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BIM Engineering Management Oriented to Curve Equation Model

Publié en ligne: 15 Jul 2022
Volume & Edition: AHEAD OF PRINT
Pages: -
Reçu: 18 Jan 2022
Accepté: 04 Mar 2022
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Format
Magazine
eISSN
2444-8656
Première parution
01 Jan 2016
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2 fois par an
Langues
Anglais
Abstract

This article uses the curve equation model to describe the initial value problem of differential equations in BIM project management. A new set of rolling curve-solving models is established for step-aligning in BIM project management. Based on the premise that the differential equation can be solved numerically, we appropriately simplify or set the required relational functions in the equation. Finally, we use mathematical software to numerically solve the differential equation and obtain the discrete function of the rolling curve. The research shows that the accuracy of the step flatness and the width of the stepped groove formed by the rolling curve of the solution in this paper is better than the original solution.

Keywords

MSC 2010

Introduction

As a new technology for forming shaft parts, cross wedge rolling has the advantages of material saving, high efficiency, and low energy consumption. The technology has broad application prospects in the machinery industry, especially the automotive industry. In the cross wedge rolling to form the inner corner steps, a transition section needs to be designed between the roll widening section and the finishing section, which we call the rolling section. In this way, the forming surface on the wedge shape can be gradually transformed into a finishing surface so that the rolling piece can gradually form an inner right-angle step. The design method of the rolling section is to use a smooth curve to cut off the corresponding wedge-shaped part in the direction perpendicular to the base surface of the roll to form the transition section [1]. The smooth curve forming the transition section is the rolling curve. The rolling curve design is the key technology for accurately forming the right-angle steps in the cross wedge rolled parts.

Firstly, the volume of the unformed zone is solved as a function of the rolling progress, and then the volume invariance principle is used to solve the rolling curve. The volume function is the key to the solution. Although the volume expression method and the specific solution method are different, the general can be summarized. First, assume a straight line, a smooth curve, or a simple rolling curve as the basic curve; we use the basic curve function with the rolling rotation relationship to derive the rolled piece Big end radius change function. Then we use the big-end radius function as the boundary condition to solve the volume model of the unformed area of the rolling piece and obtain the volume function under the corresponding parameter [2]. The advantage of this type of volume solution method is that it is relatively simple and easy to calculate. The disadvantage is that the calculation accuracy depends heavily on the assumed curve as the initial condition. This type of calculation is to correct the assumed smooth curve instead of solving a theoretical rolling curve. Although this kind of algorithm has certain practical value when dealing with simple rolling alignments such as inner right-angle steps. Still, it is ineffective for some rolling alignments with more complicated deformation processes (such as narrow step rolling alignment process). This is because it is difficult to obtain an initial curve close to the theoretical curve through simple assumptions, which leads to large errors in the calculation results. It isn't easy to guarantee the accuracy of the solution. In actual production, it is usually necessary to perform a lot of real rolling correction work on the calculated value before applying it to step forming. This leads to excessive reliance on experience in mold design, resulting in long mold design cycles, high costs, and poor design reliability.

This article starts from the basic geometric constraint relations of rolls to limit the uncertainties of various factors. We fully considered the influence of various factors on the results and established a new set of rolling curve solution models. The basic idea of this theory is to substitute the rolling curve in the result as the calculation condition of the big end radius. According to the related function of various factors, we describe the problem of solving the rolling curve as an initial value problem of a differential equation. Based on the premise that the differential equation can be solved numerically, we appropriately simplify or set each relationship function required in the equation. Finally, mathematical software is used to numerically solve the differential equation and obtain the discrete function of the rolling curve.

Functional relationship

The purpose of clarifying the relationship between the functions that affect the rolling curve is to obtain a more accurate solution model. It is generally believed that 5 initial values determine the rolling curve. They are the forming angle α, the widening angle β, the blank radius r0, the center distance RC, and the reduction of area Ψ. In addition, the radius function of the big end of the rolled piece is the boundary condition required to solve the volume of the spiral body in the unformed zone. In practice, it should be that the rolling curve is obtained through rotation constraints. In the current methods, the method of replacing the curve is adopted. We choose a simple rolling curve or set it as an Archimedes spiral with a pitch of rk tan a tan β without considering the effect of rolling. Then we solve the large-end radius function together with the rotation relationship [3]. In this type of solution, the relationship of the influencing factors in the solution process can be described in Figure 1. The common feature is that they do not consider the constraint relationship of the rolling face to the radius of the large end. We can consider the use of differential equations to express the relationship between various factors and transform the rolling curve into an initial value problem of differential equations for the solution.

Figure 1

Relation diagram of influencing factors of rolling and forming

Volume solution and the relationship with the rolling curve

The volume solution is based on the core content of the volume invariance method. We have conducted a lot of discussion and analysis. The volume solution of the unformed zone is divided into two categories: simplified model and contact zone model. The second type of model is more accurate and closer to the actual situation, but there are very complicated boundary conditions for calculating the model volume directly. The simple model can solve the volume function by simple multiple integrations [4]. Combining the characteristics of the two types of models and the conditions for solving the differential equations, we use the following general formula to express the volume V: V(φ)=0φF0(φ,ρφ)dφ+G(φ) V\left( \varphi \right) = \int_0^\varphi {{F_0}\left( {\varphi ,{\rho _\varphi }} \right)d\varphi + G\left( \varphi \right)} φ is the roll angle. ρφ is the radius of the big end. 0φF0(φ,ρφ)dφ \int_0^\varphi {{F_0}\left( {\varphi ,{\rho _\varphi }} \right)d\varphi } is the volume of the spiral cone in the simple model. G(φ) is the corrected volume. The content includes the volume of the contact area and other parts that can be expressed as elementary functions of the rotation angle. The relationship between the abscissa X of the rolling curve and the volume V of the unformed area can be expressed as: X=2V/πr12 X = 2V/\pi r_1^2 r1 is the radius after step forming.

Contact constraint relationship

Figure 2 is a schematic diagram of the restraint relationship of the rolls. We need to use the contact relationship to determine the function between the horizontal axis X of the rolling curve and the radius ρφ of the large end. If we directly use the function between the two, the expression is more complicated, so we introduce an intermediate quantity-the offset D of the straightening curve [5]. Its physical meaning is the offset displacement of the inner side of the straight line cut off the natural intersection line. The height of the rolling surface is H, and the forming angle is α. The offset D of this profile is: D=Hcotα D = H\cot \alpha

The shape of the unformed area of the rolled piece is obtained by the contact separation line rotating with the rolled piece. There is a geometric constraint relationship between the maximum radius and the height of the rolled surface. The height H of the rolled surface is obtained by designing the mold through the horizontal axis X of the rolling curve. It is a function of the roll angle. Combining equation (3), we can express the large-end radius function ρφ(φ) of the rolling stock as the following relationship: ρφ(φ)={r0φ=[0,π]r0Dtanαφ=[π,φE] {\rho _\varphi }\left( \varphi \right) = \left\{ {\matrix{ {{r_0}} & {\varphi = \left[ {0,\pi } \right]} \cr {{r_0} - D\tan \alpha } & {\varphi = \left[ {\pi ,{\varphi _E}} \right]} \cr } } \right.

φE is the corner at the end of the rolling. According to the point P in the partial unfolding diagram of the roll, the relationship function between the coordinate X, Y of the rolling curve and the offset D can be obtained: {X=D+(Y0θR1L1)tanβY=(XD)/tanβ+L1 \left\{ {\matrix{ {X = D + \left( {{Y_0} - \theta {R_1} - {L_1}} \right)\tan \beta } \hfill \cr {Y = \left( {X - D} \right)/\tan \beta + {L_1}} \hfill \cr } } \right.

Figure 2

Schematic diagram of the restraint relationship of rolls

Rotational relationship

In many rolling curve calculations, the radius of rotation rk is generally used as a parameter to express the rotation relationship. In this paper, the speed ratio δ is used instead of the rotation radius rk to express the movement relationship between the roll and the rolled piece. We use the speed ratio δ as a parameter to describe the rotation relationship [6]. After a time dt, the roll rotates by a slight angle . At the same time, the corresponding rolling piece is driven to rotate by an angle . Since these two corners are infinitely small, there must be the following relationship: dφ=ωφdt=υrkdt=ωθ(RCrk)rkdt=RCrkrkdθ d\varphi = {\omega _\varphi }dt = {\upsilon \over {{r_k}}}dt = {{{\omega _\theta }\left( {{R_C} - {r_k}} \right)} \over {{r_k}}}dt = {{{R_C} - {r_k}} \over {{r_k}}}d\theta

ωφ is a function of roller speed. ωθ is a function of rolling speed. v is the linear velocity function of the constant velocity circle. We set the speed ratio δ = ωφ / ωθ = (RCrk) / rk, so the micro-rotation relationship of the rolls is: dφ=δdθ d\varphi = \delta d\theta

Differential equations
Establishment of differential equations

According to the factor function relations and expressions, the following differential equations can be obtained by combining formulas (1), (2), (4), and formula (7): Av0φF(φ,D(φ))dφ+AvG(φ)=D(φ)+J(φ) Av\int_0^\varphi {F\left( {\varphi ,D\left( \varphi \right)} \right)d\varphi + AvG\left( \varphi \right) = D\left( \varphi \right) + J\left( \varphi \right)}

In the formula: Av=2/πr12 Av = 2/\pi r_1^2 ; J (φ) = (Y0R1δ(θ)L1) tan β. We derivate and organize the left and right sides of formula (8) to the independent variable rotation angle φ respectively to obtain: D(φ)=AVF(φ,D(φ))+AVD(φ)J(φ) D^\prime\left( \varphi \right) = {A_V}F\left( {\varphi ,D\left( \varphi \right)} \right) + {A_V}D^\prime\left( \varphi \right) - J^\prime\left( \varphi \right)

Equation solving

Since the right side of the differential equation (9) is very complicated, it is impossible to obtain a general solution. Therefore, it is necessary to use mathematical software to solve the numerical solution in the domain interval. Generally speaking, ode45 is the best choice for the first solution of most differential equations. It is a single-step solver based on the Runge-Kutta formula. First, we discretize the differential equation and then establish the recursive formula for the numerical solution. We use the single-step method to solve yn +1 in turn, the command line is: [t,Y]=ode45(odefun,tspan,y0) \left[ {t,Y} \right] = ode45\left( {odefun,tspan,{y_0}} \right)

The t parameter is an independent variable. The Y parameter is a function value. The odefun parameter is an equation expression. The first-order differential equation is in the form of an explicit function of the differential term, which is expressed as the y′ = f(t, y);tspan parameter as the range of the independent variable. The y0 parameter is the initial value of the differential equation, which is expressed as D(0) = 0. We only need the function F(φ, D(φ)), G′(φ) and J′(φ) expressions to exist, and then we can use the solver to solve the differential equation.

The best expression of the volume correction function G(φ) in formula (3) should be G(φ, D(φ)), but since the offset D is introduced in the correction volume function, the derivative function G′(φ) will have an D′ term, so the volume correction function is expressed in the form of G(φ) function. Since the correction volume is very small relative to the volume of the spiral cone, this simplification has little effect on the results.

Speed ratio

The influence of the speed ratio function δ(θ) is very complicated, and it is related to complex variables such as the shape of the contact area and friction. No theoretical formula can be given at present. Therefore, this article makes the following assumptions regarding the data obtained in the actual measurement and research of the speed ratio: We make the speed ratio in the widening process a fixed value δB = (RCr0) / r0. Suppose the speed ratio of the third stage of rolling and the finishing process is a constant value δs = (RCr1) / r1. It is assumed that the speed ratio increases linearly during the rolling process [7]. Therefore, the formula for the speed ratio of the first and second stages of rolling can be expressed as: δ(θ)=(δSδB)θEθ+δB \delta \left( \theta \right) = {{\left( {{\delta _S} - {\delta _B}} \right)} \over {{\theta _E}}}\theta + {\delta _B}

θE is the roll angle at the end of the second stage of rolling, and we take the approximate value (L1 + L2) / R0 in the application.

Solving volume function

We take the internal right-angle step rolling model as an example to analyze the problem. The differential equation can be established smoothly since the integral term in the general volume formula (1) is the upper integral function and the lower integral limit is 0. Therefore, the spiral cone volume integral function needs to be appropriately transformed to establish the differential equation. The simple volume formulas of the three stages of rolling are: V1=0φΦ11(φ,D)dφ+0πφΦ12(φ)dφV2=φπφΦ2(φ,D)dφV3=φπφEΦ3(φ,D)dφ \matrix{ {{V_1} = \int_0^\varphi {{\Phi _{1 - 1}}\left( {\varphi ,D} \right)d\varphi } + \int_0^{\pi - \varphi } {{\Phi _{1 - 2}}\left( \varphi \right)d\varphi } } \hfill \cr {{V_2} = \int_{\varphi - \pi }^\varphi {{\Phi _2}\left( {\varphi ,D} \right)d\varphi } } \hfill \cr {{V_3} = \int_{\varphi - \pi }^{{\varphi _E}} {{\Phi _3}\left( {\varphi ,D} \right)d\varphi } } \hfill \cr }

Φ1–1 and Φ1–2 respectively constitute the rotation angle function of the two parts of the first stage of rolling. Φ2 and Φ3 are the rotation angle functions of the 2nd and 3rd stages of rolling, respectively. The volume V1 of the first stage, let Φ1–2, and the volume of the contact area of the first stage be the general formula. The G function in formula (1) meets the requirements of formula (1). Since the lower limit of the volume integral in the second stage is not 0, and there are independent variable terms, we need to transform the integral formula. We transform V2 into the following form: V2=0φΦ2(φ,D)dφ+0φπΦ2(φ,D)dφ {V_2} = \int_0^\varphi {{\Phi _2}\left( {\varphi ,D} \right)d\varphi } + \int_0^{\varphi - \pi } {{\Phi _2}\left( {\varphi ,D} \right)d\varphi }

The second term of equation (13) represents the volume of the current corner φ half a week ago. Since the volume in the first stage can be solved directly, it can be inferred that the volume function is a set of piecewise recursive functions on the parameter variables [8]. So we use the half-cycle recursive substitution method to solve the second and third stages of rolling. The function algorithm program flow chart is shown in Fig. 3. More than 500 lines of programs were written using Matlab mathematics software for calculations.

Figure 3

Program flow chart

Result analysis

We use numerical simulation and real rolling experiments, respectively, to verify the correctness of the differential equation to solve the rolling curve. The rolling curve calculated by the Matlab program is shown in Figure 4. The curve in this paper has a larger value in the X direction than the precise rolling curve. This is consistent with the small design curve in previous practical experience. We use the curve to carry out Pro / E three-dimensional modeling of the mold. The article imports the model into finite element software Deform3D for finite element simulation calculation [9]. The purpose of solving the forming process with a rigid-plastic finite element model is to simulate a large amount of deformation. The basic parameters required are selected: forming angle α = 26°, widening angle β = 8°, original radius r0 = 26mm, reduction of area Ψ = 62.13%, forming radius r1 = 16mm, roll radius R0 = 250mm, widening amount LB = 30mm, roll speed n = 1rad s−1, rolling material 45# steel, rolling temperature 1150°C.

Figure 4

The result of the rolling curve solution

The purpose of limiting some minor unstable factors is to simplify the simulation process. The hypothesis is that the mold is regarded as a rigid body with a constant temperature, and the rolling friction is regarded as sheer friction. Select the constant friction coefficient.

The simulation results of the inner right-angle steps are shown in Figures 5a and 5b. Intuitively, it can be seen that the shaping effect of this article is slightly better. We select 24 points uniformly distributed on the forming step surface (Figure 5c) to quantify the forming effect. We take the axial coordinate value Z and solve the mean and variance as shown in Table 1. The mean value in the solution in this paper is closer to the target extension of 60mm, and the variance is significantly smaller than the original solution [10]. This shows that the flatness of the step surface obtained by the simulation is better.

Figure 5

Finite element simulation results (A) Accurately roll the curve. (B) The solution of this paper is to roll up the curve. (C) Schematic diagram of measuring points.

Analysis of the axial coordinate Z value of the measuring point on the forming surface

project Mean x/mm Variance s/mm2
Simulate the original solution 31.26 3.57
Simulate new solutions 30.87 1.43
The real rolling original solution 33.54 1.45
New solution for real rolling 31.02 1.36

We conducted real rolling experiments on the H500 cross wedge rolling mill. The mold parameters used in the experiment are consistent with the simulation parameters. Due to the parallel implementation with other experiments, the material head is designed to be rolled with a big end support method [11]. The rolling parts are shown in Figure 6. The right end of the part is the original solution rolling step, and the left is the paper solution rolling step. Since the actual rolled parts are small, the 24-point measurement method in the simulation results cannot be used. Therefore, we use the 8-point method of uniform distribution in a circular ring for measurement. The mean value and variance of the step groove width are shown in Table 1. It can be seen that the width of the stepped groove of the real-rolled parts is greatly affected by the rolling curve, and the planarity is not greatly affected. In addition, the actual rolling results are highly consistent with the simulation, and the curve in this paper has a more excellent effect on the inner right-angle step forming.

Figure 6

Drawing of real rolled parts

Conclusion

We take the inner right-angle step as the research object and use the Matlab software to program and solve the rolling curve. The forming effect of the precise rolling curve and the curve in this paper are compared by finite element simulation and real rolling experiments. We take 8n points uniformly distributed on the formed step surface in the result and compare the mean value and variance of the axial coordinate of the value points. We use the differential equation method to solve the rolling curve and theoretically avoid the influence of the assumed large-end radius function on the accuracy of the volume solution. This makes the mathematical model closer to the actual situation. This solution can solve the inner right-angle step rolling curve with good forming effect and provide a new theoretical system for the research on rolling of other complex steps.

Figure 1

Relation diagram of influencing factors of rolling and forming
Relation diagram of influencing factors of rolling and forming

Figure 2

Schematic diagram of the restraint relationship of rolls
Schematic diagram of the restraint relationship of rolls

Figure 3

Program flow chart
Program flow chart

Figure 4

The result of the rolling curve solution
The result of the rolling curve solution

Figure 5

Finite element simulation results (A) Accurately roll the curve. (B) The solution of this paper is to roll up the curve. (C) Schematic diagram of measuring points.
Finite element simulation results (A) Accurately roll the curve. (B) The solution of this paper is to roll up the curve. (C) Schematic diagram of measuring points.

Figure 6

Drawing of real rolled parts
Drawing of real rolled parts

Analysis of the axial coordinate Z value of the measuring point on the forming surface

project Mean x/mm Variance s/mm2
Simulate the original solution 31.26 3.57
Simulate new solutions 30.87 1.43
The real rolling original solution 33.54 1.45
New solution for real rolling 31.02 1.36

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