# Method for arranging circular blanks and calculation of number of pieces

Method for arranging circular blanks and calculation of number of pieces

Li Guixiang

The circular blank layout method discussed in this paper refers to the method of simultaneously arranging different blanks of the same row diameter on a given specification plate or other sheets under the condition of known diameter of the layout. . The calculation of the number of circular blanks is also referred to under the corresponding conditions, that is, in the case where the diameter of the circular blank is known (the diameter of the blank is equal to the blanking diameter plus the edge value) and the size of the specification plate. Calculate the process by which the board can be blanked.

The method of row blanking and the calculation of the number of pieces are extremely common in the round canning, barrel making and box making process of the whole packaging industry. If you can't correctly understand the diameter of the layout, the relationship between the length and width of the specification plate and the number of the number of samples. Then! It is not possible to find a more reasonable number of layouts relative to a given plate and the diameter of the layout. That is to say, any reasonable layout scheme of the circular blank is determined according to the specific layout conditions and methods and the corresponding number of the number of the layout. Therefore, only after studying different kinds of different layout methods and corresponding number calculation formulas, can we obtain the inevitable calculation results, and make an appropriate comparison of the calculation results, then we can get the optimal layout scheme. In short, only after the different layout methods are used to establish different calculation formulas for the number of samples, can a comprehensive layout plan be obtained through comprehensive analysis. This is the scientific method to obtain a reasonable way to arrange samples.

First, the issue of the matter

It is known from many domestic and foreign materials that there are two different understandings and opinions on the method of blanking blanks and the calculation problems. One kind of understanding and insight is that when the circular blanks are arranged on the specification board, it happens that the utilization rate of the positive row plan is relatively high. Another kind of understanding and insight is that when the circular blanks are arranged on the specification board, the misalignment scheme can achieve higher utilization than the implementation of the positive row scheme. In fact, these two kinds of understanding and insights have their own correctness. After careful research and analysis, it is concluded that when the size of the specification plate is constant, if the layout diameter d is equal to the boundary diameter d. (When the grid is fixed, the unique row diameter can be called the demarcation diameter when the number of rows and the wrong rows are equal.) Only after discussing and grasping the various calculations of the positive and the wrong rows. When it is possible to discuss this problem with sufficient space), the utilization rates of the positive and the wrong rows are completely equal; when the layout diameter d is larger than the boundary diameter mountain, the positive utilization rate is higher than the wrong row; When the sample diameter d is smaller than the boundary diameter do, the misalignment utilization ratio is higher than the positive row. Such understanding and insights are correct and reliable. This conclusion is not difficult to understand for people with rich planning experience and certain theories, but it is more difficult for people who have no layout experience and theory to understand. Therefore, it is necessary to make a topical discussion on relevant issues in this regard.

But the proof of the above conclusions involves many computational problems with positive and wrong rows. To this end, various calculation formulas related to the positive and the wrong rows must be established in order to prove the conclusion. So that people can understand the conditions under which the positive row plan is adopted; under what conditions, the staggered plan is adopted; under what conditions, the combination of the positive and the wrong rows is adopted, or the matching row based on these several solutions is obtained. Guiding ideas such as sample programs. Therefore, it is necessary to separately discuss the method of positive row, wrong row and right and wrong combination of round blanks and the corresponding number of row pieces. Then on the basis of these theories, the characteristics of the boundary diameter do and its calculation method are discussed, so that the reasonable layout scheme and critical calculation problems of the circular blank are systematically and completely solved.

The calculation of the number of pieces of the circular blank layout scheme can generally be solved by formula calculation. The calculation of the number of pieces of the arrangement of any one of the circular blanks can be understood as the number of rows of the blanks on a given plate, multiplied by the number of pieces that can be arranged in each row. result. The number of pieces of the layout is calculated as a simple positive row plan. As shown in Figure 1. In the case where the plate length L, the plate width B, and the layout diameter d are known, the following expression is solved. The number of pieces N of the positive row plan can be calculated as follows:

Number of rows to be sampled N = number of rows of rows Ã— number of rows per row.

Algebraic expression is:

N={[(Ld)/d]+1}Ã—{[(Bd)/d]+1}=LB/dd

It is also possible to use np for L/d and n for B/d. Therefore, the number N of the sample sets can be calculated as follows: N = np Ã— N. When np is regarded as the number of rows, n is regarded as the number of pieces that can be arranged in each row. On the other hand, when n is regarded as the number of rows, np is regarded as the number of pieces that can be arranged in each row.

The calculation of the number of pieces of all round blank layout schemes can be solved almost by the number calculation method similar to the positive row method. However, only the specific calculation contents of the number of rows and the expression of the number of rows in each row, or the nature of the numerical values â€‹â€‹are different, and the calculation formula is various.

Second, the positive row of circular blanks and the calculation of the number of rows:

The so-called positive blanking method of the circular blank here refers to the fact that the different blanks of the same row diameter can be oriented in the direction of the length and width of the given plate (the rows are adjacent to each other. A method in which the points are connected in parallel or perpendicular to the length of the plate or the width of the plate. As shown in Figure 1. The feature of this type of layout method is that no matter whether it is the size of the specification plate or the size of the size of the layout, as long as the relationship is positive, the layout utilization rate is 78.54%. In the case that the specification board is as large as possible, the positive arrangement is the arrangement when the diameter of the layout is large. When the layout diameter is small, the utilization rate of the row is replaced by the misalignment scheme. When the positive displacement scheme is implemented, the ratio of the plate length L and the plate width B to the layout diameter d is exactly a positive integer. That is, L/d=np, B/d=n, and np and n are both positive integers. The number of the number of pieces to be sampled is directly related to the size of the integral portion of the plate length L and the ratio of the plate width B to the row diameter d. Therefore, the calculation method of the number of the number of pieces to be arranged is straightforward and easy to perform various inverse operations.

In order to facilitate the blanking and punching of the blank for blanking, when the blank is cut and cut, the slitting line may be parallel to the longitudinal direction of the plate or the width of the plate according to the specific circumstances.

Here, we only study the basic situation when "just happening to implement the positive row plan", and do not study the positive plan in other cases, because it is necessary to determine the correct arrangement of the positive plan itself, and in a sense, It is necessary to rely on some of the misplaced i: in order to get a satisfactory solution. Therefore, only by combining the positive and the wrong rows can we make breakthrough progress in the layout theory.

In the case of a certain specification plate, if the diameter of the layout is small, the layout is replaced by the wrong arrangement, so that the number of the sample is increased, thereby improving the layout utilization of the specification plate. It is said that when analyzing the fractional length L of the gauge plate in the positive row plan and the fractional ratio of the plate width B to the row diameter d, there are still a number of reasonable layout possibilities as in the case of the staggered arrangement. Only after solving various reasonable layout possibilities of the misalignment scheme can it be possible to solve various reasonable possibilities of the positive scheme. Therefore, there is no way to discuss too much more detail here.

The layout scheme obtained by the positive row method is usually called the positive row scheme. The number N of such positive layouts can be calculated as follows:

N=LB/dd=npÂ·n

Where N- represents the number of pieces to be sampled, and the unit is a piece;

L, B- respectively indicate the length and width of the specification plate, in millimeters;

D- indicates the diameter of the layout, in millimeters;

Np, n- respectively indicate the number of rows or the number of pieces that can be arranged along the length of the plate and the width of the plate, and the unit is a row or a piece.

Example 1: A steel drum cover with a diameter of 250 mm should be placed on a 2000x1000x0.5mm specification board. Q. How many pieces can I order when I use the positive plan?

Solution: L=2000mm is known; B=lOOOOmm; d=250mm; the number of pieces to be sampled is N.

According to the positive row plan, the number N of the sampled pieces can be calculated as follows:

N=LB/dd=2000Ã—2000/250Ã—250=32 pieces

That is, when the layout is performed by the positive row plan, 32 pieces can be arranged.

From the characteristic analysis of the above-mentioned positive row plan, it can be known that it is necessary to determine whether the layout plan itself is reasonable, and also under the condition of a certain specification plate and the diameter of the layout, with the application, analysis, comparison and synthesis of various misalignment schemes. A correct conclusion. Therefore, it is inevitable that the positive formula N=LB/dd will become a reasonable initial formula for the circular blank layout scheme. Therefore, you must master it skillfully. When comparing the advantages and disadvantages of different layout schemes, it is very useful to use it to determine whether the layout scheme is reasonable.

Third, the method of staggering the blank blank and the calculation of the number of the sample

The method of staggering the circular blank mentioned here refers to the fact that the blanks of the same row diameter are sequentially displaced along the length and width directions of a given plate (the rows of adjacent misaligned tangent points are connected to the heart line). Various methods of arranging each of them obliquely or at an angle to the length of the plate or the width of the plate.

When the different blanks of the same row diameter are sequentially misaligned along the length of the plate and the width of the plate, the resulting staggered arrangement is called a two-way staggering scheme, and the habit is simply referred to as a double staggered scheme. Since the amount of misalignment in the direction of the length of the plate and the width of the plate is equal or unequal, it is divided into two different cases: when the amount of misalignment in the direction of the plate length and the width of the plate is equal, and the amount of misalignment is a constant dâˆš2/2 The two-way staggered scheme is called a bidirectional equidistant misalignment scheme. When the amount of misalignment in the board length and the board width direction is not equal, and the misalignment amount is not a certain constant, the bidirectional misalignment scheme is called two-way misalignment. The wrong row plan.

Since the misalignment of the bidirectional equidistant misalignment scheme is a constant dâˆš2/2, it is exactly 1/2 of the dâˆš2 of the distance d reject, and its size is only related to the size of the layout diameter, and the specification plate The size does not matter. As shown in Fig. 2, when it is required to cut and cut the blank of the sheet having the blank, the angle between the slit line and the length or width of the sheet is always 45 degrees. Therefore, the cutting method has better cutting process, and has the same special application significance as the positive row plan. Therefore, it is necessary to make a thematic discussion on this method of layout. The two-way unequal-distance misalignment scheme will be more appropriate when discussing the one-way misalignment scheme.

When different blanks of the same row diameter are sequentially displaced along the length of the plate or in a certain direction of the width of the plate, the obtained staggered arrangement is called a unidirectional staggering scheme, and the habit is simply referred to as a unidirectional staggering. Program. Because the range of values â€‹â€‹of misalignment is different, the layout method is slightly different, but the calculation formula of the number of layouts is completely different. The range of misalignment is generally divided into two segments: one is the misalignment B-nd or L-npd is greater than 0, but less than the number; the second is the misalignment B-nd or L-npd is equal to or '. In the specific layout, the actual amount of misalignment is related to the size of the specification plate and the size of the layout diameter. Therefore, the amount of misalignment can be expressed by an algebraic formula containing the length of the plate, or the width of the plate and the diameter of the layout. This facilitates the establishment and application of calculation formulas. As shown in Figure 3. The characteristic of this type of layout method is that the layout utilization rate will change regardless of the size of the specification plate and the different sizes of the size of the layout. In the case of the size of the specification plate, the staggered arrangement is used for the arrangement where the diameter of the layout is small. When the misalignment scheme is implemented, the calculation method of the number of the sampled parts is related to the size and nature of the displacement amount, and the amount of misalignment is determined by the ratio of the length L of the plate or the width B of the plate to the diameter d of the layout and the fractional part. The value characteristics are reasonably determined.

In order to facilitate the blanking and blanking of the blank for the blanking, it is necessary to cut the blank when the blank is cut. According to the specific situation, the slitting line and the length direction of the board, or the direction of the board width, always have a certain angle. That is, the tangential line is neither parallel to the longitudinal direction of the plate nor parallel to the width of the plate. This is the biggest difference between the positive and the wrong layout schemes. However, there is a difference between the double misalignment and the single misalignment. Generally speaking, the double misalignment is better than the single misalignment. However, under the same kind of layout conditions, the single-wrong row layout utilization rate is higher than that of the double-displacement row, and each has its own characteristics and should be used differently.

Due to the different layout properties of the double misalignment scheme and the single misalignment scheme. Therefore, the calculation method of the number of pieces N to be sampled is also different. Therefore, it is necessary to make a different discussion on them separately.

1. Calculation of the number of pieces of circular blank bidirectional equidistant misalignment scheme

The calculation principle of the number of the two-dimensional equidistant misalignment scheme of the circular blank is shown in Fig. 4. When the two-way equidistant misalignment scheme is implemented, the ratio of the plate length L and the plate width B to the layout diameter d is always coincident with a multiple of the constant rejection plus one, or a multiple of the constant cabinet plus 0.5. That is, or = n reject + 0.5, and np and n can be any positive integer equal or unequal, respectively. The only difference is that when np is not equal to n, it indicates that the bidirectional equidistant misalignment scheme is performed on the rectangular sheet. When np is equal to n, it indicates that the bidirectional equidistant misalignment scheme is implemented on the square sheet.

The layout scheme obtained by the bidirectional equidistant misplacement method is generally called a two-way equidistant misalignment scheme, which is simply referred to as a double equidistant misalignment scheme. The calculation of the number of pieces of this double equidistant misalignment scheme is quite special, and it is impossible to derive a simplified wife's wife directly to obtain the result. As shown in Figure 4. The establishment of the number of calculations for the number of pieces can be analyzed in four basic cases: the first formula is obtained when L and B are known; the second formula is obtained when L and B' are known; B, the third formula is obtained; when L and B are known, the fourth formula is obtained. However, in the specific calculation, it is usually based on the large and small trade-offs of the fractional part of the L: damage and falloff values â€‹â€‹to determine the number of the number of pieces to be sampled using that specific calculation formula. Therefore, the calculation is generally calculated in the following four cases:

(1) When the number of parts of the algebraic value of ki=L and k2=deny-base is less than 0.5, and the integer part of the numerical value can only be substituted for the calculation of the number of pieces to be sampled, a double equidistant row scheme is obtained. The number of pieces of the plan N can be calculated as follows:

Or expressed as:

N two (ki+l)(k2+1)+klk2

(2) When the number of the algebraic values â€‹â€‹of k=1 and k=2 is greater than 0.5, and only one valid fraction of the algebraic value is 0.5, together with the integer part participating in the calculation of the number of pieces, the result is still Double equidistant misalignment scheme. The number of pieces of the plan N can be calculated as follows:

Or expressed as:

N=2(ki+0.5)(k2+0.5)

(3) When the fractional part of the algebraic value of k=d is less than 0.5, and only the integer part of the numerical value is involved in the calculation of the number of layouts; at the same time, the small part of the algebraic value of k2=BA is greater than 0.5, and can only be taken When the first valid fraction of the algebraic value is 0.5 and the number of pieces to be counted is calculated, the result is still a double equidistant staggered scheme. The number of pieces of the plan can be calculated as follows:

Or expressed as:

N=(ki+1)(k2+0.5)+(k2+0.5)ki

(4) When ki=i, the fractional part of the algebraic value is greater than 0.5, and only the first decimal of the algebraic value is taken as 0.5 together with the integer part to participate in the calculation of the number of pieces, and the fractional part of the algebraic value of k2=d If it is less than 0.5, and only the integer part is involved in the calculation of the number of pieces to be sampled, the result is still a double equidistant staggered scheme. The nesting part N of the scheme can be calculated as follows:

Or expressed as:

N two (ki+0.5)(k2+1)+(ki+0.5)k2

From the four cases of the upper two-way equidistant misalignment, four different calculation formulas are obtained correspondingly. However, the calculation formula expresses the substantial calculation results of the number of bidirectional equidistant rows of samples. Therefore, the code names in the four different calculation formulas are exactly the same. N- indicates the number of pieces to be placed when double equidistant misalignment is performed, and the unit is piece; L, B- respectively indicate the length and width of a given specification plate, the unit is mm; d- indicates the diameter of the layout, the unit is mm; And k2- represent different values â€‹â€‹of the same algebraic expression to represent a concise operation relationship.

Example 2: It is now necessary to arrange a steel drum cover with a diameter of 247 mm on a 2000x1000x0.5mm specification plate. When using the solution, what is the maximum number of layouts?

Solution: The solution to this problem can only be determined by using different row calculation methods to obtain the actual number of pieces and then comparing them to determine a more reasonable layout plan.

(1) When the layout is implemented using the positive row plan, the number N of the sample sets can be calculated as follows:

(2) When using the double equidistant misalignment scheme to implement the layout, first obtain the actual algebraic value of L: harm and denier, and then determine the specific formula to calculate. It is worthwhile to use L=2000mm, B=1000mm, d=247mm for the corresponding algebraic formula:

Since the fractional parts 0.0185 and 0.1557 in the values â€‹â€‹of 5.0185 and 2.1557 are both less than 0.5 at the same time. Therefore, when only the integer part is involved in the calculation of the number of pieces of the layout plan, a double equidistant misalignment scheme is obtained. The number N of nestings for this type of solution can be calculated as follows:

(3) When the unidirectional staggering scheme is implemented along the width direction of the board, the number of samples N can be calculated as follows:

A panel discussion will be held.

According to the analysis of the calculation results of the above three kinds of layout schemes, it is found that the positive displacement scheme and the one-way misalignment scheme along the width direction of the panel are the best, and the maximum number of sampled samples that can be obtained is 32.

The actual layout calculation of the above example shows that sometimes different layout schemes will result in equivalent layout results. This conclusion provides us with a reliable basis for studying the size and nature of the layout diameter of different layout schemes under equivalent layout conditions. At the same time, it must be clearly recognized that in the process of nesting, before the specific layout plan is not specifically analyzed and calculated, it is not possible to blindly misunderstand that a certain layout scheme has obtained a large number of layouts. This makes it possible to implement an incorrect layout scheme in a realistic layout. Under the specific layout conditions, when two or more schemes are required for comparison, the calculation method of the corresponding layout scheme can directly reach an accurate conclusion.

Example 3 is to arrange a special blank with a diameter of 23 mm on a 2000x1000x1.2mm specification board. When comparing the positive and double equidistant misalignment schemes, what is the number of layouts obtained in that case? How many more?

Solution: Known 1 = 2000 mm; B = 1000 mm; d = 23 mm;

(1) When using the positive row scheme, since np=L/d=2000/23=86.957 (np=86 when only the positive integer part is taken), n=B/d=1000/23=43.478 (only integers are taken Partially, you get n=43). Therefore, the number of pieces to be sampled N can be calculated as follows:

N=npÃ—n=86x43=3698(piece)

(2) When using the double equidistant misalignment scheme, first obtain the actual algebraic value of L: harm and the dwelling house, and then calculate the specific calculation formula. It is worthwhile to use L=2000mm, B=1000mm, d=23mm for the corresponding algebraic formula:

Since the fractional parts of the values â€‹â€‹of 60.7810 and 30.0370 are greater than 0.5 and less than 0.5, respectively. Therefore, when the fractional part is taken as 0.5 and only the integer part is taken to participate in the calculation of the number of pieces to be sampled, the bidirectional equidistant misalignment scheme is still obtained. The number of pieces of the plan N can be calculated as follows:

The difference Ne of the number of pieces of the two kinds of layout schemes is:

Ne=3721 pieces-3698=23 pieces

That is, the number of sampled pieces obtained by the two-way equidistant misalignment scheme is large, and 23 pieces are more.

Under the same kind of layout conditions, if there are multiple ways of layout, the number of different types of samples should be calculated separately. Finally, the number of samples is large and the process of blanking and forming is optimal. Layout plan. (To be continued)