Investigation on variation characteristics of bent tube axis and determination of bending die motion trajectory in free bending process

In free bending process, the axis shape of bent tube is determined by motion trajectory of bending die. The bent tubes with different bending radii and bending directions can be formed by continuously adjusting the position and posture of die. To accurately form the pre-designed bent tube, the formation mechanism of bending arc in free bending process was analyzed. The mathematical model between bending radius and deviation was established and the moving direction of bending die was also determined according to the plane space azimuth where the bending arc segment is located. The formation process of transition segments with variable curvature and variable bending direction is illustrated during the movement of bending die, and the influence of transition segment on the axis shape was also explored. By comparing the axis deviation of experimental and simulated bent tubes, the length of transition segment and the moving path of bending die can be optimized. Finally, the applicability of free bending forming technology in forming bent tubes with different axis features was verified by conducting free bending experiments on bent tubes with three typical shapes. Due to the high flexibility of free bending forming technology, the work in this paper could promote the engineering application of this technology in the precise control of axis shape of bent tube.


Introduction
The bent tube components with complex axis shapes are widely used in automobiles, rail transit, ships, aerospace, and other industrial fields. For example, the weight of tube accounts for more than 60% of the weight of turbofan aeroengines [1]. As the artery and lifeline of aircraft, the tube system is spread all over the nose, fuselage, tail, and wings of aircraft. It is used for the transportation of hydraulic oil, fuel oil, lubrication oil, oxygen, water, and other media [2]. To meet the requirements of overall lightweight, flattened shape, and easy installation of equipment, the axis shape of tube is usually composed of these curved arc segments with different bending angles, bending radii, and bending directions. Furthermore, to reduce the flow resistance in the process of liquid transfer, the curvature change of tube axis should be gentle.
To obtain the bent tube components with different axis shape characteristics, many tube bending forming technologies have been proposed successively, including pushingbending [3], rolling-bending [4], rotary-bending [5], and stretch-bending [6]. However, the shape of bent tube formed by the above-mentioned traditional bending forming processes depends on the size of die, and it is difficult to manufacture the spatial tubes whose bending direction and bending radius change continuously. For example, the rotary-bending process is often used to manufacture two-dimensional or threedimensional bent tube components with multiple curved arc segments. Due to the limitation of clamping during bending Zonghui Cheng and Shaoxin Li contributed equally to this work. process, there are straight line segments between adjacent curved arc segments, resulting in that the curvature of bent tube axis cannot be smoothly transitioned.
To overcome the limitations of traditional bending processes, the flexible bending technology was proposed to manufacture the complex bent tube components with continuously changing bending curvature and direction. As an extension of push bending technology, the flexible bending process could realize the bending of spatial tube with different bending radii, bending angles, and bending directions without changing the bending dies [7]. Based on the different motion modes of die, the flexible bending technology can be divided into free-bending technology [8], torque supposed spatial (TSS) bending technology [9], and three-roll-push bending technology (TRPB) [10]. The original tube moves along the axial direction under the action of propulsion mechanism, and the deviation of forming tool (bending die, forming roll, or mobile die) from the origin can be adjusted in real time. For TSS and TRPB, the bending and torsional deformation of tube can be realized by the bending moment exerted by the forming tool and the torque exerted by the clamp. The moment and torque are simultaneously applied to the tube cross section by changing the deviation and rotation direction of bending die in free bending technology. Under the action of bending die, the spatial bent tube product can be bent and shaped.
The plastic deformation zone of bent tube is weakly constrained by forming die, many factors including material parameters and process parameters would affect the formation of bending arc. Guo et al. [11] studied that the influence of mechanical properties of material on bending radius of bent tube by free bending experiment. The bending radius decreases with the improvement of elastic modulus, density, and strain-hardening exponent. Wei et al. [12] conducted the free bending experiments on aluminum alloy, copper, and brass tubes and found that the bending radius and wrinkle factor is increased with improving the yield strength. In the free bending process, with the increase of clamping pressure, the pushing resistance increases, which leads to the obvious wrinkling and thickening of tube. To realize the free bending forming of bent tube with small bending radii, Beulich et al. [13] inserted a mandrel inside the tube to prevent geometrical instability. The influence of the material model on the free bending deformation behavior was analyzed and the material parameters obtained directly from tube were suggested. Gantner et al. [14] studied the influence of friction coefficient on the bending behavior of tube by numerical simulation method and found that the bending radius decreases with increasing the friction coefficient. Staupendahl et al. [15] found that the interaction between bending and torsion in free bending process affects the bending force and springback behavior of three-dimensional profiles, thus affecting the bending curvature.
It is a very difficult task to precisely control the shape of final metallic tubular parts because the shape errors of previous bending arc will be accumulated in the next bending arc. Furthermore, the formation of bending arc of tube is closely related to the movement of bending die; many scholars have studied the curvature distribution law of bending arc and the relationship between the motion trajectory of forming die and the shape features of bent tube axis. For example, Wu et al. [16] found that the bending and twisting can affect the springback behavior of tube, which in turn leads to the reduction of bending curvature and torsion of tube axis. The spatial tube axis can be discretized and approximated by a series of spirals. Plettke et al. [17] utilized the Frenet-Serret formulation to discretize the bending contours of bent tube with curvature and torsion and established the relationship between machine parameters and geometrical shape. To accurately form the spatial bent tube components by free bending technology, Guo et al. [18] studied the evolution law of the bending radius R with the deviation U by finite element simulation. The physical significance of each region in the motion diagram of bending die was also interpreted. Engel et al. [19] studied the effects of springback of material and deviation of bending machine on the bending radii for three-roll-push-bending and proposed a correction value model to compensate the forming-roll position. The interpolation method was used to determine the radii distribution of spline contour and the desired bending contour could be obtained. Groth et al. [20,21] analyzed the influence of process parameters of three-roll-push-bending on the transition zone and found that there is the shape deviation between CAD model and actual bent tube geometry due to the abrupt change in curvature distribution. An algorithm was presented to determine the typical bending characteristics and describe the curvature distribution of transition zone. To effectively determine the forming parameters for a complex spatial bent tube, Zhang et al. [22] established the mathematical collection for illustrating the shape features of unstable elements and acquiring the process parameter.
It was noted that the previous studies have focused on the relationship between the spatial posture of bending die and the stable arc segment. However, the evolution law of the bending direction and bending curvatures of bent tube with the motion trajectory of bending die has not been deeply explored. The formation mechanism of stable arc segment and unstable arc segment were revealed. According to the motion trajectory of bending die, the geometrical characteristic of unstable arc segment and description method were determined. Aiming at three typical complex bending tubes, the precise forming of spatial tube components could be realized by optimizing the motion trajectory of bending die. The research work accelerates the engineering application of free bending technology in forming complex spatial tube product.

Research methodology
The research framework of this paper is shown in Fig. 1. The formation mechanism of bending arc of tube in free bending process was first revealed. The evolution law of geometric characteristics of bent tube axis with the motion trajectory of die was analyzed. The deviation U and space angle θ of bending die were determined by the arc radius of bent tube and the azimuth of plane space where the arc segment is located. In addition, the transition segments with continuously varying axis curvature can be fitted by the sinusoidal half-wave equation, and the transition segments with continuously varying bending direction was optimized. Finally, three typical bent tube components with different shape characteristics were formed by free bending forming technology. A new method of re-modeling bent tube axis by introducing the transition segment was also validated by comparing the free bending experimental results with the theoretical model.

Uniaxial tensile experiment
The bent tube is weakly constrained by bending die in free bending process, and the bending behavior of tube is very dependent on the mechanical properties of material. The stainless steel tube with the diameter of 19.0 mm and the wall thickness of 1.0 mm was employed, and its tensile and compressive properties are assumed to be symmetrical [23]. The uniaxial tensile test was performed to obtain the mechanical properties of stainless steel tube. Considering that whole tube is not suitable for tensile tests, the uniaxial tensile specimens were cut along the axial direction of tube, as described in Fig. 2. The experimental mean stress-strain curve of stainless steel tube was depicted in Fig. 3. The Hollomon model described in Eq. (1) was employed to fit stress-strain curve, which is basically consistent with the experimental data. The corresponding mechanical properties are also summarized in Table 1.

Three-dimensional free bending experiment
Three-dimensional free bending experiments were performed on the free bending forming device manufactured by Nanjing University of Aeronautics and Astronautics (NUAA ) [24], as shown in Fig. 4. The device is consist of the propulsion device, bending die, and guide mechanism. The original straight tube moves uniformly along Z-axis of device under the axial thrust from the propulsion device. The bending die can move along X-axis and Y-axis of device driven by two motors. The tube is bent by the coupling effect of bending die and guide mechanism. The corresponding arc segment is formed. The deviation and moving direction of bending die could be changed in real time. The magnitude and direction of bending moment acting on the tube also change accordingly, resulting in the formation of bending arc segments with different bending directions and bending radii. In addition, to reduce the friction effect between free bending die and tube, the bending die with ceramic bushing and the guide mechanism with ceramic bushing were also used, and the tube surface was evenly coated with lubrication oil before free bending experiment. To prevent the tube from rotating along Z-axis due to its own weight during free bending process, the propulsion device with sharp claws was used, as shown in Fig. 5. When the original straight tube moves along Z-axis, the claw is cut into the end of tube under the action of axial thrust to prevent the tube from rotating. Furthermore, the clamping mechanism was used to prevent the axial instability of tube because of the excessive axial thrust in the free bending process.

Finite element simulation of free bending
To observe the formation process of bending arc segment of tube more directly, the finite element simulation for free bending was carried out. In particular, the stress-strain distribution and the change of curved tube axis can also be characterized by numerical simulation method. The finite    Fig. 6. The contact type condition between tube and die is set as the general contact. The analysis step is set as dynamic explicit. The moving path of bending die needs to be determined according to the deviation U and space angle θ for each curved arc. Mass scaling is applied in each step, and the mass scaling factor is set as 10,000 to save the computational cost. For the boundary conditions, velocity along X-axis and Y-axis is applied on spherical bearing while velocity along Z-axis was applied on tube. The stainless steel tube is modeled with the deformable shell elements, and the element size is set to 3 mm. Due to that, the bending die and guide mechanism hardly deform, they could be constrained as the rigid bodies. The filleted corners of bending die and guide mechanism bushing are in direct contact with the tube; the mesh of this area is subdivided and defined as 1 mm in size to improve the accuracy of free bending numerical simulation. The mechanical properties of stainless steel tube were imported into the finite element model, and the friction coefficient between the tube and die was set as 0.05 [25].

Stress-strain analysis
The stress-strain state and geometry parameters of bent tube in free bending process are depicted in Fig. 7. The equilibrium equation along the radial direction could be written as: where is the tangential stress, is the radial stress, and is the radius of bent tube at any position.
Based on von Mises yield criterion, the equivalent stress ̃ and strain were deduced as: Since the circumferential strain is equal to zero = 0 and the volume of material is constant ( + + = 0 ), the equivalent strain was written as: where is the tangential strain and is the radial strain. In addition, could be also expressed as: where R is the radius of strain neutral layer. Hence, the following equation was obtained by substituting (3), (4), and (5) into (1).
Combined Eqs. (2) and (6), could be derived as: where C is the constant. Since at the innermost and outermost part of bent tube is equal to zero, the radial stress and o on the intrados and extrados of bent tube were written as: where R i is the radius of innermost part of bent tube and R o is the radius of outermost part of bent tube.
Substituting Eqs. (8) and (9) into (6), the tangential stress i and o on the intrados and extrados of bent tube were expressed as respectively: In the process of free bending, the strain neutral layer would tend to shift from the intrados into the extrados under the action of axial thrust. Thus, the tangential compressive stress i is greater than the tangential tensile stress o .

Analysis on the bending behavior of tube
The force exerted by the bending die and the guide mechanism on the tube is illustrated in Fig. 8. The distance between guide mechanism and bending die was defined as the offset A, which is related to the mechanical structure of die. The distance between bending die and guide mechanism along Y-axis was called as the deviation U, which is directly determined by the position of bending die. In addition, it might be found that the boundary between bending arc segment and straight segment of tube is close to the exit of guide mechanism. It indicates that the bending deformation process of tube mainly occurs near the exit of guide mechanism. Ignoring the friction effect between tube and free bending die, the bending moment M formed by F 1 exerted by bending die, F 2 from guide mechanism, and F 3 applied by propulsion device causes the bending deformation of tube, which was expressed as: where M is related to the deviation U. The bending moment M is improved by increasing U, resulting in the decrease of bending radius of bent tube. The numerical simulation result of free bending is depicted in Fig. 9. The deviation U was set as 8.0 mm. The offset A was set as 30.0 mm. It might be summarized that the arc segment of tube is formed in the area between bending die and guide mechanism. To analyze the bending deformation behavior of bent tube in detail, the simulated tangential strain on the extrados and intrados of bent tube was extracted and is depicted in Fig. 10. It was found that there is a slight compressive strain when the tube is located inside the guide mechanism. This is because the tube is subjected to the axial thrust from the propulsion device. Subsequently, the strain on the part of tube near the exit of guide mechanism increases rapidly. The compressive strain on the inside of curved arc is obviously greater than the tensile strain on the outside of curved arc. Finally, the strain on the inside and outside of curved arc hardly changes as the free bending process progresses. According to the strain evolution law in the different areas of tube, it could be further proved that the bending deformation of tube mainly occurs near the exit of guide mechanism and the shape of curved arc between bending die and guide mechanism hardly changes.
In the process of free bending, the bending moment M exerted by free bending die on bent tube can be changed by adjusting the deviation U, then the bending radius R also  changes accordingly. Therefore, it is necessary to determine accurately the relationship between bending radius R and deviation U, which is the premise of using the free bending forming technology to accurately form the bent tube with complex spatial shape. Since the properties of stainless steel tube are uniformly distributed along the circumferential direction, it is assumed that the relationship between U and R is independent of the bending direction of tube.
The experimental stainless steel bent tubes at different deviation U are depicted in Fig. 11a. The bending radius of bent tube is decreased as the deviation U is improved.
And the relationship between U and R is also summarized in Fig. 11b. When the deviation U is smaller, the bending radius R decreases rapidly with the increase of U. However, when the deviation U is larger, the bending radius R decreases slightly with the increase of U. According to the evolution trend of U-R relationship, the following equation was used to fit the experimental U-R data.
where a and b are the fitting coefficients. According to the bending radius for the special arc segment and Eq. (13), the deviation U of bending die could be calculated and the corresponding bent tube is formed. Since the bending radius of arc segment is determined, the central angle of arc segment can also be determined by controlling the arc length.

Determination of the space angle of bending die
In the process of free bending, it is also very important to accurately determine the movement direction of bending die for forming the bent tube with complex axis characteristics. For the curved arcs with different bending directions, the corresponding movement direction of bending die are different. The movement direction of bending die can be expressed by the space angle θ, as illustrated in Fig. 12. When formulating the process parameters of free bending, the space angle θ n of the n'th curved arc may be derived from the space angle of the previous curved arc, as written in Eq. (14). where θ n-1 is the space angle of the n-1'th curved arc. The n,n+1 and n−1,n−2 are the projection angle, which are illustrated in Fig. 12. First, it is necessary to determine the control point P of each curved arc, that is, the intersection point of tangent lines at the end of arc. The space coordinate system o-xyz was established with the origin at the control point P n-1 of the n-1'th curved arc. The straight line P n-1 -P n was set as the z-axis. The projections of straight line P n-2 -P n-1 and straight line P n -P n+1 in the x-y plane are defined as L 1 and L 2 . The angles between L 1 , L 2 and the x-axis are defined as the projection angles n,n+1 and n−1,n−2 , respectively. After determining the space angle θ 1 of the first curved arc, the space angles of all other curved arcs can be iteratively calculated in turn. According to the determined space angle θ and deviation U of bending die for each curved arc, the spatial position of bending die can be calculated and derived as: where U x is the deviation of bending die along the X-axis and U y is the deviation of bending die along the Y-axis.
In the free bending process, the bending radius of bent tube is determined by the deviation U of bending die, and the bending direction of bent tube is determined by the space angle θ of bending die. According to Eq. (15), the position and posture of die is dependent on deviation U and the space angle θ of bending die, which is the premise of using the free bending forming technology to accurately form the predesigned bent tube.

Geometric characteristics of bent tube axis
When forming the single bending arc with the constant curvature, the position of die is constant. When forming two adjacent arc segments with different bending radii or bending directions, the bending die needs to be transferred from one position to another. Since the offset A is fixed, the bending die can move in the plane where X-axis and Y-axis are located. According to the changing characteristics of bending radii and bending directions of the adjacent arcs, the movement trajectories of bending die could be divided into two categories. As depicted in Fig. 13, when the bending radii of adjacent arc segments change, the bending die would move along the radial direction. However, when the bending directions of adjacent arc segments change, the bending die would move along the circumferential direction. As we can find, the movement of bending die along different directions on X-Y plane corresponds to the changes of curvature radius and bending direction of adjacent arc segments, respectively. During the movement of bending die, the original straight tube is pushed continuously along the axial direction. Hence, the transition segment with gradually changing bending radius or bending direction would be formed, which is located between two adjacent arc segments.
The formation process of transition segment with the variable bending radius is shown in Fig. 14, which was obtained by the numerical simulation method. The arc segment b with the bending radius R b is connected to another arc segment a with the bending radius R a . When forming an arc segment b, the bending die needs to be transferred from original deviation U a to pre-determined deviation U b . Since the tube is always fed along the axial direction, the transition segment a-b would be formed and its bending radius R a-b gradually decrease from R a to R b . Subsequently, the stable arc segment The formation process of transition segment with the variable bending direction is shown in Fig. 15. The bending radii of adjacent curved arcs c and d are the same, but their bending directions differ by 90°. When the bending die is transferred from U θy to U θx , the spatial bending direction of transition segment c-d also changes from the vertical direction to the horizontal direction. Then, the stable arc segment d with the fixed bending direction is formed when the movement direction of bending die U θx does not change.
To sum up, due to the changes in the shape of bent tube axis, such as the bending radius or bending direction, the spatial position of bending die needs to be adjusted accordingly. The transition segment of bent tube caused by the movement of die would affect the shape of bent tube component, and its curvature radius or bending direction is constantly changing. In the free bending process, the impact of geometric characteristics of bent tube axis on shape of bent tube needs to be further considered.

Transition segment with the continuous change in bending radius
Based on the analysis in Sect. 3.4, the transition segment caused by the movement of bending die may affect the axis shape of bent tube. According to the motion trajectory of die, the shape of transition segment can be divided into two types: continuous change in the axis bending radius and continuous change in the axis bending direction. First, the influence of transition segment with the continuous change in bending radius on the shape of bent tube was analyzed. The geometric model of bent tube is depicted in Fig. 16a. The bending radius of bent tube changes suddenly between the straight line segment and the arc segment. The simulated bent tube and the evolution of curvature of curved arc are In the transition segment e-f, the bending radius of tube gradually decreases from infinity to R = 116 mm. As the length of transition segment increases, its proportion in the arc segment e becomes larger. The axis error between the theoretical model and the simulated result is summarized in Fig. 17. It was found that the simulated bent tube axis deviates from the geometric model as the length of transition segment increases. When the length of transition segment is equal to the length of entire arc segment e, the bending radius of transition segment changes slowly from infinity to 116 mm. Due to the existence of transition segment, the simulated central angle is always smaller than that of the geometric model when the arc length is the same. According to the research of Xiong et al. [26], the crosssectional distortion of transition segment of tube is more significant than that of the stable segment. To optimize the cross-sectional distortion of transition segment, the evolution law of cross-sectional distortion rate with the length of transition segment was analyzed and is summarized in Fig. 18. As the length of transition segment increases, the distortion rate of transition segment decreases gradually. This may be due to the fact that the bending die is always in motion during the formation of transition segment. With the combined action of bending moment and dynamic effect, the degree of cross-sectional distortion of transition segment is larger than that of stable segment. As the length of transition segment increases, the bending die moves slower. The weaker dynamic effect of bending die movement would lead to the smaller cross-sectional distortion. When the length of transition segment exceeds 30 mm, the distortion rate is almost no longer reduced. According to the analysis results in Fig. 17 and Fig. 18, when the length of transition segment is 30 mm and the bending radius is 115 mm, both the axis shape and cross-sectional distortion of bent tube are where s is the length of transition segment. Since the transition segment would affect the axial shape of bent tube, it is necessary to explore the axis evolution law and the modeling method of transition segment based on the characteristic of free bending process. Currently, the method of describing axis shape of transition segment has not been studied yet. Taking into account the continuous change in the axis bending radius of transition section, the transition curve is adopted to characterize the axis shape of transition segment. The curvature of transition curve changes continuously, enabling the smooth transition between the stable arc segments with the different bending radii. In this paper, the sinusoidal half-wave equation was used to model the transition segment with the continuous change in bending radius, as depicted in Eq. (17).
where x and y are the tangential direction and vertical direction in the tangent coordinate system. R is the bending radius of stable arc segment connected by the transition curve and s is the length of transition curve that could be calculated by Eq. (16). Figure 19a shows the connected transition curve between the stable arc segment and the straight line segment. It was concluded that the sudden change of bent tube axis curvature is well characterized by the transition curve. Due to the continuous change of bending radius in the transition segment, the corresponding deviation U of bending die also changes continuously, which may be calculated according to U-R relationship curve. Finally, the transition segment of bent tube can be precisely shaped. The geometric model of bent tube with the designed transition curve and the experimental result are shown in Fig. 19b and c. The shape of experimental bent tube is very close to the axis of geometric model. Transition segment with the continuous change in bending radius can be well represented by the sinusoidal half-wave curve.

Transition segment with the continuous change in bending direction
Similarly, the effect of transition segment with the continuous change in bending direction on the axis shape of curved arc was analyzed. The geometric model of bent tube is depicted in Fig. 20a. The bending direction of bent tube changes suddenly between the adjacent stable arc segments. The spatial azimuth angles of the planes where the adjacent arc segments are located differ by 90°. The simulated bent tube and the evolution of bending direction of curved arc are also depicted in Fig. 20b. Since the bending directions of adjacent arc segments differ by 90°, the space angle θ of bending die are also different, which can be determined according to Eqs. (14) and (15). When forming these two adjacent stable arc segments, the transition segment is formed due to the movement of bending die. The bending Fig. 17 Axis error between the geometric model and the simulated result Due to the fact that the bending direction of transition segment changes continuously, it is necessary to explore its influence on the axis shape of bent tube. For these two adjacent stable arc segments, the corresponding space positions of bending die were shown in Fig. 21a. The bending die can be moved from P 1 to P 2 via route 1, route 2, or route 3. For route 1, the bending die is moved from P 1 to P 2 along the circumferential direction. Although the space angle θ of bending die is constantly changing, its deviation U is the same, so the bending radius of formed transition segment is unchanged. For route 2, the bending die is moved directly from P 1 to P 2 along the straight line. Both the deviation U and the space angle θ of bending die are changing in real time, resulting in the continuous changes in the bending radius and the spatial azimuth angle of transition segment. The bending die first returns to the origin and then moves towards P 2 along the radial direction when the route 3 is adopted. Since the bending die is moved along the radial direction, the spatial azimuth angle of transition segment does not change. The simulated bent tubes are shown in Fig. 21b under three route conditions. The length of transition segment is equal, which can be determined by Eq. (16). The shape of simulated bent tube always deviates from the geometric model regardless of the moving path of bending die.
The simulated bent tube based on route 3 is closest to the geometric model, and the simulated bent tube obtained by route 2 deviates significantly from the geometric model. The deviation degree of simulated bent tube based on route 1 is between those of the simulated bent tubes based on route 2 and route 3. This may be because when the bending Fig. 19 Geometric model and the experimental result of bent tube die is moved from P 1 to P 2 , the spatial azimuth angle of transition segment based on route 1 and route 2 gradually changes, resulting in the spatial orientation of the formed arc segment 2 significantly deviating from the geometrical model. In addition, the bending radius of transition segment with Route 2 is also changing, which further leads to the increase of axis error. However, the change in the spatial azimuth angles of arc segments 1 and 2 only occurs when the bending die is located at the origin of route 3. For the transition segment with route 3, the bending radius gradually changes and the spatial azimuth angle remains unchanged.
According to the above phenomenon, it was summarized that the impact of transition segment with the continuous change in bending direction on the axis shape of curved arc is more obvious. It could be concluded from Fig. 21a that the cross-sectional distortion rate does not depend on the movement route of die. In general, when the bending direction of adjacent curved arc changes, the transition arc caused by the movement of bending die would affect the axis shape of bent tube. The transition segment formed by the movement of bending die along route 3 has the least effect on the shape error of bent tube.  Three typical bent tubes including the arc segment with involute axis, the arc segment with spiral axis, and the spatial multi-bend arc segments with different bending radii and bending direction were employed to study the applicability of free bending forming technology in forming bent tubes with different shape features. The curvature of arc segment with involute axis varies continuously, while the bending direction of arc segment with spiral axis varies continuously. The conventional bending forming technologies are difficult to manufacture the spatial bent components with continuous changes in bending direction and bending radius, which are dependent on the shape and size of bending die. The bending direction and bending radius of an arc segment in the spatial multi-bend arc segments are constant. The traditional rotary-bending technology can realize the integral forming of multi-bend arc components by changing bending dies.

Typical bent tube with involute axis
Considering that the position of forming die might be changed in real time, the free bending forming technology has advantages in forming the bent tube with continuous change in bending radius. Here, the bent tube with involute axis was employed and the bending radius of axis varies continuously. The shape and geometric parameters of curved tube is depicted in Fig. 22a. The stainless steel tubes were used in this paper. The bending radius of bent tube varies continuously along the axis of involute and the bending radius of each segment can be calculated by its curvature. As can be found, the bending radius gradually increases from 70 mm to infinity. According to Eq. (13), the deviation U of bending die corresponding to each section of bent tube axis can be obtained. Regardless of the bending direction of tube, the change of the spatial position of bending die is described in Fig. 22b. It could be seen that the motion of bending die is very smooth and there is no sudden change of spatial position. Therefore, when the bent tube with continuous variable curvature is formed by free bending forming technology, no additional transition segment needs to be introduced. The experimental bent tube with involute axis is shown in Fig. 23. Its axis shape is almost identical to the geometric model. To quantitatively analyze the deviation between the experimental bent tube and geometric model, the axis of experimental bent tube was measured and extracted by three coordinate measuring system. The axis deviation between the experimental curved tube and the geometric model is less than 1.0 mm. Finally, the bent tube with continuous variable curvature was accurately formed by the free bending forming technology without considering transition segment.

Typical bent tube with spiral axis
To explore the capability of free bending forming technology in forming bent tube with continuous variable bending direction, the typical bent tube with spiral axis was employed in this paper. The geometric model of curved arc is depicted in Fig. 24a. Since the bending radius of spiral curved arc is the same at each point along its axis, the deviation U of bending die was determined based on U-R relationship curve. Nevertheless, the tangential direction of spiral bent tube changes continuously along its axis. According to the analysis in Sect. 3.3, the space angle θ of bending die corresponding to each point on the axis can be calculated. The change of the spatial position of bending die is described in Fig. 24b.
As we can find, the bending die is rotated around the origin in the X-Y plane as the free bending forming process proceeds. The continuous change of bending direction of bent tube can be achieved by modifying the deflecting direction of bending die. The motion of bending die is very smooth and there is no sudden change of spatial position. Therefore, when the bent tube with continuous variable bending direction is formed by free bending forming technology, no additional transition segment needs to be introduced. The experimental spiral curved tube is given in Fig. 25. Its axis is extracted and compared with the geometric model. The geometric shape of experimental curved tube is almost identical to the geometric model. The axis error between the experimental curved tube and the geometric model is less than 1.0 mm. The above results indicated that the bent tube with continuous variable bending direction can be formed by the free bending forming technology. In general, the bent tubes with complex spatial shape are composed of the axis with continuous variable curvature, the axis with continuous variable bending direction or the combination of the above two characteristic axes, which are all suitable for the free bending forming process. Furthermore, the curved tubes with continuous variable bending radius and bending direction can be integrally formed by the free bending forming technology without introducing transition segment and considering the influence of the transition segment on the axis shape.

Typical bent tube with spatial multi-bend arc
The typical bent tubes with spatial multi-bend arc are widely used in various fields of industry, which might be manufactured by CNC rotation-drawing bending technology. The shape and specific dimensional parameters of bent tube are illustrated in Fig. 26. The axis of the entire bending tube is composed of the straight line segment and the circle arc segment with the constant bending radius and bending direction. Different from these bent tubes whose axis curvature and bending direction change continuously, the curvature and bending direction of the spatial multi-bend arc change suddenly. For example, there is the sudden change in the axis curvature between the arc segment S 2 and the straight line segment L 3 . The bending direction of arc segment S 6 is different from that of arc segment S 8 by 90°. Before conducting the free bending forming experiment, it is necessary to first determine the deviation U and the space angle θ of bending die according Eqs. (13)- (14). The spatial position of bending die corresponding to each segment is shown in Fig. 26. The experimental bent tube is shown in Fig. 27. The axis shape of experimental bent tube deviates significantly from the geometric model. Due to the existence of the transition segment, the tangential direction at the endpoint of each bending arc deviates from the geometric tangential direction. The experimental bending radius is obviously larger than that of geometric model. The central angle of experimental bent tube is obviously smaller than that of geometric model. Starting from the first arc segment, the axis error becomes more and more obvious as the forming process progresses. This is because the axis error of previous arc segment would be accumulated in the next arc segment. According to the analysis in Sect. 4, it is essential to analyze the impact of the transition segment on the axis shape of curved tube, especially for bent tube axis with abrupt changes in curvature and bending direction. Equation (16) and route 3 were used to construct the axis of transition segment. In addition, reducing the bending radius of stable arc segment is of great significance to keep the spatial tangential direction of each arc segment unchanged. To accommodate the characteristics of the free bending forming process, the modified axis shape of spatial multi-bend arc is shown in Fig. 28. The relative positions of the two ends of modified curved arc are similar to the original geometric model. However, the overall axis of the modified curved tube is not completely consistent  with the spatial multi-bend arc. The experimental bent tube based on the modified axis shape is shown in Fig. 29a. Its axis was extracted and compared in Fig. 29b. The axis of the experimental curved tube was almost identical to the geometric model. The axis error between the experimental curved tube and the geometric model is less than 1.0 mm. Compared with the experimental bent tube without considering the transition segment, the experimental bent tube redesigned by using the transition curve is closer to the original geometric model, especially the spatial position relationship between the ends of bent tube. Furthermore, it should be noted that the bending radius of each stable arc segment is slightly smaller than that of the original geometric model. For the space bent tube with multi-bend arc segment, an approximate bent tube can be formed by the free bending forming technology. It is necessary to introduce the transition segment in the process of geometric model reconstruction. Moreover, when designing the shape of bent tube axis, it is essential to consider the characteristics that the curvature and bending direction of axis cannot be changed suddenly in free bending process.
Compared with the conventional bending forming technology, the free bending forming process is more flexible, and the arc segment with continuous variable curvature and bending direction can be bent and formed. In addition, the arc segments with constant curvature and bending direction can be also precisely formed by the free bending forming technology.

Discussion and conclusion
In this research, the formation mechanism of bending arc segments in the free bending forming process was fully revealed and three typical bent tubes with different shape features were used to illustrate the applicability of free bending forming technology and the modeling rule of bending tube axis.
(1) The bending radius and bending direction of bent tube are determined by the deviation U and the space angle θ of bending die. Obtaining U-R relationship curve and the plane space azimuth is the premise of precisely forming complex spatial bent tube.
(2) According to the movement trajectories of bending die, the shape features of bent tube axis can be divided into the stable arc segment and transition segment. When the bending die moves along the radial direction, the transition segment with continuous change in bending radius would be formed, which can be described by the sinusoidal halfwave equation. The experimental bent tube with spatial multi-bend arc (3) For the adjacent arc segments with different bending directions, the transition segment with continuous change in bending direction would be introduced in the free bending process. The corresponding movement route of bending die in X-Y plane was optimized to reduce its influence on the shape of bent tube axis.
(4) The bent tubes with continuous change in curvature and bending direction could be precisely shaped via the free bending forming technology without considering transition segment. For the bent tube with spatial multi-bend arc segment, the transition segment needs to be introduced into the reconstruction of geometric model.