CN105414246A - Prediction method of bending angle of titanium alloy laser bending forming part - Google Patents

Abstract

The invention discloses a prediction method of a bending angle of a titanium alloy laser bending forming part, so as to solve the technical problem that the accuracy of a predicted bending angle by the conventional bending angle prediction method is poor. The prediction method is characterized in that 12 membership functions and 81 fuzzy rules that are related to the energy density, the plate width, the plate thickness and the scan path curvature during titanium alloy laser bending forming are determined; a mathematical model of the bending angle during the titanium alloy laser bending forming is established; by employing titanium alloy laser bending forming experimental data, weight coefficients and weight numbers of the fuzzy rules are selected out, and are substituted into the mathematical model of the bending angle during the titanium alloy laser bending forming; and a prediction model of the bending angle during the titanium alloy laser bending forming is obtained. According to the prediction method, the maximum error between a predicted test sample result and an experimental result of the bending angle during TC4 alloy laser bending forming is less than 4%.

Description

The Forecasting Methodology at titanium alloy laser bend forming spares bend angle

Technical field

The present invention relates to a kind of Forecasting Methodology of spares bend angle, particularly relate to a kind of Forecasting Methodology of titanium alloy laser bend forming spares bend angle.

Background technology

Titanium alloy is the combination property with its excellence, is applied widely at Aero-Space marine field.At present, the equipment such as Domestic Aircraft, satellite, rocket have employed large-scale titanium alloy straight line bending component and the kink of curve part of larger specific gravity.Because titanium alloy plasticity is at room temperature poor, cold deformation is very difficult, and adopt heat distortion manufacturing technology to need to manufacture a large amount of high-temperature resistance die and heating furnace, its cycle is long, cost is high, and serviceability is difficult to ensure.For large-scale sheet titanium alloy crooked part, its manufacture difficulty is larger.Structural metallic materials laser bend forming belongs to without mould, flexible plastic New Machining Technology, can under the condition without the need to frock or mould, the curved surface part of rapid figuration manufacture any shape.Laser bend forming technology eliminates a large amount of high-temperature resistance die and heating furnace manufacture, and under the prerequisite ensureing part serviceability, can reduce manufacturing cost, shorten manufacturing process, accelerate part development, be one of advanced manufacturing technology of 21st century.

The same with traditional bending forming technology, in laser bend forming technology, the angle of bend of crooked part is the basic parameter of technological design and optimization, and plate geometric parameter, laser forming parameter are very remarkable on the impact of crooked part angle of bend.

It is linear approximate relationship that document 1 " laser bending characteristics [J] of Ti-6Al-4V titanium alloy; Li Liqun; Chen Yanbin; Zhang Liwen; Feng little Song; China YouSe Acta Metallurgica Sinica, 2005,15 (6): 842-847 " reports Ti-6Al-4V titanium alloy plate angle of bend and scanning times after repeatedly laser straight line sweep.

Document 2 " titanium alloy laser bend forming precision controlling and numerical simulation study [M]; Liu Chang; Central China University of Science and Technology's master thesis, 2012,1-72 " reports sweep speed, laser power, scanning times, energy density to the affecting laws of TC4 titanium alloy laser straight line angle of bend.Titanium alloy is a kind of difficult deformable metal structural material, its dynamic characteristic is responsive especially to impressed field variable, therefore titanium alloy laser bend forming process is considered the complex process of a multi-scenarios method, nonlinearity, the crooked part angle of bend data that its physical simulation experiment obtains are few, very limited to the directive function of technological design and optimization.

Document 3 " titanium-alloy thin-plate laser bend forming numerical simulation and optimal design research [M], explain good, HeFei University of Technology's master thesis, 2009,1-74 " reports the artificial nerve network model of prediction TA15 titanium-alloy thin-plate laser straight line angle of bend.During titanium alloy laser bend forming, energy density is the comprehensive effect of laser power, spot diameter, sweep speed, and sweep interval is relevant to sweep speed again; Adopt energy density more accurate on the impact of titanium alloy laser bending dynamic characteristic as outfield physical quantity, document 3 is not using appropriate as the principal element affecting titanium alloy laser bending angle to laser power, spot diameter, sweep speed, sweep interval yet.Meanwhile, the forecast model of document 3 does not consider the impact on titanium alloy laser bending angle of sheet material geometric parameter, particularly sheet metal thickness, and therefore the description of its Mathematical Modeling to titanium alloy laser bending characteristic is inaccurate.

In fact, in titanium alloy laser bend forming process, sheet material geometric parameter, energy density are very remarkable on the impact of angle of bend, are the Important Parameters of titanium alloy laser bending technological design.Therefore, the angle of bend of the titanium alloy crooked part of existing model prediction is adopted can not to meet the requirement of bending process design.

Summary of the invention

In order to overcome the deficiency of existing angle of bend Forecasting Methodology prediction angle of bend low precision, the invention provides a kind of Forecasting Methodology of titanium alloy laser bend forming spares bend angle.12 membership functions of energy density, strip width, sheet metal thickness and scanning pattern curvature when first the method determines titanium alloy laser bend forming and 81 fuzzy rules, then the Mathematical Modeling of angle of bend when setting up titanium alloy laser bend forming; Adopt titanium alloy laser bend forming experimental data, optimize fuzzy rule weight coefficient and weights, then substitute into the Mathematical Modeling of angle of bend during titanium alloy laser bend forming, obtain the forecast model of angle of bend during titanium alloy laser bend forming.Worst error during the TC4 Alloy by Laser bending forming of the present invention's prediction between the test sample book result of angle of bend and experimental result is less than 4%.

The technical solution adopted for the present invention to solve the technical problems is: a kind of Forecasting Methodology of titanium alloy laser bend forming spares bend angle, is characterized in comprising the following steps:

Step one, employing laser carry out scanning and irradiation process to titanium alloy plate surface, measure the angle of bend of titanium alloy laser bending part;

Step 2, to energy density I during titanium alloy laser bend forming, unit J/mm ², strip width b, unit mm, sheet metal thickness t, unit mm and scanning pattern curvature unit mm ^-1be normalized;

Step 3, by energy density x during titanium alloy laser bend forming ₁, strip width x ₂, sheet metal thickness x ₃with scanning pattern curvature x ₄be set to input variable, above-mentioned four input variables are divided into large, medium and small three subintervals, be expressed as UL, UM, US}, angle of bend is set to function-output φ, unit °;

Step 4, energy density, strip width, sheet metal thickness and scanning pattern curvature are respectively at the membership function in three subintervals when determining titanium alloy laser bend forming,

For energy density: ${\mathrm{UL}}_1\left(x\right)=\left\{\begin{array}{cc}1& x>I_L\\ \exp \[\frac{-{(x-I_L)}^2}{B_I}\]& x\≤I_L\end{array}\right.---\left(1a\right)$

${\mathrm{UM}}_1\left(x\right)=\exp \[\frac{-{(x-(I_L+I_S)/2)}^2}{B_{I0}}\]---\left(1b\right)$

${\mathrm{US}}_1\left(x\right)=\left\{\begin{array}{cc}1& x<I_S\\ \exp \&lsqb;\frac{-{(x-I_S)}^2}{B_I}\&rsqb;& x\&GreaterEqual;I_S\end{array}\right.---\left(1c\right)$

For strip width: ${\mathrm{UL}}_2\left(x\right)=\left\{\begin{array}{cc}1& x>b_L\\ \exp \&lsqb;\frac{-{(x-b_L)}^2}{B_b}\&rsqb;& x\&le;b_L\end{array}\right.---\left(2a\right)$

${\mathrm{UM}}_2\left(x\right)=\exp \&lsqb;\frac{-{(x-(b_L+b_S)/2)}^2}{B_{b0}}\&rsqb;---\left(2b\right)$

${\mathrm{US}}_2\left(x\right)=\left\{\begin{array}{cc}1& x<b_S\\ \exp \&lsqb;\frac{-{(x-b_S)}^2}{B_b}\&rsqb;& x\&GreaterEqual;b_S\end{array}\right.---\left(2c\right)$

For sheet metal thickness: ${\mathrm{UL}}_3\left(x\right)=\left\{\begin{array}{cc}1& x>t_L\\ \exp \&lsqb;\frac{-{(x-t_L)}^2}{B_t}\&rsqb;& x\&le;t_L\end{array}\right.---\left(3a\right)$

${\mathrm{UM}}_3\left(x\right)=\exp \&lsqb;\frac{-{(x-(t_L+t_S)/2)}^2}{B_{t0}}\&rsqb;---\left(3b\right)$

${\mathrm{US}}_3\left(x\right)=\left\{\begin{array}{cc}1& x<t_S\\ \exp \&lsqb;\frac{-{(x-t_S)}^2}{B_t}\&rsqb;& x\&GreaterEqual;t_S\end{array}\right.---\left(3c\right)$

For scanning pattern curvature: ${\mathrm{UL}}_4\left(x\right)=\left\{\begin{array}{cc}1& x>{\stackrel{\&OverBar;}{r}}_L\\ \exp \&lsqb;\frac{-{(x-{\stackrel{\&OverBar;}{r}}_L)}^2}{B_{\stackrel{\&OverBar;}{r}}}\&rsqb;& x\&le;{\stackrel{\&OverBar;}{r}}_L\end{array}\right.---\left(4a\right)$

${\mathrm{UM}}_4\left(x\right)=\exp \&lsqb;\frac{-{(x-({\stackrel{\&OverBar;}{r}}_L+{\stackrel{\&OverBar;}{r}}_S)/2)}^2}{B_{\stackrel{\&OverBar;}{r}0}}\&rsqb;---\left(4b\right)$

${\mathrm{US}}_4\left(x\right)=\left\{\begin{array}{cc}1& x<{\stackrel{\&OverBar;}{r}}_S\\ \exp \&lsqb;\frac{-{(x-{\stackrel{\&OverBar;}{r}}_S)}^2}{B_{\stackrel{\&OverBar;}{r}}}\&rsqb;& x\&GreaterEqual;{\stackrel{\&OverBar;}{r}}_S\end{array}\right.---\left(4c\right)$

In formula, B _jfor the variance of membership function, I _l, I _sbe respectively maximum and the minimum of a value of energy density, b _l, b _sbe respectively maximum and the minimum of a value of strip width, t _l, t _sbe respectively maximum and the minimum of a value of sheet metal thickness, be respectively maximum and the minimum of a value of scanning pattern curvature.

Step 5, the fuzzy rule of energy density, strip width, sheet metal thickness, scanning pattern curvature is when determining titanium alloy laser bend forming,

Fuzzy rule i: if x ₁uL, x ₂uL, x ₃uL, x ₄uL,

Then, output function value $y^i=p_0^i+p_1^i{x}_1+p_2^i{x}_2+p_3^i{x}_3+p_4^i{x}_4---\left(5a\right)$

Fuzzy rule weights $w^i={\mathrm{\&mu;}}_1^i\&CenterDot;{\mathrm{\&mu;}}_2^i\&CenterDot;{\mathrm{\&mu;}}_3^i\&CenterDot;{\mathrm{\&mu;}}_4^i---\left(5b\right)$

In formula, w ⁱbe the weights of i-th fuzzy rule, it is the weight coefficient of i-th fuzzy rule;

Step 6, the Mathematical Modeling setting up titanium alloy laser bend forming spares bend angle be,

$\mathrm{\&phi;}=\frac{\underset{i=1}{\overset{m}{\&Sigma;}}\&lsqb;({\mathrm{\&mu;}}_1^i\&CenterDot;{\mathrm{\&mu;}}_2^i\&CenterDot;{\mathrm{\&mu;}}_3^i\&CenterDot;{\mathrm{\&mu;}}_4^i)(p_0^i+p_1^i{x}_1+p_2^i{x}_2+p_3^i{x}_3+p_4^i{x}_4)\&rsqb;}{\underset{i=1}{\overset{m}{\&Sigma;}}\&lsqb;{\mathrm{\&mu;}}_1^i\&CenterDot;{\mathrm{\&mu;}}_2^i\&CenterDot;{\mathrm{\&mu;}}_3^i\&CenterDot;{\mathrm{\&mu;}}_4^i\&rsqb;}---\left(6\right)$

In formula, m is number of fuzzy rules, divides, m=3 according to fuzzy region ⁴;

Step 7, the energy density, strip width, sheet metal thickness, scanning pattern curvature and the angle of bend that obtain from the experiment of titanium alloy laser bend forming are chosen multi-group data in combining and are combined as teacher's sample.Adopt teacher's sample to train mathematics modular form (6), when the cumulative errors of angle of bend are less than 2%, determine the weight coefficient of fuzzy rule with the weight w of fuzzy rule ⁱ.The fuzzy rule weight coefficient determined and fuzzy rule weights are substituted into formula (6), is the forecast model at titanium alloy laser bend forming spares bend angle.

The invention has the beneficial effects as follows: 12 membership functions of energy density, strip width, sheet metal thickness and scanning pattern curvature when first the inventive method determines titanium alloy laser bend forming and 81 fuzzy rules, then the Mathematical Modeling of angle of bend when setting up titanium alloy laser bend forming; Adopt titanium alloy laser bend forming experimental data, optimize fuzzy rule weight coefficient and weights, then substitute into the Mathematical Modeling of angle of bend during titanium alloy laser bend forming, obtain the forecast model of angle of bend during titanium alloy laser bend forming.Worst error during the TC4 Alloy by Laser bending forming of the present invention's prediction between the test sample book result of angle of bend and experimental result is less than 4%.

Below in conjunction with detailed description of the invention, the present invention is elaborated.

Detailed description of the invention

The Forecasting Methodology concrete steps at titanium alloy laser bend forming spares bend angle of the present invention are as follows:

Example is predicted as with TC4 Alloy by Laser bending forming spares bend angle.

(1) supply state TC4 sheet alloy is carried out Linear cut, obtain thickness be 0.8,1,1.5,2.0mm, width is 20,30,40,50,40mm, length is the bar-shaped sample of 50mm;

(2) with washes of absolute alcohol TC4 alloy sample, at its clean surface coating Ti-β coating;

(3) be fixed on fixture by TC4 sheet alloy, carry out laser scanning bending forming to plate surface, after laser bend forming, air cooling is to room temperature, the maximum deflection angle after measurement laser bend forming on the center line of specimen length direction;

(3) to energy density (I, J/mm during TC4 sheet alloy laser bend forming ²), strip width (b, mm), sheet metal thickness (t, mm), scanning pattern curvature ( mm ^-1) be normalized; If energy density is input variable x ₁, strip width is input variable x ₂, sheet metal thickness is input variable x ₃, scanning pattern curvature is input variable x ₄, angle of bend (φ, °) is functional value (i.e. function-output, φ, °);

(4) when determining TC4 sheet alloy laser bend forming, energy density, strip width, sheet metal thickness, scanning pattern curvature are respectively at the membership function in three subintervals,

For energy density: ${\mathrm{UL}}_1\left(x\right)=\left\{\begin{array}{cc}1& x>4.39\\ \exp \&lsqb;\frac{-{(x-4.39)}^2}{3}\&rsqb;& x\&le;4.39\end{array}\right.---\left(1a\right)$

${\mathrm{UM}}_1\left(x\right)=\exp \&lsqb;\frac{-{(x-2.585)}^2}{1.2}\&rsqb;---\left(1b\right)$

${\mathrm{US}}_1\left(x\right)=\left\{\begin{array}{cc}1& x<0.78\\ \exp \&lsqb;\frac{-{(x-0.78)}^2}{3}\&rsqb;& x\&GreaterEqual;0.78\end{array}\right.---\left(1c\right)$

For strip width: ${\mathrm{UL}}_2\left(x\right)=\left\{\begin{array}{cc}1& x>50\\ \exp \&lsqb;\frac{-{(x-50)}^2}{200}\&rsqb;& x\&le;50\end{array}\right.---\left(2a\right)$

${\mathrm{UM}}_2\left(x\right)=\exp \&lsqb;\frac{-{(x-35)}^2}{80}\&rsqb;---\left(2b\right)$

${\mathrm{US}}_2\left(x\right)=\left\{\begin{array}{cc}1& x<20\\ \exp \&lsqb;\frac{-{(x-20)}^2}{200}\&rsqb;& x\&GreaterEqual;20\end{array}\right.---\left(2c\right)$

For sheet metal thickness: ${\mathrm{UL}}_3\left(x\right)=\left\{\begin{array}{cc}1& x>2\\ \exp \&lsqb;\frac{-{(x-2)}^2}{0.3}\&rsqb;& x\&le;2\end{array}\right.---\left(3a\right)$

${\mathrm{UM}}_3\left(x\right)=\exp \&lsqb;\frac{-{(x-1.4)}^2}{0.12}\&rsqb;---\left(3b\right)$

${\mathrm{US}}_3\left(x\right)=\left\{\begin{array}{cc}1& x<0.8\\ \exp \&lsqb;\frac{-{(x-0.8)}^2}{0.3}\&rsqb;& x\&GreaterEqual;0.8\end{array}\right.---\left(3c\right)$

For scanning pattern curvature: ${\mathrm{UL}}_4\left(x\right)=\left\{\begin{array}{cc}1& x>0.033\\ \exp \&lsqb;\frac{-{(x-0.033)}^2}{0.0002}\&rsqb;& x\&le;0.033\end{array}\right.---\left(4a\right)$

${\mathrm{UM}}_4\left(x\right)=\exp \&lsqb;\frac{-{(x-0.0165)}^2}{0.0001}\&rsqb;---\left(4b\right)$

${\mathrm{US}}_4\left(x\right)=\left\{\begin{array}{cc}1& x<0\\ \exp \&lsqb;\frac{-x^2}{0.0002}\&rsqb;& x\&GreaterEqual;0\end{array}\right.---\left(4c\right);$

(5) when determining TC4 sheet alloy laser bend forming, the fuzzy rule of energy density, strip width, sheet metal thickness, scanning pattern curvature is respectively,

Fuzzy rule 1: if x ₁uL, x ₂uL, x ₃uL, x ₄uL,

Then, output function value $y^1=p_0^1+p_1^1{x}_1+p_2^1{x}_2+p_3^1{x}_3+p_4^1{x}_4---\left(5a\right)$

Fuzzy rule weights $w^1={\mathrm{\&mu;}}_1^1\&CenterDot;{\mathrm{\&mu;}}_2^1\&CenterDot;{\mathrm{\&mu;}}_3^1\&CenterDot;{\mathrm{\&mu;}}_4^1---\left(5b\right)$

Fuzzy rule 2: if x ₁uM, x ₂uM, x ₃uM, x ₄uM,

Then, output function value $y^2=p_0^2+p_1^2{x}_1+p_2^2{x}_2+p_3^2{x}_3+p_4^2{x}_4---\left(6a\right)$

Fuzzy rule weights $w^2={\mathrm{\&mu;}}_1^2\&CenterDot;{\mathrm{\&mu;}}_2^2\&CenterDot;{\mathrm{\&mu;}}_3^2\&CenterDot;{\mathrm{\&mu;}}_4^2---\left(6b\right)$

Fuzzy rule 3: if x ₁uS, x ₂uS, x ₃uS, x ₄uS,

Then, output function value $y^3=p_0^3+p_1^3{x}_1+p_2^3{x}_2+p_3^3{x}_3+p_4^3{x}_4---\left(7a\right)$

Fuzzy rule weights $w^3={\mathrm{\&mu;}}_1^3\&CenterDot;{\mathrm{\&mu;}}_2^3\&CenterDot;{\mathrm{\&mu;}}_3^3\&CenterDot;{\mathrm{\&mu;}}_4^3---\left(7b\right)$

┇

Fuzzy rule 81: if x ₁uS, x ₂uS, x ₃uS, x ₄uS,

Then, output function value $y^{81}=p_0^{81}+p_1^{81}{x}_1+p_2^{81}{x}_2+p_3^{81}{x}_3+p_4^{81}{x}_4---\left(8a\right)$

Fuzzy rule weights $w^{81}={\mathrm{\&mu;}}_1^{81}\&CenterDot;{\mathrm{\&mu;}}_2^{81}\&CenterDot;{\mathrm{\&mu;}}_3^{81}\&CenterDot;{\mathrm{\&mu;}}_4^{81}---\left(8b\right);$

(6) Mathematical Modeling at TC4 Alloy by Laser bending forming spares bend angle is set up,

$\mathrm{\&phi;}=\frac{\underset{i=1}{\overset{81}{\&Sigma;}}\&lsqb;({\mathrm{\&mu;}}_1^i\&CenterDot;{\mathrm{\&mu;}}_2^i\&CenterDot;{\mathrm{\&mu;}}_3^i\&CenterDot;{\mathrm{\&mu;}}_4^i)(p_0^i+p_1^i{x}_1+p_2^i{x}_2+p_3^i{x}_3+p_4^i{x}_4)\&rsqb;}{\underset{i=1}{\overset{81}{\&Sigma;}}\&lsqb;{\mathrm{\&mu;}}_1^i\&CenterDot;{\mathrm{\&mu;}}_2^i\&CenterDot;{\mathrm{\&mu;}}_3^i\&CenterDot;{\mathrm{\&mu;}}_4^i\&rsqb;}---\left(9\right);$

(7) teacher's sample when choosing TC4 sheet alloy laser bend forming is as shown in table 1.Adopt teacher's sample to train mathematics modular form (9), when the cumulative errors of angle of bend are less than 2%, optimize fuzzy rule weight coefficient with fuzzy rule weights (w ⁱ) each value.

Teacher's sample during table 1.TC4 sheet alloy laser bend forming

Sequence number	Energy density (J/mm 2)	Strip width (mm)	Sheet metal thickness (mm)	Scanning pattern curvature (mm -1)	Angle of bend (°)
						1	3.66	30	0.8	0	3.2
2	1.22	40	1.5	0.014	0.2
						3	0.78	50	2	0.017	0
4	4.1	50	1.5	0	2.5
						5	1.43	30	0.8	0.02	0.5
6	3.9	40	2	0	1
						7	3.33	40	1	0.033	1.7
8	3.9	30	2	0.014	0.5
						9	3.51	20	2	0.02	0.2
10	4.39	30	1.5	0	2
						11	1.43	40	1	0	0.6
12	3.43	50	0.8	0.033	2
						13	1.71	20	1	0.017	0.4
14	4.18	50	1	0.02	2.2
						15	3.33	40	1.5	0	2.8
16	3.33	40	1	0.014	2.9

(8) the fuzzy rule weight coefficient determined and weights are substituted into formula (9), be the forecast model at TC4 Alloy by Laser bending forming spares bend angle.During TC4 Alloy by Laser bending forming, the test sample book of angle of bend predicts the outcome as shown in table 2 with Comparison of experiment results, and in table, worst error is less than 4%.

Test sample book during table 2.TC4 sheet alloy laser bend forming

Thus prove that the Forecasting Methodology of TC4 Alloy by Laser crooked part angle of bend provided by the present invention has higher accuracy and reliability.

Claims (1)

1. the Forecasting Methodology at titanium alloy laser bend forming spares bend angle, is characterized in that comprising the following steps:

Step one, employing laser carry out scanning and irradiation process to titanium alloy plate surface, measure the angle of bend of titanium alloy laser bending part;

Step 2, to energy density I during titanium alloy laser bend forming, unit J/mm ², strip width b, unit mm, sheet metal thickness t, unit mm and scanning pattern curvature unit mm ^-1be normalized;

Step 3, by energy density x during titanium alloy laser bend forming ₁, strip width x ₂, sheet metal thickness x ₃with scanning pattern curvature x ₄be set to input variable, above-mentioned four input variables are divided into large, medium and small three subintervals, be expressed as UL, UM, US}, angle of bend is set to function-output φ, unit °;

Step 4, energy density, strip width, sheet metal thickness and scanning pattern curvature are respectively at the membership function in three subintervals when determining titanium alloy laser bend forming,

For energy density: ${\mathrm{UL}}_1\left(x\right)=\left\{\begin{array}{cc}1& x>I_L\\ \exp \left[\frac{-{(x-I_L)}^2}{B_I}\right]& x\&le;I_L\end{array}\right.---\left(1a\right)$

${\mathrm{UM}}_1\left(x\right)=\exp \left[\frac{-{(x-(I_L+I_S)/2)}^2}{B_{I0}}\right]---\left(1b\right)$

${\mathrm{US}}_1\left(x\right)=\left\{\begin{array}{cc}1& x<I_S\\ \exp \left[\frac{-{(x-I_S)}^2}{B_I}\right]& x\&GreaterEqual;I_S\end{array}\right.---\left(1c\right)$

For strip width: ${\mathrm{UL}}_2\left(x\right)=\left\{\begin{array}{cc}1& x>b_L\\ \exp \left[\frac{-{(x-b_L)}^2}{B_b}\right]& x\&le;{\text{b}}_L\end{array}\right.---\left(2a\right)$

${\mathrm{UM}}_2\left(x\right)=\exp \left[\frac{-{(x-(b_L+b_S)/2)}^2}{B_{b0}}\right]---\left(2b\right)$

${\mathrm{US}}_2\left(x\right)=\left\{\begin{array}{cc}1& x<b_S\\ \exp \left[\frac{-{(x-b_S)}^2}{B_b}\right]& x\&GreaterEqual;b_S\end{array}\right.---\left(2c\right)$

For sheet metal thickness: ${\mathrm{UL}}_3\left(x\right)=\left\{\begin{array}{cc}1& x>t_L\\ \exp \left[\frac{-{(x-t_L)}^2}{B_t}\right]& x\&le;{\text{t}}_L\end{array}\right.---\left(3a\right)$

${\mathrm{UM}}_3\left(x\right)=\exp \&lsqb;\frac{-{(x-(t_L+t_S)/2)}^2}{B_{t0}}\&rsqb;---\left(3b\right)$

${\mathrm{US}}_3\left(x\right)=\left\{\begin{array}{cc}1& x<t_S\\ \exp \&lsqb;\frac{-{(x-t_S)}^2}{B_t}\&rsqb;& x\&GreaterEqual;t_S\end{array}\right.---\left(3c\right)$

For scanning pattern curvature: ${\mathrm{UL}}_4\left(x\right)=\left\{\begin{array}{cc}1& x>{\stackrel{\&OverBar;}{r}}_L\\ \exp \&lsqb;\frac{-{(x-{\stackrel{\&OverBar;}{r}}_L)}^2}{B_{\stackrel{\&OverBar;}{r}}}\&rsqb;& x\&le;{\stackrel{\&OverBar;}{r}}_L\end{array}\right.---\left(4a\right)$

${\mathrm{UM}}_4\left(x\right)=\exp \&lsqb;\frac{-{(x-({\stackrel{\&OverBar;}{r}}_L+{\stackrel{\&OverBar;}{r}}_S)/2)}^2}{B_{\stackrel{\&OverBar;}{r}0}}\&rsqb;---\left(4b\right)$

${\mathrm{US}}_4\left(x\right)=\left\{\begin{array}{cc}1& x<{\stackrel{\&OverBar;}{r}}_S\\ \exp \&lsqb;\frac{-{(x-{\stackrel{\&OverBar;}{r}}_S)}^2}{B_{\stackrel{\&OverBar;}{r}}}\&rsqb;& x\&GreaterEqual;{\stackrel{\&OverBar;}{r}}_S\end{array}\right.---\left(4c\right)$

In formula, B _jfor the variance of membership function, I _l, I _sbe respectively maximum and the minimum of a value of energy density, b _l, b _sbe respectively maximum and the minimum of a value of strip width, t _l, t _sbe respectively maximum and the minimum of a value of sheet metal thickness, be respectively maximum and the minimum of a value of scanning pattern curvature;

Step 5, the fuzzy rule of energy density, strip width, sheet metal thickness, scanning pattern curvature is when determining titanium alloy laser bend forming,

Fuzzy rule i: if x ₁uL, x ₂uL, x ₃uL, x ₄uL,

Then, output function value $y^i=p_0^i+p_1^i{x}_1+p_2^i{x}_2+p_3^i{x}_3+p_4^i{x}_4---\left(5a\right)$

Fuzzy rule weights $w^i={\mathrm{\&mu;}}_1^i\&CenterDot;{\mathrm{\&mu;}}_2^i\&CenterDot;{\mathrm{\&mu;}}_3^i\&CenterDot;{\mathrm{\&mu;}}_4^i---\left(5b\right)$

In formula, w ⁱbe the weights of i-th fuzzy rule, it is the weight coefficient of i-th fuzzy rule;

Step 6, the Mathematical Modeling setting up titanium alloy laser bend forming spares bend angle be,

$\mathrm{\&phi;}=\frac{\underset{i=1}{\overset{m}{\&Sigma;}}\&lsqb;({\mathrm{\&mu;}}_1^i\&CenterDot;{\mathrm{\&mu;}}_2^i\&CenterDot;{\mathrm{\&mu;}}_3^i\&CenterDot;{\mathrm{\&mu;}}_4^i)(p_0^i+p_1^i{x}_1+p_2^i{x}_2+p_3^i{x}_3+p_4^i{x}_4)\&rsqb;}{\underset{i=1}{\overset{m}{\&Sigma;}}\&lsqb;{\mathrm{\&mu;}}_1^i\&CenterDot;{\mathrm{\&mu;}}_2^i\&CenterDot;{\mathrm{\&mu;}}_3^i\&CenterDot;{\mathrm{\&mu;}}_4^i\&rsqb;}---\left(6\right)$

In formula, m is number of fuzzy rules, divides, m=3 according to fuzzy region ⁴;

Step 7, the energy density, strip width, sheet metal thickness, scanning pattern curvature and the angle of bend that obtain from the experiment of titanium alloy laser bend forming are chosen multi-group data in combining and are combined as teacher's sample; Adopt teacher's sample to train mathematics modular form (6), when the cumulative errors of angle of bend are less than 2%, determine the weight coefficient of fuzzy rule with the weight w of fuzzy rule ⁱ; The fuzzy rule weight coefficient determined and fuzzy rule weights are substituted into formula (6), is the forecast model at titanium alloy laser bend forming spares bend angle.