CN100401257C - Method for fast simulating tension diameter-reducing procedure of seamless steel tube - Google Patents

Abstract

The present invention discloses a method for fast simulating process of tension reducing of a seamless steel tube. The present invention takes distribution of rotation speed of a roll which takes part in rolling a machine frame as input and variation of the longitudinal wall thickness of the steel tube in the process of rolling as output, and takes the process of tension reducing of each pair of adjacent machine frames as a subsystem. The present invention expresses each subsystem by partial differential equations, denotes characteristics of steel tubes and a tension reducing mill and the mutual action among the subsystems by boundary condition, boundary condition and restraint condition of equations, establishes a mathematical model of variation of the longitudinal wall thickness of the seamless steel tubes in the process of tension reducing, and converts the model to a recursion form by a difference method. The recursion form is conveniently solved by a computer. The present invention predicts the situation of the variation of the wall thickness in the process of the steel tube passing through each machine frame under different distribution of rotation speed of rolling and according to characteristic data of pierced billet and the shape of holes, and provides a compact and rapid simulating method. The computation speed and the precision of the present invention can satisfy controlled requirements of the wall thickness of the seamless steel tubes.

Description

The tension diameter-reducing procedure of seamless steel tube emulation mode

Technical field

The present invention relates to a kind of mathematical modeling and emulation mode of tension diameter-reducing procedure of seamless steel tube, relate in particular to a kind of about vertical wall thickness change of setting up weldless steel tube in the tension diameter-reducing procedure the distribution parameter mathematical model and carry out the tension diameter-reducing procedure of seamless steel tube emulation mode that the computer numerical of model is found the solution in conjunction with the tension diameter-reducing procedure characteristics.

Background technology

Stretch reducing is rolling to be a kind of hot rolling technology process of making weldless steel tube or welded still pipe.Its technological principle is not adopt internal strain instrument such as plug, makes the hollow billet that rolls out through tapping machine and tandem mill or hollow forging successively by having the series connection frame close to each other of sub-circular cross section pass, and the caliber of hollow forging is reduced to desired size.Modernization stretch reducing milling train moulding process is to carry out in non-adjustable type three roller cages.On three-high mill, three rolls become 120 ° of configurations around the mill train center line, and the roll of adjacent support becomes 60 ° of dislocation to arrange.

Stretch-reducing mill is last group equipment that seamless steel pipe hot-rolling is produced, be the link that tube quality is had the greatest impact, control the wall thickness of stretch reducing fished pipe well the raising of roll seamless steel tube recovery rate and the important practical sense that reduced production costs.One accurately tension diameter-reducing procedure wall thickness model and fast the computer solving algorithm be to implement the tension diameter-reducing procedure thickness of steel pipe key of control in real time.Tension diameter-reducing procedure of seamless steel tube has different rolling characteristics in rolled steel tube head section, stage casing during with rear, and " pipe end thickens " phenomenon is the intrinsic characteristic of tension diameter-reducing procedure.The speed system of given one group of each breast roller of weldless steel tube stretch-reducing mill and relevant technological parameter, traditional static mathematical model according to the used tension diameter-reducing procedure of stretch reducing technological design, the average wall thickness of fished pipe can only be obtained, and the length of the pipe end section of thickening of fished pipe can't be understood.

In order to understand the whole distribution situation of stretch reducing fished pipe wall thickness, foundation is very necessary with the distributed parameter model that thickness of steel pipe in the operation of rolling is changed to output.The research method of stretch-reducing mill distributed parameter model mainly contains two kinds at present: a kind of is according to the complex distributions parameter model of detailed mechanism foundation based on finite element analysis; Another kind is to utilize the method for data fitting that tension diameter-reducing procedure is carried out modeling by neural network etc.These two kinds of methods all have weak point, and are huge based on the calculated amount of the computation model of finite element, need a large amount of computational resources, can't adapt to the requirement of the on-the-spot real-time of actual production; Computation model based on data fitting has then broken away from technological principle, and the parameter physical significance of model is indeterminate, and the validity of model can not get guaranteeing.

Summary of the invention

The objective of the invention is at the deficiencies in the prior art, set up a kind of simple relatively, express with the analytical form of partial differential equations, be described in the tension diameter-reducing procedure, thickness of steel pipe plants the dynamic mathematical models of variation at various roll rotational speed branches, and provides the tension diameter-reducing procedure of seamless steel tube emulation mode of the high-speed computer numerical solution of model.

The tension diameter-reducing procedure of seamless steel tube emulation mode is to be distributed as input with the roll rotational speed that participates in rolling machine frame, with the vertical wall thickness change of steel pipe in the operation of rolling is output, the tension diameter-reducing procedure that each adjacent frame is right is as a subsystem, with each subsystem of partial differential equation formal representation, starting condition with equation, boundary condition and constraint condition are represented the feature of steel pipe and stretch-reducing mill and the interaction between subsystem, set up the mathematical model of the vertical wall thickness change of weldless steel tube in the tension diameter-reducing procedure, adopt difference method the recursive form of model conversation for ease of computer solving.

Set up the vertical wall thickness change mathematical model of weldless steel tube in the tension diameter-reducing procedure: the quantitative relationship of keeping weighing apparatus and steel tube section distortion and steel pipe axial tensile force according to the steel pipe volume in the operation of rolling is set up the partial differential equations of the vertical wall thickness change model of steel pipe in each subsystem, the starting condition that provides equation according to the original outer diameter and the wall thickness situation of hollow forging, utilize relation and the steel pipeline speed and the roll angular velocity of tension force and steel pipeline velocity contrast, the relation of the roll radius of clean-up provides boundary condition and the constraint condition that model need satisfy, and the tension diameter-reducing procedure dynamic mathematical models of representing with the interactive subsystem form are

$\left\{\begin{array}{c}{v}_i(x,t)\frac{\∂}{\∂x}{A}_i(x,t)+\frac{\∂}{\∂t}{A}_i(x,t)=-k_i{P}_i\left(t\right)\\ A_i(x,t)\frac{\∂}{\∂x}{v}_i(x,t)=k_i{P}_i\left(t\right)\end{array}\right.$

Model satisfies constraint condition

$\left\{\begin{array}{c}{v}_i(L_{i+1},t)=v_i(L_i,t)+k_i{P}_i\left(t\right){\∫}_{L_i}^{L_{i+1}}\frac{1}{A_i(x,t)}\mathrm{dx}\\ A_i(x,t)=\mathrm{\π}{S}_i(x,t)(D_i(x,t)-S_i(x,t))\end{array}\right.$

The starting condition of model and boundary condition are

The steel tube section initial distribution: $A_i(x,0)=\left\{\begin{array}{cc}{A}_0,& i=1,x=0\\ 0,& i\≠1\end{array}\right.$

The initial roll line velocity distribution of steel pipe: $v_i(x,0)=\left\{\begin{array}{cc}{\mathrm{\&omega;}}_1,{\mathrm{<figure-callout\; id="1"\; label="R"\; filenames="C20051004913700173.png,C20051004913700183.png"\; state="\{\{state\}\}">R</figure-callout>}}_1,& i=1,x=0\\ 0,& i\&NotEqual;1\end{array}\right.$

Steel pipe rolling linear velocity position distribution: v _i(0, t)=ω _iR _i

Wherein, i=1,2 ..., N-1, x is the position of steel pipe with respect to first frame, and t is a rolling time, A, and D and S are respectively the sectional area of steel pipe, external diameter and wall thickness, v is the linear velocity of steel pipe, and P is the tension force that acts on the steel pipe, and R represents the radius of clean-up of breast roller, and N is for participating in rolling frame number.

Adopt difference method with the recursive form of model conversation: at first, to utilize difference method with the model discretize for ease of computer solving; Then, adopt the mode of circulation recursion to calculate the sectional area of steel pipe at each point, step is: the first step, and calculate the radius of clean-up of roll according to the characteristic of roll angular velocity and steel pipe, and calculate the linear velocity of steel pipe at each frame place; Second step, at first utilize the constraint condition of model rotating speed to calculate the tension force correlation, utilize the linear velocity of the each point of Model Calculation steel pipe then; This step be steel pipe axially on calculate; The 3rd goes on foot, and utilizes the sectional area of Model Calculation steel pipe, and this step is to calculate on time orientation; In the 4th step, repeat the aforementioned calculation step and leave institute's organic frame until steel pipe.

The present invention has following technique effect: the emulation mode that the present invention provides is succinct, quick, and its computing velocity and precision can satisfy the weldless steel tube wall thickness needs of control in real time.The present invention can be according to the characteristic of hollow forging and pass, the situation of change of wall thickness when prediction plants steel pipe by each frame at the rolling rotating speed branch of difference; Can distribute according to each roller speed input and obtain vertical Thickness Distribution of fished pipe; Can study the average wall thickness of stretch reducing pipe and the relation between the pipe end section of thickening Thickness Distribution rule and supplied materials hollow forging and rolling rotating speed quantitatively according to model, as stretch-reducing mill breast roller rotating speed being feedovered and the foundation of FEEDBACK CONTROL.

Description of drawings

Fig. 1 keeps the weighing apparatus synoptic diagram for the steel pipe volume;

Fig. 2 is three-high mill roll and pass parameter synoptic diagram;

Fig. 3 is a mathematical model synoptic diagram of the present invention;

Fig. 4 is a tension diameter-reducing procedure model structure synoptic diagram of the present invention;

Fig. 5 is a whole thickness of steel pipe distribution plan of the embodiment of the invention;

Fig. 6 is the steel pipe head end Thickness Distribution figure of the embodiment of the invention;

Fig. 7 is the steel pipe tail end Thickness Distribution figure of the embodiment of the invention.

Embodiment

The present invention sets up the tension diameter-reducing procedure dynamic mathematical models based on the thermoplasticity principle, and has provided the Fast numerical method for solving of this model.These dynamic mathematical models are distributed as input, are changed to output with thickness of steel pipe in the operation of rolling with the roll rotational speed that participates in rolling institute's organic frame, can simulate pipe end and thicken phenomenon.

The step of dynamic mathematical models of setting up the tension diameter-reducing procedure of seamless steel tube thickness of steel pipe is as follows:

1. set up the long-pending relational expression with steel pipeline speed of steel tube section

In the operation of rolling, the changing value of steel volume equals to flow into this interval steel volume and flows out the poor of this interval steel volume in a segment length interval.Extension in the tube reducing process as shown in Figure 1.Set positions with first frame of stretch-reducing mill is a true origin, and x is for more arbitrarily, note V (x, t) be the t moment by the volume of position 0 to the steel pipe the static position x, keeping weighing apparatus by volume can get

$\frac{\&PartialD;V(x,t)}{\&PartialD;t}=A(0,t)v(0,t)-A(x,t)v(x,t)---\left(1\right)$

Wherein, v (x, t) and A (x, t) expression t is constantly at the instantaneous velocity and the sectional area of position x place steel pipe.The steel pipe volume can obtain by integration

$V(x,t)={\&Integral;}_0^{x}A(s,t)\mathrm{ds}$

By following formula t being carried out partial differential can get

$\frac{\&PartialD;V(x,t)}{\&PartialD;t}=\frac{\&PartialD;}{\&PartialD;t}{\&Integral;}_0^{x}A(s,t)\mathrm{ds}={\&Integral;}_0^x\frac{\&PartialD;}{\&PartialD;t}A(s,t)\mathrm{ds}$

And then in conjunction with (1) formula

${\&Integral;}_0^x\frac{\&PartialD;}{\&PartialD;t}A(s,t)\mathrm{ds}=A(0,t)v(0,t)-A(x,t)v(x,t)$

By following formula x is asked local derviation again, have

$\frac{\&PartialD;}{\&PartialD;t}A(x,t)=-\frac{\&PartialD;}{\&PartialD;x}\left(A(x,t)v(x,t)\right)$

Expansion obtains

$\frac{\&PartialD;}{\&PartialD;t}A(x,t)=-v(x,t)\frac{\&PartialD;}{\&PartialD;x}A(x,t)-A(x,t)\frac{\&PartialD;}{\&PartialD;x}v(x,t)---\left(2\right)$

2. set up the relational expression of steel tube section distortion and steel pipe axial tensile force

Deformation of steel is observed plasticity, and its extension strength and suffered tension force are proportional.Sectional area rate of change with steel pipe is basic extension strength variable, represents the vary in diameter in this cross section with tangential deformation, represents the wall thickness change in this cross section with radial deformation.Dynamically treat the particular cross section of steel pipe, its distortion (comprising tangential deformation and radial deformation) is proportional with suffered axial tensile force.Sectional area rate of change with steel pipe is represented above-mentioned distortion, when the steel pipe radius changing rate when tension force is not too big, steel tube section amasss rate of change

(x, relation t) roughly can be considered linear relationship with suffered tension force P.

The area of section rate of change that steel pipe is positioned at the x place is

$\frac{\mathrm{dA}(x,t)}{\mathrm{dt}}=-\mathrm{kP}(x,t)---\left(3\right)$

Its physical significance is meant that steel tube section amasss the decrease in the unit interval

Be directly proportional with suffered tension force (P).Scale-up factor k is relevant with factors such as steel pipe material, heating-up temperatures.

3. set up the partial differential equation of tension diameter-reducing procedure steel tube section distortion

For a certain cross section x=x (t), steel pipe in the linear velocity in this cross section is

$v(x,t)=\frac{\mathrm{dx}}{\mathrm{dt}}$

(3) formula is launched

$\frac{\&PartialD;A}{\&PartialD;x}\&CenterDot;\frac{\mathrm{dx}}{\mathrm{dt}}+\frac{\&PartialD;A}{\&PartialD;t}=-\mathrm{kP}$

Obtain

$v(x,t)\&CenterDot;\frac{\&PartialD;}{\&PartialD;x}A(x,t)+\frac{\&PartialD;}{\&PartialD;t}A(x,t)=-\mathrm{kP}---\left(4\right)$

Simultaneous (2) formula and (4) formula obtain

$\left\{\begin{array}{c}v(x,t)\frac{\text{\&PartialD;}}{\&PartialD;x}A(x,t)+\frac{\&PartialD;}{\&PartialD;t}A(x,t)=-\mathrm{kP}\\ A(x,t)\frac{\&PartialD;}{\&PartialD;x}v(x,t)=\mathrm{kP}\end{array}\text{---}\left(5\right)\right.$

4. set up the relational expression of tension force and steel pipeline velocity contrast

The extension strength of steel pipe is relevant with suffered tension force, and the size of tension force is relevant at the linear differential at adjacent frame place with steel pipe.Synchronization, the tension force between two frames are definite value along its length, are obtained by second equation by boundary integration of (5) formula

$v(L_{n+1},t)=v(L_n,t)+\mathrm{kP}{\&Integral;}_{L_n}^{L_{n+1}}\frac{1}{A(x,t)}\mathrm{dx}---\left(6\right)$

5. set up the equation that calculates the roll radius of clean-up

The rotating speed of roll, the physical parameter of steel pipe and stretch-reducing mill frame pass, rolled piece nip angle etc. are to the influence of tension diameter-reducing procedure, can be synthetically variation by the roll radius of clean-up reflect, the radius of clean-up of directly considering roll can avoid finding the solution the complex relationship between roll angular velocity and the steel pipe tension force.The linear velocity of putting on the pairing roll of roll radius of clean-up R is consistent in the linear velocity at this frame place with steel pipe.Usually have

R _min≤R≤R _max

R wherein _MinBe the radius of roll axis to the pass bottom, R _MaxFor roll axis is the radius at roll gap place to the pass edge.Fig. 2 has provided the synoptic diagram of three-high mill roller parameter and circular hole, oval groove parameter.Among the figure, a, b are respectively pass width and height, and θ represents the pairing angle radian of the radius of clean-up.Owing to along the linear velocity difference of rolling face, therefore produce advancing slip phenomenon to roll radius of clean-up place steel pipe with respect to roll, then produce the sliding phenomenon in back from the roll radius of clean-up to roll gap in the pass bottom.

During technological design, always be with θ Determine the roll radius of clean-up, but in fact holding residing roll end to end, the θ of radius of clean-up correspondence with

Gap is bigger.For three roller rolling machines, always near the bottom of pass, promptly θ approaches 0 to the roll radius of clean-up corresponding point of the head end frame of living in of steel pipe, and limiting case is θ=0; And the roll radius of clean-up corresponding point of the residing frame of tail end of steel pipe are always near the edge of pass, and promptly θ approaches

Limiting case is

For single roll, along with the rising of forward pull, the roll radius of clean-up is shifted to the pass edge; Otherwise, rising along with backward pull, the roll radius of clean-up is shifted at the bottom of the pass, after the radius of clean-up has moved on to limit point, be that the radius of clean-up equals radius at the bottom of pass edge radius or the pass, then further improve two speed discrepancies between frame again and will cause the institute of roll and steel pipe to have point of contact and all slides, and do not have the effect of increase axial tensile force.

Be the maximin of determining that rolling angle is reached, introduce a variable---rolling angle modified value ζ. when the rolling frame number of participation reaches maximal value m, the rolling angle at m frame place is arranged ${\mathrm{\&theta;}}_m=\frac{\mathrm{\&pi;}}{6}-\mathrm{\&xi;},$ The rolling angle at No. 1 frame place ${\mathrm{\&theta;}}_1=\frac{\mathrm{\&pi;}}{6}+\mathrm{\&xi;},$ Because the variation range at rolling angle arrives between 0 Between, so the variation range of ζ is

$0\&le;\mathrm{\&xi;}\&le;\frac{\mathrm{\&pi;}}{6}$

Can determine ζ according to the actual process parameter empirical value that has obtained, the volumetric flow rate conservation of being imported and exported by tension diameter-reducing procedure obtains

${\mathrm{\&omega;}}_1(R_d-\frac{{\mathrm{<figure-callout\; id="1"\; label="D"\; filenames="C20051004913700173.png,C20051004913700183.png"\; state="\{\{state\}\}">D</figure-callout>}}_1}{2}\cos (\frac{\mathrm{\&pi;}}{6}+\mathrm{\&xi;}))A_1={\mathrm{\&omega;}}_m(R_d-\frac{{\mathrm{<figure-callout\; id="2"\; label="D\; m"\; filenames="C20051004913700183.png"\; state="\{\{state\}\}">D</figure-callout>}}_m}{2}\cos (\frac{\mathrm{\&pi;}}{6}-\mathrm{\&xi;}))A_m---\left(7\right)$

Wherein m is for participating in rolling total frame number, ω _iBe the roll rotational speed at i frame place, R _dBe the desirable radius of the standard of milling train, D _iBe i frame place the pass diameter, A ₁The steel tube section that is No. 1 frame place is long-pending, in order to substitute inlet sectional area, A _mBe the sectional area at m frame place, in order to substitute discharge area.

Therefore, adopt following method to determine the roll radius of clean-up of institute's organic frame.Suppose that the angular velocity of steel pipe at each frame place is ω _k, determine the angle theta of the radius of clean-up at each frame place _kConcrete grammar be:

(1),, be the m frame just, as can be known in last rolling frame with frame headed by the 1st the rolling frame just if the frame number that is participating in rolled steel tube is m ${\mathrm{\&theta;}}_1=\frac{\mathrm{\&pi;}}{6}+\mathrm{\&xi;},$ ${\mathrm{\&theta;}}_m=\frac{\mathrm{\&pi;}}{6}-\mathrm{\&xi;}.$

(2) angle of the radius of clean-up of k frame of following calculating

${\mathrm{\&theta;}}_k=\frac{({\mathrm{\&omega;}}_{k+1}-{\mathrm{\&omega;}}_k){\mathrm{\&theta;}}_{k-1}+({\mathrm{\&omega;}}_k-{\mathrm{\&omega;}}_{k-1}){\mathrm{\&theta;}}_{k+1}}{{\mathrm{\&omega;}}_{k+1}-{\mathrm{\&omega;}}_{k-1}}---\left(8\right)$

Further, can calculate the radius of clean-up of roll

$R_k=R_d-\frac{D_{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="C20051004913700183.png"\; state="\{\{state\}\}">k</figure-callout>}}}{2}\cos \left({\mathrm{\&theta;}}_k\right)---\left(9\right)$

Steel pipe in the linear velocity at frame place is

v _k＝ω _kR _k (10)

6. set up the dynamic mathematical models of complete thickness of steel pipe

Each adjacent frame is handled as a relatively independent subsystem (subsystem i) the tension diameter-reducing procedure of (as i-1 frame and i frame).The tension force of this subsystem is mainly produced by the roll rotational speed difference of frame.The structure of the dynamic mathematical models of each subsystem is referring to shown in Figure 3.

Have reciprocation between adjacent subsystems i-1 and the subsystem i, the radius of clean-up by influencing roll and rolling starting condition show.

Fig. 4 shows the synoptic diagram of model structure.At first provide the meaning of mathematic sign in the tension diameter-reducing procedure of seamless steel tube dynamic mathematical models structure:

X: along the position coordinates of steel pipe working direction, be true origin among the subsystem i with the i frame; Unit is a rice (m);

T: the time parameter of tension diameter-reducing procedure, entering the 1st frame with steel pipe is true origin; Unit is second (s);

A _i(x, t): the cross-sectional area of steel pipe among the subsystem i is x, the distribution variable on the t; Unit is a square metre (m ²);

S _i(x, t): the wall thickness of steel pipe among the subsystem i is x, the distribution variable on the t; Unit is a rice (m);

D _i(x, t): the external diameter of steel pipe among the subsystem i is x, the distribution variable on the t; Unit is a rice (m);

v _i(x, t): the roll line speed of steel pipe among the subsystem i is x, the distribution variable on the t; Unit is meter per second (m/s);

P _i(t): among the subsystem i at t tension force constantly; Unit is ox/square metre (N/m ²);

ω _i(t): the i frame is at t roll rotational speed constantly; Unit is radian per second (rad/s);

R _i(t): the i breast roller is in the t radius of clean-up constantly; Unit is a rice (m);

L _i: the frame spacing among the subsystem i; Unit is a rice (m);

k _i: the tension set coefficient among the subsystem i; Unit is (m ⁴/ (Ns));

N: stretch-reducing mill is participated in rolling frame sum;

The tension diameter-reducing procedure dynamic mathematical models of representing with the interactive subsystem form are the form of nonlinear partial differential equation group.

$\left\{\begin{array}{c}{v}_i(x,t)\frac{\&PartialD;}{\&PartialD;x}{A}_i(x,t)+\frac{\&PartialD;}{\&PartialD;t}{A}_i(x,t)=-k_i{P}_i\left(t\right)\\ A_i(x,t)\frac{\&PartialD;}{\&PartialD;x}{v}_i(x,t)=k_i{P}_i\left(t\right)\end{array}\right.---\left(11\right)$

Wherein, X ∈ [0, L], t ∈ [0, ∞), i=1,2 ..., N-1. correspondingly also should satisfy constraint condition

$\left\{\begin{array}{c}{v}_i(L_{i+1},t)=v_i(L_i,t)+k_i{P}_i\left(t\right){\&Integral;}_{L_i}^{L_{i+1}}\frac{1}{A_i(x,t)}\mathrm{dx}\\ A_i(x,t)=\mathrm{\&pi;}{S}_i(x,t)(D_i(x,t)-S_i(x,t))\end{array}\right.$

Corresponding boundary condition is

The steel tube section initial distribution: $A_i(x,0)=\left\{\begin{array}{cc}{A}_0,& i=1,x=0\\ 0,& i\&NotEqual;1\end{array}\right.$

The initial roll line velocity distribution of steel pipe: $v_i(x,0)=\left\{\begin{array}{cc}{\mathrm{\&omega;}}_1{\mathrm{<figure-callout\; id="1"\; label="R"\; filenames="C20051004913700173.png,C20051004913700183.png"\; state="\{\{state\}\}">R</figure-callout>}}_1,& i=1,x=0\\ 0,& i\&NotEqual;1\end{array}\right.$

Steel pipe rolling linear velocity position distribution: v _i(0, t)=ω _iR _i

Nonlinear partial differential equation group (11) can not get analytic solution, and finding the solution of it need be undertaken by numerical evaluation.The Fast numerical computing method of the mentioned strain tube reducing mathematical model that the present invention proposes are as follows:

1. the numerical evaluation route of stretch reducing model

Getting time step during with the model discretize is Δ t, is designated as τ, and the space step-length is Δ x, is designated as h, selects the space step-length to satisfy L _K+1-L _k=mh. is between two adjacent frames, and the tension force of arbitrary moment diverse location is identical, promptly works as x _iWhen being between n frame and the n+1 frame, have

kP(x _i，t _j)＝kP ⁽ⁿ⁾(t _j)

Therefore, utilize forward difference, (11) formula is approximately

$\left\{\begin{array}{c}v(x_i,t_j)\frac{A(x_{i+1},t_j)-A(x_i,t_j)}{h}+\frac{A(x_i,t_{j+1})-A(x_i,t_j)}{\mathrm{\&tau;}}=-{\mathrm{kP}}^{\left(n\right)}\left(t_j\right)\\ A(x_i,t_j)\frac{v(x_{i+1},t_j)-v(x_i,t_j)}{h}=k{P}^{\left(n\right)}\left(t_j\right)\end{array}\right.$

Constraint condition becomes

$v(L_{n+1},t_j)=v(L_n,t_j)+k{P}^{\left(n\right)}\left(t_j\right)\underset{k=1}{\overset{m}{\mathrm{\&Sigma;}}}\frac{h}{A(x_k,t_j)}$

With A[i, j] and v[i, j] expression binary function A (x, t) and v (x, t),, j with KP[k] represent the tension force KP of differential tension sizing reduction section _k(t). promptly

A[i, j] corresponding to A (i Δ x, j Δ t)

V[i, j] corresponding to v (i Δ x, j Δ t)

KP[k, j] corresponding to KP _k(j Δ t)

Wherein Δ x, Δ t are that discretize is calculated the sampling point of variable on x, t at interval.K represents k section stretch reducing section, i.e. that section between k roll and k+1 the roll.If participating in the rolling machine frame number is K, every section stretch reducing section is divided into the N section, then the span of i is 1 to (K-1) * N.

When steel pipe does not enter frame, the A[i that has ready conditions, 0]=0, and the sectional area A of known hollow forging ₀With the steel pipeline speed at each milling train place and the rotating speed of roll relation is arranged

v[kN，j]＝ω[k，j]×R _k

Here, use one-dimension array w[k, j] represent to participate in N rolling running roller at j Δ t rotating speed constantly, its value is known, R _kBe the radius of clean-up.Model is for being input with the roll rotational speed, is the Thickness Distribution dynamic changing process of output with the sectional area of steel pipe.

Adopt the recursion iterative algorithm to calculate.The process of recursion is: at first utilize the constraint condition of model rotating speed to calculate tension force correlation KP[k, j]; Utilize model second formula to calculate the linear velocity v[i of the each point of steel pipe, j then]; This two step all be steel pipe axially on calculate.At last, utilize model first formula to calculate A[i, j], this step is to calculate on time orientation.

Utilize difference equation, above-mentioned recursive process can be described as

(1) fixing j, by A[i, j] calculating K P (k, j);

(2) by KP (k, j) calculate v (i, j);

(3) by A[i, j] and v[i, j] calculating A[i, j+1];

(4) repeating step 1 to 3.

2. the detailed process of numerical evaluation

For residing that section of a certain end of steel pipe, obviously do not have tension force and exist, be i.e. KP[k, j]=0. for the steel pipe between two frames, the discrete approximation that can obtain model constrained condition is expressed as

${\mathrm{KP}}_k\left(\mathrm{j\&Delta;t}\right)=\frac{{\mathrm{\&omega;}}_{k+1}{R}_{k+1}-{\mathrm{\&omega;}}_k{R}_k}{\underset{i=(k-1)N+1}{\overset{\mathrm{kN}}{\mathrm{\&Sigma;}}}\frac{1}{A(\mathrm{i\&Delta;x},\mathrm{j\&Delta;t})}\mathrm{\&Delta;x}}$

Corresponding discrete calculation formula is

$\mathrm{KP}[k,j]=\frac{w[k+1,j]R_{k+1}-w[k,j]R_k}{\underset{i=(k-1)N+1}{\overset{\mathrm{kN}}{\mathrm{\&Sigma;}}}\frac{1}{A[i,j]}\mathrm{\&Delta;x}}---\left(12\right)$

Find the solution v[i, j by model second formula]. approximate have

$v(\mathrm{i\&Delta;x},\mathrm{j\&Delta;t})={\mathrm{\&omega;}}_k{R}_k+K{P}_k\left(\mathrm{j\&Delta;t}\right)\&CenterDot;\left(\underset{n=(k-1)N+1}{\overset{i}{\mathrm{\&Sigma;}}}\frac{1}{A(\mathrm{n\&Delta;x},\mathrm{j\&Delta;t})}\mathrm{\&Delta;x}\right),(k-1)N+1\&le;i\&le;\mathrm{kN}$

Corresponding discrete calculation formula is

$v[i,j]=w{[k,j]R}_k+KP[k,j]\left(\underset{n=(k-1)N+1}{\overset{i}{\mathrm{\&Sigma;}}}\frac{1}{A(n,j)}\mathrm{\&Delta;x}\right),(k-1)N+1\&le;i\&le;\mathrm{kN}---\left(13\right)$

Promptly

$\mathrm{\&Delta;v}=v[i,j]-v[i-1,j]=\mathrm{KP}[k,j]\frac{1}{A(i,j)}\mathrm{\&Delta;x}$

Can get according to model first formula

$\frac{A(\mathrm{i\&Delta;x},(j+1)\mathrm{\&Delta;t})-A(\mathrm{i\&Delta;t},\mathrm{j\&Delta;t})}{\mathrm{\&Delta;t}}=-\frac{\mathrm{Av}((i+1)\mathrm{\&Delta;x},\mathrm{j\&Delta;t})-\mathrm{Av}(\mathrm{i\&Delta;x},\mathrm{j\&Delta;t})}{\mathrm{\&Delta;x}}$

Corresponding discrete calculation formula

$\frac{A[i,j+1]-A[i,j]}{\mathrm{\&Delta;t}}=-\frac{A[i+1,j]v[i+1,j]-A[i,j]v[i,j]}{\mathrm{\&Delta;x}}$

Therefore, A[i, j] can followingly calculate

$A[i,j+1]=A[i,j]-\left(A\right[i+1,j\left]v\right[i+1,j]-A[i,j\left]v\right[i,j\left]\right)\frac{\mathrm{\&Delta;t}}{\mathrm{\&Delta;x}}---\left(14\right)$

Further, need to handle boundary condition, arrive the situation of next frame as yet for pipe end, promptly of living in section of steel pipe head end, tail end, (x, t) value is zero to A in the model on some position, or is step function.(x, t) be zero this moment to Tension Distribution P.

The distribution of corresponding calculating variate-value is then as follows:

The steel pipe head end: exist I to satisfy (k-1) N+1≤I≤kN,

$A[i,j]=\left\{\begin{array}{cc}\&NotEqual;0& (k-1)N+1\&le;i\&le;I\\ 0& I<i\&le;\mathrm{kN}\end{array}\right.$

At this moment, the calculating formula of tube head section linear velocity is

$v[i,j]=\left\{\begin{array}{cc}w[k-1]R_{k-1}& A[i,j]\&NotEqual;0\\ 0& A[i,j]=0\end{array}\right.$

Tube head section wall thickness about the recursion calculating formula of time is

A[i，j+1]＝A[i-w[k-1，j]R _k-1Δt，j]，(k-1)N+1＜i≤kN

The steel pipe tail end: exist I to satisfy (k-1) N+1≤I≤kN,

$A[i,j]=\left\{\begin{array}{cc}0& (k-1)N+1\&le;i\&le;I\\ \&NotEqual;0& I<i\&le;\mathrm{kN}\end{array}\right.$

At this moment, the calculating formula of pipe rear linear velocity is

$v[i,j]=\left\{\begin{array}{cc}w[k,j]R_k& A[i,j]\&NotEqual;0\\ 0& A[i,j]=0\end{array}\right.$

Pipe rear wall thickness about the recursion calculating formula of time is

A[i，j+1]＝A[i-w[k，j]R _kΔt，j]，(k-1)N+1＜i≤kN

For keeping computational accuracy, when discretize parameter Δ x, Δ t, need satisfy condition:

w[k，j]R _kΔt≤Δx，k＝0，1，2…K-1

The pass of sectional area and wall thickness is

$A=\mathrm{\&pi;}({\mathrm{<figure-callout\; id="0"\; label="r"\; filenames="C20051004913700171.png,C20051004913700183.png"\; state="\{\{state\}\}">r</figure-callout>}}_0^2-r_i^2),r_i=\sqrt{{\mathrm{<figure-callout\; id="0"\; label="r"\; filenames="C20051004913700171.png,C20051004913700183.png"\; state="\{\{state\}\}">r</figure-callout>}}_0^2-\frac{A}{\mathrm{\&pi;}}}$

$d=r_0-r_i=r_0-\sqrt{{\mathrm{<figure-callout\; id="0"\; label="r"\; filenames="C20051004913700171.png,C20051004913700183.png"\; state="\{\{state\}\}">r</figure-callout>}}_0^2-\frac{A}{\mathrm{\&pi;}}}$

The external diameter that has just gone out the running roller place is determined, therefore can obtain external diameter, the wall thickness distribution in time of this position.Go out after last running roller, do not had the effect of tension force, the size of steel pipe no longer changes, and external diameter all is the internal diameter of last rolling hole.The process of recursion is that the porch from stretch-reducing mill begins to carry out.

Embodiment

With Baosteel steel pipe branch office hot rolling mill stretch-reducing mill production run is practical object.The stretch-reducing mill of Baosteel steel pipe branch office has 28 frames, and each frame is the three-roller type roll.Roll standard ideal noise diode is 330 millimeters, and distance is 310 millimeters between frame.Each frame has separately independently governing system.The typical model steel pipe φ 73.00 * 5.51 that chooses steel pipe branch office carries out simulation calculation research, and the thickness of steel pipe value of simulation result and manual measurement is compared.The hollow forging specification of model φ 73.00 * 5.51 is 152.50 millimeters of external diameters, 6.00 millimeters of wall thickness, and trimmed size pipe specification is 73.00 millimeters of external diameters, 5.51 millimeters of wall thickness, its pass schedule is as shown in table 1.

The pass schedule of table 1, steel pipe φ 73.00 * 5.51

Shelf number i	Pass internal diameter (millimeter)	Roll rotational speed (rev/min)
			1	149.51	214.70
2	145.16	225.71
			3	139.57	238.82
4	131.50	259.06
			5	124.01	270.48
6	117.06	282.99
			7	110.60	296.58
8	104.57	311.31
			9	98.94	327.20
10	93.69	344.24
			11	88.76	362.57
12	84.16	382.11
			13	79.83	397.42
14	76.60	408.13
			15	74.91	417.29
16	74.02	422.32
			17	73.73	425.51
18	73.73	428.01

Because participating in rolling roll stand number is 18, comprises 17 subsystems, also promptly in equation (11), N is taken as 18, therefore, obtains the following model equation of representing with the interactive subsystem form

$\left\{\begin{array}{c}{v}_i(x,t)\frac{\&PartialD;}{\&PartialD;x}{A}_i(x,t)+\frac{\&PartialD;}{\&PartialD;t}{A}_i(x,t)=-k_i{P}_i\left(t\right)\\ A_i(x,t)\frac{\&PartialD;}{\&PartialD;x}{v}_i(x,t)=k_i{P}_i\left(t\right)\end{array}\right.$

Model satisfies constraint condition

$\left\{\begin{array}{c}{v}_i(L_{i+1},t)=v_i(L_i,t)+k_i{P}_i\left(t\right){\&Integral;}_{L_i}^{L_{i+1}}\frac{1}{A_i(x,t)}\mathrm{dx}\\ A_i(x,t)=\mathrm{\&pi;}{S}_i(x,t)(D_i(x,t)-S_i(x,t))\end{array}\right.$

The starting condition of model and boundary condition are

The steel tube section initial distribution: $A_i(x,0)=\left\{\begin{array}{cc}\mathrm{\&pi;}\&times;6.00(152.50-6.00),& i=1,x=0\\ 0,& i\&NotEqual;1\end{array}\right.$

The initial roll line velocity distribution of steel pipe: $v_i(x,0)=\left\{\begin{array}{cc}214.70\&times;\frac{(330.00-149.51)}{2},& i=1,x=0\\ 0,& i\&NotEqual;1\end{array}\right.$

Steel pipe rolling linear velocity position distribution: v _i(0, t)=ω _iR _i

Wherein, i=1,2 ..., 17, x is the position of steel pipe with respect to first frame, t is a rolling time, A, D and S are respectively the sectional area of steel pipe, and external diameter and wall thickness, v are the linear velocity of steel pipe, and P is the tension force that acts on the steel pipe, and R represents the radius of clean-up of breast roller.

The first step, the radius of clean-up of calculating roll according to the characteristic of roll angular velocity and steel pipe shown in (7) to (9) formula, and calculate steel pipe in the linear velocity at each frame place shown in (10) formula;

Second step, at first utilize the constraint condition of model rotating speed to calculate the tension force correlation shown in (12) formula, the linear velocity of each point of utilizing the Model Calculation steel pipe then is shown in (13) formula; This step be steel pipe axially on calculate;

In the 3rd step, the sectional area that utilizes the Model Calculation steel pipe is shown in (14) formula, and this step is to calculate on time orientation;

In the 4th step, repeat the aforementioned calculation step and leave institute's organic frame until steel pipe.

The result of the manual measurement value of simulation result and actual steel pipe shown in Fig. 5～7, wherein, the wall thickness value that the solid line representative model calculates, the actual thickness of steel pipe value of manual measurement represented in asterisk.

With 7% of average wall thickness is the standard of the pipe end section of thickening, and it is as follows to calculate comparing result: the manual measurement value of average wall thickness is 5.52 millimeters, and the result of model emulation is 5.51 millimeters; The manual measurement value of head section thickened end is 1.25 meters, and the result of model emulation is 1.17 meters; The manual measurement value of rear thickened end is 1.30 meters, and the result of model emulation is 1.20 meters.

Artificial measured data and The model calculation contrast show, the relative error of average wall thickness＜1%, relative error＜8%. actual computation that pipe end thickens segment length show that this model can better simulate the stretch reducing operation of rolling, and can explain the phenomenon that pipe end thickens in the stretch reducing production run.

Claims (2)

1. tension diameter-reducing procedure of seamless steel tube emulation mode, it is characterized in that: be distributed as input with the roll rotational speed that participates in rolling machine frame, with the vertical wall thickness change of steel pipe in the operation of rolling is output, the tension diameter-reducing procedure that each adjacent frame is right is as a subsystem, with each subsystem of partial differential equation formal representation, starting condition with equation, boundary condition and constraint condition are represented the feature of steel pipe and stretch-reducing mill and the interaction between subsystem, set up the mathematical model of the vertical wall thickness change of weldless steel tube in the tension diameter-reducing procedure, adopt difference method with the recursive form of model conversation for ease of computer solving, describedly set up the vertical wall thickness change mathematical model of weldless steel tube in the tension diameter-reducing procedure: the quantitative relationship of keeping weighing apparatus and steel tube section distortion and steel pipe axial tensile force according to the steel pipe volume in the operation of rolling is set up the partial differential equations of the vertical wall thickness change model of steel pipe in each subsystem, the starting condition that provides equation according to the original outer diameter and the wall thickness situation of hollow forging, utilize relation and the steel pipeline speed and the roll angular velocity of tension force and steel pipeline velocity contrast, the relation of the roll radius of clean-up provides boundary condition and the constraint condition that model need satisfy, and the tension diameter-reducing procedure dynamic mathematical models of representing with the interactive subsystem form are

$\left\{\begin{array}{c}{v}_i(x,t)\frac{\&PartialD;}{\&PartialD;x}{A}_i(x,t)+\frac{\&PartialD;}{\&PartialD;t}{A}_i(x,t)=-k_i{P}_i\left(t\right)\\ A_i(x,t)\frac{\&PartialD;}{\&PartialD;x}{v}_i(x,t)=k_i{P}_i\left(t\right)\end{array}\right.$

Model satisfies constraint condition

$\left\{\begin{array}{c}{v}_i(L_{i+1},t)=v_i(L_i,t)+k_i{P}_i\left(t\right){\&Integral;}_{L_i}^{L_{i+1}}\frac{1}{A_i(x,t)}\mathrm{dx}\\ A_i(x,t)=\mathrm{\&pi;}{S}_i(x,t)(D_i(x,t)-S_i(x,t))\end{array}\right.$

The starting condition of model and boundary condition are

The steel tube section initial distribution: $A_i(x,0)=\left\{\begin{array}{cc}{A}_0,& i=1,x=0\\ 0,& i\&NotEqual;1\end{array}\right.$

The initial roll line velocity distribution of steel pipe: $v_i(x,0)=\left\{\begin{array}{cc}{\mathrm{\&omega;}}_1{R}_1,& i=1,x=0\\ 0,& i\&NotEqual;1\end{array}\right.$

Steel pipe rolling linear velocity position distribution: v _i(0, t)=ω _iR _i

Wherein, i=1,2 ..., N-1, x is the position of steel pipe with respect to first frame, and t is a rolling time, A, and D and S are respectively the sectional area of steel pipe, external diameter and wall thickness, v is the linear velocity of steel pipe, and P is the tension force that acts on the steel pipe, and R represents the radius of clean-up of breast roller, and N is for participating in rolling frame number, L _iBe the frame spacing among the subsystem i, k _iBe the tension set coefficient among the subsystem i.

2. a kind of tension diameter-reducing procedure of seamless steel tube emulation mode as claimed in claim 1 is characterized in that, described rice with difference method with the recursive form of model conversation for ease of computer solving: at first, utilize difference method with the model discretize; Then, adopt the mode of circulation recursion to calculate the sectional area of steel pipe at each point, step is: the first step, and calculate the radius of clean-up of roll according to the characteristic of roll angular velocity and steel pipe, and calculate the linear velocity of steel pipe at each frame place; Second step, at first utilize the constraint condition of model rotating speed to calculate the tension force correlation, utilize the linear velocity of the each point of Model Calculation steel pipe then; This step be steel pipe axially on calculate; The 3rd goes on foot, and utilizes the sectional area of Model Calculation steel pipe, and this step is to calculate on time orientation; In the 4th step, repeat the aforementioned calculation step and leave institute's organic frame until steel pipe.

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