CN105241964B - The delay calculating method of cylindrical surface workpiece phase-control focusing ultrasound detection - Google Patents

Abstract

The invention discloses a time-delay calculation method for phase-controlled focused ultrasonic detection of a cylindrical curved surface workpiece, which is characterized in that it includes the time-delay calculation method of each array element during direct-ray detection and the time delay of each array element during focus detection after reflection from the inner wall of a cylinder Calculation method. The time delay calculated by the delay calculation method proposed by the present invention can form a better detection sound beam, which can effectively detect shafts or tubular workpieces with small outer diameters, including transverse defects in butt welds of small diameter pipes, and can also be used for this Supplementary detection of other defects in similar workpieces to assist in the characterization of defects.

Description

Time-delay calculation method for phase-controlled focused ultrasonic testing of cylindrical curved surface workpieces

technical field

The invention relates to a time-delay calculation method for phase-controlled focused ultrasonic detection of cylindrical curved surface workpieces, and belongs to the technical field of ultrasonic detection.

Background technique

Butt welds of small-diameter tubes are widely used in various large-scale boilers such as petroleum, chemical industry, and heating, especially thermal power boilers, and are used for heat exchange and material transmission. For example, there are more than 80,000 welds in the installation of a 1 gigawatt boiler. In 2014, the installed capacity of thermal power generation in my country was about 915.7 gigawatts, and the number of small diameter pipe butt welds is huge. If there are undiscovered harmful defects in the weld, it is easy to cause tube explosion, which may lead to unplanned shutdown of the boiler, resulting in huge economic losses and social problems. The service environment of this type of weld is harsh, so the requirements for detection reliability and detection efficiency are extremely high, and detection is difficult. Especially the transverse defects are easy to be missed. Transverse defects refer to the defects in the weld whose trend is perpendicular to the centerline of the weld and the detection surface. The most common transverse defect is a transverse crack.

At present, the non-destructive testing techniques commonly used in weld inspection include ray, ultrasonic, magnetic particle, penetration and eddy current, among which only ray and ultrasonic can detect the surface and internal defects of small-diameter pipe butt welds at the same time.

Radiographic inspection is to detect defects by using the difference in the absorption ability of the defective part and the non-defective part when the ray passes through the object to be inspected, resulting in a difference in the brightness of the imaging film. Generally, a single transillumination double-wall double-projection detection method is used. It is insensitive to more harmful area-type defects such as cracks and lack of fusion; it is difficult to detect thicker workpieces through transillumination, and there are blind areas in the detection, which may easily lead to missed inspections; there are also deficiencies in radiation, pollution, and low efficiency.

At present, conventional ultrasonic testing generally adopts oblique incident shear wave (SV wave) pulse-echo method for single-sided and double-sided detection. The probe is manually moved back and forth for zigzag scanning, and the echo amplitude and echo changes are observed to locate quantitative and qualitative defects. When this method is detected, the sound propagation direction is in the same plane as the transverse defect, so it is difficult to detect the transverse defect, and the self-generating and self-retracting method is adopted, and the leading edge of the wedge is large, making it difficult to detect thinner workpieces. In addition, the detection blind zone of the ultrasonic diffraction time-of-flight method is too large, even larger than the pipe wall thickness, which is not suitable for detecting small-diameter pipe butt welds. Finally, the phased array ultrasonic testing technology commonly used in industrial testing at present uses the focused pulse shear wave echo method sector scanning, electronic scanning and multi-scan imaging for testing. The defect is coplanar, and the echo signal reflected by the defect is very weak, even submerged in the noise signal, which can easily cause missed detection. Two-dimensional phased array ultrasonic testing technology is used for testing. The number of probe array elements is large, requiring a large number of equipment channels, and the testing equipment is expensive and not suitable for field application.

The current ultrasonic detection of transverse defects can only detect transverse defects of plate butt welds. Mainly adopt conventional ultrasonic single-chip probes, which are arranged on both sides of the weld. The angle formed by the oblique probe and the center line of the weld is not more than 10°, and the oblique parallel scanning in two directions is performed. If the weld reinforcement is ground flat, the probe Parallel scanning in two directions can be performed on the weld seam and the heat-affected zone. However, oblique parallel scanning cannot be used for pipe butt welds, mainly because the wedge-workpiece curve causes serious sound beam divergence, which makes the detection sound field chaotic and cannot be detected.

In order to overcome the above problems, a transverse defect detection device for workpieces with cylindrical curved surfaces has been developed. For details, refer to patent 201510498212.1. When using this device for detection, it is necessary to set the delay of each array element. The existing method is not suitable for this device, and the above patent The specific calculation method is not disclosed in , which will lead to unreasonable delay setting and the inability to form an effective detection sound beam.

Contents of the invention

In order to solve the above-mentioned technical problems, the present invention provides a time-delay calculation method for phase-controlled focused ultrasonic testing of cylindrical curved workpieces.

In order to achieve the above object, the technical scheme adopted in the present invention is:

The time-delay calculation method for the phase-controlled focused ultrasonic inspection of cylindrical curved surface workpieces, including the calculation method of the time-delay calculation method of each array element during the direct inspection and the calculation method of the time-delay calculation method of each array element during the focus inspection after reflection from the inner wall of the cylinder;

The process of calculating the delay of each array element during the direct detection is as follows:

A1) Build a defect detection model, and make a cross-section along the center of the probe;

A2) define the point where the cross section intersects with the pipe axis as the origin O, and establish a polar coordinate system;

A3) The position of the rear end point B of the wedge slope is B(R _B ,θ _B ); the position of the rear end point A of the wedge lower surface is A(R _out ,0), and R _out is the outer diameter of the pipe;

A4) Calculate the position of the center of each array element;

Define the position of the i-th array element center I as I(R _i ,θ _i );

Among them, AI is the distance between point A and the center I of the i-th array element, and BI is the distance between point B and the center I of the i-th array element;

H ₀ is the distance between point A and point B, α ₀ is the inclination angle of the wedge;

BI=p×(i-1)+s ₀

s ₀ is the distance from point B to the center of the nearest array element, p is the array element spacing, that is, the distance between adjacent array element centers, p=g+e, where g is the array element gap size, and e is the array element width ;

A5) according to Snell's law, obtain the equation of incident point position;

Define the position of the incident point D of the sound wave excited by the i-th array element as D(R _out , θ _D ); the position of the acoustic focus point P as P(R _P , θ _P );

The equation for θ _D is,

Among them, c _w is the sound velocity of the wedge material, and c _s is the sound velocity of the tube material;

A6) Obtain the value range of the incident point position;

Connect I and P with a straight line, the point where the straight line intersects the arc is E, the position of E is E(R _out , θ _E ), and the value range of θ _D is between θ _i and θ _E ;

A7) According to the value range and equation of the incident point position, the value of the incident point position is obtained by numerical calculation using the dichotomy method;

A8) Obtain the delay of the array element according to the position value of the incident point; the delay Δt _i of the i-th array element is,

Δt _i =max(t)-t _i +t ₀

in,

It is the sound time used for transmitting the sound wave to the point P at the center I of the i-th array element;

max(t) is the maximum value in t _i ;

t ₀ is the initial delay in the delay setting, which is a fixed constant;

Calculation method of the time delay of each array element during focus detection after reflection from the inner wall of the cylinder:

B1) Build a defect detection model, and make a cross-section along the center of the probe;

B2) define the point where the cross section intersects with the tube axis as the origin O, and set up a polar coordinate system;

B3) The position of the rear end point B of the wedge slope is B(R _B , θ _B ); the position of the rear end point A of the lower surface of the wedge is A(R _out , 0), and R _out is equal to the outer diameter of the pipe;

B4) Calculate the position of the center of each array element;

Define the position of the i-th array element center I as I(R _i , θ _i );

Among them, AI is the distance between point A and the center I of the i-th array element, and BI is the distance between point B and the center I of the i-th array element;

H ₀ is the distance between point A and point B, α ₀ is the inclination angle of the wedge;

BI=p×(i-1)+s ₀

s ⁰ is the distance from point B to the center of the nearest array element, p is the array element spacing, that is, the distance between adjacent array element centers, p=g+e, where g is the array element gap size, and e is the array element width ;

B5) according to Snell's law and the law of sine and cosine, obtain the equation of incident point position;

Define the position of the incident point H of the sound wave excited by the i-th array element as H(R _out , θ _H ); the position of the acoustic focal point P' as P'(R _P' , θ _P' ); The position of point F is F(R _in , θ _F ), and R _in is the inner diameter of the tube;

The equation for θ _H is,

B6) Obtain the value range of the incident point position;

Connect I and P' with a straight line, the point where the straight line intersects the arc is K, the position of K is K(R _out , θ _K ), and the value range of θ _H is between θ _i and θ _K ;

B7) According to the value range and equation of the incident point position, the value of the incident point position is obtained by numerical calculation using the dichotomy method;

B8) Obtain the delay of the array element according to the position value of the incident point;

The delay Δt _i ′ of the i-th array element is,

Δt _i '=max(t)'-t _i '+t ₀

in,

It is the sound time used by the center I of the i-th array element to transmit the sound wave to point P′;

max(t)' is the maximum value among t _i '.

The beneficial effects achieved by the present invention: the delay calculated by the delay calculation method proposed by the present invention can form a better detection sound beam, and can effectively detect shafts or tubular workpieces with smaller outer diameters (generally less than 159mm), including small-diameter tubes Transverse defects in butt welds can also be used for supplementary detection of other defects in this type of workpiece to assist in the characterization of defects.

Description of drawings

Fig. 1 is a schematic structural diagram of a transverse defect detection device for a workpiece with a cylindrical curved surface.

Fig. 2 is a schematic cross-sectional view of direct wave detection.

Figure 3 is a partially enlarged view.

Figure 4 is a cross-sectional schematic diagram of reflected wave detection

detailed description

The present invention will be further described below in conjunction with the accompanying drawings. The following examples are only used to illustrate the technical solution of the present invention more clearly, but not to limit the protection scope of the present invention.

As shown in Figure 1, it is a transverse defect detection device for cylindrical curved surface workpieces, which can realize direct detection and focus detection after reflection from the inner wall of the cylinder. In general, direct detection is sufficient for areas far away from the detection surface. It is multi-directional. When the direction of the defect is relatively close to the direct wave, it is difficult to detect the defect, and it is necessary to use the focus detection after reflection from the inner wall of the cylinder.

The time-delay calculation method for phase-controlled focused ultrasonic testing of cylindrical curved surface workpieces includes the calculation method for the time-delay calculation of each array element in the direct-ray inspection and the calculation method for the time-delay calculation of each array element in the focused inspection after reflection from the inner wall of the cylinder.

The process of calculating the delay of each array element during direct detection is as follows:

A1) Build a defect detection model, and make a cross-section along the center of the probe, as shown in Figures 2 and 3.

A2) Define the point where the cross section intersects the tube axis as the origin O, and establish a polar coordinate system.

A3) The position of the rear end point B of the wedge slope is B(R _B , θ _B ); the position of the rear end point A of the wedge lower surface is A(R _out , 0), and R _out is equal to the outer diameter of the pipe.

A4) Calculate the position of each array element.

Define the position of the i-th array element center I as I(R _i , θ _i );

In ΔABI, define the distance AB=H ₀ between point A and point B, ∠ABI=α ₀ +π/2, α ₀ is the inclination angle of the wedge;

The distance BI between point B and the center I of the i-th array element is

BI=p×(i-1)+s ₀ ; (1.1)

Among them, s ₀ is the distance from point B to the center of the nearest array element, p is the array element spacing, that is, the distance between adjacent array element centers, p=g+e, where g is the array element gap size, and e is the array element spacing element width;

From this, the distance AI between point A and the center I of the i-th array element can be calculated,

So θ _B is,

θ _B ＝π-β ₀ -∠BAI (1.4)

Then, in ΔIAO,

Calculated by the law of sines,

The parameters used in the above calculations are summarized, as shown in Table 1.

Table 1 Summary of parameters used

Calculate the position of the center of each array element, as shown in Table 2.

Table 2 Center position of each array element

A5) Obtain the equation for the position of the incident point.

Define the position of the incident point D of the sound wave excited by the i-th array element as D(R _out , θ _D ), the incident angle is α _i ; the position of the acoustic focal point P is P(R _P , θ _P ), the refraction The angle is α _r .

In ΔIOD,

In ΔPOD,

By Snell's law,

Among them, c _w is the sound velocity of the wedge material, and c _s is the sound velocity of the tube material;

After simplification, the equation about θ _D is obtained as,

In order to avoid the situation where the denominator is zero and facilitate programming calculations, the formula is transformed into,

A6) Obtain the value range of the incident point position.

Connect I and P with a straight line, the intersection point of the straight line and the arc is E, the position of E is E(R _out , θ _E ), and the value range of θ _D is between θ _i and θ _E.

In ΔIOP,

∠PIO＝∠EIO, for,

In ΔIEO, ∠IEO is,

so

θ _E = π-∠EIO-∠IEO+θ _i . (2.10)

A7) According to the value range of the incident point position and Equation (2.6), use the bisection numerical calculation to obtain the incident point position value.

DI and DP can be obtained by substituting the value of the incident point position into (2.1) and (2.3).

A8) Obtain the delay of the array element according to the position value of the incident point.

is the center I of the i-th array element, and the acoustic time used for transmitting the sound wave to point P is,

Then the delay of the i-th array element is,

Δt _i =max(t)-t _i +t ₀ (2.12)

in,

max(t) is the maximum value in t _i ;

t ₀ is the initial delay in the delay setting, which is a fixed constant and can be set to 0.

Assuming that the position of point P is (30mm, 30°), the time delay of each array element is shown in Table 3.

Table 3 Delay of each array element during direct wave detection

It can be seen from Table 3 that under the listed parameter system, the incident angle of the radiated sound wave of each array element to the wedge-workpiece interface changes within a small range, and its acoustic transmittance does not change much.

Calculation method of the time delay of each array element during focus detection after reflection from the inner wall of the cylinder:

B1) Build a defect detection model, and make a cross-section along the center of the probe, as shown in Figure 4.

B2) Define the point where the cross-section intersects the pipe axis as the origin O, and establish a polar coordinate system.

B3) The position of the rear end point B of the wedge slope is B(R _B , θ _B ); the position of the rear end point A of the wedge lower surface is A(R _out , 0), and R _out is equal to the outer diameter of the pipe.

B4) Calculate the position of the center of each array element (same as step A4).

B5) Obtain the equation of the incident point position.

Define the position of the incident point H of the acoustic wave excited by the i-th array element as H(R _out , θ _H ), and the incident angle as β _i ; the position of the acoustic focal point P' is P'(R _P' , θ _{P ′} ); the position of the reflection point F on the inner wall of the tube is F(R _in , θ _F ), the refraction angle is β _r , and R _in is the inner diameter of the tube.

In ΔIOH,

According to the law of sine and HI, get β _i as,

In ΔFOH, FH is calculated by the law of cosines as,

From the law of sines,

Using Snell's law and formula (3.2) to solve β _r , we get,

Simplify formula (3.5) to get,

In ΔFOH, calculate ∠OFH by the law of sine as,

In ΔFOP', calculate ∠P'OF by the law of sine as,

Combining formulas (3.7) and (3.8) to solve the formula, we get,

In ΔFOP', P'F is calculated by the law of cosines,

Substituting formulas (3.3) and (3.10) into (3.9), we get,

B6) Obtain the value range of the incident point position.

Connect I and P with a straight line, the point where the straight line intersects the arc is K, the position of K is K(R _out , θ _K ), the value range of θ _H is between θ _i and θ _K , adopt the method in step A6 method to calculate θ _K .

B7) According to the value range and equation of the position of the incident point, the value of the position of the incident point is obtained by numerical calculation using the dichotomy method.

Substituting the value of the incident point position into formulas (3.1), (3.3) and (3.10) can obtain HI, HF and P'F.

B8) Obtain the delay of the array element according to the position value of the incident point.

is the center I of the i-th array element, and the sound time used for transmitting the sound wave to the point P′ is,

Then the delay of the i-th array element is,

Δt _i '=max(t)'-t _i '+t ₀ (3.13)

Wherein, max(t)' is the maximum value in t _i '.

Assuming that the position of point P' is (34mm, 45°), the time delay of each array element is shown in Table 4.

Table 3 Delay of each array element during reflected wave detection

It can be seen from Table 4 that under the listed parameter system, the incident angle of each array element radiated sound wave incident on the wedge-workpiece interface changes within a small range, and its acoustic transmittance does not change much.

The delay calculated by the above delay calculation method can form a better detection sound beam, which can effectively detect shafts or tubular workpieces with small outer diameters (generally less than 159mm), including transverse defects in butt welds of small-diameter pipes, and can also detect It is used for supplementary detection of other defects in this type of workpiece, and assists in the characterization of defects.

The above is only a preferred embodiment of the present invention, and it should be pointed out that for those of ordinary skill in the art, without departing from the technical principle of the present invention, some improvements and modifications can also be made. It should also be regarded as the protection scope of the present invention.

Claims (1)

1. The time-delay calculation method for phase-controlled focused ultrasonic detection of cylindrical curved surface workpieces, which is characterized in that: it includes the time-delay calculation method of each array element during direct-ray detection and the time-delay calculation method of each array element during focus detection after reflection from the inner wall of the cylinder; The process of calculating the delay of each array element during the direct detection is as follows: A1) Build a defect detection model, and make a cross-section along the center of the probe; A2) define the point where the cross section intersects with the pipe axis as the origin O, and establish a polar coordinate system; A3) The position of the rear end point B of the wedge slope is B(R _B ,θ _B ); the position of the rear end point A of the wedge lower surface is A(R _out ,0), and R _out is the outer diameter of the pipe; A4) Calculate the position of the center of each array element; Define the position of the i-th array element center I as I(R _i ,θ _i ); <mrow> <msub> <mi>R</mi> <mi>i</mi> </msub> <mo>=</mo> <msqrt> <mrow> <msubsup> <mi>R</mi> <mrow> <mi>o</mi> <mi>u</mi> <mi>t</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <msup> <mi>AI</mi> <mn>2</mn> </msup> <mo>-</mo> <mn>2</mn> <msub> <mi>R</mi> <mrow> <mi>o</mi> <mi>u</mi> <mi>t</mi> </mrow> </msub> <mo>&amp;CenterDot;</mo> <mi>A</mi> <mi>I</mi> <mo>&amp;CenterDot;</mo> <msub> <mi>cos&amp;theta;</mi> <mi>B</mi> </msub> </mrow> </msqrt> </mrow> <mrow> <msub> <mi>sin&amp;theta;</mi> <mi>i</mi> </msub> <mo>=</mo> <mfrac> <mrow> <mi>A</mi> <mi>I</mi> </mrow> <msub> <mi>R</mi> <mi>i</mi> </msub> </mfrac> <msub> <mi>sin&amp;theta;</mi> <mi>B</mi> </msub> </mrow> Among them, AI is the distance between point A and the center I of the i-th array element, and BI is the distance between point B and the center I of the i-th array element; <mrow> <msup> <mi>AI</mi> <mn>2</mn> </msup> <mo>=</mo> <msup> <mi>BI</mi> <mn>2</mn> </msup> <mo>+</mo> <msubsup> <mi>H</mi> <mn>0</mn> <mn>2</mn> </msubsup> <mo>+</mo> <mn>2</mn> <mi>B</mi> <mi>I</mi> <mo>&amp;CenterDot;</mo> <msub> <mi>H</mi> <mn>0</mn> </msub> <mo>&amp;CenterDot;</mo> <msub> <mi>sin&amp;alpha;</mi> <mn>0</mn> </msub> </mrow> H ₀ is the distance between point A and point B, α ₀ is the inclination angle of the wedge; BI=p×(i-1)+s ₀ s ₀ is the distance from point B to the center of the nearest array element, p is the array element spacing, that is, the distance between adjacent array element centers, p=g+e, where g is the array element gap size, and e is the array element width ; A5) according to Snell's law, obtain the equation of incident point position; Define the position of the incident point D of the sound wave excited by the i-th array element as D(R _out , θ _D ); the position of the acoustic focus point P as P(R _P , θ _P ); The equation for θ _D is, <mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>c</mi> <mi>s</mi> </msub> <msub> <mi>R</mi> <mi>i</mi> </msub> <mi>sin</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;theta;</mi> <mi>D</mi> </msub> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <msqrt> <mrow> <msubsup> <mi>R</mi> <mi>P</mi> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>R</mi> <mrow> <mi>o</mi> <mi>u</mi> <mi>t</mi> </mrow> <mn>2</mn> </msubsup> <mo>-</mo> <mn>2</mn> <msub> <mi>R</mi> <mi>P</mi> </msub> <msub> <mi>R</mi> <mrow> <mi>o</mi> <mi>u</mi> <mi>t</mi> </mrow> </msub> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;theta;</mi> <mi>P</mi> </msub> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mi>D</mi> </msub> <mo>)</mo> </mrow> </mrow> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <msub> <mi>c</mi> <mi>w</mi> </msub> <msub> <mi>R</mi> <mi>P</mi> </msub> <mi>sin</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;theta;</mi> <mi>P</mi> </msub> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mi>D</mi> </msub> <mo>)</mo> </mrow> <msqrt> <mrow> <msubsup> <mi>R</mi> <mi>i</mi> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>R</mi> <mrow> <mi>o</mi> <mi>u</mi> <mi>t</mi> </mrow> <mn>2</mn> </msubsup> <mo>-</mo> <mn>2</mn> <msub> <mi>R</mi> <mi>i</mi> </msub> <msub> <mi>R</mi> <mrow> <mi>o</mi> <mi>u</mi> <mi>t</mi> </mrow> </msub> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;theta;</mi> <mi>D</mi> </msub> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> </mrow> </msqrt> </mrow> </mtd> </mtr> </mtable> </mfenced> Among them, c _w is the sound velocity of the wedge material, and c _s is the sound velocity of the tube material; A6) Obtain the value range of the incident point position; Connect I and P with a straight line, the point where the straight line intersects the arc is E, the position of E is E(R _out , θ _E ), and the value range of θ _D is between θ _i and θ _E ; A7) According to the value range and equation of the incident point position, the value of the incident point position is obtained by numerical calculation using the dichotomy method; A8) Obtain the delay of the array element according to the position value of the incident point; the delay Δt _i of the i-th array element is, Δt _i =max(t)-t _i +t ₀ in, It is the sound time used for transmitting the sound wave to the point P at the center I of the i-th array element; <mrow> <mi>D</mi> <mi>I</mi> <mo>=</mo> <msqrt> <mrow> <msubsup> <mi>R</mi> <mi>i</mi> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>R</mi> <mn>0</mn> <mn>2</mn> </msubsup> <mo>-</mo> <mn>2</mn> <msub> <mi>R</mi> <mi>i</mi> </msub> <msub> <mi>R</mi> <mn>0</mn> </msub> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;theta;</mi> <mi>D</mi> </msub> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> </mrow> </msqrt> <mo>;</mo> </mrow> 1 <mrow> <mi>D</mi> <mi>P</mi> <mo>=</mo> <msqrt> <mrow> <msubsup> <mi>R</mi> <mi>P</mi> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>R</mi> <mrow> <mi>o</mi> <mi>u</mi> <mi>t</mi> </mrow> <mn>2</mn> </msubsup> <mo>-</mo> <mn>2</mn> <msub> <mi>R</mi> <mi>P</mi> </msub> <msub> <mi>R</mi> <mrow> <mi>o</mi> <mi>u</mi> <mi>t</mi> </mrow> </msub> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;theta;</mi> <mi>P</mi> </msub> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mi>D</mi> </msub> <mo>)</mo> </mrow> </mrow> </msqrt> <mo>;</mo> </mrow> max(t) is the maximum value in t _i ; t ₀ is the initial delay in the delay setting, which is a fixed constant; Calculation method of the time delay of each array element during focus detection after reflection from the inner wall of the cylinder: B1) Build a defect detection model, and make a cross-section along the center of the probe; B2) define the point where the cross section intersects with the tube axis as the origin O, and set up a polar coordinate system; B3) The position of the rear end point B of the wedge slope is B(R _B , θ _B ); the position of the rear end point A of the lower surface of the wedge is A(R _out , 0), and R _out is equal to the outer diameter of the pipe; B4) Calculate the position of the center of each array element; Define the position of the i-th array element center I as I(R _i , θ _i ); <mrow> <msub> <mi>R</mi> <mi>i</mi> </msub> <mo>=</mo> <msqrt> <mrow> <msubsup> <mi>R</mi> <mrow> <mi>o</mi> <mi>u</mi> <mi>t</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <msup> <mi>AI</mi> <mn>2</mn> </msup> <mo>-</mo> <mn>2</mn> <msub> <mi>R</mi> <mrow> <mi>o</mi> <mi>u</mi> <mi>t</mi> </mrow> </msub> <mo>&amp;CenterDot;</mo> <mi>A</mi> <mi>I</mi> <mo>&amp;CenterDot;</mo> <msub> <mi>cos&amp;theta;</mi> <mi>B</mi> </msub> </mrow> </msqrt> </mrow> <mrow> <msub> <mi>sin&amp;theta;</mi> <mi>i</mi> </msub> <mo>=</mo> <mfrac> <mrow> <mi>A</mi> <mi>I</mi> </mrow> <msub> <mi>R</mi> <mi>i</mi> </msub> </mfrac> <msub> <mi>sin&amp;theta;</mi> <mi>B</mi> </msub> </mrow> Among them, AI is the distance between point A and the center I of the i-th array element, and BI is the distance between point B and the center I of the i-th array element; <mrow> <msup> <mi>AI</mi> <mn>2</mn> </msup> <mo>=</mo> <msup> <mi>BI</mi> <mn>2</mn> </msup> <mo>+</mo> <msubsup> <mi>H</mi> <mn>0</mn> <mn>2</mn> </msubsup> <mo>+</mo> <mn>2</mn> <mi>B</mi> <mi>I</mi> <mo>&amp;CenterDot;</mo> <msub> <mi>H</mi> <mn>0</mn> </msub> <mo>&amp;CenterDot;</mo> <msub> <mi>sin&amp;alpha;</mi> <mn>0</mn> </msub> </mrow> H ₀ is the distance between point A and point B, α ₀ is the inclination angle of the wedge; BI=p×(i-1)+s ₀ s ₀ is the distance from point B to the center of the nearest array element, p is the array element spacing, that is, the distance between adjacent array element centers, p=g+e, where g is the array element gap size, and e is the array element width ; B5) according to Snell's law and the law of sine and cosine, obtain the equation of incident point position; Define the position of the incident point H of the sound wave excited by the i-th array element as H(R _out , θ _H ); the position of the acoustic focal point P' as P'(R _P' , θ _P' ); The position of point F is F(R _in , θ _F ), and R _in is the inner diameter of the tube; The equation for θ _H is, <mrow> <mtable> <mtr> <mtd> <mrow> <msup> <mrow> <mo>&amp;lsqb;</mo> <msub> <mi>c</mi> <mi>s</mi> </msub> <msub> <mi>R</mi> <mi>i</mi> </msub> <mi>sin</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;theta;</mi> <mi>H</mi> </msub> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> </mrow> <mn>2</mn> </msup> <mrow> <mo>(</mo> <msubsup> <mi>R</mi> <mrow> <mi>i</mi> <mi>n</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>R</mi> <mrow> <mi>o</mi> <mi>u</mi> <mi>t</mi> </mrow> <mn>2</mn> </msubsup> <mo>-</mo> <mn>2</mn> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mo>(</mo> <mrow> <msub> <mi>&amp;theta;</mi> <mi>F</mi> </msub> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mi>H</mi> </msub> </mrow> <mo>)</mo> <msub> <mi>R</mi> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msub> <msub> <mi>R</mi> <mrow> <mi>o</mi> <mi>u</mi> <mi>t</mi> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <msup> <mrow> <mo>&amp;lsqb;</mo> <msub> <mi>c</mi> <mi>w</mi> </msub> <msub> <mi>R</mi> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msub> <mi>sin</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;theta;</mi> <mi>F</mi> </msub> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mi>H</mi> </msub> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> </mrow> <mn>2</mn> </msup> <mrow> <mo>(</mo> <msubsup> <mi>R</mi> <mi>i</mi> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>R</mi> <mrow> <mi>o</mi> <mi>u</mi> <mi>t</mi> </mrow> <mn>2</mn> </msubsup> <mo>-</mo> <mn>2</mn> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mo>(</mo> <mrow> <msub> <mi>&amp;theta;</mi> <mi>H</mi> </msub> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mi>i</mi> </msub> </mrow> <mo>)</mo> <msub> <mi>R</mi> <mi>i</mi> </msub> <msub> <mi>R</mi> <mrow> <mi>o</mi> <mi>u</mi> <mi>t</mi> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>;</mo> </mrow> <mrow> <mtable> <mtr> <mtd> <mrow> <msup> <mrow> <mo>&amp;lsqb;</mo> <msub> <mi>R</mi> <msup> <mi>P</mi> <mo>&amp;prime;</mo> </msup> </msub> <mi>sin</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;theta;</mi> <msup> <mi>P</mi> <mo>&amp;prime;</mo> </msup> </msub> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mi>F</mi> </msub> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> </mrow> <mn>2</mn> </msup> <mrow> <mo>(</mo> <msubsup> <mi>R</mi> <mrow> <mi>i</mi> <mi>n</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>R</mi> <mrow> <mi>o</mi> <mi>u</mi> <mi>t</mi> </mrow> <mn>2</mn> </msubsup> <mo>-</mo> <mn>2</mn> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mo>(</mo> <mrow> <msub> <mi>&amp;theta;</mi> <mi>F</mi> </msub> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mi>H</mi> </msub> </mrow> <mo>)</mo> <msub> <mi>R</mi> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msub> <msub> <mi>R</mi> <mrow> <mi>o</mi> <mi>u</mi> <mi>t</mi> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <msup> <mrow> <mo>&amp;lsqb;</mo> <msub> <mi>R</mi> <mrow> <mi>o</mi> <mi>u</mi> <mi>t</mi> </mrow> </msub> <mi>sin</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;theta;</mi> <mi>F</mi> </msub> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mi>H</mi> </msub> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> </mrow> <mn>2</mn> </msup> <mrow> <mo>(</mo> <msubsup> <mi>R</mi> <msup> <mi>P</mi> <mo>&amp;prime;</mo> </msup> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>R</mi> <mrow> <mi>i</mi> <mi>n</mi> </mrow> <mn>2</mn> </msubsup> <mo>-</mo> <mn>2</mn> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mo>(</mo> <mrow> <msub> <mi>&amp;theta;</mi> <msup> <mi>p</mi> <mo>&amp;prime;</mo> </msup> </msub> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mi>F</mi> </msub> </mrow> <mo>)</mo> <msub> <mi>R</mi> <msup> <mi>P</mi> <mo>&amp;prime;</mo> </msup> </msub> <msub> <mi>R</mi> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>;</mo> </mrow> B6) Obtain the value range of the incident point position; Connect I and P' with a straight line, the point where the straight line intersects the arc is K, the position of K is K(R _out , θ _K ), and the value range of θ _H is between θ _i and θ _K ; B7) According to the value range and equation of the incident point position, the dichotomy method is used for numerical calculation to obtain the incident point position value; B8) Obtain the delay of the array element according to the position value of the incident point; The delay Δt _i ′ of the i-th array element is, Δt _i '=max(t)'-t _i '+t ₀ in, It is the sound time used for transmitting the sound wave to the point P′ at the center I of the i-th array element; <mrow> <mi>H</mi> <mi>I</mi> <mo>=</mo> <msqrt> <mrow> <msubsup> <mi>R</mi> <mi>i</mi> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>R</mi> <mrow> <mi>o</mi> <mi>u</mi> <mi>t</mi> </mrow> <mn>2</mn> </msubsup> <mo>-</mo> <mn>2</mn> <msub> <mi>R</mi> <mi>i</mi> </msub> <msub> <mi>R</mi> <mrow> <mi>o</mi> <mi>u</mi> <mi>t</mi> </mrow> </msub> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;theta;</mi> <mi>H</mi> </msub> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> </mrow> </msqrt> <mo>;</mo> </mrow> <mrow> <mi>F</mi> <mi>H</mi> <mo>=</mo> <msqrt> <mrow> <msubsup> <mi>R</mi> <mrow> <mi>i</mi> <mi>n</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>R</mi> <mrow> <mi>o</mi> <mi>u</mi> <mi>t</mi> </mrow> <mn>2</mn> </msubsup> <mo>-</mo> <mn>2</mn> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;theta;</mi> <mi>F</mi> </msub> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mi>H</mi> </msub> <mo>)</mo> </mrow> <msub> <mi>R</mi> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msub> <msub> <mi>R</mi> <mrow> <mi>o</mi> <mi>u</mi> <mi>t</mi> </mrow> </msub> </mrow> </msqrt> <mo>;</mo> </mrow> <mrow> <msup> <mi>FP</mi> <mo>&amp;prime;</mo> </msup> <mo>=</mo> <msqrt> <mrow> <msubsup> <mi>R</mi> <msup> <mi>P</mi> <mo>&amp;prime;</mo> </msup> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>R</mi> <mrow> <mi>i</mi> <mi>n</mi> </mrow> <mn>2</mn> </msubsup> <mo>-</mo> <mn>2</mn> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;theta;</mi> <msup> <mi>P</mi> <mo>&amp;prime;</mo> </msup> </msub> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mi>F</mi> </msub> <mo>)</mo> </mrow> <msub> <mi>R</mi> <msup> <mi>P</mi> <mo>&amp;prime;</mo> </msup> </msub> <msub> <mi>R</mi> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msub> </mrow> </msqrt> <mo>;</mo> </mrow> max(t)' is the maximum value in t _i '; t ₀ is the initial delay in the delay setting, which is a fixed constant.