Design of three-roll bending machine tool and research on compensation algorithm

With the rapid development of profile processing industry, the processing shape, efficiency, and consistency of profiles are more and more demanding. In this case, the progress of bending machine tools was very important. In this study, the disadvantages and shortcomings of traditional three-roll bending machine are investigated, and a new design of three-roll bending machine is proposed, including a new mechanical structure, hydraulic pressure, and control system. ABAQUS was used to analyze the constrained mode of the table, and the first six order natural frequencies and natural vibration modes were obtained. A mathematical model was established to analyze the relationship between the pressure displacement and the forming radius under ideal working conditions, and the limit position of the downward movement of the lower press roll and the speed relationship between the lower press roll and the support roll are discussed. Based on the experimental data, the nonlinear curve fitting was carried out on the rebound compensation algorithm of asymmetric rolling bending forming. The fitting rebound compensation algorithm was carried out on the symmetric rolling bending forming experiment. The experimental results show that this algorithm can be applied to both symmetric rolling bending and asymmetric rolling bending.


Introduction
In recent years, the application of bending machine tool in profile processing industry has become more and more extensive, playing an important role in railway construction, aerospace, and other fields. 1 Most of the bending machine tools are based on the three-point circular principle, and through the change of the relative position between the work rolls and the rotating motion, the profiles produce continuous elastic-plastic bending so as to obtain the metal forming equipment of predetermined shape.The processing methods can be divided into symmetric processing and asymmetric processing. 1,2Common processing profiles are Angle steel, square pipe, aluminum profiles, and so on.
At present, the common bending processes in the market are bending, stretch bending, press bending, roll bending, etc. 3,4 Bending processing is often used in the field of steel bar bending, such as Yin and Li 5 designed a steel bar bending machine with adjustable bending 1 School of Mechanical and Automotive Engineering, Liaocheng University, Liaocheng, China 2 Angle, whose bending Angle can be adjusted between 0°and 45°.The motor drives the bending eccentric wheel to rotate through belt drive and two-stage gear deceleration drive.It provided power for the bending steel bar and filled the gap in the market.Niu et al. 6 designed a small electronic hydraulic pipe bender using the pressure bending process, which can bend large diameter pipes.Compared with stretch bending and press bending, the flexibility of roll bending is obviously higher.By adjusting the relative position of the work roll, it can be processed into different shapes such as circular arc, oval, spiral line, and so on.Authors such as Chen and Tan 3 developed aluminum profile rolling bending machine with large rolling modulus, and its mechanical structure adopted the form of 3 + 1 roller.In terms of hydraulic drive system, Su et al. 7 analyzed the shortcomings of the hydraulic system of the original three-roll plate rolling machine, and designed an improvement scheme featuring complete oil filling and smooth buffering, thus improving the service life of the hydraulic motor.In terms of calculation, the mathematical relationship between the pressure displacement of the lower roll and the forming radius of the threeroll bending machine is rich.For example, Deng et al. 8 analyzed the relationship between the forming radius and the displacement of symmetrical three roll plate winding machine, and deduced the functional relationship between the pressing radius and the forming radius based on the mathematical model.Yao 9 derived the functional relation between the pressure and the forming radius under asymmetric roll bending according to the mathematical model.The above is only under ideal conditions to deduce the relationship, in the actual processing process, the profile often because of its own material, shape, and other factors produce rebound phenomenon, which leads to the above relationship can not be directly used for production and processing.In this context, Shu and Wang 10 found that the relationship between pressure reduction and molding radius was in line with the power function relationship according to the experimental data.Gong and Wang 11 adopted the curve fitting optimization method with the minimum curvature radius to obtain the relationship function between the pressure and the forming radius in asymmetric roll bending forming, and verified it through experiments.It can be seen that the function relationship derived on the basis of experiment has high reliability in practical production.
Considering the problems of traditional three-roll bending machine in machining accuracy, machining method, and energy consumption, a three-roll bending machine is designed in this paper.The innovation design of mechanical structure, hydraulic and control system to improve the above problems.The mathematical model of three-roll motion is established, and the relationship between the pressure of the lower roll and the forming radius of the profile is analyzed under the ideal working condition.The limit position of the lower roll moving down and the speed coordination between the lower roll and the support roll are discussed.Based on the experimental data, the nonlinear fitting of the asymmetric roll bending rebound compensation algorithm was carried out.After the fitting was completed, the symmetric roll bending forming experiment was carried out on the algorithm.The experimental results show that the algorithm can be applied to both symmetric roll bending and asymmetric roll bending.

Working principles and processes
The three-roll bending machine tool designed in this paper adopts the working principle of three-point circle forming, 12 and the processing process is divided into loading stage and unloading stage. 13,14As shown in Figure 1: Support rolls A and B rotate in a clockwise direction, and lower press roll C rotates in a counterclockwise direction at the same time.In this way, the profile will undergo continuous elastic-plastic bending to obtain a predetermined shape.
The relative position relationship between the lower roll and the support roll will affect the machining effect, but has no effect on the machining process.When the lower displacement of the press roll is larger, the forming radius of the profile is smaller.The smaller the pressing displacement of the current roller, the larger the forming radius of the profile.
The specific working process is as follows: first, the profiles to be machined are put into the nylon molds of the support rollers A and B, and then the hydraulic station works to provide working energy for the Bursa accumulator and the hydraulic cylinder.The Bursa accumulator will store the energy temporarily, and the hydraulic cylinder will push the lower roller C to the position that just presses the machined profiles but does not make them deform.At this time, the signal of the grating ruler is cleared by the control system, and the motion detection mechanism pushes forward and compacts the processed profile.Finally, the supporting rollers A, B, and C rotate at the same speed and realize the processing of profiles with slow pressing of the lower roller C. The motion detection mechanism detects the real-time position of the profile movement, and the grating ruler detects the pressing distance of the lower roller C. When the processing is complete, the cylinder and hydraulic cylinder are retracted to the initial position, waiting for the next processing.Figure 2 is the flow chart of the three-roll bending machine.

Mechanical structure
The mechanical structure of three-roll bending machine mainly includes three parts: 1, work roll mechanism, 2, motion detection mechanism, 3, table mechanism.Its structure diagram is shown in Figure 3.The two support rolls of the traditional bending machine are in A fixed state, which can not take into account both symmetric and asymmetric processing modes.In this design, the support rolls A and B are connected with the hand screw.The processing personnel can adjust the position of the support rolls A and B by the hand screw, so that the two processing modes of symmetric and asymmetric can be realized.The lower press roll C is connected to the piston rod in the hydraulic cylinder and is driven forward or backward by the expansion of the piston rod.
Considering that the contact side of the square tube and the lower press roll C will be indented in the processing process, and the non-contact side will be stretched to a certain extent, resulting in an axial force parallel to the working shaft, the design uses tapered roller bearings to support the rotating parts 15 and bear the axial force generated in the processing process to reduce friction.
Considering the high requirements of the equipment for the rotation speed control and position accuracy of the work roll system, to ensure the stable operation of its own weight as far as possible, so this design adopts the combination of servo motor and cycloidal pin wheel reducer as the power device of the work roll system.
Cycloidal pin wheel reducer is a novel transmission device which applies planetary transmission principle and uses cycloidal pin tooth meshing.The whole transmission device of cycloidal reducer can be divided into three parts: input part, reduction part, and output part.The input shaft is equipped with a 180°dislocation of the double eccentric sleeve, and the eccentric sleeve is equipped with two roller bearings called rollers, forming H structure.The center hole of the two cycloidal wheel is the raceway of the roller bearing on the eccentric sleeve, and the cycloidal wheel meshes with a group of circular arranged needle teeth on the needle gear, in order to form an internal meshed speed reduction mechanism where the teeth difference is one tooth.

Motion detection institutions. Motion detection mechan-
ism is an important auxiliary mechanism for accurate control of three-roll bending machine.It is mainly used to detect the motion of profiles in the process of processing, to judge whether there is skid phenomenon, and to detect the length of profile motion (arc length of processing).This design makes up for the traditional three-roll bending machine can not accurately control the actual arc length of the profile, and improves the machining accuracy of arc length.Its structure is mainly composed of cylinder, Z-shaped connecting frame, auxiliary encoder, synchronous belt module, nylon press roll, moving table, frame, linear sliding rail module, L-shaped connecting frame, and other components.Figure 5 is a schematic diagram of motion detection mechanism.
The main working principle of the motion detection mechanism is: the nylon press roll and auxiliary encoder installed on the mobile platform, through the synchronous belt module to achieve belt drive.The mobile platform is connected to the frame through the linear  sliding rail module, the cylinder is fixed on the frame, and the piston rod and the mobile platform are rigidly connected, through the telescopic movement of the piston rod to drive the mobile platform forward or backward.When the mobile platform moves forward, the nylon press roller will press the profile, and it will rotate with the profile under the action of friction.Under the condition of belt drive, it will drive the auxiliary encoder to rotate at the same time, so that the movement of the profile can be detected in real time.
Workbench mechanism.The worktable mechanism is a platform for carrying work roll mechanism and motion detection mechanism, which is installed on the square pipe support frame.As shown in Figure 6, the table mechanism is composed of front baffle, side baffle, and casting.In order to reduce the pressure of the square tube support frame and to achieve lightweight of the whole machine structure, four quadrilateral weight reducing holes are arranged on the casting.According to the working stroke of the hydraulic cylinder and the structural size of the work roll, the shapes and sizes of the casting, front baffle, and side baffle are reasonably designed.The lower press roll chute is arranged inside the casting, and it is combined with the front baffle and side baffle to form the support roll chute.The advantage of this design is that the work roller mechanism can be installed on the table mechanism in the form of a sleeve, reducing the difficulty of installation.

Modal analysis
Modal analysis is a method to study the dynamic characteristics of structures.It is an application of system identification method in engineering vibration field.Mode is the natural vibration characteristic of mechanical structure, and each mode has a specific natural frequency, damping ratio, and mode shape. 16In order to study the anti-vibration performance of the workbench, ABAQUS was used to conduct constrained modal analysis of the workbench.The first six modes of the threeroll bending machine are shown in Figure 7.
The device operates at a frequency of approximately 66.7Hz.The minimum frequency of the 1-6 mode is 440.64Hz, which is much higher than the operating frequency of the equipment, so there is no resonance phenomenon.
From the analysis of the first six modes, it can be seen that the mode shape of the first mode is represented by the longitudinal stretching of the table.The modes of order 2 and 3 are represented by the transverse bending and torsion of the table and the transverse stretching, respectively.The mode shapes of the fourth order show that the right side of the table is lengthwise stretched and the left side is lengthwise compressed.The mode shape of order 5 shows that the left and right sides of the front baffle of the workbench have large longitudinal tensile deformation.The mode shapes of the sixth order show that the right side of the front baffle of the workbench has a large longitudinal compression deformation, and the left side has a large longitudinal tensile deformation.

Hydraulic system design
According to the working principle of the three-roll bending machine, it only needs to extend or retract the  piston rod of the hydraulic cylinder to control the pressing position of the pressing roll C. 7 The difficulty lies in the control of the piston rod of the hydraulic cylinder can be changed continuously and according to a certain program, whether the expansion distance or expansion speed should be accurately controlled.
Considering the single processing action, the oil supply mode of a single gear pump is selected. 6In order to reduce the consumption of electric energy, this design uses the cystic accumulator for the temporary storage of energy, uses the relief valve to adjust the rated working pressure of the hydraulic system, uses the pressure sensor to detect the pressure value of the whole hydraulic system, when the actual pressure reaches the maximum rated working pressure, the control system stops the operation of the gear pump.At this time, the energy required by the telescopic movement of the piston rod is provided by the sac accumulator.When the actual pressure is less than the minimum rated working pressure, the control system starts the gear pump operation.Considering the control difficulty of the piston rod, this design uses the direct acting proportional reversing valve to control the movement of the piston rod.The direct acting proportional direction valve uses the proportional solenoid, in which the spool can not only perform transposition, but the transposition stroke can also be continuous, proportional, or vary according to a certain program command.The valve can control the speed of the piston which is in line with the control requirements of this article.
Compared with the traditional bending machine hydraulic system, this design has the advantage of improving the accuracy and reducing the consumption of electric energy.According to 12 h of processing per day, about 23.4 kWh of energy can be saved per day.Figure 8 shows the working principle of the hydraulic system.

Control system design
The control mode of the three-roll bending machine designed in this paper is PID control.Its control system mainly includes: 1. motion detection subsystem, 2. hydraulic drive subsystem, and 3. motor drive subsystem.
In the motion detection subsystem, the solenoid valve is controlled by the motion controller, which can control the extension or retraction of the cylinder, so as to make the auxiliary encoder advance or retreat.At the same time, the auxiliary encoder feeds back signals to the motion controller in real time, forming the first closed-loop control.
In the hydraulic drive subsystem, GTS motion controller to drive a pulse signal to control the work or stop of the hydraulic pump, hydraulic pump output pressure to the accumulator, accumulator can be temporarily stored, pressure sensor can measure the working pressure value in real time and feedback to GTS motion controller, forming a second closed-loop control.When the pressure value just reaches the specified working pressure (5 MPa), GTS motion controller will close the hydraulic pump, then the pressure used by the work is provided by the accumulator.
GTS motion controller sends voltage signals to drive 4 to control the hydraulic valve to open and close, so In the motor drive subsystem, the GTS motion controller sends pulse signals to driver 1, driver 2, driver 3, and then control the motions of motor 1, 2, and 3. Thus the rotational motions of the support roll A, support roll B, and press roll C are achieved.
GTS motion controller is the brain of the whole control system, all the control procedures of the system are placed in GTS motion controller, GTS motion controller and the terminal board is connected, controlled equipment and acquisition equipment and the terminal board corresponding signal interface connection.GTS motion controller through the terminal board analog and switch output to complete the operation of the control object, acquisition device acquisition site data and through the terminal board feedback to GTS motion controller acquisition data.
The touch screen is connected with the controller through the RS-232 communication protocol, which displays the man-machine operation interface of the control software.Operators master the operation of the entire equipment through the data state displayed on the touch screen, and control the electrical components through the man-machine interface, modify and adjust the mechanical and control parameters and other operations.Figure 9 is the wiring diagram of the control system.

Mathematical model analysis
Three-roll bending machine realizes rolling bending forming of the profile through the three points where the lower press roll C and the support roll A and B contact the profile.Because the positions of the support roll A and B are fixed in the processing process, it is necessary to adjust the position of the lower press roll C to determine the third point of the arc processing, so as to process the arc with different radii.The machining radius of profiles is related to the depth of pressing roll C. 9 In the actual production process, most factories lack theoretical guidance on the use of bending machines.Workers need to refer to the theoretical drawings and adjust the downward pressure displacement of the lower press roll for many times, 17 so as to process profiles consistent with the shape of the theoretical drawings.This process often causes a large waste of time and processed materials.Therefore, it is the main research direction of three-roll bending machine to explore the relationship between the pressure of the lower roll and the forming radius of the profile.The three roll bending machine is divided into two kinds of processing that is, symmetrical and asymmetric. 18hese two kinds of processing can both be done by the equipment demonstrated in this article, so the two kinds of processing are analyzed and calculated.

Analysis of the relationship between pressure and molding radius
Before the three-roll bending machine starts working, workers will adjust the positions of support rolls A and B. When the positions of rollers A and B are symmetrical in the downward direction of lower roll C, it is symmetrical rolling bending, as shown in Figure 10.
As shown in Figure 10, the vertical distance between the center of profile and the center of support roll is, The vertical distance between the center of the lower press roll and the center of the support roll is, The pressing depth of the lower roller is, Substituting ( 1) and ( 2) into (3), below can be obtained, 9 This equation is the relation between the pressure of the press roll C under symmetric rolling bending and the molding radius of the profile, and is only the theoretical equation under the ideal model.

Analysis of maximum displacement of lower roll
From equation ( 4), it can be seen that the curvature radius R of the profile is a function of the downward  pressure displacement D of the downward pressure roll, and the derivative of equation ( 4) on the downward pressure displacement D can be obtained, 8 When D = a, dR dD = 0, and the minimum value of curvature radius R of the profile is, 8 Because the radius of curvature R of the profile is also limited by the radius of the lower roller r 0 , the minimum radius of curvature that the profile can be processed is, 8

Speed matching analysis of lower roll and support roll
As shown in Figure 11, arc length of the lower press roll is, Arc length of support roll rotation is, Since a 1 = a 0 , simultaneous equation ( 8), ( 9) can be obtained, 15 The equation is the relation equation of the angular velocity between the lower press roll and the support roll.According to the equation, the speed coordination between the lower press roll and the support roll can be realized.

Experimental study of algorithm
The experimental prototype was built according to the design, as shown in Figure 12.The diameter r 0 of the lower press roll C is 155 mm, and the diameter r of the support roll A and B is 150 mm.Adjust the two support rolls to the symmetrical position of lower press roll C by hand wheel, the distance a = 191 mm.At present, common carbon steel square tube section size on the market is 50 3 12, 50 3 15, 30 3 30 (length 3 width, unit: mm), thickness between 1.5 and 4 mm.This paper will be used on the market commonly used 30 3 30 3 2 mm carbon steel square tube (L = 30 mm) for the experiment.Substituting a, L, r into equation ( 6), R a = 11 mm can be obtained.Because r 0 .R a , it can be seen from equation ( 7) that the minimum value of the curvature radius of the machined profile under this condition is 155 mm.Under this condition, the curvature radius of the five groups of profiles to be processed is defined.According to equation (4), the required downward displacement of the curvature radius of each group can be calculated respectively.Define the angular velocity of the support roll as v 0 = 0.52 rad/s, According to equation (10), the angular velocity of the lower roller under each group of experiments is v 1 .The details are shown in Table 1.R d is the defined radius, and D L is the pressure displacement calculated according to equation (4).The pressure displacement in Table 1 and the angular velocity of the working roll were respectively input into the experimental prototype system for symmetric rolling forming.Figure 13(a) is the experimental process diagram, and Table 2 is the experimental radius of the pressure displacement in each group.Figure 13(b) Comparison of the results of defined radius and experimental radius.
The experimental results show that there is a large error between the experimental radius and the defined radius of the profile processed by the theoretical formula (4).This is because in the actual processing process, when the load of the press roll C is withdrawn, part of the deformation of the profile will recover, which is referred to as rebound phenomenon. 11herefore, in actual processing, D value in equation (11) needs to be compensated and corrected.
The mathematical model of the traditional three-roll bending machine is established on the basis of the experimental data, and its R-D curve equation is D = a 3 R b .As the experimental results are affected by various factors, there is a considerable difference between them and the actual results.Direct application in processing will cause many adverse consequences.In order to avoid this phenomenon, researchers obtained the curve equation of R-D through theoretical calculation, 11 specifically, as follows: This equation is also used for asymmetric roll bending forming. 16ince the three-roll bending machine designed in this paper can satisfy both symmetric and asymmetric processing modes, the following experiments are done in this paper to explore whether the equation can satisfy both symmetric and asymmetric processing modes.
In order to obtain this formula, seven groups of new downward pressure displacements were defined, and seven groups of new experimental radii were measured after symmetrical rolling bending forming.By combining the seven groups of data with the five groups of data in Table 2, 12 groups of experimental data could be obtained, as shown in Table 3.
Because the arrangement of data is nonlinear distribution, the nonlinear curve fitting method will be adopted to solve this equation.According to the experimental data in Table 3, equation ( 11) was nonlinear fitted.After six iterations, the fitting converges and the results are shown in Figure 14.
According to the fitting result shows that the variable coefficient of the rebound compensation a = 112.90662,b = 20.00696,c = 33.57437,d = 20.000814487.The rebound compensation equation under the mechanical condition can be obtained by substituting equation (11).In order to further verify the reliability of this equation, this equation is introduced into the control system for symmetric roll bending forming verification.
On the basis of meeting the conditions, five groups of curvature radii are defined again, and the five groups of data are tested.Figure 15 shows the experimental results, and Table 4 shows the circular arc experimental data table.

Discussion
To intuitively reflect the machining accuracy, calculate the error rate h according to equation (12), and the results are shown in Table 5.With reference to the bending accuracy of existing three-roll bending machine tools on the market and considering the mechanical errors and measurement errors of experimental results the construction of the prototype, the error rate is set as \0.5% in this paper.
It can be seen from the error rate data table that the maximum error rate of this experiment is 0.452%, and the machining accuracy meets the requirements within the set error rate range.It can be seen that this rebound compensation equation is applicable not only to asymmetric rolling bending machining but also to symmetric rolling bending machining.

Conclusion
In this paper, considering the shortcomings of the existing three-roll bending machine, the mechanical structure, hydraulic system, and control system of the three- roll bending machine were innovated.Compared with the traditional three-roll bending machine, the design can not only realize the two processing modes of symmetrical and asymmetric processing, but also detect the arc length of the profile in real time and improve the processing accuracy.The mathematical model was established to analyze the relationship between the pressure of the lower roll and the forming radius of the profile under the ideal working condition, the maximum displacement of the lower roll, and the speed matching relationship between the lower roll and the support roll.The experimental prototype was built and the square tube was used as the experimental profile for the preliminary experiment.According to the experimental data, the non-linear fitting was carried out on the rebound compensation algorithm of asymmetric roll bending forming.After the fitting was completed, five groups of symmetric roll bending forming experiments were carried out on the algorithm, and the error rates of the experimental results of each group were 0.263%, 0.103%, 0.200%, 0.061%, and 0.067%, respectively.The experimental results are all lower than the error rate set before the experiment, indicating that the algorithm can be applied to both symmetric and asymmetric rolling bending.

Declaration of conflicting interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Figure 1 .
Figure 1.Working principle of three-roll bending machine.
Work roll mechanism.The work roll mechanism is the core of the mechanical system of the three-roll bending machine.The whole machine is processed into the required shape through the cooperative movement of the work roll mechanism.The work roll mechanism includes support roll A, support roll B, and lower press roll C. Figure 4(a) shows the position diagram of the work roll.All three are driven by the combination of the same model servo motor and reducer.Because this design uses 30 3 30 3 2 mm general carbon square steel pipe bending test, so the nylon mold on the work roll slot height is 30 mm, slot depth is 15 mm.

Figure 2 .
Figure 2. Working flow chart of three-roll bending machine.

Figure 3 .
Figure 3. Three-dimensional assembly drawing of three-roll bending machine.

Figure 4 .
Figure 4. Schematic diagram of work roll mechanism: (a) work roll position drawing and (b) work roll surface polishing diagram.

Figure 7 .
Figure 7. Cloud diagram of sixth order modal vibration mode of the bending machine workbench: (a) the first mode, (b) the second mode, (c) the third mode, (d) the fourth mode, (e) the fifth mode, and (f) the sixth mode.

Figure 8 .
Figure 8. Working principle of hydraulic system.

Figure 10 .
Figure 10.Symmetrical rolling and bending diagram.R: profile radius; r: radius of support roll; a: center distance of support roll; r 0 : lower roll radius; b: vertical distance between lower press roll and support roll; D: displacement of lower roller; L: profile thickness.

Figure 11 .
Figure 11.Speed relation between lower roll and support roll.O: center of profile curvature; C 1 : lower roll rotation arc length; a 1 : angle corresponding to arc length of lower roll rotation; C 0 : arc length of support roll rotation; a 0 : support roll rotation arc length corresponding angle; r: radius of support roll; r 0 : lower roll radius; Let this period of motion be t; The angular velocity of the lower roller is v 1 ; The angular velocity of the support roll is v 0 .

Figure 13 .
Figure 13.Experimental diagram under ideal equation: (a) experimental process diagram and (b) define the radii and the experimental radii.

Table 1 .
Theoretical calculation data sheet.

Table 2 .
An experimental data sheet.

Table 4 .
Data sheet of three experiments.Figure 15.Experimental results.

Table 5 .
Error rate data table.