CN104923573A - Method for predicting waving form of thin cold-rolled strip steel in width direction - Google Patents

Abstract

The invention provides a method for predicting the waving form of thin cold-rolled strip steel in the width direction. The method is implemented on the basis that the residual stress value and the coordinate values of the thin cold-rolled strip steel are obtained and comprises the following steps that (1), a left auxiliary vector and a right auxiliary vector are calculated; (2), the residual stress positive maximum value vector including the endpoint value and the coordinate vector are calculated; (3), the residual stress negative minimum value vector including the endpoint value and the coordinate vector are calculated; (4), the width vector is calculated; (5), the amplitude vector is calculated; and (6), a function for the waving form in the width direction is determined. By the adoption of the method, a waving area and the waving form in the width direction of the thin cold-rolled strip steel can be predicted, and the capacity of evaluating the strip steel quality during cold rolling production is improved.

Description

A kind of Forecasting Methodology of cold-rolled thin steel strip width waviness form

Technical field

The invention belongs to board rolling field, relate to during a kind of cold-rolled thin steel strip is produced and utilize the method detecting the residual-stress value predicted width direction waviness form obtained.

Background technology

The stability problem rolling rear band is the important branch of cold rolled sheet shape problem, is the mechanical foundation of the good discrimination standard of plate shape.In cold rolled strip steel production, if rolling direction plastic elongation skewness in the width direction, band steel will produce residual stress in face, and after the inhomogeneities of this plastic elongation acquires a certain degree, band steel will produce visible shape wave.

At present, following several method is had about the case study of cold-strip steel waviness:

1, document 1 (Yang Quan. the research of cold-strip steel Post-buckling Theory and band steel shape control target. Ph.D. Dissertation. University of Science & Technology, Beijing .1995.5) report, by surveying the deflection functions and residual stress of being with steel, summarize its form by polynomial function and trigonometric function, finally adopt energy variation method to determine concrete undetermined coefficient.

2, document 2 (Dai Jietao, Zhang Qingdong, Qin Jian. thin wide cold-strip steel local plate shape Buckling modes analysis research. engineering mechanics .2011, (10): 236-242) report, Gaussian function is adopted to describe the inhomogeneities of cripling situation lower boundary load, introduce local influence coefficient to consider in cripling action process with the impact of steel straight portion on flex region, obtain Critical Buckling condition and the relation between local influence coefficient and local flex region width, analysis and solution has been carried out in the application post-buckling path of perturbation method to cripling.

3, document 3 (Zhang Qingdong, Lu Xingfu, Zhang Xiaofeng. there is initial warpage defect thin strip cold steel plate shape cobble deformation behavioral study. engineering mechanics, 2014, 31 (8): 243-249) report, for the bent flatness defect of the warpage compound wooden dipper occurred in thin strip cold rolling production process, the golden principle of virtual displacement of application shallow shell theory and gal the Liao Dynasty, establish bending deformation analytical calculation model under non-uniform load effect, solve and obtain Buckling Critical Load and the critical wavelength that cobble deformation occurs the band steel with initial warpage defect again, the Discuss and analyse affecting laws of initial warpage to cobble deformation.

4, document 4 (Wang Lan, Cao Jianguo, Jia Shenghui, Deng. the finite element analysis of cold-rolled strip steel shape bending deformation unstability limit. Central South University's journal: natural science edition, 2007,38 (6): 1157-) report, adopt ANSYS Finite Element Simulation Analysis technique study cold-rolled wide strip steel plate shape flexing buckling behaviors.Utilize the geometrical non-linearity of ANSYS to solve module, set up wide Thin Strip Steel two-dimensional finite element model, analyze the flexing Instability under multi-form load, thick wide ratio and tensile strain, computing board shape bending deformation unstability is limit.

5, document 5 (Lin Zhenbo, Zhang Bo, connect family's wound, Duan Zhenyong. the analysis and thinking of Cold Rolled Strip shape discrimination model. iron and steel, 1995,30 (8): 39-43) report, application finite strip method solves the shape discrimination model of cold-strip, for typical residual stress distribution pattern, discuss residual stress distribution characteristic sum deviation size and the change of band flakiness ratio to the affecting laws of shape discrimination and plate shape state.

This shows, the research of cold-strip steel waviness problem is using certain ideal form residual stress and deflection functions as starting point, the Г а л ё р к и н method of the applied energy calculus of variations, perturbation method, parsing, classical Finite Element or finit-strip method etc. carry out the Analyse dlasto of entirety or local to it, obtain the relevant geometric parameter of unstability, Instability limit, post-buckling field variable or flexing path.But be made with following two problems like this: first, according to field measurement data, cold-strip steel residual stress has very complicated distribution form, some local characteristics of ubiquity, as stress spike, change is violent in the width direction, also changes along rolling direction, and often have pro forma qualitative change, as middle wave tendency changes limit wave tendency into; The second, the waviness form of cold-strip steel is also comparatively complicated, shows periodically, do not have evident regularity in the width direction along rolling direction piecewise.Therefore, above-mentioned way with certain subjectivity, or needs to survey amount of deflection form, and arranging especially in strip width direction deflection functions, is difficult to the deformation rule reflecting its complexity.

Summary of the invention

In order to solve the problem, the object of the invention is to obtain on the basis of cold-strip steel residual-stress value σ and coordinate value r thereof in detection, proposing a kind of new method can predicting cold-strip steel width waviness form of off-line.

Basic calculation of the present invention is as follows:

(1) left redundant vector σ is calculated ^leftwith right redundant vector σ ^right;

(2) the residual stress positive maximum coordinate vector c comprising endpoint value is calculated;

(3) the negative minimum vector s and coordinate vector b thereof comprising the residual stress of endpoint value is calculated;

(4) molded breadth vector a;

(5) amplitude vector ε is calculated;

(6) width waviness form function δ (x) is determined.

1, left redundant vector σ is calculated ^leftwith right redundant vector σ ^right

According to the cold-rolled thin steel strip residual-stress value σ of actual measurement, 1 calculate left redundant vector σ with the formula ^leftwith right redundant vector σ ^right, further to analyze.

$\left\{\begin{array}{c}{\mathrm{\σ}}^{\mathrm{left}}=({\mathrm{\σ}}_2,{\mathrm{\σ}}_1,{\mathrm{\σ}}_2,...,{\mathrm{\σ}}_{n-3},{\mathrm{\σ}}_{n-2},{\mathrm{\σ}}_{n-1})\\ {\mathrm{\σ}}^{\mathrm{right}}=({\mathrm{\σ}}_2,{\mathrm{\σ}}_3,{\mathrm{\σ}}_4,...,{\mathrm{\σ}}_{n-1},{\mathrm{\σ}}_n,{\mathrm{\σ}}_{n-1})\end{array}\right.---1$

In formula:

σ ₁, σ ₂, σ ₃..., σ _n-2, σ _n-1, σ _nfor detecting the band residual stress of steel component of tensor σ obtained _ysame transversal

The detected value of difference on face, the namely component of vectorial σ;

σ ^leftfor left redundant vector;

σ ^rightfor right redundant vector;

N is test point number.

2, the residual stress positive maximum coordinate vector c comprising endpoint value is calculated

When certain is a bit all large than its both sides point, this point is a maximum point, the position of positive maximum point in vector can be obtained further again according to operator U, therefore can be obtained the positive maximum coordinate vector c of residual stress by formula 2, and this formula can than consecutive points, large and numerical value be that positive end points retains by two of a σ end points.

$\left\{\begin{array}{c}u=N\left\{U\right[U(\mathrm{\σ}-{\mathrm{\σ}}^{\mathrm{left}}).*U(\mathrm{\σ}-{\mathrm{\σ}}^{\mathrm{right}}).*\mathrm{\σ}].*(r+o_r)\}\\ c=u-o_u\end{array}\right.---2$

In formula:

Computing U represents the component being greater than 0 in a vector is put 1, and the component being less than or equal to 0 sets to 0, and then obtains a new vector;

Computing .* represents and is multiplied by the vectorial corresponding element of two same dimension, and then obtains a new vector, and its operator precedence degree is lower than operator U;

R is the vector that on a certain cross section of cold-rolled thin steel strip, residual stress test point coordinate is corresponding;

O _rfor the vector that and component identical with r dimension is 1 entirely;

Computing N represents that residual components forms a new vector by after null component all removes in a vector;

U is an intermediate vector in computational process;

C is the residual stress positive maximum coordinate vector comprising endpoint value;

O _ufor the vector that and component identical with u dimension is 1 entirely.

3, the negative minimum vector s and coordinate vector b thereof comprising the residual stress of endpoint value is calculated

By σ, σ ^leftand σ ^rightget opposite number, utilize the thinking in 2 can obtain the negative minimum point of σ, calculate negative minimum vector s and coordinate vector b thereof by formula 3, and this formula can little and numerical value be that negative end points retains than consecutive points by two of a σ end points.

$\left\{\begin{array}{c}s=N\left\{U\right[U(-\mathrm{\σ}+{\mathrm{\σ}}^{\mathrm{left}}).*U(-\mathrm{\σ}+{\mathrm{\σ}}^{\mathrm{right}}).*(-\mathrm{\σ})].*\mathrm{\σ}\}\\ w=N\left\{U\right[U(-\mathrm{\σ}+{\mathrm{\σ}}^{\mathrm{left}}).*U(-\mathrm{\σ}+{\mathrm{\σ}}^{\mathrm{right}}).*(-\mathrm{\σ})].*(r+o_r)\}\\ b=w-o_w\end{array}\right.---3$

In formula:

S is that the residual stress comprising endpoint value bears minimum vector;

W is an intermediate vector in computational process;

B is that the residual stress comprising endpoint value bears minimum coordinate vector;

O _wfor the vector that and component identical with w dimension is 1 entirely.

4, molded breadth vector a

For determining waviness form, needing to calculate its size on strip width direction, representing with width vector a, and calculated by formula 4.

$a=\min \left[\mathrm{abs}(o_c^{T}b-c^T{o}_b)\right]---4$

In formula:

After min represents the minimum of a value taking-up by each row of matrix, the row vector formed by the arrangement of original row order;

Abs represents and is taken absolute value by each element of matrix;

A is width vector;

O _cfor the vector that and component identical with c dimension is 1 entirely;

for o _ctransposition;

C ^tfor the transposition of c;

O _bfor the vector that and component identical with b dimension is 1 entirely;

and do matrix multiplication between b;

C ^twith o _bbetween do matrix multiplication.

5, amplitude vector ε is calculated

For determining waviness form, needing to calculate it being with the size in steel short transverse, representing with amplitude vector ε, and being calculated by formula 5.

$\mathrm{\ϵ}=-\frac{3\mathrm{\π}}{4E}a.*s---5$

In formula:

π is pi;

E is the elastic modelling quantity 2.1 × 10 of steel ⁵mPa;

ε is amplitude vector.

6, width waviness form function δ (x) is determined

After obtaining the vector about waviness position and shape, strip width direction waviness forms can be calculated by 6 formulas.

$\mathrm{\δ}\left(x\right)=\sqrt{\underset{j=1}{\overset{m}{\mathrm{\Σ}}}\frac{{\mathrm{\ϵ}}_j}{a_j^4}H(a_j-b_j+x)H(a_j+b_j-x){[a_j^2-{(x-b_j)}^2]}^2}---6$

In formula:

M is the dimension of vectorial a, b, ε;

ε _jfor the component of vectorial ε;

A _jfor the component of vectorial a;

B _jfor the component of vectorial b;

X is the independent variable along strip width direction, and scope is from zero to strip width;

δ (x) is strip width direction waviness form function;

H (x) is Heaviside function, $H\left(x\right)=\left\{\begin{array}{cc}1,& x>0\\ \frac{1}{2},& x=0.\\ 0,& x<0\end{array}\right.$

Utilize the method can predict waviness region and the waviness form of cold-rolled thin steel strip width, improve the evaluation capacity to strip quality in cold rolling production.

Accompanying drawing explanation

The calculated value of Fig. 1 width waviness form function.

The actual waviness form of Fig. 2 band steel.

The contrast of Fig. 3 residual stress measured value and approximation.

Detailed description of the invention

Below in conjunction with embodiment, technical scheme of the present invention is described further.Table 1 is the residual stress measured value of certain factory one volume cold-strip steel on a certain cross section, comprises 76 test points altogether.

The residual stress measured value of certain factory one volume cold-strip steel of table 1 on a certain cross section

Therefore have

σ＝(-7.7702,-8.0203,-3.3241,…,-21.7170,-26.7423,-28.5301)

r＝(0,0.0150,0.0300,…,1.5450,1.5600,1.5750)

1, left redundant vector σ is calculated ^leftwith right redundant vector σ ^right

According to the cold-rolled thin steel strip residual-stress value σ of actual measurement, 1 calculate left redundant vector σ with the formula ^leftwith right redundant vector σ ^right, further to analyze.

$\left\{\begin{array}{c}{\mathrm{\&sigma;}}^{\mathrm{left}}=({\mathrm{\&sigma;}}_2,{\mathrm{\&sigma;}}_1,{\mathrm{\&sigma;}}_2,...,{\mathrm{\&sigma;}}_{n-3},{\mathrm{\&sigma;}}_{n-2},{\mathrm{\&sigma;}}_{n-1})\\ {\mathrm{\&sigma;}}^{\mathrm{right}}=({\mathrm{\&sigma;}}_2,{\mathrm{\&sigma;}}_3,{\mathrm{\&sigma;}}_4,...,{\mathrm{\&sigma;}}_{n-1},{\mathrm{\&sigma;}}_n,{\mathrm{\&sigma;}}_{n-1})\end{array}\right.---1$

In formula:

σ ₁, σ ₂, σ ₃..., σ _n-2, σ _n-1, σ _nfor detecting the band residual stress of steel component of tensor σ obtained _ythe detected value of difference on same cross section, the namely component of vectorial σ;

σ ^leftfor left redundant vector;

σ ^rightfor right redundant vector;

N is test point number.

Therefore have

σ ^left＝(-8.0203,-7.7702,-8.0203,…,-17.7509,-21.7170,-26.7423)

σ ^right＝(-8.0203,-3.3241,-2.3883,…,-26.7423,-28.5301,-26.7423)

2, the residual stress positive maximum coordinate vector c comprising endpoint value is calculated

When certain is a bit all large than its both sides point, this point is a maximum point, the position of positive maximum point in vector can be obtained further again according to operator U, therefore can be obtained the positive maximum coordinate vector c of residual stress by formula 2, and this formula can than consecutive points, large and numerical value be that positive end points retains by two of a σ end points.

$\left\{\begin{array}{c}u=N\left\{U\right[U(\mathrm{\&sigma;}-{\mathrm{\&sigma;}}^{\mathrm{left}}).*U(\mathrm{\&sigma;}-{\mathrm{\&sigma;}}^{\mathrm{right}}).*\mathrm{\&sigma;}].*(r+o_r)\}\\ c=u-o_u\end{array}\right.---2$

In formula:

Computing U represents the component being greater than 0 in a vector is put 1, and the component being less than or equal to 0 sets to 0, and then obtains a new vector;

Computing .* represents and is multiplied by the vectorial corresponding element of two same dimension, and then obtains a new vector, and its operator precedence degree is lower than operator U;

R is the vector that on a certain cross section of cold-rolled thin steel strip, residual stress test point coordinate is corresponding;

O _rfor the vector that and component identical with r dimension is 1 entirely;

Computing N represents that residual components forms a new vector by after null component all removes in a vector;

U is an intermediate vector in computational process;

C is the residual stress positive maximum coordinate vector comprising endpoint value;

O _ufor the vector that and component identical with u dimension is 1 entirely.

Therefore have

c＝(0.1650,0.3450,1.4100)

3, the negative minimum vector s and coordinate vector b thereof comprising the residual stress of endpoint value is calculated

By σ, σ ^leftand σ ^rightget opposite number, utilize the thinking in 2 can obtain the negative minimum point of σ, calculate negative minimum vector s and coordinate vector b thereof by formula 3, and this formula can little and numerical value be that negative end points retains than consecutive points by two of a σ end points.

$\left\{\begin{array}{c}s=N\left\{U\right[U(-\mathrm{\&sigma;}+{\mathrm{\&sigma;}}^{\mathrm{left}}).*U(-\mathrm{\&sigma;}+{\mathrm{\&sigma;}}^{\mathrm{right}}).*(-\mathrm{\&sigma;})].*\mathrm{\&sigma;}\}\\ w=N\left\{U\right[U(-\mathrm{\&sigma;}+{\mathrm{\&sigma;}}^{\mathrm{left}}).*U(-\mathrm{\&sigma;}+{\mathrm{\&sigma;}}^{\mathrm{right}}).*(-\mathrm{\&sigma;})].*(r+o_r)\}\\ b=w-o_w\end{array}\right.---3$

In formula:

S is that the residual stress comprising endpoint value bears minimum vector;

W is an intermediate vector in computational process;

B is that the residual stress comprising endpoint value bears minimum coordinate vector;

O _wfor the vector that and component identical with w dimension is 1 entirely.

Therefore have

s＝(-8.0203,-5.7144,-6.2545,-28.5301)

b＝(0.0150,0.5775,0.8775,1.5750)

4, molded breadth vector a

For determining waviness form, needing to calculate its size on strip width direction, representing with width vector a, and calculated by formula 4.

$a=\min \left[\mathrm{abs}(o_c^{T}b-c^T{o}_b)\right]---4$

In formula:

After min represents the minimum of a value taking-up by each row of matrix, the row vector formed by the arrangement of original row order;

Abs represents and is taken absolute value by each element of matrix;

A is width vector;

O _cfor the vector that and component identical with c dimension is 1 entirely;

for o _ctransposition;

C ^tfor the transposition of c;

O _bfor the vector that and component identical with b dimension is 1 entirely;

and do matrix multiplication between b;

C ^twith o _bbetween do matrix multiplication.

Therefore have

a＝(0.1500,0.2325,0.5325,0.1650)

5, amplitude vector ε is calculated

For determining waviness form, needing to calculate it being with the size in steel short transverse, representing with amplitude vector ε, and being calculated by formula 5.

$\mathrm{\&epsiv;}=-\frac{3\mathrm{\&pi;}}{4E}a.*s---5$

In formula:

π is pi;

E is the elastic modelling quantity 2.1 × 10 of steel ⁵mPa;

ε is amplitude vector.

Therefore have

ε＝(1.3498×10 ^-5,1.4907×10 ^-5,3.7369×10 ^-5,5.2818×10 ^-5)

6, width waviness form function δ (x) is determined

After obtaining the vector about waviness position and shape, strip width direction waviness form functions can be calculated by 6 formulas.

$\mathrm{\&delta;}\left(x\right)=\sqrt{\underset{j=1}{\overset{m}{\mathrm{\&Sigma;}}}\frac{{\mathrm{\&epsiv;}}_j}{a_j^4}H(a_j-b_j+x)H(a_j+b_j-x){[a_j^2-{(x-b_j)}^2]}^2}---6$

In formula:

M is the dimension of vectorial a, b, ε;

ε _jfor the component of vectorial ε;

A _jfor the component of vectorial a;

B _jfor the component of vectorial b;

X is the independent variable along strip width direction, and scope is from zero to strip width;

δ (x) is strip width direction waviness form function;

H (x) is Heaviside function, $H\left(x\right)=\left\{\begin{array}{cc}1,& x>0\\ \frac{1}{2},& x=0.\\ 0,& x<0\end{array}\right.$

Therefore have

$\left\{\begin{array}{ccc}{\mathrm{\&epsiv;}}_1=1.3498\&times;{10}^{-5}& a_1=0.1500& b_1=0.0150\\ {\mathrm{\&epsiv;}}_2=1.4907\&times;{10}^{-5}& a_2=0.2325& b_2=0.5775\\ {\mathrm{\&epsiv;}}_3=3.7369\&times;{10}^{-5}& a_3=0.5325& b_3=0.8775\\ {\mathrm{\&epsiv;}}_4=5.2818\&times;{10}^{-5}& a_4=0.1650& b_4=1.5750\end{array}\right.$

Namely 6 formulas that substituted into by above coefficient obtain width waviness form function.

This example utilizes said method, coding, obtain width waviness form function δ (x), as shown in Figure 1, width waviness form function δ (x) has reflected the middle wave of band steel and monolateral wave, especially wave in comparatively complicated " one high and one low bimodal " form being with steel to have, with actual observation on-the-spot shown in Fig. 2 to band steel waviness form contrast, waviness position and form are substantially identical.

For further analytical calculation precision, utilize formula 7 inverse residual stress approximation, and contrast with measured value, as shown in Figure 3, relative error is 4.3%.

$\mathrm{\&sigma;}\&ap;-\underset{j=1}{\overset{m}{\mathrm{\&Sigma;}}}\frac{{3s}_j}{4}[(\frac{{(x-b_j)}^2}{a_j^2}-1)\left(\frac{x-b_j}{a_j}\ln \right|\frac{x-b_j-a_j}{x-b_j+a_j}|+2)+\frac{2}{3}]---7$

In formula:

S _jfor the component of vectorial s.

Claims (1)

1. a Forecasting Methodology for cold-rolled thin steel strip width waviness form, is characterized in that:

The corresponding vectorial σ=(σ of residual stress detected value on a certain cross section of the cold-rolled thin steel strip that production scene plate profile instrument records ₁, σ ₂, σ ₃..., σ _n-2, σ _n-1, σ _n), adopt formula 1 to obtain left redundant vector σ ^leftwith right redundant vector σ ^right

$\begin{array}{c}{\mathrm{\&sigma;}}^{\mathrm{left}}=({\mathrm{\&sigma;}}_2,{\mathrm{\&sigma;}}_1,{\mathrm{\&sigma;}}_2,...,{\mathrm{\&sigma;}}_{n-3},{\mathrm{\&sigma;}}_{n-2},{\mathrm{\&sigma;}}_{n-1})\\ {\mathrm{\&sigma;}}^{\mathrm{right}}=({\mathrm{\&sigma;}}_2,{\mathrm{\&sigma;}}_3,{\mathrm{\&sigma;}}_4,...,{\mathrm{\&sigma;}}_{n-1},{\mathrm{\&sigma;}}_n,{\mathrm{\&sigma;}}_{n-1})\end{array}---1$

In formula:

σ ₁, σ ₂, σ ₃..., σ _n-2, σ _n-1, σ _nfor detecting the detected value of the band residual stress of steel difference on same cross section obtained, the namely component of vectorial σ;

σ ^leftfor left redundant vector;

σ ^rightfor right redundant vector;

N is test point number,

Formula 2 is adopted to obtain the residual stress positive maximum coordinate vector c comprising endpoint value

$\begin{array}{c}u=N\left\{U\right[U(\mathrm{\&sigma;}-{\mathrm{\&sigma;}}^{\mathrm{left}}).*U(\mathrm{\&sigma;}-{\mathrm{\&sigma;}}^{\mathrm{right}}).*\mathrm{\&sigma;}].*(r+o_r)\}\\ c=u-o_u\end{array}---2$

In formula:

Computing U represents the component being greater than 0 in a vector is put 1, and the component being less than or equal to 0 sets to 0, and then obtains a new vector;

Computing .* represents and is multiplied by the vectorial corresponding element of two same dimension, and then obtains a new vector, and its operator precedence degree is lower than operator U;

R is the vector that on a certain cross section of cold-rolled thin steel strip, residual stress test point coordinate is corresponding;

O _rfor the vector that and component identical with r dimension is 1 entirely;

Computing N represents that residual components forms a new vector by after null component all removes in a vector;

U is an intermediate vector in computational process;

C is the residual stress positive maximum coordinate vector comprising endpoint value;

O _ufor the vector that and component identical with u dimension is 1 entirely,

In the same way, the residual stress adopting formula 3 to obtain comprising endpoint value bears minimum vector s and coordinate vector b thereof

$\left\{\begin{array}{c}s=N\left\{U\right[U(-\mathrm{\&sigma;}+{\mathrm{\&sigma;}}^{\mathrm{left}}).*U(-\mathrm{\&sigma;}+{\mathrm{\&sigma;}}^{\mathrm{right}}).*(-\mathrm{\&sigma;})].*\mathrm{\&sigma;}\}\\ w=N\left\{U\right[U(-\mathrm{\&sigma;}+{\mathrm{\&sigma;}}^{\mathrm{left}}).*U(-\mathrm{\&sigma;}+{\mathrm{\&sigma;}}^{\mathrm{right}}).*(-\mathrm{\&sigma;})].*(r+o_r)\}\\ b=w-o_w\end{array}\right.---3$

In formula:

S is that the residual stress comprising endpoint value bears minimum vector;

W is an intermediate vector in computational process;

B is that the residual stress comprising endpoint value bears minimum coordinate vector;

O _wfor the vector that and component identical with w dimension is 1 entirely,

Formula 4 is adopted to obtain width vector a

$a=\min \left[\mathrm{abs}(o_c^{T}b-c^T{o}_b)\right]---4$

In formula:

After min represents the minimum of a value taking-up by each row of matrix, the row vector formed by the arrangement of original row order;

Abs represents and is taken absolute value by each element of matrix;

A is width vector;

O _cfor the vector that and component identical with c dimension is 1 entirely;

for o _ctransposition;

C ^tfor the transposition of c;

O _bfor the vector that and component identical with b dimension is 1 entirely;

and do matrix multiplication between b;

C ^twith o _bbetween do matrix multiplication,

Formula 5 is adopted to obtain amplitude vector ε

$\mathrm{\&epsiv;}=-\frac{3\mathrm{\&pi;}}{4E}a.*s---5$

In formula:

π is pi;

E is the elastic modelling quantity 2.1 × 10 of steel ⁵mPa;

ε is amplitude vector,

Formula 6 is adopted to determine width waviness form function δ (x)

$\mathrm{\&delta;}\left(x\right)=\sqrt{\underset{j=1}{\overset{m}{\mathrm{\&Sigma;}}}\frac{{\mathrm{\&epsiv;}}_j}{a_j^4}\text{H}(a_j-b_j+x)H(a_j+b_j-x)[a_j^2-{(x-b_j)}^2{]}^2}---6$

In formula:

M is the dimension of vectorial a, b, ε;

ε _jfor the component of vectorial ε;

A _jfor the component of vectorial a;

B _jfor the component of vectorial b;

X is the independent variable along strip width direction, and scope is from zero to strip width;

δ (x) is strip width direction waviness form function;

H (x) is Heaviside function, $H\left(x\right)=\left\{\begin{array}{cc}1,& x>0\\ \frac{1}{2},& x=0\\ 0,& x<0\end{array}\right..$