Design and development of highprecision fourroller CNC plate rolling machine and automatic control model
Bending force analysis
The stress diagram of the continuous rolling and bending process of the profile is shown in Figure 5.
The force acting on the left scroll wheel \(F_{c}\)
$$ F_{c} = \frac{M}{{\left( {R + t} \right)\sin \alpha }} $$
(4)
Force acting on the lower roller \(F_{b}\)
$$ F_{b} = \frac{M}{{\left( {R + t} \right)\sin \left( {\delta + \theta { – }\alpha } \right)}} $$
(5)
Force acting on the upper roller \(FA}\)
$$ F_{a} = \frac{M}{{\left( {R + t} \right)}}\left( {\frac{1}{\tan \alpha } + \frac{1}{{ \tan \left( {\delta + \theta – \alpha } \right)}}} \right) $$
(6)
Calculation of bending moment of metal profiles
The functional relationship between the stress distribution along the height of the steel plate and the true stress on the plate section during linear pure plastic bending can be expressed as:
$$ \sigma = \sigma_{s} + E_{2} \varepsilon $$
(7)
Where \(\Sigma\) is the stress of the workpiece,\(\sigma_{s}\) is the yield limit of the material, \(\Varisilon\) is the strain of the workpiece, and \(E_{2}\) It is the linear strengthening modulus of the material, which can be obtained by referring to the relevant manual.
The bending moment M on the section is:
$$ M = \int_{A} {\sigma_{y} } dA = 2\int_{0}^{{\frac{\delta }{2}}} {\sigma bydy} $$
(8)
In the formula, b is the maximum width of the steel plate rolled by the plate rolling machine.Deformation initial bending moment \(M_{0}\) yes
$$ M_{0} = \sigma_{s} \frac{{b\delta^{2} }}{4} $$
(9)
Upper roller drive torque
The upper roller of the fourroller plate bending machine is the driving roller. The total driving torque acting on the upper roller is the sum of the torque consumed during the deformation process and the torque required to overcome the friction. The friction moment includes overcoming the frictional resistance consumed by the roller rolling on the curved plate and the frictional moment consumed by the roller bearing.
The torque consumed during the deformation process can be determined by assuming that the work done by the internal bending force is equal to the work done by the external force exerted on the upper roller:
$$ W_{n} = \frac{{M_{0} + M}}{2} \cdot \frac{L}{{R{\prime} }} $$
(10)
$$ W_{w} = M_{n1} \cdot \frac{L}{{{{D_{a} } \mathord{\left/ {\vphantom {{D_{a} } 2}} \right. \kern0pt} 2}}} $$
(11)
Where \(W_{n}\) is the work done by bending the internal force, \(W_{w}\) is the work of the external force acting on the upper roller, \(L\) is the length of the plate corresponding to the bending angle.
Let equation (10) be equal to equation (11).The torque consumed during the deformation process can be written as
$$ M_{0} = \sigma_{s} \frac{{b\delta^{2} }}{4} $$
(12)
The torque required to overcome friction can be determined using (13).Friction torque when shaft rollers are arranged asymmetrically
$$ M_{n2} = f\left( {F_{a} + F_{b} + F_{c} } \right) + \frac{\mu }{2}\left( {F_{a} d_{ a} + F_{b} d_{b} \cdot \frac{{D_{a} }}{{D_{b} }} + F_{c} d_{c} \cdot \frac{{D_{a} }}{{D_{c} }}} \right) $$
(13)
Where \(F\) is the rolling friction coefficient, take \(f = 0.8\;{\text{mm}}\); \(\exist\) is the sliding friction coefficient at the journal, \(\mu = 0.05\sim 0.1\) \(d_{a}\), \(D b}\), \(d_{c}\) are the journal diameters of the upper roller, lower roller and side roller respectively.
The total driving torque acting on the upper roller is:
$$ M_{n} = M_{n1} + M_{n2} $$
(14)
Upper roller driving power
The calculation formula of driving power is:
$$ P = \frac{{M_{n} v}}{60r\eta } $$
(15)
Where \(v\) represents the contour speed, \(r\) Represents the radius of the drive roller, take \(r = {{D_{a} } \mathord{\left/ {\vphantom {{D_{a} } 2}} \right.\kern0pt} 2}\); \(\and\) Indicates transmission efficiency.
Servo electric cylinder
Servo electric cylinder principle.
The servo electric cylinder is a new modular product that integrates the design of a servo motor and a ball screw or roller screw. By converting the rotary motion of the servo motor into linear motion, the product can precisely control speed, rotational speed and torque. In addition, the advantages of precise control of servo motors can also be used to achieve precise speed, position and thrust control, making this product a revolutionary solution for highprecision linear motion applications.
Characteristics of servo electric cylinder

(1)
The servo electric cylinder system is controlled by a servo motor and utilizes the closedloop control characteristics of the servo motor to achieve precise control of thrust, speed and position.

(2)
Compared with traditional hydraulic cylinders, electric cylinders offer several advantages, such as higher response reliability, higher precision stability, and advanced control functions.

(3)
The servo electric cylinder has the characteristics of long service life, strong environmental adaptability, and strong start and stop capabilities. Through the integration of servo motor and electric cylinder technology, this novel product can achieve excellent environmental performance, energy efficiency and highprecision motion control capabilities.
Servo electric cylinder structure
The structure of the servo electric cylinder is relatively simple, mainly composed of four parts: a driving mechanism, a reduction device, a linear transmission mechanism, and a transmission mechanism. There are two structural forms: servo motor and electric cylinder. One is a directconnected electric cylinder, as shown in Figure 6a. Directconnected servo motors are directly connected to the motor through a coupling. The other is a parallel structure, as shown in Figure 6b. The electric motor is installed in parallel with the electric cylinder through a highstrength timing belt, which is also called a parallel structure or a folding structure. In addition to the characteristics of the series electric cylinder, the overall length of the parallel electric cylinder is also shorter, making it more suitable for installation in limited spaces.
Research on servo electric cylinder selection
Selecting the appropriate servo electric cylinder parameters mainly depends on many factors, such as cylinder load, cylinder life, number of cycles, stroke distance and installation space.

(1)
load
Accurate knowledge of the load is critical to determining the most costeffective and reliable electric cylinder solution.
The relationship between the motor output torque and the electric cylinder output force is
$$ F = \frac{T \cdot \eta \cdot 2\pi \cdot R}{L} $$
(16)
Where \(F\) is the output force of the electric cylinder, \(T\) For the motor output torque, \(R\) is the transmission ratio, \(L\) is the lead of the screw (mm), and \(\and\) is the mechanical efficiency, usually 8590%.
Equation (17) can be used to provide a preliminary estimate of the motor size required to meet the load demand. In addition to the maximum force exerted during operation, force variation throughout the stroke is also a key factor to consider. The average load can be derived from the force variation over the entire operating cycle and serves as the basis for calculating the cylinder service life.
The average load calculation formula of the electric cylinder is:
$$ F_{m} = \sqrt[3]{{\frac{{F_{1}^{3} \cdot V_{1} \cdot t_{1} + F_{2}^{3} \cdot V_{2} \cdot t_{2} + F_{ 3}^{3} \cdot V_{3} \cdot t_{3} }}{{V_{1} \cdot t_{1} + V_{2} \cdot t_{2} + V_{3} \cdot t_{3} }}}} $$
(17)

(2)
life.
In the context of electric cylinders, the term “life” usually refers to the service life of the screw used within the cylinder. This life can be divided into two distinct components: the fatigue life of the screw (which can be quantified through calculations) and the service life (which depends on various service conditions, including temperature, average load, type of lubrication, frequency of relubrication, and other relevant factor.
The formula for calculating the service life is.
$$ L_{10} = \left( {\frac{{C_{a} }}{{F_{m} }}} \right)^{3} L $$
(18)
Where \(L_{10}\) For the service life of the electric cylinder,\(C_{a}\) is the basic dynamic load rating of screw pavement,\(FM}\) is the average load borne by the electric cylinder, and L is the lead load of the screw.

(3)
Cycle counting.
The optimal electric cylinder can be selected by accurately specifying the actuator’s acceleration and speed or providing the cycle time and required distance.

(4)
Running distance and installation space.
The selection of the electric cylinder, including its working stroke and installation space, is highly relevant. During operation of the electric cylinder, it must be ensured that it does not reach its mechanical limits. Therefore, safety strokes must be added to both ends of the working stroke S, resulting in a longer stroke section.The sum of these two strokes constitutes the operating distance S of the electric cylinder, that is\(S = S_{{{\text{Work}}}} + 2S_{{{\text{Safety}}}}\)as shown in Figure 7.
Whole machine assembly
Figure 81 shows the installation process of the slide rail on the bed, Figure 83 shows the installation process of the spindle motor and coupling, Figure 83 shows the installation process of the servo cylinder and side roller shaft, and finally the entire machine is installed.
The plate rolling machine is mainly composed of three parts: mechanical equipment, pneumatic system, and CNC system. The mechanical structure of this machine consists of four rollers: upper roller, left and right rollers, lower middle roller, servo electric cylinder, spindle motor, and frame. The upper roller is fixed on the frame, and the left and right rollers are driven by servo electric cylinders to carry out feeding movement along the guide rails. The threedimensional model of the fourroller CNC plate rolling machine is shown in Figure 9a, and the actual object is shown in Figure 9b.
discuss
From the comparison of Figure 10a and b, it can be found that the mechanical structure using the servo electric cylinder solution is cleaner, and the use of pure electric control increases the safety of the machine during operation. The specific comparative analysis is shown in Table 1.