# Springback Prediction of Free Bending Based on Experimental Method
## 2 Free-Bending Technology  
Free-bending technology is different from traditional bending technologies such as winding, rolling and pressing. It is characterized by dieless bending, as there is no need to change the die to accommodate tubes with different bending radii. The tube bender adjusts the bending die pose in real time so as to realize the forming of tubes with different bending radii. As shown in Figure 1, \(U\) is the offset of the bending die, which is the height of the center of the bending die from the axis of the guide mechanism in the plane formed by the bending tube. The distance between the center of the bending die and the foremost end of the guide mechanism is called \(A\). The length of the arc segment formed during the movement of the bending die from 0 to \(U\) is called '\(\prime\)'. With the bending die tangent to the spherical bearing, the spherical bearing installed on the bearing seat is driven by the motor to move in the plane XOY perpendicular to the feeding direction of the \(Z\) axis. The bending die is assumed to be at the origin. Before processing, the bending die is set in the original position, and its axis coincides with the axis of the guide mechanism, which is called the equilibrium position. During processing, the spherical bearing is pushed by a servomotor which drives the bending die to move in the XOY plane from the equilibrium position to the processing position \(U\). The tube blank deforms under the combined action of the force applied by the inner wall of the bending die and the propulsion force to form an arc-shaped tube with a bending radius \(R\) and an axis arc length \(l\). After the tube with the predetermined arc length is formed, the bending die moves to the next processing position or returns to the equilibrium position. Conversely, the straight section is processed when the bending die is kept at the initial equilibrium position. As shown in Figure 2, there are five stages of a single bend: the first straight section, first transition section, arc section, second transition section, and second straight line. (1) First transition section  
Figure 3a shows the first transition section of the bending die movement to \(U\). During the first transition section, the tube is in a state of dynamic combination and transition from a straight line and an arc section. At the moment the bending die starts to move, the tube within the guide distance \(A\) are straight segments, and the arc section is firstly generated at the outlet of the guide mechanism and gradually spreads to the whole guide distance along with the uniform delivery of the tube blank. The straight section within the guide distance becomes shorter and shorter during the feeding movement, and the radius of the arc section decreases from infinity to the target radius. Figure 1: Diagram of free-bending structure. Figure 2: Diagram of the stages of free-bending. Set the feeding time as \(t_{1}\), the real-time sending length is \(v_{Z}t_{1}\), the bending die offset \(U_{1}\) of straight section, the bending die offset \(U_{2}\) of arc section is can be obtained using Equations (1) and (2):  
\[U_{1}=\tan\frac{180v_{Z}t_{1}}{\pi R}\times\left(A-R\sin\frac{180v_{Z}t_{1}}{ \pi R}\right) \tag{1}\]  
\[U_{2}=R\left(1-\cos\frac{180v_{Z}t_{1}}{\pi R}\right) \tag{2}\]  
The total bending die offset \(U\) is the sum of \(U_{1}\) and \(U_{2}\), its value is in dynamic change. The relationship between \(U\), radius \(R\) and feeding time \(t_{1}\) can be calculated by Equation (3):  
\[U=U_{1}+U_{2}=R-R\cos\frac{180v_{Z}t_{1}}{\pi R}+\tan\frac{180v_{Z}t_{1}}{\pi R }\left(A-R\sin\frac{180v_{Z}t_{1}}{\pi R}\right) \tag{3}\]  
The total length \(L_{1}\) of the tube blank in the first transition section can be calculated by Equation (4):  
\[L_{1}=\frac{\pi R\mathrm{arcsin}\frac{A}{R}}{180^{\circ}} \tag{4}\]  
(2) Arc section  
In the arc section, the bending die reaches the predetermined \(U\) and holds on.