### Algorithm for transmission and reception of ultrasonic beams for the dual grating method

The relative position of the defect and the source of the ultrasonic beam has a great influence on the detection result^{16}. As shown in Fig. 3, assuming that the deflection angle of the central axis of the main sound beam is (theta)the angle of inclination of the plane defect with respect to the horizontal surface is (tau)and therefore the angle between the incident wave and the fault plane is (tau – theta). The sound pressure of the reflected wave at the center point of the transducer can be expressed as Eq. (1)^{5}:

$$ P_{t} = frac{{p_{0} r_{p} D_{L}^{2} left( theta right)sin left[ {k_{L} {text{csin}}left( {theta – tau } right)} right]}}{{k_{L} {text{ctan}}left( {theta – tau } right)}} $$

(1)

where ( r_{p} ) is the acoustic reflection coefficient of the ultrasonic wave at the air-sample interface, (D_{L} left( theta right)) is the directivity coefficient of the wave, (k_{L} = frac{2pi }{lambda }) is the wave vector, (lambda) is the acoustic wavelength. When ({uptheta } approx {uptau}), the reflected ultrasonic pressure of the defects is at its maximum value. The axis of the main ultrasonic wave is perpendicular to the fault plane and the acoustic pressure of the fault echo received by the transducer is at its maximum value. When ( {uptau } – {uptheta } > 0), that is, when the angle of inclination of the defect with respect to the horizontal surface is greater than the angle between the central axis and the main beam, the transducer cannot receive the ultrasonic wave reflected after the main ultrasonic wave interacts with the defect. In this case, the peak diffraction signal from the planar defect is too weak to be used for defect inspection. Therefore, it is important to consider the structure of the sample and the method of transmission and reception of ultrasonic waves when designing an ultrasonic testing algorithm.

Figure 4 shows the integral structure of the superalloy turbine disk and the arrangement of the dual-array ultrasonic transducers. Since the defect plane is parallel to the welded interface, when the grating-A ultrasonic transducer is transmitted, the defect reflection signal can be effectively received by the grating-B ultrasonic transducer. However, due to the short horizontal distance between the array ultrasonic transducer and the inspection area, and the vertical distribution of the welds, according to Eq. (1), the reflected sound pressure of the defect is low. If the deflection of the acoustic beam is used directly for detection, the fault signal will be very weak, even overwhelmed by other signals. In addition, since the vertical welds of the superalloy turbine disk are distributed over the entire sample, the deflection and focusing parameters of the acoustic beam must be specifically designed to achieve detection of the entire weld area. . Therefore, in this algorithm, all the welds on the cylindrical surface of the superalloy turbine disc will be considered as the inspection object. To realize the full coverage inspection of welding interface, the acoustic beam control scheme and imaging method based on dual array ultrasonic transducers are designed.

The interaction between the ultrasonic transducers in the array and the large deflection angle of the sound beam have a great influence on the detection result^{15}. To avoid this, first, the specific focus delay law is used for the transmission of the grating-A ultrasonic transducer and the reception of the grating-B ultrasonic transducer. In this way, the lower part of the welds near the transducer-B can be detected. Next, the transmit and receive transducer to detect the upper part of the welds, Figure 5 shows the ultrasonic test diagram of the pitch capture network for the superalloy disc.

Assuming that the amount of grating elements for each transducer is N/2, taking grating-A ultrasonic transducer as an example, the deflection angle ({uptheta }) of the synthesized acoustic beam is in (left[ {arctan left( {frac{{d times left( {N – 1} right)/2 + e + L_{S} }}{h}} right),arctan left( {frac{{d times left( {N – 1} right)/2 + e + L_{S} }}{h/2}} right)} right]).

Where (D) is the step of the elements of the array, *e* is the width of the array elements, (L_{s}) is the distance from the outer shape of the transducer, this distance is also shown in Fig. 8, and *h* is the vertical height of the weld. Given the angle of inclination of the plane defect with respect to the horizontal surface, ({uptau})is fixed at (90^circ)the angle of incidence acoustic waves to detect defects at different locations can be calculated.

For different types of incidence wave and angles of incidence, different waveform conversions will occur after being reflected from the interface and affect the evaluation of the fault signal. It is important to further confirm the type of focused acoustic wave. Figure 6 shows the waveform conversion with longitudinal waves and transverse waves at a certain angle of incidence. Figure 6a shows the waveform conversion of the reflected sound wave with the longitudinal wave at an angle of incidence ({mathrm{alpha}}_{L}). As the incident angle ({mathrm{alpha}}_{L}) increases, the reflected ultrasonic wave will contain both longitudinal and transverse waves. Figure 6b shows that, for a transverse wave, as the angle of incidence ({mathrm{alpha}}_{s}) increases, the longitudinal wave will disappear and only the reflected transverse waves will exist. Considering eq. (2) and the geometric size of the superalloy disk, the minimum incident angle of the ultrasonic wave in this detection pattern is (65.3^circ ). If the longitudinal wave is used for focusing, most of the energy after the interaction with the defect will be converted into reflected transverse waves while reflected longitudinal waves exist at the same time, leading to a complex reflected signal and low energy. Therefore, to ensure a simple pure shear wave reflected signal, the focused shear wave is used for detection.

### Calculation of the timing law

Different from the conventional array ultrasonic scanning method, for this sample, the synthetic ultrasonic beam from the array ultrasonic transducer should change the current channel focus depth and focus deviation angle. As shown in Fig. 7, taking the intersection point O of the weld and the lower surface of the sample as the origin of the coordination, it is assumed that the width of the network element is (w)the step of the elements of the array is (e )the height of the weld is (h)the transverse speed of sound in the sample is (c_{s}). When the ultrasonic beam is focused on the welding position (pleft( {0,z_{p} } right))according to Fermat’s principle, the wave propagation time of the element (j) at the focal point (p) is obtained by Eq. (2):

$$ T_{j} = min sqrt {frac{{x_{j}^{2} + left( {h – z_{p} } right)^{2} }}{{c_{s }^{2} }}} left( {z_{p} le frac{h}{2}} right) $$

(2)

where

$$ x_{j} = – frac{{2L_{S} + hd + w}}{2} + jd $$

(z_{p}) is located on the Z axis at a specific focal point. According to this, the value (T_{j}) can be calculated. For a fixed focal point on the solder interface, Eq. (3) can be used to calculate the propagation time of ultrasonic waves by all elements of the excitation network in the current channel. The maximum propagation delay is defined as (T_{max}) Therefore, the delay time ({Delta }T_{j}) corresponding to the element (j) is calculated by Eq. (3):

$$ vartriangle T_{j} = T_{max} – T_{j} $$

(3)

When the array-A ultrasonic transducer works for transmission, a number of focal points in the range of (Z_{0} – Z_{n}) are discrete at the bottom half of the weld near the B-array ultrasonic transducer. According to the timing law, the timing time for the (left( {n + 1} right)) focal point can be calculated. Also, when the B-grating ultrasonic transducer works for transmission, depending on the symmetry of the structure, the A-grating ultrasonic focusing law can be directly applied to detect the upper half of the weld.

At the same time, the grating ultrasonic transducer will also receive the side wall reflection signals. Therefore, the total ultrasonic propagation paths of the fault signal from different positions of the weld to the receiving transducer are different, i.e. (left| {x_{t} z_{j} } right| + left| {z_{j} x_{r} } right|) as shown in Figure 7. Therefore, after determining the discrete focus points, the total travel time of the ultrasonic beam transmitted to each discrete point and then reflected back to the receiving transducer must be calculated. Correspondingly, for each channel, specific gate parameters must be designed to accurately receive reflected signals from different defects.

Assuming that the superalloy turbine disk is an isotropic medium, the acoustic velocity of the transverse wave remains unchanged as the angle of propagation changes. If the abscissa of the center of the transducer is (x_{t})when the synthetic ultrasound beam from the array-A ultrasound transducer propagates towards the focal point (z_{i}) and is reflected by the defect, the propagation delay to the receive-B transducer is calculated by Eq. (4):

$$ t_{j} = frac{{sqrt {left( {h – z_{j} } right)^{2} + x_{t}^{2} } + sqrt {z_{j} ^{2} + left( {frac{{z_{i} x_{t} }}{{h – z_{j} }}} right)^{2} } }}{{c_{s} }} $$

(4)