Mathematical Simulating of the Process of Friction Stir Welding for the Manufacture of Tank Bottoms in Aerospace Construction

In modern aerospace technology, one of the perspective ways of assembly parts of thin metal plates is friction stir welding. The advantage of this method is the higher strength of the welds of aluminum alloy joints, compared with the traditional MIG welding for the assembly of rockets and space constructions. However, the lack of friction stir welding is a necessity of formation of the system "machine –tool part" significantly more effort, which is caused by the need to stir the material in the weld zone in a plastic state. Friction stir welding is used to connect the individual elements of tanks and bodies, in particular, panels, elements of the bottoms with each other. In this paper, is consider the typical design for rocket and space technology bottoms of large tanks. For industrial robot equipped with a special head for friction welding with mixing, a mathematical model of obtaining a spherical bottom from individual segments is proposed. The paper presents a mathematical model describing the geometry of a spherical bottom with a flap articulation and a working zone of an industrial robot, which allows defining constraints on process operation continuous seam welding for standard designs of bottoms and hulls of rocket and space technology. The dependences allowing determining the initial position of the robot relative to the bottom for performing friction stir welding of the continuous seam are proposed.


Introduction
Currently, one of the most promising ways to connect thin metal plates is friction stir welding [1][2][3][4]. The advantage of this method is higher strength of welds of aluminum alloy joints used in rocket and space engineering constructions compared to MIG welding [4][5][6][7][8][9]. At the same time, the feature of the method is the presence of large force factors in the system "machinetool-part".
For the manufacture of structures of tanks and hulls of rocket and space technology, which, as a rule, have a thin shell spherical shape, it is necessary to develop specialized equipment for friction stir welding [4,8,9]. To obtain the longitudinal and transverse (circumferential) welds of the tanks developed specialized equipment. For production of longitudinal seams of shells installations with a vertical arrangement of a welded seam are used. In such installations, the welding head is moved on a special support from top to bottom. To obtain circular joints, machines are used with the basing of internal power elements in a basket that maintains the constancy of the cross-section profile of the tank.
For the manufacture of tank bottoms and hulls, the domestic industry is developing specialized technological equipment. At production of bottoms it is necessary to consider features of geometry of the bottom, presence on its surface of branch pipes, hatches. In the mathematical model of limiting geometrical parameters of welding of branch pipes to the bottoms of rockets stated in [8], the geometry of the bottom, branch pipe, head for friction stir welding is considered. At the same time welding of elements of bottoms among themselves demands further studying.
In this paper, is consider the typical design of the bottoms of rocket and space technology, describes the basic technological equipment for the manufacture of friction stir welding [10,11], and, for an industrial robot equipped with a head for friction stir welding [12], developed a mathematical model of obtaining the bottom of the individual elements. On the basis of the model, the geometric possibilities of friction stir welding of the bottoms of different designs are estimated.
The large dimensions of tanks and hulls of rocket and space technology lead to the need for dividing the bottoms into separate elements. The most characteristic is the petal (Fig.1, a) and segmental division ( Fig. 1, b) [9]. The bottom, as a rule, has a spherical or ellipsoidal shape. The diameter of the bottom can reach 7.7 meters (fuel tank "Energy"), 8.5 m (fuel tank system "Space Shuttle") [9]. For welding of such bottoms specialized large-size equipment were used. This paper discusses the technical possibilities of manufacturing bottoms and petal partitioning in the dimensions of modern launch vehicles (diameter up to 4.0 m).

Overview of friction stir welding equipment
To assess the possibility of manufacturing bottoms by friction stir welding, it will conduct a brief review of industrial process equipment produced by leading manufacturers [9][10][11][12].
Depending on the purpose, industrial equipment for friction stir welding is divided into: • Small-sized 2-3 coordinate machines • Portal type machines • Machines for welding of panels • Industrial robot Small-sized 2-3 coordinate machines (Fig.2) designed for welding products of small dimensions, have a limited working area. Portal type machines (5 coordinate) are used for welding parts of complex shape, with a large thickness of the welded edges (Fig.3). Large efforts arising in the process of welding of such seams require the use of particularly rigid structures of machine equipment and devices, which is implemented in the portal scheme. In addition, such welding plants are equipped with a rotary welding head to expand their capabilities. Equipment for welding of the panels (Fig.4) are used for the manufacture of large panels with a large length of the resulting weld and have large dimensions and significant stiffness. Industrial robots have wide kinematic possibilities and some models of robots of ESAB, ABB, KUKA firms are equipped with replaceable heads for friction stir welding (Fig.5). Limitations of the use of such equipment are the rigidity of the robot structure associated with the applied forces in the welding zone, and geometric restrictions on the working area of the robot. The efforts in the area of the weld impact strength and plastic properties of a welded material, its thickness, temperature of material, process parameters -speed, head movement, frequency of rotation and other parameters.  Figure 6 shows the geometric constraints of the ESAB Rosio robot working area [12]. The working area of manipulators can be represented as a three-dimensional working area. Now the robots is a very effective way to manufacturing parts of aerospace constructions [13][14][15][16][17][18]. However, in addition to geometrical problem, can be solved and thermic, stresses, quality's problem [19][20][21][22][23][24]. For solve this problem used simulation for friction stir welding in various representation [8,9,13,25].

Methods
When analyzing industrial equipment, it was noted that the domestic industry does not produce specialized plants for the manufacture of bottoms. In this paper, is consider the possibility of using an industrial robot equipped with a friction welding head with mixing, to obtain the bottoms of rocket and space technology from individual elements. For this purpose, a mathematical model of the working area of the robot, a model of the geometry of the weld at different ways of dividing the bottom and the geometric limitations of the friction stir welding are determined (Fig.7).  On Fig. 8 presents a geometric interpretation of the mathematical model, which shows the working area of the robot [6], the profile of the bottom (Fig.8, a) with the coordinates of the centers of rotation of the individual parts of the manipulator, determining the position of the working area. On Fig. 8, b layers are shown in colors and indicated by numbers from 1 to 6 areas of the working area, which use different equations and boundary conditions. In the model (Fig.8, a) the x-axis is directed to the right, the y-axis is up. The center of intersection of axes (coordinate center) is the intersection of the vertical axis of rotation of the robot with the lower reference plane of the robot. In certain nodal points relative to the center of coordinates there are centers of spherical and toroidal surfaces characterizing the near and far boundary positions of the manipulator head relative to this nodal point with coordinates (x01, y01), (x02, y02), (x04, y04). At different heights relative to the floor, the robot's working area is described by different equations, since there are several centers of toroidal surfaces describing the near and far boundaries of the working zone.
In overall, the working area is described by a system of inequalities where x, y, zcoordinates of the points of the working area, r(y) -is the radius of the near zone boundary, R(y) -is the radius of the far boundary of the zone, f(x,y,z) -is a function describing the position of the curve.
For the specific robot model shown in Fig. 6, the following system of inequalities describing the position of the working area is proposed.
where y1 , y2 , y3, y4, y5coordinates of boundary values of y, x01, x02, x04, y01, y02, y04position of centers of axes of toroidal surfaces, R1, R2, R3, R4radii of toroidal surfaces. The system of inequalities (2) describes the lowest part of the robot's working area at yϵ[0;y1], located directly at the lower boundary of the robot and the bottom, and painted in blue (Fig.8b). The height of this area of the work area is y1. The first inequality of the system (2) describes the position of the far boundary of the working zone, and the second inequality describes the position of the near boundary of the working zone. The system of inequalities (3) describes the next region (sequence from bottom to top) of the robot's working area at yϵ(y1;y2] and in this system, just as in (2) the first inequality describes the far boundary and the second the near boundary. Area 2 is painted pink in Fig.8b. The following areas have similar descriptions of the working area in inequality systems and are shown in Fig.8b: area 3 is green at yϵ(y2;y3], area 4 is brown at yϵ(y3;y4], area 5 is light-green at yϵ(y4;y5], area 6 is blue at yϵ(y5;+ ∞).  (8) Given that (x-x05)=(y-y05)• tgand z05=0 are obtain:

Results
On the basis of the proposed model, the results of calculations for the boundaries of the near and far zone of the robot welding head and the coordinates of the weld bottoms with petal division were obtained.
The boundaries of the working area of the ESAB Rosio robot were used as initial data (Fig.8, a): х01 = х03 = 430 mm, х02 = 1400 mm, х04 = 540 mm, у01 = у03 = 800 mm, у02 = 950 mm, у04 = 1300 mm, у1 = 100 mm, у2 = 1150 mm, у3 = 1500 mm, у4 = 2200 mm, у5 = 2900 mm, R1 = 700 mm, R2 = 1000 mm, R3 = 2000 mm, R4 = 1000 mm (the centers of the axes of torus circles with radii R1 and R3 coincide, so х01 = х03 и у01 = у03). Table 1 shows the raw data for calculating the coordinates of the weld for the bottom reed articulation: the diameters of the lower edge of the bottom D, the radius of the spherical surface is R, the distance from the center of the sphere to the bottom surface of the bottom y05, bottom height h. In the calculation was varied with the angle of inclination of the seam  in the range from 0 0 to 53 0 and the distance between the vertical axes of the robot's workspace and the bottom x05 in the range from 1600 to 2500 mm.  1950  1240  600  480  2300  1470  720  590  2700  1700  845  695  3100  1930  900  870  3500  2160  1090  900  4000 2200 1130 910 Figure 10, a shows the far boundary of the robot's working area (rotated, the vertical axis of rotation of the robot in the figure is horizontal) and the far and near boundaries of the working area (Fig.10, b).   Welding is possible if the weld is located in the space between the near and far boundaries of the working area of the robot. If the seam crosses one of the boundaries, the welding head will not be able to continuously, without moving the workpiece, weld the seam area beyond the working area. Figure 11, a shows (red rectangle) that part of the trajectory of some welds cross the far boundary of the working area and, therefore, at this position of the bottom welding of the seam is impossible. In figure 11, b, you can see that those welds that do not cross the far border of the working area can cross the near border (red rectangle), which also does not allow welding in the shown position. Figure 12 shows the results of the calculation of valid values position of the bottom relative to the robot, shows the workspace of the robot and weld, passing completely inside the working area and not across its borders, allowing continuous welding of the parts. The possible distance between the robot axis and the bottom axis for different diameters with different angles of inclination  of the weld is determined for the bottoms with petal division (Fig.13).
For a bottom with a local specified weld angle and diameter to ensure guaranteed welding, the minimum distance from the robot axis to the bottom axis is equal to the value on the lower branch of the graph curve, and the maximum distance is on the upper branch. For example, for a bottom with a weld angle =20 0 and a diameter of 2700 mm, the distance between the bottom and robot axes should be between 1875-2150 mm.

Discussion and application in the education
A further development of the proposed model for the description of spherical surfaces can be the use of vector calculus for problems of spherical geometry, and for the description of torus surfaces-quite simple analytical dependencies.
The considered model is widely used in the educational process of the Department of technologies of rocket and space engineering in the study of technological equipment of the welding process of the bottoms of rocket and space technology. The model is implemented in the software product MathLab and allows you to visualize the position of the robot, the bottom, the weld between them. Students build on the given geometric parameters of the working area of the robot, set the parameters of the bottom and determine the possibility of obtaining a continuous weld. By results build schedules and prepare the report.

Conclusion
The review of technological equipment for friction stir welding of sheet materials is carried out. The use of universal industrial robots equipped with a head for friction stir welding is proposed.
A mathematical model describing the geometry of the bottom and the working area of the robot is developed, which allows to determine the possibility and impossibility of welding a continuous seam for typical designs of the bottoms of rocket and space technology.
For bottoms with petal division, dependencies are proposed to determine the position of the robot relative to the bottom, providing friction stir welding of the continuous seam. This model is used in educational process in Bauman Moscow State Technical University for studying students on aerospace manufacturing.