Force Analysis and Curve Design for Laying Pipe in Loop Laying Head of Wire Rod Mills
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Abstract
Laying head is a high-precision engineering device in hot-rolled high speed wire rod production line. Previously research works are focused on the laying pipe wear-resisting. Laying pipe curve design method based on wire rod kinematics and dynamics analyses are not reported before. In order to design and manufacture the laying pipe, the motion and force process of the wire rod in the laying pipe should be studied. In this paper, a novel approach is proposed to investigate the force modeling for hot-rolled wire rod in laying pipe. An idea of limited element method is used to analysis and calculates the forces between laying pipe inner surface and wire rod. The design requirements of laying pipe curve for manufacturing are discussed. The kinematics and dynamics modeling for numerical calculation are built. A laying pipe curve equation is proposed by discussing design boundary conditions. Numerical results with different laying pipe curves design parameters are plotted and compared. The proposed approach performs good result which can be applied for laying pipe curve design and analysis for engineering application.
Keywords
Wire rod mills Laying head Laying pipe curve design Wire rod force analysis1 Introduction
Laying head is one of a key and precision device in high speed production line of hot-rolled wires. Laying head device is located in water cooling section after the finishing rolling mill. The finishing mill is designed for a speed of 90‒120 m/s for wire rod having a minimum diameter. Normally the mill produces series for steel wire rods diameter is between ϕ4.5 and ϕ16.0 mm [2, 3]. Thus, the speed of linear wire rods pass through the laying head is very fast while the laying head is rotating. The hot wire rods are formed into loops out of the laying head and continuously laid on a Stelmor conveyor to send to the next processes [4, 5].
The output shaft continually rotates, meanwhile the linear wire rod pass into the rotated laying pipe from the entry and throughout from the exit. In this process, the wire rod is formed to be looped shape as coils and fall down on the conveyer. The dimension of the formed coils is decided by the radius of the laying pipe exit [6, 7, 8].
Laying pipe is an important part in laying head device for hot rolled high speed wire rod production line. Most of the research works of laying pipe are presented by patents. Some mechanical structures for high speed laying head devices are proposed in Refs. [9, 10, 11]. The speed relationship between laying head and rolling mill is presented in Ref. [12] for control. Fiorucci tried different materials used for laying pipe in Refs. [13, 14], and some new design schemes to improve the wearing resistance property for laying pipes are reported in Refs. [15, 16, 17]. The manufacturing device and operating process for laying pipe is presented in Ref. [18]. A curve design and load analysis method is presented in Ref. [19], which is benefit for laying pipe applies theoretically. An optimal design of the Morgan Construction Companies’ laying pipe is investigated in Ref. [20] to deal with uneven wear. A pipe design approach is developed to minimize and evenly distribute wear on the inner surface of laying pipe. Some researchers have provided ideas to improve the laying hand performance in modeling, design and application [9, 21, 22, 23, 24, 25, 26, 27].
Most of the mentioned research work focused on the wear-resisting for laying pipe application. The wire rod kinematics and dynamics performance while passing through the laying pipe is not referred. Therefore, the design method based on wire rod kinematics and dynamics analysis for laying pipe curve cannot be carried out. Thus, the study on the topic of wire rod force analysis in laying pipe is a research gap before.
In this paper, a method for laying pipe design, as well as the kinematics and dynamics modeling for the wire rod passing in the laying pipe are proposed. The first section introduces the hot rolled high speed wire rod production line. The function and structure of the laying head machine is described. It is pointed out that the laying pipe is the key issue for the laying head. In the second section, design requirements for the laying pipe are discussed. The problems for pipe curve, laying head working stability and speed relationships are mentioned. In the third section, a method of kinematics and dynamics analysis for wire rod in the pipe is proposed to characterize the process of wire rod in the laying pipe. We also proposed a new method for laying pipe design in fourth section. The numerical computation results for the designed laying pipe are compared with existing laying pipes in the fifth section. The numerical results indicated the feasibility of the proposed design method and kinematics and dynamics modeling. It is also provided a method for laying pipe parameters choosing and discussing.
2 Design Requirements for the Laying Head Pipe
The wire rod is pinched and feed into rotated laying pipe by pinch roll as shown in Figure 1. The wire rod in the laying pipe is effect by the pipe inner surface and its shape is changed from a straight line into coils. There are friction force, centrifugal force, inertial force and supporting force impacted on wire rod [28]. The entire forces act on the wire rod together and the wire is bended according the laying pipe curve and formed into coils to achieve the desired radius at the exit as Figure 2. The requirements for the laying head pipe in a stable working condition can be explain from three aspects.
2.1 Speed Requirement
If the balance of Eq. (1) is disturbed, the laying head will work unstable and the coils will be formed in different dimensions.
2.2 Laying Pipe Curve Requirement
2.3 Wear-Resisting Requirement
The laying pipe should be used for long term and cannot be worn easily. The worn is caused by the friction force on the inner surface of laying pipe. The curve should be designed smoothly to prevent the wire rod be worn sharply in some position. An ideal laying pipe curve can make the wire rod pass through comfortably. In this situation, the friction force will be decreased and will not be concentrated in small section of the laying pipe. As the result, the laying pipe will be used for a long term and the laying head working stability is improved.
3 Kinematics and Dynamics Modeling
What we desire is an arm exoskeleton which is capable of following motions of the human upper-limb accurately and supplying the human upper-limb with proper force feedback if needed. In order to achieve an ideal controlling performance, we have to examine the structure of the human upper-limb.
3.1 Kinematics Analysis for the Wire Rod Motion in the Laying Pipe
When the laying pipe curve Eq. (2) is given, the relative velocity between wire rod and laying pipe \(\vec{\varvec{v}}_{\text{g}}\) can be calculated by Eq. (6) and the relative acceleration \(\vec{\varvec{a}}_{\text{g}}\) can be obtained by substituting Eq. (11) into Eq. (10).
Equation (15) express the absolute acceleration components in the three directions of the cylindrical coordinate in Figure 8. The parameters of (a_{r})_{a}, (a_{θ})_{a}, (a_{z})_{a} are the target of the kinematics calculation.
3.2 Dynamics Analysis for the Wire Rod Motion in the Laying Pipe
A force state of the wire rod is modeling and analyzed in this part. In order to analysis and calculate the force affection of the wire rod, the entire wire rod in the laying pipe is divided into limited sectional parts. Each part unit can be seen as an independent object with several kinds of forces acted on it. Thus, a limited elements method is proposed and applied to build the dynamics modeling of sectional wire rod unit.
As shown in Figure 9, the small section unit A–B, which defined as section number N, is selected to build the force modeling.
In Figure 9, \(\vec{\varvec{f}}_{mi}\) is the friction force on the sectional rod unit N, acted by inner surface of laying pipe; \(\vec{\varvec{P}}_{ni}\) is the supporting force on the section rod unit N, acted by inner surface of laying pipe; \(\vec{\varvec{N}}_{i}\) is the axial push force on the section rod unit N, acted by the section rod unit (N − 1); \(\vec{\varvec{N}}_{i + 1}\) is the pull force on the section rod unit N, acted by the section rod unit (N + 1); \(\vec{\varvec{f}}_{i}\) is the inertia force on the section rod unit N.
In Eqs. (16) and (17), m_{i} is the mass of the section rod unit N (A−B); f_{ri}, f_{θi} and f_{zi} are the component forces in each direction; (a_{r})_{ai}, (a_{θ})_{ai} and (a_{z})_{ai} are the acceleration in each directions, also mentioned in Eq. (15).
In Eq. (19), the direction of the supporting force \(\vec{\varvec{P}}_{ni}\) is the same as the normal of vector of section rod unit N (vector \(\overrightarrow {{\varvec{AB}}}\)), the direction of the friction force \(\vec{\varvec{f}}_{mi}\) is the same as the tangential of vector \(\overrightarrow {{\varvec{AB}}}\).
Thus, Eq. (19) can be solved if N_{i+1} is given as a known parameters from laying pipe working boundary condition.
3.3 Boundary Conditions
It can be concluded that the inertia force \(\vec{\varvec{f}}_{{i}}\) can be calculated by providing the equation of laying pipe curve and the rotation speed of the laying head. Thus, the contact force \(\vec{\varvec{P}}_{ni}\), friction force \(\vec{\varvec{f}}_{{mi}}\) and push or pull force \(\vec{\varvec{N}}_{{i}}\) for the sectional unite in the laying pipe can be calculated by Eqs. (27), (28) and (29), if the vector force \(\vec{\varvec{N}}_{{i+1}}\) is known as an initial parameter.
3.3.1 Boundary Conditions at the Entrance of the Laying Pipe
There is a speed relationship between the wire rod feeding speed v_{f} and the rotated speed of laying pipe ω_{0} at the entry of the laying pipe, which has been formulated as v_{f} = ω_{0}R_{0} in Eq. (1) of Section 2. The wire rod at the entry of the laying pipe should keep the relationship in Eq. (1) and it can be a boundary conditions at the entrance of the laying pipe.
3.3.2 Boundary Conditions at the Exit of the Laying Pipe
Let’s select the sectional unit at the exit of laying pipe to study, at this unique moment \(\vec{\varvec{N}}_{N + 1}\) is a boundary condition for Eq. (19) as expressed in Eq. (30). Thus, the force balance Eq. (19) of the final unit can be solved. For rod unit N, the result of push force \(\vec{\varvec{N}}_{N}\) in Eq. (19) can be calculated and the result \(\vec{\varvec{N}}_{N}\), seen as the pull force for rod unit N − 1, can be used as a given parameter in the rod unit N − 1 force balance equation. Thus, all the forces on each sectional unit can be calculated by providing computed result \(\vec{\varvec{N}}_{i}\) as a given parameter for i − 1 unit. Thus, the sectional units force equilibrium equations (19) will be solved one by one from the final sectional unit to the first sectional unit.
3.4 Numerical Procedure for the Wire Rod Dynamics Analysis
A numerical calculation procedure for the proposed modeling can be carried out based on the above-mentioned kinematics and dynamics analysis to compute the velocity and forces of the wire rod in the laying pipe.
In order to solve the velocity and forces of the wire rod, some parameters should be given as the initial conditions for the numerical procedure calculation. The curve equation \(\varvec{\delta}(R,\theta ,Z)\), the feed speed of the linear wire rod v_{f} and the rotation speed of the laying pipe ω_{0} should also be given as the parameters for the kinematics and dynamics modeling.
4 Boundary Conditions Discussion and Laying Pipe Curve Design
4.1 Boundary Conditions for the Laying Pipe Curve Design
Some known conditions can be concluded at the laying pipe entry and exit:
When the wire rod passing through the entry (t = 0), the position components in R, θ and Z directions are all zero; the velocities components in R, θ and Z directions are 0, v_{f} and 0 respectively.
When the wire rod passing through the exit (t = T_{0}), the position components in R, θ and Z directions are R_{0}, θ_{0} and Z_{0}; The velocities components in R, θ and Z are 0, 0 and ω_{0} respectively.
The boundary conditions in Eqs. (33) and (34) should satisfy the proposed laying pipe curve equation. These known conditions will help us to choose the form and parameters for laying pipe curve equation.
4.2 Curve Equation Design for the Laying Pipe Curve
In order to design the laying pipe curve equation, the parameters equations θ(t), R(t), Z(t) with variable t should be considered independently.
4.2.1 Relationship between Rotation Angle θ and Time t
4.2.2 Relationship between Radial Displacement R and Time t
4.2.3 Relationship between Axis Displacement Z and Time t
It can be seen that from the functions characters in Figure 12(a). The θ(t) is a the linear function, the function characters for R(t) and Z(t) are similar as sine function or cosine function.
A solution for design parameters of laying pipe
Design parameter | Value | Design considerations |
---|---|---|
v_{f} (m/s) | 92 | Decided by rolling production process flow |
R_{0} (mm) | 525 | Decided by wire coils dimension |
θ_{0} (°) | 350 | Decided by mechanical structure of laying head |
Z_{0} (mm) | 1623.4 | Decided by mechanical structure of laying head |
ω_{0}(rad/s) | 175.24 | Computed by v_{f}/R_{0} |
T_{0}(s) | 0.035 | Computed by θ_{0}/ω_{0} |
k* | 1.0349 | Computed by Eq. (40) set α _{0} ^{* } = 2° |
5 Numerical Results and Comparison for Laying Pipes
5.1 Laying Pipe Curves Obtain
The coordinate data for R_{2}(t) and R_{3}(t) function
Point in spline curve | Parameter | ||
---|---|---|---|
t (×10^{−3}) | R_{2} (mm) | R_{3} (mm) | |
1 | 0 | 0 | 0 |
2 | 3.28 | 67.4 | 16.25 |
3 | 6.9 | 196.6 | 106.2 |
4 | 10.4 | 301.0 | 248.4 |
5 | 13.9 | 395.7 | 372.9 |
6 | 17.4 | 443.7 | 452.5 |
7 | 20.9 | 479.1 | 485.1 |
8 | 24.5 | 505.5 | 494.0 |
9 | 28.0 | 518.7 | 508.3 |
10 | 31.5 | 523.5 | 520.9 |
11 | 35.0 | 525 | 525.5 |
5.2 Numerical Computation and Comparison with Kinematics and Dynamics Results
Figure 15 shows the numerical results for wire rod in laying pipes. It can be seen that the wire rod relative velocities in three laying pipes have the same tendency. The relative velocities in axial direction v_{z} are all equal to the feed speed v_{f} at the entry, and decrease meanwhile the wire rod is passing in the laying pipe. Finally it is nearly zero at the exit. The relative velocities in tangential direction are v_{θ} are all zero at the entry, and change to nearly feed speed v_{f} at the exit of the laying pipe. The relative velocities in radial direction are v_{r} are all zero at the entry and change to the maximum in the middle part of the laying pipe, then it is decreased to zero at the exit of laying pipe.
Comparing with δ_{2} and δ_{3}, the maximum value of v_{θ} for designed laying pipe is appeared nearly the exit of the pipe. It is caused by the smooth curve between the pipe entry and middle part, and the wear of the laying pipe inner surface will be reduced in this position. The variable characters of the wire rod relative velocity in Figure 15 ensure the laying pipes has only axial velocity at entry and has only tangential velocity at exit. The resultant velocity of the three velocities components is equal to the feed speed v_{f} at any time in Figure 15.
5.3 Numerical Results and Comparison with Different Design Parameters
In this section, some different values of parameters are changed for the designed laying pipe curve. Numerical results are carried out to discuss the dynamic performance with different design parameters. The initial design parameters are list in Table 1, and the wire rod dimension is ϕ6.5 mm.
It can be seen that the maximum value of axial force in Figure 19(a) are 30 MPa, 43 MPa and 72 MPa, respectively for different feed speeds. The axial force is increased obviously when the feed speed is 110 m/s. In Figure 19(b), the maximum friction force in 110 m/s is about twice as much as 90 m/s. Comparing with other feed speed, when the feed speed is 110/m, the laying pipe will be worn out first. Actually, the higher feed speed will lead to fast wearing out, it has been verified in other wire rod rolling plant.
It should be noticed that, the tensile strength limitation for most hot-rolled rod materials are less than 70 MPa in the temperature range of 750‒900 °C. Thus, the wire rod will be pulled broken into pieces if the laying pipe curve working with v_{f} = 110 m/s. Thus, there should be a new designed laying pipe curve for the higher feed speed and rotation speed.
It is indicated from the numerical results from Figure 19 to Figure 22 that the requirement for higher speed laying pipe curve design should increase the laying pipe axial dimension Z_{0} and choose a suitable pipe curve rotated angle θ_{0} between 350° and 400°.
6 Conclusions
The paper focused on the laying pipe of laying head device in the hot-rolled high speed wire rod production line. The kinematics and dynamics modeling are first proposed and investigated in this paper with the aim to obtain the wire rod performance while passing through the laying pipe. The statics study and calculation for the wire rod in the laying pipe are carried out by using sectional divided method. A laying pipe curve equation is formulated by considering its design boundary conditions. Thus, numerical results of wire rod velocity and forces can be calculated by the obtained laying pipe equations. With the proposed modeling, kinematics and dynamics results comparison between the designed laying pipe curve and two existing laying pipes are carried out. Some variable parameters of laying pipe curve are calculated with different value, and the results are discussed to find suitable design pipe curve parameters for different working conditions. The results of calculation and discussion indicate the proposed method of laying pipe curve design and the research for kinematics and dynamics are feasible for laying pipe application.
Notes
Authors’ Contributions
SY conceived and studied the laying pipe design method and the force computations. MC contributed the force analysis of the laying pipes. Bin Ma performed the experiments for laying pipe curves. SY and MC wrote the paper. MC, GC and BM reviewed and edited the manuscript. All authors read and approved the final manuscript.
Authors’ Information
Shuangji Yao, born in 1981, he is currently an associate professor at Yanshan University, China. He received his PhD degree from Beihang University, China, in 2010. He has joined the study program at LARM during 2007‒2008 in Italy. His research interests include the theory of mechanisms, robots, and mechanical research in rolling machine. He is a member of IEEE and ASME.
Marco Ceccarelli, born in 1958, received his Ph.D. degree in Applied Mechanics in 1988. He is a Full-Time Professor of Mechanics of Machinery and Director of LARM, Laboratory of Robotics and Mechatronics at University of Cassino and South Latium, Italy. He is a Member of Robotics Commission of IFToMM (The International Federation for the Promotion of Machine and Mechanism Science). He is the IFToMM President. He has written the books ‘‘Fundamentals of Mechanics of Robotic Manipulation’’ in 2004 and ‘‘Mecanismos’’ in 2008 and 2014. His research interests are in mechanics of mechanisms and robots. He is the author/co-author of more than 700 papers, presented at conferences or published in journals, and he has edited 23 books as for conference proceedings and specific topics.
Giuseppe Carbone, born in 1972, is currently an associate professor at University of Cassino and South Latium, Italy. He received his PhD degree from University of Cassino and South Latium, Italy. He has carried out several periods of research abroad, such as in Germany, Japan, Spain, and China. His research interests include stiffness of multibody robotic systems, robotic hands and grippers, mechatronic designs, design of experimental test-beds. He has published more than 250 peer reviewed papers on the above-mentioned topics.
Bin Ma, born in 1981, is currently a mechanical engineer at HBIS Group Co., Ltd, China. He received his master degree on mechatronidcs from Beijing Institute of Technology, China, in 2006.
Acknowledgements
We would especially like to thank Dingxuan Zhao, Wantang Fu and Qingtian Zhou for the fruitful discussions, leading to very interesting formulations of problems and deeper insight. Furthermore but not least, we would like to thank Zhifeng Pang, Yongqian Zhao and Renquan Wang for their great help and kindness.
Competing interests
The authors declare that they have no competing interests.
Funding
Supported by China Postdoctoral Science Foundation Project (Grant No. 2017M611184).
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