2.1. Modeling of the Interlayer Gap with Bidirectional Clamping Forces
In order to eliminate the interlayer burr by controlling the interlayer gap, the modeling of the interlayer gap in stack drilling is performed. As shown in
Figure 2a, the drilling process of stacked aluminum plates can be decomposed into three stages. In stage 1, the upper plate is drilled and undergoes the drilling thrust force while the lower plate bends downward due to the pressure transmitted from the upper plate. In this stage, the initial interlayer gap is eliminated. In stage 2, the upper plate is drilled through and the drill bit reaches the lower plate, then spring-back occurs on the upper plate while the deflection of the lower plate increases due to the drilling thrust force. The interlayer gap begins to form and increase in stage 2. In stage 3, the drill bit is fully contacted with the lower plate. The drilling thrust force is exclusively applied on the lower plate while the upper plate endures a negligible force. The different force conditions of the upper and lower plates result in the maximum interlayer gap during the whole drilling process. The interlayer burr situation in stage 3 is shown in
Figure 2b. The interlayer gap includes the upper plate exit burr and lower plate entrance burr, and both are strictly related to the interlayer gap formation, so diminishing the interlayer gap could be an efficient way to eliminate the interlayer burr.
In the automatic drilling and riveting system, as shown in
Figure 3a, the bidirectional presser feet are utilized to apply clamping forces on both sides of the drilling area. The schematic of the stack drilling with bidirectional clamping forces is demonstrated in
Figure 3b, where:
represents the drilling thrust force;
and
represent the pressures applied on the clamping areas of the upper side and lower side, respectively;
,
,
, and
represent the inner radius and outer radius of the upper presser foot and lower presser foot, respectively; a, b and h are the length, width and height of the plate, respectively; and
is the initial interlayer gap.
According to the theory of plates and shells [
28], a simplified model of the stack drilling with bidirectional clamping forces is established and displayed in
Figure 4. The model mainly consists of two stacked rectangular plates with drilling thrust force and bidirectional axially symmetric clamping forces distribution. The thrust force is considered as a concentrated force at the drilling center, while the bidirectional clamping forces in the ring-shaped area are reasonably simplified into circular equivalent forces [
29]. The
substitutes the upper clamping force, which uniformly distributed at a circle with radius
around the drilling center. Similarly, the
substitutes the lower clamping force applied at a circle with radius
. The calculation formulas of the equivalent clamping forces are as follows:
Furthermore, some critical hypotheses are given to ensure the validity of the mathematical model [
18]:
The stacked plates are both thin plates, with the length and thickness ratio larger than 0.5;
Only elastic deformation and small deflection occur to the plates, the maximum deflection does not exceed 1/5 of plate thickness.
The edges of the plates are considered as built-in or fixed.
During the drilling process, the interlayer gap has a positive correlation with the interlayer burr size, and the largest interlayer gap occurs at the drilling center [
12]. It is obvious that the interlayer gap decreases as the clamping force increases until the two plates are entirely in contact with each other. Therefore, when the interlayer gap at the drilling center is reduced to zero, the clamping forces can be considered to be optimal. With the simplified stack drilling model, we can calculate the total deflection of each plate by superposing the deflection under each load. So, the relationship between the interlayer gap at the drilling center and the bidirectional clamping forces can be established. Then the optimal clamping forces can be obtained when the interlayer gap equals to zero.
To achieve the above objective, firstly, we need to calculate the deflection of each plate under each load. We assume that
represents the deflection at the circle of radius
(
) caused by a unit force applied at position
. It should be noticed that
could be a point or a uniformly distributed circle. For the upper plate, it only bears the upper clamping force
, so the deflection at the circle of radius
can be represented as Equation (3):
The lower plate is subject to both drilling thrust force and lower clamping force. The deflection of the lower plate caused by the thrust force
is determined as Equation (4):
Similarly, the deflection of the lower plate caused by the lower clamping force
can be represented as Equation (5):
Then, the interlayer gap
at the circle of radius
around the drilling center
can be calculated with Equation (6):
where
is the initial interlayer gap.
After the drilling process parameters and workpiece parameters are determined, the interlayer gap will become a function of three variables
, can be expressed as zero, which means
when
. Then, the Equation (6) would only have two variables
and
. In other words, the relationship between the upper clamping force and lower clamping force when the interlayer gap is zero can be obtained. However, on the premise that the interlayer gap is zero, we still need to find the optimal group of the bidirectional clamping forces. Because different groups of the bidirectional clamping forces will give rise to different deflections of the stacked plates, which will result in the non-coaxiality of the holes [
23]. Therefore, the optimization objective of the bidirectional clamping forces is proposed, which is to reduce the degree and non-uniformity of the deflections of the stacked plates.
According to Equation (6), when the interlayer gap equals to zero at the drilling center, the deflections of the upper plate
and lower plate
can be expressed as Equations (7) and (8), respectively:
If one-sided clamping is adopted, which means , then would be zero. As can be seen from Equation (7) and Equation (8), the deflections of the upper and lower plates and will both increase. The main reason is that when one-sided clamping is used, the downward bending of the lower plate is increased due to the lack of lower clamping force. As a result, the upper plate has to bend more to ensure that the interlayer gap remains zero. The increase of the deflections of the stacked plates will affect the coaxiality of the hole and ultimately reduce the fatigue performance of the joint. Therefore, compared with one-sided clamping, the bidirectional clamping way is more capable because it can reduce not only the interlayer burr but also the deformation of the stacked plates.
Then, the specific calculation method of the deflection of the thin plate is presented. According to the theory of plates and shells [
28], the differential equation of the thin plate bending is shown in Equation (9):
where
is the deflection of the plate,
is the force applied on the plate,
is a point in the plane coordinate,
is elastic modulus,
is the plate thickness and
is Poisson’s ratio.
In the case of the rectangular plates are constrained on four edges, the boundary conditions can be expressed as Equation (10):
where the parameter
can be expressed in the form of double trigonometric series, the detailed calculation process can be referred to our previous job [
27].
Then, the Naiver solution of the deflection of the rectangular thin plate can be represented as Equation (11).
Define
as the position of the drilling thrust force applied at the lower plate, and
,
are the coordinates of the upper and lower clamping forces applied at the plates, respectively. Their positional relationship can be represented as Equations (12) and (13):
With Equations (11)–(13), we can calculate the deflection at any point
of each plate under each load in a uniformed coordinate system. Eventually, the deflection caused by unit force at drilling center and clamping areas can be obtained as follows:
2.2. Optimization of Two-Side Clamping Force
Based on the above mathematical model of the interlayer gap, the optimization of the bidirectional clamping forces for the automatic drilling and riveting system can be performed. Firstly, we need to acquire the relationship formula between the upper and lower clamping forces when the interlayer gap is zero.
The 2024-T3 aluminum alloy is widely used in airplane structures such as fuselage and wings because of its high strength to weight ratio and excellent fatigue properties [
21]. In this investigation, the 2024-T3 aluminum alloy plates are used for stack drilling. The materials of the presser foot and the drill are stainless steel and cemented carbide, respectively. Their material properties are shown in
Table 1.
The initial interlayer gap
is set to 0.35 mm and the drilling thrust force
is set to 130 N according to the actual manufacturing condition [
24]. The geometric dimensions of the plates and the presser feet of the automatic drilling and riveting system are shown in
Table 2.
Substituting the parameters in
Table 1 and
Table 2 into Equation (14), the calculation results of the deflections caused by the unit force applied at the drilling center and clamping areas can be obtained. Due to the low convergence rate of the double trigonometric series, when calculating the Naiver solution, the first 20 items of the series are taken as an approximate solution. The calculation results are shown in
Table 3.
When the interlayer gap is zero (
), substitute the above calculation results into Equation (6), the relationship between the bidirectional clamping forces can be obtained as follows:
where
and
.
Theoretically, when the bidirectional clamping forces satisfy the Equation (15), the interlayer gap would be zero, and the interlayer burr could be eliminated. However, there is an ocean of groups of the bidirectional clamping forces that meet Equation (15), and if not correctly selected, it may lead to excessive and non-uniform deformations of the stacked plates. Therefore, it is necessary to find the optimal group of the bidirectional clamping forces to reduce the interlayer burr and total deflections of the stacked plates.
In order to quantitatively evaluate the uniformity of the deflections of the two stacked plates, an evaluation index T is defined in Equation (16), it represents the average deviation of each plate from the position where the total deformation of the stacked plates is the minimum. The smaller the deflection index T is, the smaller the total deformation of the stacked plates.
According to Equation (7) and Equation (8), on the premise that the interlayer gap is zero, the deflections of the upper and lower plate with different bidirectional clamping forces can be obtained, with the results represented in
Figure 5. The X-axis indicates that the lower clamping force
increases from zero while the corresponding upper clamping force
can be calculated from Equation (15).
With the increase of , the deflections of the stacked plates can be divided into four stages listed below:
Stage1: When the lower clamping force , the one-sided clamping force is adopted. The upper clamping force is the largest among all groups of bidirectional clamping forces at this stage. The deflection of the lower plate is induced by the drilling thrust force , while the upper plate deforms downward due to the upper clamping force until the interlayer gap is eliminated. At this stage, the deflection of the upper plate is more significant than the lower plate , and their difference is equal to the initial gap .
Stage2: As increases, the deflections of both plates decrease until the effect of counteracts the effect of the drilling force on the lower plate. At the end of stage 2, the deflection of the upper plate is equal to the initial gap while the deflection of the lower plate is zero.
Stage3: When is greater than the drilling thrust force , the lower plate begins to deform upward, and the downward deformation of the upper plate decreases accordingly. In this stage, the sum of the deflections of the two stacked plates is always equal to the initial gap . When the deflections of the upper and lower plates are the same (), the non-uniformity of the deformation is the smallest. The bidirectional clamping forces at this time can be considered as optimal.
Stage4: As continues to increase, finally reaches zero. At this moment, the deflection of the upper plate is zero, the lower clamping force is the maximum, and the deflection of the lower plate is equal to the initial gap.
According to the above analysis, when the deflections of the upper and lower plates are equal to half of the initial gap (), the optimal bidirectional clamping forces can be obtained, the calculation results are and .