Engineering mathematical calculation method for curved surface die design of metal materials

1. Introduction

Metal material curved mold is an important part of machining, which is widely used in chemical industry, light industry, food and other industries [1]. For example, the blades of large water turbines and large wind turbines, large tank containers loaded with concrete, tank body of oil tank containing oil (gasoline, etc.) [2]. Small metal sheet components are generally stamped with metal sheet, while large metal sheet curved surface components are often pressed with split curved surface components due to the limitation of sheet size and pressure, or made by cold beating method, and then welded into an overall process scheme [3, 4].

In practical engineering applications, the calculation methods of metal material curved surface die are different according to different parts, and there are few general engineering mathematical calculation methods [5]. Based on practice, this paper puts forward the mathematical model of die design from the perspective of mathematics, and summarizes the general engineering mathematical calculation methods [6].

2. Mathematical model

Metal casting, commonly known as hard mold casting, is a process of manufacturing castings with metal materials and pouring molten metal into the mold under gravity to obtain castings [7, 8]. Because a pair of metal mold can be poured hundreds to tens of thousands of times, metal mold casting is also called permanent mold casting [9]. Take the mold in Fig. 1 as an example.

The die is composed of a pressing die, which is composed of an upper die and a lower die [10]. The upper die is divided into convex panels and the rib plates, which are welded by steel plates. The material of the formwork is a rectangular steel plate, which is simple in shape and easy to manufacture [11]. The panel is in melon petal shape, the shape of rib plate is shown in Fig. 1, and its curved edges are ellipsoidal generatrix. It is a standard template. The rib plate can be manufactured according to the setting out of the template [12]. The height of the large end of the rib plate is h ⁢ 1 , h ⁢ 2 , ⋯ ⁢ l ⁢ 1 , l ⁢ 2 , ⋯ and the height of the small end is H ⁢ 1 , H ⁢ 2 , ⋯ , L ⁢ 1 , L ⁢ 2 , ⋯ . The height varies with the position of each rib plate on the die, so it is difficult to determine its value, and so is the height in the middle of the curved edge of the natural rib plate In the production and manufacture of ribbed plates, the values are determined by experience, and then repaired when welding the die, resulting in large dimensional error [13, 14].

OPEN IN VIEWER

As can be seen from Fig. 1, the problem is solved as long as the rib plate height is calculated. In order to accurately calculate the height dimensions of the large and small ends of the rib plate and other points on the curved edge, it can be envisaged that there is an ideal plane w parallel to the upper and lower formwork, 4 high points of die (that is, the four low points of the punch) E , F , G and K are in this plane. For example, if the height hi of the large end of the rib plate is required, only H ⁢ 1 , H ⁢ 2 , ⋯ , L ⁢ 1 , L ⁢ 2 , ⋯ can be obtained. As shown in Fig. 1, h ⁢ i = H ⁢ 0 - H ⁢ i ′ and I = 1 , 2 , 3 , ⋯ , according to the above assumption, we establish the following mathematical model: take a melon petal surface s on the ellipsoid shown in Fig. 1, E , F , G and K are the four vertices of the melon petal, which generally should divide the ellipsoidal surface n equally and weld several melon petal surfaces s to obtain the ellipsoidal head. The plane composed of points E , F , G and K is the ideal plane W . To calculate the height of points on the edge of the stiffened plate, only calculate the distance from these points to plane W .

The equation of the known ellipsoid is:

x 2 a 2 + y 2 b 2 + z 2 c 2 = 1(1)

The equation of large circle at the bottom of the ellipsoidal head is:

{ x 2 + y 2 = a 2 z = - g(2)

The equation of small circle at the top of the ellipsoidal head is:

{ x 2 + y 2 = d 2 z 0 = c a ⁢ a 2 - d 2(3)

Next, we establish the equation of the ideal plane W , assuming that the ellipsoidal head is divided into n melon petals, then the center angle of each melon petal surface s is α = 360 ∘ / n . As shown in Fig. 1, E , F , G and K are the four vertices of melon petal surface, and their coordinates are e ⁢ ( a , 0 , - G ) and F ⁢ ( 𝑎𝑐𝑜𝑠𝑎 , 𝑎𝑐𝑜𝑠𝑎 , - g ) , K ⁢ ( 𝑑𝑐𝑜𝑠 ⁢ α , 𝑑𝑠𝑖𝑛 ⁢ α , z 0 ) .

Let M ⁢ ( x , y , z ) be a fixed point on plane W . from analytical geometry, the equation of plane W is:

| E ⁢ M → ⋅ E ⁢ G → × F ⁢ K → = 0 X - a Y Z 0 + g ( d - a ) ⁢ cos ⁡ α ( d - a ) ⁢ cos ⁡ α Z 0 + g |(4)

Expand the above equation:

X + 1 - cos ⁡ α sin ⁡ α ⁢ Y + a - b z 0 + g ⁢ Z - a ⁢ z 0 + d ⁢ g z 0 + g = 0(5)

Then the standard form of plane W is:

A ⁢ X + B ⁢ Y + C ⁢ Z - D = 0(6)

Therefore, the distance from any point ( x , y , z ) of melon petal surface s to plane W is:

P = | A X + B Y + C Z - D = 0 | A 2 + B 2 | + C 2(7)

In order to obtain the maximum distance from surface S to plane W , the conditional extreme value of the following equation is required:

{ P = | A ⁢ X + B ⁢ Y + C ⁢ Z - D | A 2 + B 2 | + C 2 x 2 a 2 + y 2 b 2 + z 2 c 2 = 1 0 ⩽ z ⩽ z 0(8)

Taking a curved surface part of metal material as an example, the above engineering calculation method is demonstrated. The dimensions of curved surface parts of metal materials are shown in Fig. 2.

It is known that a = 125 mm, C = 54 mm, g = 40 mm, d = 125 mm, n = 8.

From Eqs (3) and (5), Z 0 ≫ 698 , a = 1 , b ≫ 0.215 , c ≫ 1.515 , d = 1328 . Then the plane W equation is:

X + 0.515 ⁢ Y + 1.3147 ⁢ Z - 1328 = 0(9)

The calculation criteria are determined according to the possible failure forms of parts, and the basic dimensions of parts are determined according to the calculation criteria, as shown in Table 1.

OPEN IN VIEWER

It can be seen from Table 1 that in the processing and manufacturing of metal curved surface molds, the feature processing mainly includes rounding, chamfering and perforation. When rounding, the matching technology should be used according to the mold conditions to extract the features from the dimensions of the two-sided flow body. Generally, rounded corners can be divided into cylindrical rounded corners and ordinary cylindrical corners. These two kinds of round corners can be defined according to the roundness and surface radius of the die [15]. Each fillet has a smooth edge, and the excessive features of this smooth edge and its adjacent faces can be used as identification elements. When processing chamfer, it can be processed according to cone or plane, because chamfer often presents geometric structure, and the width of each face and edge can be regarded as the main features [16]. When working with holes, you need to pay attention to the direction of the holes. Usually, when extracting the inner ring data of the hole, we should also pay attention to the extension of the supporting surface of the hole. When simplifying various operations, it can be optimized as a whole or step by step [17]. The sequential deletion strategy in the step-by-step method is more suitable for complex situations, and the overall deletion strategy is suitable for simple scenarios. This needs to be selected according to the actual situation of the mold.

The following points should be paid attention to in the engineering mathematical algorithm of metal material curved surface die design: (1) Write the equation of the curved surface. If the equation of the curved surface cannot be written in one, divide the curved surface into several equations according to the manifold theory. Design the die for each curved surface equation, and finally weld the machined components into an integral component. (2) For each curved surface equation, the split mold design is carried out, and the ideal plane equation is established for each melon petal. The operation is carried out by imitating the method in the previous section. For any curved surface component, the mold design calculation can be carried out according to the above method to manufacture the required mold. (3) The number of melon petals n is made according to the material and press regulations, preferably 8 or 12. In this way, the size calculation is more accurate, and it is easy to meet the requirements of high precision. (4) The size calculated by the equation is the size of the geometric center layer after melon flap forming. In actual operation, appropriate correction shall be made according to the specification of the plate to improve the accuracy of the calculated data. If the known number is composed of M significant digits, B, C, t Δ M-1 significant digits after the decimal point shall be reserved.