Design of hole matrix unit for high-precision batch hole making of acoustic lining assembly of aero-engine

Acoustic lining assemblies with millions of high-density array holes are key structures for high-performance and low-noise aero-engines. However, the material specificity, variable curvature shape structure, high-order number of holes, and small-sized hole spacing of acoustic lining assemblies pose great challenges to the hole-making technology. In this paper, in view of the machining requirements of millions of acoustic lining holes, a hole matrix unit design method for batch hole making is proposed. The optimal hole matrix unit dimension is calculated with the angle deviation of normal vector as the constraint by matching the structural features of the acoustic lining assembly, dividing and interpolating the hole-making surface, and solving the normal vector of the hole position. The experimental results show that the proposed hole matrix unit design method can effectively control the angle deviation of normal vector of the acoustic lining holes and ensure the processing accuracy of the holes.


Introduction
The invention and application of aviation aircraft are one of the greatest scientific and technological achievements in the twentieth century [1]. As the "heart" of the aircraft, the aeroengine is the critical propulsion system for providing power, shown in Fig. 1 [2]. The aero-engine with high thrust and good fuel efficiency plays a crucial role in the development of aircraft and the aviation industry [3]. Nowadays, increasingly stringent international regulations on noise pollution put forward higher performance requirements for noise reduction of aero-engines [4]. To reduce noise generated from engines during take-off, landing, and flight, a number of noise reduction approaches have been developed. Placing acoustic lining assembly on the internal wall of nacelle is a common noise attenuation method [5,6]. Acoustic lining assembly is typically fabricated by perforated panel, honeycomb, and back panel, as shown in Fig. 2, behaving like a quarter-wavelength resonator [7]. To achieve noise reduction, millions of high-density array holes need to be made on the carbon fiber perforated panel to form anechoic resonant cavities. On the acoustic lining assembly, the hole diameter is about 0.75-1.6 mm, and the hole spacing is only 1.5-3.0 mm [8,9]. The troubles of variable curvature structure of acoustic lining assembly, huge number of holes, and ultra-small hole diameter and spacing have created new challenges for classical hole-making technology.
For making multi-holes, various techniques such as mechanical drilling, laser beam, punching, ultrasonic machining, and electrical discharge machining (EDM) have been developed. Punching has high efficiency, stable dimensions, and unlimited hole patterns, but it's hard for punching to process non-plastic materials such as carbon fiber [10,11]. EDM supplemented by spray cooling can realize the processing of carbon fiber materials and effectively solves the dust problem in traditional machining [12,13]. However, the high temperature during EDM processing inevitably causes burrs, flanging, and delamination at the entrance and exit of carbon fiber holes [14]. Drilling is the most commonly used hole processing method [15].
Compared with other techniques, drilling has high processing accuracy and better quality in carbon fiber panel processing, which is regarded as the most suitable process technology for carbon fiber hole cutting [16]. When drilling carbon fiber holes with CNC equipment, the variable feed technique can effectively prevent the delamination around the drilled holes and to improve drilling efficiency at the same time [17]. The American company Electroimpact has developed a horizontal automated wing drilling equipment with five axes for the drilling of A380's wing [18,19]. Robot automatic hole-making system is also a common drilling equipment [20]. Electroimpact has teamed up with Boeing to design a robotic drilling system for drilling the holes of ailerons and fuselage surfaces on the Boeing 737 [21]. To further increase the speed of drilling, Creno have developed the spindle head with integrated multi-drills, which can process carbon fiber-grouped holes in batches [22]. Chengdu aircraft industry company proposed to use the batch hole-making tool integrated with multi-drills to process the holes in batches, making the efficiency double [23]. However, the batch hole-making tool introduced normal vector deviation of holes which affects the precision and the shape of holes. There are few reports on the method to control the normal vector deviation of batch hole-making tool. Therefore, the method to control the normal vector deviation becomes a technical problem that restricts the processing efficiency of acoustic lining holes.
To control the normal vector deviation, the structural characteristics of the perforated surface have to be matched to find a hole plane unit that approximate the surface and meet the requirements of the normal vector deviation. For the discrete and planar approximation of free-form surfaces, scholars study the types of planar patches and approximation algorithms. Brunet, P. [24] presented an algorithm for the piecewise linear approximation of trimmed surfaces. The algorithm generated a triangulation that approximates the initial surface within a predefined tolerance. Marc Vigo Anglada [25] introduced the concept of directed triangulation and further improved the estimation formula. Zhang [26] provided a new estimation formula for approximating curved surfaces with interpolated triangular plane slices, which reduced the number of subdivisions and improved the computational efficiency. However, these studies divided the parametric surface into triangular planes. If the hole plane unit of the batch hole-making tool is designed as a triangular plane, the number of holes to be machined simultaneously will be limited and make the programming complicated. In contrast, by dividing the parametric surface into rectangular planes and using rectangular plane units to drilling holes, multi-holes can be drilled simultaneously, and the programming complexity will be reduced. Therefore, to design the optimal hole matrix unit that satisfies the normal vector deviation has great significance to ensure the processing quality and improve the efficiency of the acoustic lining holes. Fig. 1 The aero-engine of the aircraft Fig. 2 The structure of the acoustic lining assembly [7] In order to improve the processing accuracy of acoustic lining holes, this paper studies the dimension of hole matrix unit to control the normal vector deviation. In view of the machining requirements of millions of acoustic lining holes, a hole matrix unit design method for batch hole making is proposed. The optimal hole matrix unit dimension is calculated with the angle deviation of normal vector as the constraint by matching the structural features of the acoustic lining assembly, dividing and interpolating the hole-making surface, and solving the normal vector of the hole position. The feasibility of the design method is verified by experiments.

Normal vector characteristics of batch hole making of acoustic lining assembly
In the perforated panel of the acoustic lining assembly, millions of holes need to be drilled to form anechoic resonant cavities for noise reduction. The perforated panels are mostly free-form surfaces with complex and variable curvature, large diameters, and thin walls, as shown in Fig. 3.
The structural characteristics of perforated panels are listed in Table 1.
The batch hole-making tool integrated multiple drills can batch process acoustic lining holes. The drills of the batch hole-making tool are arranged in a parallel array, and the drill tips are located on the same datum plane. In hole drilling, the tool feeds with the spindle to form an independent plane unit on the perforated panel, translating successively to drilling holes. Each drill feeds along the direction of the normal vector of the center drill to make multi-holes simultaneously. The normal vector of the made holes is parallel, and the drilling depth is the same.
Since the perforated panel of the acoustic lining assembly is characterized by free curvature in space, the positions of the holes are not in the same plane, and the normal vectors of the holes are not parallel. When using the batch holemaking tool to drill holes on the perforated panel, except for the center drill, there is a certain angle between the feeding direction of the drills and the direction of the normal vector of holes. This results in the angle deviation ( ) between the theoretical normal vector and the actual normal vector of the holes. The angle deviation will lead the actual hole spacing of acoustic lining holes to be different from the theoretical spacing. Figure 4 shows the angle deviation of the normal vector due to batch hole making.

Design of hole matrix unit
The hole matrix unit is an independent plane unit formed when the batch hole-making tool is fed, and its hole position distribution matches that of the acoustic lining holes. When drilling the acoustic lining holes, the batch hole-making tool feeds and the hole matrix unit are translated successively on the perforated panel to drilling holes. In order to control the angle deviation of the normal vector of holes, thus leading the acoustic lining holes to meet the processing requirements, this paper will match the structural characteristics of perforated surface to design the hole matrix unit. The design process includes the following steps: firstly, the geometric information of the perforated surface is converted into data information via dividing the surface into blocks and collecting discrete point data based on the digital model of the perforated surface. Secondly, the cumulative chord  length cubic spline function is used to interpolate the perforated surface, and the normal vector of the discrete points is solved. Finally, the dimension of hole matrix unit is calculated with the angle deviation of normal vector as the constraint. The design process is shown in Fig. 5.

Blocking of the perforated surface
The perforated surfaces of the acoustic lining assemblies in aero-engines are mostly free-form surfaces, and the curvature distribution is complex, leading to difficulty for fitting model. In order to ensure the accuracy of the model and to obtain comprehensive geometric information, the coordinate data of discrete points should be as much as possible. However, fitting with a large number of data directly is difficult and would breed infamous results. To reduce the amount of data, the perforated surface is divided into different blocks according to the curvature characteristics, and the coordinates of discrete points are collected for curve fitting. The curvature characteristics of the perforated surface of a typical acoustic lining assembly are shown in Fig. 6. Taking the curvature variation range as the basis for division, the direction with a larger curvature variation range is defined as the meridian direction ( u ), and the direction with a smaller curvature variation range is defined as the latitudinal direction ( v ). Then, the strips are divided equidistantly with the spacing of the holes in two directions, respectively, and the perforated surface is divided into discrete blocks, as shown in Fig. 7.

Coordinate collection of discrete points
The intersection point of the block's boundary curve is defined as a discrete point after the perforated surface dividing into blocks. Then the coordinates of the discrete points are extracted and stored through the data structure as s i, j = (x i,j , y i,j , z i,j ) , i = 0 ∼ m , and j = 0 ∼ n , where m is the number of discrete points in u-direction of the perforated surface and n is the number of discrete points in v-direction, as shown in Fig. 8. The coordinates of the discrete points are the data basis for subsequent curve fitting.

Interpolation of the perforated surface
Considering the accuracy of interpolation, three cubic spline functions x u (t) , y u (t) , and z u (t) are constructed in the meridian and latitudinal directions with the accumulative chord length as the parameter. Then the coordinates of the discrete points are used as the interpolation data to obtain the equation of interpolation curve.
On the perforated surface, a strip along the u direction has m × 1 data points. The chord length L i between two adjacent data points is The cubic spline function with chord length as parameter can be constructed as where t is the accumulative chord length of each node. Taking the interval of parameter t as 0 < t 1 < t 2 < t 3 < ⋯ < t n , t i can be obtained by For each node t i , the spline function P u (t) = x u (t), y u (t), z u (t) is constructed with x i,j , y i,j ,z i,j as the interpolation data, which is accumulative chord length cubic parameter spline. The construction process of the parametric spline is as follows (take x u (t) as an example): Fig. 4 The angle deviation of the normal vector (1) Interpolation node: the accumulative chord length t i is the interpolation node, which satisfies 0 < t 1 < t 2 < t 3 < ⋯ < t n . (2) Interpolation condition: (3) Continuity cCondition: tThere is functional continuity, first derivative continuity and second derivative continuity at node t i , i.e.,    The interpolation processes of y u (t) and z u (t) are same as that of x u (t) . Similarly, the interpolation process of the interpolation functions of the remaining strips along the u-direction and the v-direction is also the same as that of x u (t) . So the interpolation functions of the perforated surface are obtained: Thus, the interpolation functions of the perforated surface are obtained by fitting the coordinates of the discrete points, and then the normal vector of the discrete points will be solved according to the interpolation functions.

Solution of the normal vector of discrete points
The normal vector of discrete points is obtained by the method of differential geometry, as shown in Fig. 9. Via taking the partial derivative of the interpolation function P u (t) and substituting the coordinate of the discrete points, the tangential vector of the discrete points along u direction is obtained: The tangent vector along v direction of discrete points is obtained by the same method: The normal vector of the discrete points is obtained by cross-multiplying the tangent vectors in two directions: Through the above method, the normal vector of each discrete point can be solved, and then the normal vector is used to design the hole matrix unit.

Calculation of the hole matrix unit dimension
In order to meet the requirements of normal vector deviation of the acoustic lining holes, the dimension of the hole matrix unit is obtained by calculating the angle of normal vector between the discrete points in turn and judging whether the requirements of normal vector deviation ̂ is met: (1) Definitions: the boundary of the perforated surface is defined as the boundary edge. The space plane formed by the initial point, the points in u direction, and the points in v direction is defined as the hole matrix unit. (2) Design steps Step 1: A discrete point from the boundary edge is selected as the initial point S i,j , and the normal vector of the initial point is calculated.
Step 2: Along u direction, the next adjacent point S i+1,j is selected, and the normal vector � ⃗ n u is calculated.
Step 3: Along v direction, the next adjacent point S i,j+1 is selected, and the normal vector � ⃗ n s is calculated.
Step 4: The points S i,j , S i+1,j , and S i,j+1 are connected to generate a triangle T. The vector � ⃗ u between S i,j and S i+1,j and the vector � ⃗ u between S i,j and S i+1,j are calculated. As shown in Fig. 10, the normal vector � ⃗ n s of the triangle T is obtained by cross product: calculated sequentially, taking the spacing of the acoustic lining hole as the step size. Then, the dimension of the process hole matrix unit is determined with the hole-making normal vector deviation as the constraint. The minimum process unit is determined as the dimension of the hole matrix unit by traversing the discretization points of the perforated surface in the whole domain. Finally, the optimal hole matrix unit matching the curvature characteristics of the perforated surface is formed. The calculation process is shown in Fig. 12.

Batch hole-making experiment of acoustic lining assembly
Taking the inner wall plate assembly as an example, the hole matrix unit is designed according to the requirements of the diameter and normal vector deviation of the acoustic lining holes. Then, according to the hole matrix unit, a tool of batch hole making is designed to process the acoustic lining holes, and the feasibility of the design method is verified by the experiment results. The inner wall plate is a typical acoustic lining assembly of the nacelle, and about 10,0000 acoustic lining holes need to be made in it for noise reduction. The processing of inner wall plate is highly related to our research topic, so inner wall plate is selected as the verification example. The assembly is an integral rotating thin-walled structure, composed of three layers of heterogeneous materials, carbon fiber skin, honeycomb, and carbon fiber skin, as shown in Fig. 13. The innermost carbon fiber panel is the perforated panel with complex curvature and irregular features. The maximum diameter of the perforated panel perpendicular to the airflow direction is 2 m, and the minimum diameter is 1.4 m. The acoustic lining holes to be drilled are shown in Table 2.  Step 5: The angle between the normal vector of the triangle T and the normal vectors of S i+1,j and S i,j+1 1 and 2 are calculated as follows: Step 6: If 1 meets the requirements of angle deviation ̂ , the design step turns to Step 2. If not, turn to Step 8.
Step 7: If 2 meets the requirements of angle deviation ̂ , the design step turns Step 3. If not, turn to Step 8.
Step 8: The distance between S i,j and S i+1,j is defined as the dimension of process hole matrix unit in u-direction M i , and the distance between S i,j and S i,j+1 is defined as the dimension of process hole matrix unit in v-direction N j .
Step 9: The initial point jumps to a point down from u -direction and turns to Step 2 until it returns to the most initial point.
Step 10: The initial point jumps to the downward point of v-direction and turns to Step 2 until it returns to the most initial point. The calculation of the dimension of the hole matrix unit is completed.
Step 11: The minimum value of M i is defined as the dimension of the hole matrix unit in u-direction M , and the minimum value of N j is defined as the dimension of the hole matrix unit in v-direction N . So the dimension of the hole matrix unit is M × N . Figure 11 shows the iterative calculation of hole matrix unit.
The angle between the unit cell of the process hole matrix and the normal vector of the discretization points is The curvature changes sharply in the direction perpendicular to the airflow and the curvature changes parallel to the airflow direction are relatively gentle, as shown in the curvature distribution of the inner wall panel assembly in Fig. 13. Therefore, the direction perpendicular to the airflow is defined as the warp direction ( v direction), and the direction parallel to the airflow direction is defined as the latitudinal direction ( u direction). According to the design process in Sect. 3.4, the dimension of the hole matrix unit that matches the curvature characteristics of the perforated panel is calculated as M = 57.6mm , N = 43mm with the constraint of whether the normal vector deviation requirement (± 5°) is satisfied. The hole matrix unit is shown in Fig. 14, where the dots are the positions of the acoustic lining holes.
According to the dimension of the hole matrix unit and the parameters of the acoustic lining holes, a batch holemaking tool that clamps multi-drills is designed. The structure of the tool and the arrangement of the drill are shown in Fig. 15. The batch hole-making tool is used to conduct the acoustic lining hole-making experiment of the inner wall panel assembly, and the diameter and spacing of the holes are measured. GTF3010-9000 is used in this experiment, and the twist drill used in this experiment has a diameter of 1.2 mm. The processing method is single-feed drilling with batch hole-making tool. The 9 drill bits of the tool are drilling simultaneously during processing. The spindle speed is 8000 r/min, and the feed rate is 150 mm/min.
In the experiment, a total of 101,340 acoustic lining holes of Ø1.2 mm were made on the carbon fiber perforated panel, as shown in Fig. 16. Taking 100 × 100 mm as a single area, the diameter and spacing of the acoustic lining holes were measured. The results are shown in Table 3. The diameter and spacing of the acoustic lining holes processed by the batch hole-making tool meet the tolerance requirements according to the measurement results. As the actual spacing of the holes is within the tolerance range, it is proved that the deviation of the angle deviation of normal vector of the holes meets the design requirements. The feasibility of batch hole-making tool is verified.

Conclusion
Acoustic lining assemblies are the crucial structures of noise reduction in aero-engines. The processing of millions of acoustic lining holes poses great challenges to the manufacturing of acoustic lining assembly. The classical hole-making technology has problems such as low efficiency and normal vector deviation. In view of the machining requirements of millions of acoustic lining holes, a hole matrix unit design method for batch hole making is proposed. The optimal hole matrix unit dimension is calculated with the angle deviation of normal vector as the constraint by matching the structural features of the acoustic lining assembly, dividing and interpolating the hole-making surface, and solving the normal vector of the hole position. A batch hole-making tool designed according to the dimension of the hole matrix unit is used for processing experimental. A total of 101,340 acoustic lining holes of Ø1.2 mm is made on the inner wall plate. Ten sets of sampling  Funding The work is supported by the National Natural Science Foundation of China (No.51975288).
Data availability Not applicable.
Code availability Not applicable.

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