Approaches to Simulate the Operation of the Bending Machine

Transcription

1 Approaches to Simulate the Operation of the Bending Machine Ornwadee Ratanapinunchai Chanunchida Theerasilp Teerut Suksawai Project Advisors: Dr. Krit Chongsrid and Dr. Waree Kongprawechnon Department of Electrical Engineering, Sirindhorn International Institute of Technolog, Thammasat Universit (Rangsit Campus), Pathumthani, Thailand Abstract The stud is aimed to simulate the operation of a bending machine. It creates a core concept used to invent an automatic bending machine. Being a prototpe model, the task is to generate its duplication b simulating the operation of the bending machine via MATLAB. Two approaches are introduced for the simulation. The first approach is to move the blade so that an entirel new segment is bended each time, while the second approach moves the blade b a smaller amount each time. Therefore the second approach involves a more complicated situation, since there is an initial curve of the blade before bending. Lastl, the results from these two approaches will be compared (in order to create) the best algorithm. 1. Introduction Knowing the basic operation of the automatic bending machine is essential for understanding this report. Referring to Figure 1.1, two DC servomotors are used to perform the operation. The feeder motor feeds the work piece in the direction shown b the arrows, while the bender motor controls the movement of the bender and the supporter. These two motors must work correspondingl. The bender motor has to wait until the feeder motor stops feeding the work piece. When the feeder motor stops at the desired position, the bender motor will now start to bend the work piece. Thus, position control is needed to control the rotation of both motors. The shape of the work piece depends on the orientation of the bender as well as its pressing distance. Figure 1.2 shows the direction of the bender and its corresponding shape. One application of the automatic bending machine is to generate various shapes of the cutting blade. Since cutting blades are widel used in Footwear and Leather industries, the insufficient capacit of the blade production often delas the entire production line. Therefore the automatic bending machine becomes an important tool to increase the blade productivit. The purpose of this stud is to simulate the bending operation for an automatic bending machine. The whole bending operation can be viewed as a function that bends the blade b one segment at a time. Two approaches are introduced and their results will be compared. The difference between the two approaches is the amount that the blade moves after getting pressed. The first approach moves the blade for an entire segment until there is onl a straight part of the blade that comes in between the supporter. Within this first approach, two models are proposed to represent the shape of the bended blade; the quadratic curve and V- curve. The second approach reduces the amount that the blade moves each time. Its

2 result is expected to show a higher accurac and smoothness of the curve. Recentl, the complexit of the hardware and control algorithm has not been considered. It is assumed that once the blade is bended, it stas that shape without an springback deformation. Another assumption is that, even though the length of the blade segment increases after bending, its effect is neglected. We ve introduced this idea onl briefl, however the next sections will describe each approach (more thoroughl). The detailed explanation of the first approach and its result is provided in Section 2, while the overall algorithm for the second approach is discussed in Section 3. The conclusion and recommendation for further research on this stud are provided in Section 4 and 5, respectivel. MATLAB programming can be found in Appendix C. 2. First Approach The first approach is to simulate the bending of the blade, one segment at a time. Once a segment of the blade is bent, an entirel new segment is moved in between the two supporters and is read for the next press. This wa there will be no initial curve on the blade before each press. The shape of each segment after bending will be modeled into two shapes. One of the proposed models assumes that the blade will have a quadratic shape after bending. Another one assumes that the blade will have a V shape with a round tip at the contact spot with the bender, as shown in Figure 2.1. The overall algorithm of the approach and the description of each step are explained in Section 2.1 to 2.4. The results and comparison between the two models are discussed in Section 2.5. Figure 1.1 Automatic bending machine Concave curve Convex curve Figure 1.2 Orientation of the bender and its corresponding shape. Figure 2.1 V-shaped model of the bended blade. 2.1 Overall Algorithm Initiall, the prototpe model of the blade must be represented mathematicall. The entire model is divided into small segments. After obtaining the coordinates for each segment of the prototpe, the straight blade is pressed down until the proposed model of the bended blade best fits the prototpe model. The amount that the blade is pressed down is represented b k. It is aimed to find the best value of k, which would minimize the error between the proposed model and the prototpe model. The two proposed models, quadratic and V-curve, are modeled via a mathematical mean. Lastl, the two models of each segment are plotted on the same graph as the prototpe model for comparison. After finishing with one segment, the same process is done to its adjacent segment to complete the duplication of the prototpe. The flow chart of this bending algorithm is shown in Figure 2.2.

3 Figure 2.2 shows an overall bending algorithm. The variable n represents the segment that is undergoing the operation. The distance in which the bender is pushed down is represented b the variable k. The first task is to analze the data from the prototpe, whose steps are explained in Section 2.2. After that, the methods of modeling the bended blade are described in Section 2.3. After obtaining the model, it is essential to validate the model b comparing it to the prototpe. Section 2.4 provides this comparison method. the position of these points with respect to the supporters. Therefore the points in each segment must be transformed to the new axes, where its x-axis lies on the two ends of each supporter, as shown with a dash line in Figure 2.3. It is shown that the new axis is rotated b θ from the original axis and that its origin is translated b vector (x o, o ) from the old origin. With this information, a transformation matrix (T) can be easil established, as shown in Equation 1. This matrix is used to transform the set of coordinates into the new axis as shown in Equation 2. [1] cosθ sinθ xo T = sinθ cosθ o xnew xoriginal 1 new = T original 1 1 (1) (2) Figure 2.2 Algorithm for the first approach. 2.2 Analzing the Prototpe The information needed from the prototpe is given in the set of coordinates. The model can be drawn in the AutoCAD, since the program allows us to fetch these coordinates to be used in MATLAB. The prototpe model in Figure 2.3 is divided into man segments. Each segment is comprised from a different numbers of points. One segment has a width equal to the width between the two supporters (w). Coordinates in each segment are measured according to a common axis as shown with the bold line in Figure 2.3. However, it is necessar to know Figure 2.3 Prototpe model. 2.3 Modeling the Bended Blade After transforming the coordinates of the prototpe, the value of k, which is the length that the blade is pushed down, must be found. The best value for k is the one that minimizes the error between the proposed model of the bended blade and the prototpe model.

4 The task will be simpler if the profile is modeled with a quadratic curve. MATLAB has a tool that directl provides the quadratic curve, so that this best fits the given prototpe data [2]. In this case, the value of k is simpl the -value of the middle point in the quadratic curve. The coordinates of the result can be obtained from this quadratic curve. It would be more complicated if the profile were modeled with a V curve. The tip of this V-curve is modeled with a lower part of circle, whose radius is equal to the radius (r) of the bender s head as shown in Equation 3. In Figure 2.4, the two straight parts of the V- curve are tangent to this circle at points (-x c, c ) and (x c, c ), respectivel. These coordinates can be found via Equations 3 and 4. Since these points la on the circle, Equation 3 can be used to find x c and c. Equation 4 shows that the slope of the tangent line is perpendicular to the slope of the circle s radius at point (x c, c ). Since the width (w) between the two supporters is known, another point for a straight line is known. With some computation, the straight-line equations (Equation 5 and 6) for the two sides of a V-curve can be found [3]. k r x 2 + ( ) = 1 (3) Figure 2.4 Model of a bended blade. 2.4 Comparing the Result and the Prototpe After the mathematical models of the result (bended blade) are computed, it is essential to compare them to the prototpe. Since the equations of the result are computed with respect to the new axis, the coordinates obtained from these equations must be transformed to use the common axis as the original coordinate of the prototpe. In order to do so, the coordinates of the result must undergo the transformation matrix (T), which is used earlier in Section 2.2. The equation is shown in Equation 7. These transformed coordinates of the result are then plotted on the same graph as the original coordinates of the prototpe. 2 c ( r k) c = 1 (4) w 2xc x c slope of tangent slope of radius xoriginal xnew original = T new 1 1 (7) 2 c = ( 2 w 2xc x+ w) (5) 2 c = ( 2x+ w) (6) w 2xc 2.5 Results and Analsis for First Approach Figure 2.5 shows the result of the quadratic curve model and Figure 2.6 shows the result from the V-curve model. The result is plotted with a blue line, while the prototpe model is plotted with a red line. In both figures, the two colors cannot be clearl distinguished, since the error is onl a fraction of a millimeter. However, from the comparison of the error in each segment between the one modeled with a quadratic curve as shown in Figure 2.7 and the other one with a V-curve as shown in Figure 2.8, it can be seen that the quadratic curve gives a closer result to the prototpe model. This is because the prototpe

5 model has a smooth curve, just like the quadratic one. However, it doesn t mean that a quadratic model should be used. Since the proposed model is aimed to represent the output of a sensor, the desired modeling profile should be the one that best reflects the real bending phenomenon of the blade. The actual behavior of the blade can be obtained from Finite Element Analsis (FEM) [4]. FEM can be used to simulate the actual bending of the metal blade. It is possible that the best model ma be a combination of the quadratic and V- curve model. Since the bending operation is done b moving an entire segment at a time, the joint between each segment can have a deviation in slope when compared with the prototpe model. Since each segment is bent b a ver small amount, it is difficult to detect these rugged connections at the joint. However to make the algorithm suitable for a more complicated shape of the prototpe, this problem must be solved. Moving the blade b a smaller amount each time can reduce this problem and make the result more accurate. This idea has led to the invention of the second approach, which will be explained in Section 3. Figure 2.6 Comparison of the prototpe model and the result from V-curve modeling Figure 2.7 The error of each segment for the quadratic modeling Figure 2.5 Comparison of the prototpe model and the result from quadratic modeling Figure 2.7 The error of each segment for the V-curve modeling

6 3. Second Approach After experimenting with the actual behavior of the blade after bending, it was shown that the blade resembled more to a V- shape curve with a ver small round tip, so it becomes insignificant. Therefore this approach will use a V-curve to represent the shape of the blade. This second approach is focused on moving the blade b 5 millimeters each time. To correspond with the actual machine, the width between the two supporters (w) is chosen to be 20 mm. A new algorithm must be introduced since there is an initial curve of the blade before the next press. 3.1 Basic Idea The prototpe model is divided into segments; each has a displacement of 5 mm. A line is used to connect these points and to make a polgon version of the prototpe model, as shown with a dash line in Figure 3.1. The blade is first pressed to reach the point on the polgon model, as shown in Figure 3.2. For the next press, the blade is moved in such a wa that the next pressing point has a displacement of 5 mm awa from the first press point, as shown in Figure 3.3. The new model of the prototpe must also be moved correspondingl with the blade, so that its next point aligns the bender head. The amount of the blade being pressed (k) must be calculated so that the blade is pressed down until the point of press reaches the prototpe, as shown in Figure 3.4. With a brief investigation of the bending behavior using the a real machine, it is assumed that the red section in Figure 3.4 will not change its shape during the third press. Therefore this section is read for plotting. Figure 3.2 Blade model after the first press. Figure 3.3 Blade and prototpe model after translating for the second press. Figure 3.4 Blade model after the second press. 3.2 Overall Algorithm The algorithm of this second approach will be explained according the flowchart in Figure 3.5. Figure 3.1 Prototpe model with a polgon.

7 The distance (t ) that the blade has to shift up is found via Equation 10, which is the equation of the line after translated to the right b x b. Lastl, the model should be rotated so that point (x r, r ), referring to Figure 3.8, lies perfectl on the left supporter. The angle θr is found via Equation 11. Plotting starts after the second press, denoted b n=2. The section that will be plotted is the one in between point (x 1, 1 ) and point (x 2, 2 ) as shown with a red line in Figure 3.9. Point (x 1, 1 ) can be found via Equation 12, while point (x 2, 2 ) is simpl (0, k 2 ).The plotted section can now be modeled via Equation 13. Coordinates from Equation 13 must be rotated back to the original axis via the rotation matrix before plotting. ( k ) + ( x ) = 5 (8) b 2 2 b b 2kx b + = k (9) w Figure 3.5 Algorithm for the second approach. Similar to the first approach, coordinates of the prototpe are transformed to a new axis for each segment. The polgon model is formed from these coordinates as shown with a dash line in Figure 3.1. For the first press (n=1), the pressing distance (k 1 ) is equal to the -coordinate of the polgon when x=0, as shown in Figure 3.2. For the next press, the polgon model must be moved so that the next vertex of the polgon lies on the -axis. First of all the polgon must be moved to the right b Tx, referring to Figure 3.2. Then it has to move up b T so that it lies on the right supporter, referring to Figure 3.5. Lastl it has to rotate so that point A (referring Figure 3.6) lies on the left supporter. The pressing distance (k n ) is also equal to the -coordinate of the polgon at x=0, after the rotation. After receiving the value of k n, the blade is pressed down and the resulting curve is similar to the one shown in Figure 3.7. Before moving the blade, the new point (x b, b ) must be found via Equation 8 and 9. The method of moving the blade is similar to the prototpe s method. The x-coordinate of the new bending point (x b ) suggests the distance that the blade has to move in x-direction for the next press. t 2k w = x b + k (10) w 2 θ w 1 r r = sin (11) x Cosθ Sinθ x n 1 r r b = k n 1 Sinθr Cosθ r n 1 t (12) k x n 1 n = + n 1 x k n (13) Figure 3.5 Blade model after translated b Tx.

8 4. Conclusion Figure 3.6 Blade model after translated b Tx and T. Figure 3.7 Blade model after first press. Since the simulation part of the second approach has not been et accomplished, the conclusion will mainl discuss the first approach. The most important factor in deciding which model should be used is whether that model can best represent the actual shape of the bended blade. With further investigation of the bended blade from a manual bending machine, it is shown that the blade resembles a V-shape. The slight curve at the tip is almost insignificant. Even though the quadratic modeling of the blade gives a more accurate result than the V-curve modeling does, it would be more appropriate to carr on our stud using a V-shaped model. Since the second approach is aimed to produce a more accurate result and a smoother connection at the joint of each segment, less error should occur when comparing the result to the prototpe model. However, even though moving a blade b a smaller amount can increase the accurac, the whole operation will take a longer time since pressing is done more frequentl. With this trade off between the two approaches, careful consideration must be used to match the correct approach with the prototpe model. Combining the two approaches is another alternative. The second approach ma be applied to the more complicated part of the model, and the sstem switches back to the first approach when encountering a simple part of the model. 5. Further Research Figure 3.8 Blade model after x and translation. Figure 3.9 Blade model after the second press. In the future, the algorithm for the second approach must be implemented for programming. A further contribution will make progress on the programming and simulate the second approach. After that its result will be compared with to first approach. Therefore, the objective of this stud can be fulfilled. Apart from progressing on with the simulation, a more realistic algorithm must be considered. In the real bending operation, the deformation of a blade sheet ma occur. The blade sheet will tr to turn back to the original shape if the applied force is not high to overcome its plastic propert. This is an important factor that should be taken into account. The stud of FEM would help to solve this problem. In the addition, the

9 deviation in the length of the bended segment should also be noted. The calculation for the length of blade before the bending operation should be considered. The feeder motor can feed the blade extra amounts, to compensate for the length of the blade, which will be pulled down after it bends. This wa the new segment will not overlap the previous one. Therefore, in further developing the project, man assumptions must be eliminated in order to correspond with realit. Spring-back deformation and affects of ε (difference in length of blade before and after the bend) should be considered. Man hardware and metal bending phenomena will be thoroughl studied in order to understand their behavior. This would then allow an appropriate design of the automatic bending algorithm. Reference [1]. Khalil, W. and Dombre, E., Modeling, Identification and Control of Robots, Hermes Penton Science Ltd, England, [2]. Palm, William J. the third, Matlab for Engineering Application, McGraw-Hill, USA, [3]. Anton, Howard, Calculus with Analtic Geometr. John Wile&Sons, USA, [4]. Fagan, M.J., Finite Element Analsis, Pearson Education Limited, England, 1992.