Hence, they are not machine-tool-speci®c. In addition, ®nite element analyses used to predict springback accurately [14], take too long for real time control. A number of AI techniques like the knowledge based systems, hybrid intelligent systems may be used to control springback. Compared to the various AI techniques, Neural networks are fault tolerant and robust, are amenable to parallel implementation and are faster than conventional computing.

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M.V. Inamdar et al. / Journal of Materials Processing Technology 108 (2000) 45±54

Nomenclature E K n N rd rp R(n, d) t w W Wji Y

error in ANN prediction, Young's modulus (MPa) constant for updating momentum term strain hardening exponent, number of hyperplanes number of training sets die radius punch tip radius number of regions created by n hyperplanes in a d-dimensional space sheet thickness die gap (mm) total number of synaptic weights Weight from the ith neuron of the previous layer to the jth neuron of the present layer yield strength under uniaxial tension (MPa)

Greek symbols a momentum term b constant for updating learning rate e error permissible over a training set Z learning rate y angle of bend l constant used in updating the momentum term and the learning rate Given the large number of factors that determine springback and allowing for some inevitable variation in each of them, a robust process control tool is called for. Knowledge based systems are rule based and the system might break down completely on violation of a rule. In contrast, in case of an ANN, no catastrophic effect is seen despite breaking down a few interconnections. ANNs are known to generalise better. Unlike expert systems, they can be trained using a set of data to predict to a reasonable accuracy the result for a new set of process parameters. Variations in the test data set can be accommodated by extrapolation. In short, robustness (fault tolerance), speed of computation and better generalisation makes an ANN a viable tool to control springback. It would be a machine speci®c tool which would account for the tool and machine compliances and machine dependent parameters not considered by analytical models. Also, for an ANN trained on one machine, it might probably need few samples to train it on another similar machine. Like adaptive control it would be capable of controlling springback in real time. Unlike adaptive control however, it would complete the operation in a single stroke leading to an increase in productivity. (Adaptive control will need a number of iterations as springback is sensed every time the punch is withdrawn.) Machine setting time required when a material or die geometry is changed will be saved as the new parameters fed

to the ANN will be suf®cient to keep angular dimensions under control. A multi-layer perceptron (MLP) neural network was developed and used for the prediction of springback. The backpropagation (BP) algorithm was used for training the MLP network. The BP algorithm is well tested for training the MLP network. The Newton's method though extensively used in other areas, involves inverting the Hessian matrix and to take care of this inversion Levenberg±Marquardt (L±M) algorithm is required [16], which is completely avoided by the BP algorithm. The ANN based on BP was initially trained using an analytical model (that correlated well with the experimental data) and subsequently with experimental data. The ANN architecture was evolved with the help of an analytical model. Testing of the ANN was performed on experimental data. The sections that follow describe the development of the ANN and bring out its capabilities. 2. Experimental procedure The ANN was trained as well as tested with the experimental data generated using the setup shown in Fig. 1. The tool permitted easy changes in the die gap. The sheet was not clamped and was free to swivel with increasing angle of bend. The angle under load (before springback) was measured on the external surface of the bend arm (Fig. 1) and that after springback was measured using a bevel protractor. The difference between the two angles was used to determine springback of the included angle. The attempted included angle ranged between 69 and 126.58. The punch travel was measured using a dial indicator having a count of at least 0.01 mm. Five materials, namely, commercially pure aluminium, aluminium alloy, high tensile steel, mild steel and deep drawing quality steel sheet were used in the study. They were characterised by the tensile properties, which are given in Table 1. Experiments on bending were performed. In all, about 432 sets of data were generated from 35 mm wide specimens from the ®ve materials along the rolling direction bent over

Fig. 1. Schematic of the experimental setup illustrating the procedure for the measurement of the bend angle under load.

M.V. Inamdar et al. / Journal of Materials Processing Technology 108 (2000) 45±54

47

Table 1 Characteristics of materials used for experimental investigations Material

Thickness (mm)

E (GPa)

sy (MPa)

K (MPa)

n

High tensile steel (HTS) Mild steel (MS) Aluminium alloy (NA) Deep drawing steel (DDS) Commercially pure aluminium (CA)

0.75 0.8 0.83 1.0 1.35

220 201 65 201 69

207 141.4 126.21 220.33 77.5

696.61 564.7 601.16 674.4 155.3

0.24 0.27 0.327 0.28 0.0705

three die radii (2, 4 and 8 mm), three die gaps (20, 30 and 40 mm) and two punch radii (2 and 4 mm). Representative data points generated for commercially pure aluminium are shown in Table 2. 3. Inputs to the ANN The major parameters in¯uencing springback were identi®ed to be [14,15]: (a) yield strength (Y); (b) Young's modulus (E); (c) strain hardening exponent (n); (d) die gap (w); (e) die radius (rd); (f) punch radius (rp); (g) sheet thickness (t); and (h) angle of bend (y). The effect of other parameters having comparatively minor in¯uence on springback like normal anisotropy, friction, etc. were not considered as inputs to the ANN. The parameters of interest, namely, the punch displacement and springback angle were the output from the ANN. The eight input parameters were combined into ®ve dimensionless parameters, namely: (i) ratio of yield strength to Young's modulus, Y=E; (ii) strain hardening exponent, n; (iii) ratio of punch radius to sheet thickness, rp =t; (iv) die gap to die radius ratio, w=rd ; (v) the angle of bend, y. In view of the fact that the input and the output parameters are related by a continuous function, only one hidden layer

was considered to be adequate. The method for calculating the number of hidden nodes was based on the method of separating (classifying) M clusters of points into a d-dimensional space using hyperplanes [2]. The method is brie¯y described in Section 4. 4. Architecture and training of the ANN 4.1. Calculation of the number of hidden nodes Each neuron in the hidden layer is equivalent to a hyperplane in a d-dimensional space. The number of hidden nodes is a function of the number of clusters to be classi®ed [17]. In the worst case, Mÿ1 hidden nodes might be required to separate M classes in a two-dimensional space so that every adjacent point belongs to a different class. The number of clusters were assumed to be equal to the number of training sets (i.e., number of experiments). The number of regions R(n, d) created by n hyperplanes in a d-dimensional space is given by [18]: R…n; d† ˆ

d X iˆ0

n! i!…n ÿ i†!

(1)

Table 2 Sample experimental springback data for commercial aluminium Punch radius (mm)

Die gap (mm)

Die radius (mm)

Attempted bend angle (degree)

Punch displacement (mm)

Springback angle (degree)

4 4 4 4 4 4 4 4 4 4 4 4 2 2 2 2 2 2

20 20 30 30 40 40 20 20 30 20 30 40 20 20 30 40 40 40

2 2 2 2 2 2 4 4 4 8 8 8 2 4 4 4 8 8

107 92 105.5 82 105.5 86 119 86.5 100 107.5 81.5 89.5 117.5 113 84.5 85 86.5 91.5

8.07 9.99 12.51 18.01 17.00 23.00 8.25 12.00 13.5 9.98 20.00 24.10 7.07 8.00 18.00 24.01 25.56 23.5

8.7 11.38 7.93 8.75 10.92 11.17 6.56 6.89 8.13 7.8 10 10.91 7.12 7.15 11.99 11.63 12.76 11.89

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M.V. Inamdar et al. / Journal of Materials Processing Technology 108 (2000) 45±54

where n would be equivalent to the number of hidden nodes. The size of the training set to obtain good generalisation from the network is related to the fraction of permissible error e over a training set. The expression was suggested by Haykin [18] as     32W 32W ln (2) N e e where N is the number of training sets, 2e the fraction of error permissible over a training set and W the total number of synaptic weights. It was found that Eq. (2) represents the worst case formula (suggesting too large number of training patterns for a given level of accuracy) and the necessary level of accuracy could in fact be obtained with fewer number of training patterns given by [19]: N>

W e

(3)

In view of Eqs. (2) and (3), the number of hidden nodes must be consistent with (a) the maximum number of regions to be generated by n hidden nodes and (b) having suf®ciently large number of synaptic weights for achieving the acceptable level of the error fraction from N training data. The number of regions R(n, d) to be generated by n hyperplanes (hidden nodes) may thus be seen as the number of training patterns to be separated by the n hidden nodes. The number of hidden nodes is therefore calculated from Eqs. (1) and (3) so that d X iˆ0

n! W ˆ i!…n ÿ i†! e

(4)

Solving Eq. (4) using the Newton±Raphson method, the number of hidden nodes was found to be 10, for accuracy e ˆ 0:1 (10% error) and 700 experiments. Hence, the ANN architecture of 5-10-2 was arrived at. 4.2. Training of the ANN An ANN code based on the BP learning algorithm with the momentum term was developed. The learning rate and the momentum term were updated every next iteration using the following equations: DZ ˆ KZ eÿbZ DE

(5)

Da ˆ ÿla a

(6)

Da ˆ Ka eÿba DE

(7)

DZ ˆ ÿlZ Z

(8a)

where values KZ ˆ 3, bZ ˆ 10 000, lZ ˆ 1%, Ka ˆ 1, ba ˆ 5000, la ˆ 0:5% were chosen by trial and error. Eqs. (5) and (6) were used when E…n† < E…n ÿ 1† and Eqs. (7) and (8a) were used when E…n† > E…n ÿ 1†. E(n) and E(nÿ1) are the mean squared error at the nth and the (nÿ1)th

iteration, respectively, de®ned as Eˆ

o X …dk ÿ Ok †2

(8b)

kÿ1

where dk is the desired output and Ok the actual output at the output node k and o is the number of output nodes. Values of Z and a are changed in the opposite directions. Increasing both simultaneously saturates the neurons, while decreasing both leads to a low learning rate. The ANN was subsequently trained to give springback and punch travel predictions determined by an analytical model developed [20] and using values of the input parameters from the literature [11,14]. This data which would also be devoid of noise was randomised to avoid biasing the ANN. The weights were initialised between ÿ0.5 and ‡0.5 using the following relation [22]: Wji

50500:45 ÿ rand…† 100001 ÿ 999:9

(9)

where rand() is the C function which generated a random number. 4.3. Batch mode vs. pattern mode of training Both pattern and batch modes of training were attempted [21,22]. The batch mode was found to converge faster than the pattern mode (in fewer iterations) for training using the model data. The generalisation of the model data was found to be better in pattern mode than in batch mode (Fig. 2a±c). An analytical model [20] was used to generate the training as well as the testing data. The tool geometry and the material properties used as inputs to the model were obtained from [14] and are listed in Tables 3 and 4. For each of the 10 types of bend (V1±V10, Table 3), 52 sets of data were generated. This yielded a total of 520 points for training with the BP algorithm. The BP algorithm was considered to have converged when the average squared error Eav was less than e2, the square of the permissible error. The speed of convergence was found to depend upon the initial weights and the range of normalisation. To improve the speed of convergence after the weights were initialised (using Eq. (9)) the following heuristic was used [23]: Wji …NEW† ˆ

Wji …OLD† jWj j

(10a)

Here, |Wj| is the average of the absolute values of weights to the jth neuron, i.e., ( ) I 1 X jWji j (10b) jWj j ˆ I iˆ1 where I ˆ number of nodes in the previous layer and Wji the synaptic weight between ith neuron from the previous layer to the jth neuron of the current layer.

M.V. Inamdar et al. / Journal of Materials Processing Technology 108 (2000) 45±54

The input data was normalised between 0.2 and 0.8 using the following equation: x…norm† ˆ

X ÿ X…min†  0:7 ‡ 0:2 X…max† ÿ X…min†

(11)

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A comparison between predictions of the network trained using the pattern mode and batch mode based on the test inputs to the analytical model (unknown to the network) is given in Tables 5 and 6. It may be observed that the network trained by the pattern mode of learning could map the input

Fig. 2. Generalisation of data in pattern and batch modes of training: (a) V1 type bend, (b) V5 type bend, (c) V8 type bend (also see Tables 3 and 4).

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Fig. 2. (Continued ) Table 3 Geometric parameters as input to the analytical model for training the ANN [14]

Table 5 Comparison of pattern mode and batch mode of learning of the ANN based on test data from analytical model: springback

Serial No.

Type of bend

rp (mm)

rd (mm)

Material

Pattern mode

Batch mode

1 2 3 4 5 6 7 8 9 10

V1 V2 V3 V4 V5 V6 V7 V8 V9 V10

6.0 6.0 3.0 3.0 3.0 6.0 3.0 6.0 3.0 6.0

15.0 15.0 15.0 15.0 15.0 6.0 6.0 15.0 15.0 15.0

M2 M3 M2 M3 M2 M2 M2 M4 M4 M5

Springback (degree) analytical model [20] 2.072 2.066 2.054 2.042 2.003 1.98 1.92 1.884 1.81

1.945 1.937 1.923 1.909 1.867 1.841 1.768 1.725 1.645

1.927 1.923 1.914 1.904 1.864 1.839 1.769 1.729 1.66

onto the output with a higher level of accuracy for springback as well as punch displacement. 4.4. Effect of network parameters The effects of the network parameters namely, momentum (a), learning rate (Z) and the number of nodes in the

hidden layer (n) on the mean square error (MSE) were studied in the pattern mode of training for 10 000 iterations using the training data from the analytical models. Four initial values of momentum a, namely, 0.2, 0.4, 0.6 and 0.8 were considered for Z ˆ 0:5 and a 5-10-2 ANN architecture. The results are given in Fig. 3(a). The effect of the learning rate on the MSE was studied using initial values of 0.2, 0.4,

Table 4 Material characteristics as input to analytical model for training the ANN [14]a Material

Thickness (mm)

Yield stress (MPa)

R

Stress±plastic strain relations

M2 M3 M4 M5

3 6 3 3

246.7 264.3 246.7 246.7

1.0 1.0 1.0 1.8

s ˆ 550:1…0:0116 ‡ ep †0:18 s ˆ 549:97…0:009 ‡ ep †0:15534 s ˆ 550:1…0:0116 ‡ ep †0:36 s ˆ 550:1…0:0116 ‡ ep †0:18

a

E ˆ 205 800 MPa.

M.V. Inamdar et al. / Journal of Materials Processing Technology 108 (2000) 45±54 Table 6 Comparison of pattern mode and batch mode of learning of the ANN based on test data from analytical model: punch displacement Punch displacement (mm) analytical model [20]

Pattern mode

Batch mode

17.512 16.009 14.979 14.005 13.377 13.034 11.989 11.256 9.801

17.858 16.011 14.876 13.854 13.201 12.843 11.734 10.944 9.38

17.993 16.853 15.95 15.009 14.365 14.003 12.867 12.06 10.5

0.6 and 0.8 for Z with a ˆ 0:1 for a 5-10-2 ANN architecture. The results are given in Fig. 3(b). The effect of the number of hidden layer nodes was studied for n ˆ 6, 8, 10, 12 and 14 with Z ˆ 0:5 and a ˆ 0:1. Fig. 3(c) shows the result. It may be inferred from Fig. 3(a) that an increase in momentum leads to a faster decrease in the MSE. All curves (except that for a ˆ 0:2) are seen to approach a certain minimum level of the MSE after 10 000 iterations. The number of iterations required to reach this MSE level is found to decrease linearly with an increase in the momentum. The MSE for a ˆ 0:2 would then need more than 10 000 iterations and hence a higher MSE was observed in this case. A higher learning rate on the other hand leads to a lower MSE for 10 000 iterations. A greater learning rate speeds up the reduction in MSE. This is observed in Fig. 3(b). Fig. 3(c) shows how the number of hidden layer nodes in¯uence the rate of decrease in the MSE. The MSE for n ˆ 10 was found to be marginally lower than the others after 10 000 iterations. It is known that neither too few, nor too many nodes achieve a low MSE since too few nodes would lead to inadequate training while too many of them would cause poor generalisation. Based on the results above the ANN architecture of 5-102 was chosen for training and testing using the experimental data.

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Table 7 Sample experimental data learned by the ANN (commercial aluminium) Experimental punch displacement (mm)

Experimental springback (degree)

Punch displacement from trained ANN (mm)

Springback from trained ANN (degree)

16.18 8.01 14.05 22.72 25.56 9.51 12.05 18

10.3 7.15 10 12.18 12.76 8.53 9.6 11.99

15.96 8.124 14.23 19.38 24.87 9.98 11.83 18.39

11.33 8.376 10.99 12.1 13.5 8.94 10.54 12.79

4.5. Training of the ANN with experimental data The ANN was thereafter trained with the experimental data which was not ®ltered. It was normalised between 0.2 and 0.8. The parameter 1=…rd ‡ w† was found to be more appropriate than the dimensionless rd =w as the effect of the two tool parameters on springback are similar. Hence, ratio …rd =w† was replaced by 1=…rd ‡ w†. While training with the experimental data some stray points occurred which were removed from the data ®le and the training was continued further. Fig. 4 shows the accuracy of learning of the ANN after the fourth stage (i.e, after a total of 0.3 million iterations, each stage consisting of 75 000 iterations followed by the removal of stray points). Table 7 shows sample of experimental training data learned by the ANN for commercial aluminium. Fresh experimental data (about 24 patterns) were generated for testing. A comparison of the punch displacement and the springback angle predicted by the ANN with the experimentally measured values is given in Table 8. 5. Conclusions The present work demonstrates the application of an ANN to the problem of controlling springback in air-vee bending.

Table 8 Comparison of predictions of the ANN with experimental observations: test data Die gap (mm)

Die radius (mm)

Punch radius (mm)

Punch d isplacement, ANN (mm)

Springback angle ys, ANN (degree)

Observed punch displacement (mm)

Observed included bend angle, y (degree)

Observed springback angle (degree)

20 24 24 24 20 30 34 34 34

2 4 6 4 6 6 6 4 4

4 4 4 2 2 2 2 4 2

10.22 13.10 14.10 13.37 12.4 17.62 21.05 19.47 19.91

9.35 8.68 8.8 8.2 8.04 9.8 10.83 10.26 10.6

10.22 13.00 14.01 13.00 12.01 17.2 21.0 19.1 19.00

88.5 88.0 89.5 90.00 90.00 91.00 89.0 89.5 89.0

9.78 8.39 8.77 8.17 8.37 9.06 11.1 8.77 9.84

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M.V. Inamdar et al. / Journal of Materials Processing Technology 108 (2000) 45±54

Fig. 3. Effect of network parameters on MSE: (a) momentum (a); (b) learning rate (Z); (c) number of hidden layer nodes (n).

The ANN developed based on the BP algorithm with updating of the momentum term and the learning rate is capable of predicting springback (with relatively few training patterns) within 28 for a desired bend angle of 908. Better

predictions (much lower deviation from the desired angle) could be obtained using a larger number of training patterns. As for the tool geometry, 1=…rd ‡ w† was found to be a more appropriate form of input than rd =w (despite being

M.V. Inamdar et al. / Journal of Materials Processing Technology 108 (2000) 45±54

53

Fig. 4. Error in mapping for punch displacement after fourth stage of training (see text).

non-dimensional). An increase in either rd or w has a similar effect on springback and hence the form 1=…rd ‡ w† would re¯ect the effect (on springback) of an increase in rd or w or both better than the form rd =w. The former was used for training with experimental data as well as for testing. During the development of the ANN the following observations could be made: 1. The pattern mode of training was found to be in¯uenced by randomisation of data and also gave smaller prediction error than the batch mode of training, though it took a little longer to train. The in¯uence of randomisation on the pattern mode of training is well understood as randomisation would affect the process of updating of weights, an event occurring once for every pattern. 2. Empirically it was observed that one got better prediction by increasing the number of training patterns, rather than modifying the ANN architecture. This is clear from Fig. 3(c) which shows a relatively small effect of the number of nodes in the hidden layer on the MSE. 3. Network parameters like Z and a were found to in¯uence the MSE: Larger initial values of learning rate Z led to a faster decrease in the MSE. Similar was the case with the momentum parameter a. Moreover, larger values of a (for a ®xed value of Z) and large values of Z (for ®xed values a) led to lower MSE values at the end of 10 000 iterations.

4. Change in the values of the learning rate Z and momentum a should be small (1±2% of the initial value) during training. Larger values disturb the training process. 5. The values of punch displacement (for a ®nal included bend angle of 908) predicted by the ANN were found to be very close to the observed punch displacement for a test tool geometry completely different from that used for generating the experimental training data. Thus, the proposed ANN is capable of generalisation. Acknowledgements The stimulating discussions with Prof. R. Raghunathan, Department of Chemical Engineering, IIT Bombay, and encouragement from Prof. K. Narasimhan, Department of Metallurgical Engineering and Materials Science, are thankfully acknowledged. References [1] N.K. Bose, P. Liang, Neural Network Fundamentals With Graphs, Algorithms and Applications, Tata McGraw-Hill, New Delhi, 1998, pp. 407±440. [2] K. Mehrotra, C.K. Mohan, S. Ranka, Elements of Arti®cial Neural Networks, MIT Press, Cambridge, MA, 1997. [3] K. Osakada, Y. Guobin, Application of neural networks to expert system for cold forging, Int. J. Mach. Tool Manuf. 31 (1991) 577±587.

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