The optimal clamping force option for robotic drilling of stacked aluminum sheets based on shell theory

Abstract

In aircraft assembly, the influence of interlayer burr formation in drilling of stacked metal sheets is an important problem. This article aims to develop a deeply understanding of the relationship between the clamping force and interlayer burr formation. The minimum clamping force, which would just make the stacked sheets entirely contact each other, is considered as the optimal clamping force. Hence, the additional deformation would appear at the contact region when the clamping force keeps rising up. The optimal clamping force is obtained first through theoretical calculation. Then, numerical simulations and experimental verifications are conducted. The results demonstrate the effectiveness of the theoretical analysis based on shell theory.

Keywords

Optimal clamping force stacked aluminum sheets interlayer burr robotic drilling shell theory

Introduction

Mechanical connection is one of the important connection ways to fasten aircraft structure permanently. There are as many as 1,500,000–2,000,000 rivets and bolts in a modern large aircraft.¹ Traditional manual drilling is time-consuming and the hole quality is difficult to guarantee. Therefore, the robot drilling system has been widely applied in modern aircraft manufacturing industry.^2–4

In order to improve the assembly quality and efficiency, one-side drilling operation is applied to drill different stacked combinations instead of drilling each layer material separately.^5,6 The stacked structure makes the drilling process very difficult. One of the challenges is eliminating interlayer burrs.⁶ Burr removing needs to disassemble the connection and reassemble after deburring. These additional operations are typically difficult to automate and are mostly done manually.⁷ Gillespie⁸ indicates that deburring can account for as much as 30% of the total part cost. Hence, burr formation control is an important part during aircraft assembly.

Numerous investigations have been conducted concentrated on burr formation and minimization in drilling stacked metal materials. Many researches point out that workpiece parameters, tool parameters, and process parameters have significant influence on burr formation. Most of the parameters are summarized in Table 1.^4,9–17 Two-dimensional or three-dimensional finite element (FE) methods are introduced to study the mechanism of burr formation and evaluate the influence of the relevant parameters as well.^7,18,19

Table 1.

Parameters related to burr formation.

No.	Parameter type	No.	Parameter type
1	Clamping distance	9	Feed rate
2	Clamping type	10	Interlayer gap
3	Cooling type	11	Sheet geometry
4	Cutting time	12	Specific powder
5	Drill coating	13	Spindle speed
6	Drill geometry	14	Temperature
7	Drill material	15	Thrust force
8	Drill type	16	Tool wear

Although a lot of work has done to control burr initiation during drilling stage, few researches focus on the effect of the clamping force on eliminating interlayer burrs. The inhibit mechanism of the clamping force on interlayer burr is still not clear.²⁰ Hellstern²¹ conducted stacked sheets drilling test with specific clamping force. They concluded that the larger is the clamping force, the smaller is the burr. But they did not explain how to select the appropriate clamping force. At present, the value of clamping force is mainly dependent on empirical data. If the clamping force is too small, the effect of eliminating interlayer burrs would be undesired. On the other hand, if the clamping force is too large, local oversize deformation of the sheets may lead to quality problem.

This article aims to develop a deep understanding of the relationship between the clamping force and interlayer burr formation based on shell theory. The optimal clamping force is obtained after theoretical calculation, numerical simulation, and experimental verification.

Clamping force theoretical analysis

The stacked aluminum sheets drilling process can be generally divided into three stages. Stage 1, the drill bit works on the upper sheet, while the lower sheet undergoes interlayer force from upper sheet. Stage 2, the drill bit breaks through the upper sheet and reaches the lower sheet. The interlayer gap begins to rise gradually. Stage 3, the drill bit is engaged in the lower sheet. Then, the hole is completed. The burrs are mainly formed when a drill enters and exits the hole in stage 2,⁹ as illustrated in Figure 1.

Figure 1.

The sketch of stacked aluminum sheets’ drilling process.

During aircraft assembly, the sheets can be divided into numerous thin plates with four sides simply supported.⁶ When drilling, the drilling thrust force and one-side clamping force will act on the stacked sheets. The forces will lead to different bendings of each sheet which would result in interlayer gap formation and variation. As shown in Figure 1, the clamping foot is a metal ring with external diameter $R_{e}$ and inner diameter $R_{i}$ . P is the average clamping pressure distributed on the sheets. The total clamping force, F, is calculated in equation (1). The clamping force is controlled by the pressure device of the end effector.² Since the effect area is small, the clamping force can be approximately substituted by the equivalent force, $F_{p} = F$ , distributed linearly around the center of the hole.¹⁷ The equivalent distribution radius is expressed in equation (2). At the same time, the drilling thrust force is considered as point force because the drilling area is much smaller than the total sheets surface. The simplified model is displayed Figure 2

F = π P (R_{e}^{2} - R_{i}^{2})

(1)

R_{s} = \frac{(R_{e} + R_{i})}{2}

(2)

Figure 2.

The simplified model with four sides simply supported.

Due to the application of the fixture, the deformation of the thin sheets with four sides simply supported is small during the manufacturing process. The solution of the bending of metal sheet belongs to plate theory.²² L Jie⁶ points out that the only difference between the beam theory and the plate theory is dimension. Therefore, the beam theory can be used to describe the relationship of interlayer gap and acting forces. As a result of the simplification, the deformations of the sheets can be described in terms of the deformations of the beams, as shown in Figure 3.⁶

Figure 3.

Stacked sheets’ beams.

In Figure 3, the unit force, $F_{u}$ , is applied at the circle of radius $R_{u}$ . $R_{d}$ is the radius of the other circle. Since the sheets will deform under the effect of acting force, we assume that $ω (R_{u} \to R_{d})$ represents the deflection at the circle of radius $R_{d}$ caused by the unit force, $F_{u}$ , applied at the circle of radius $R_{u}$ . Clearly, the value of $ω (R_{u} \to R_{d})$ declines with the increment of $R_{d}$ .

In stage 2, the upper and lower sheets do not contact each other. As illustrated in Figure 3, for lower sheet, the drilling thrust force, $F_{d}$ , acts on the drilling center O. The deflection at the circle of radius r caused by the drilling force is as follows

ω_{1} (r) = F_{d} \cdot ω (O \to r)

(3)

where $r \in [0, b]$ , and b is the sheet boundary.

For upper sheet, the clamping force, $F_{p}$ , is applied on the circle of radius $R_{s}$ . The deflection at the circle of radius r caused by the clamping force is as follows

ω_{2} (r) = F_{p} \cdot ω (R_{s} \to r)

(4)

where $r \in [0, b]$ , and b is the sheet boundary.

Then, the interlayer gap, $Δ ω (r)$ , at the circle of radius r around the drilling center O is as follows

Δ ω (r) = F_{d} \cdot ω (O \to r) - F_{p} \cdot ω (R \to r) + μ

(5)

where $μ$ is the initial interlayer gap.

When the workpiece structure and the drilling parameters are decided, the interlayer gap, $Δ ω (r)$ , is a function of two variables $F (r, F_{p})$ . Assume that the interlayer gap is zero, $F (r, F_{p}) = 0$ , then

F_{p} (r) = \frac{μ + F_{d} \cdot ω (O \to r)}{ω (R \to r)}

(6)

where $F_{p} (r)$ is the clamping force applied at the circle of radius R. r is the parameter, $r \in [0, b]$ , and b is the sheet boundary. Obviously, $F_{p} (r)$ , is a monotone increasing function. When $r = 0$ , $F_{p} (r)$ gets the minimum value.

As shown in Figure 1, the deformation at drilling center is usually the largest. And, the interlayer gap at drilling center is the largest as well. When the interlayer gap at drilling center is eliminated, the upper and the lower sheets will entirely contact each other. The purpose of this article is to find out the minimum clamping force which can just eliminate the largest interlayer gap. When $F_{p} (r)$ achieves the proper value, the interlayer gap at the drilling center will disappear. Since $F_{p} (r)$ is a monotone increasing function, $F_{p} (r)$ can get the minimum value at drilling center $(r = 0)$ . Then, additional deformation may occur at the contact region with the continuously increment of the clamping force. Hence, in this article, the minimum clamping force is considered as the optimal clamping force.

The deformation of the thin sheet is small since the application of the fixture and support. According to shell theory,²³ for elastic thin sheet, the stress function $φ (x, y) \equiv 0$ . So, the basic differential equation of the thin sheet can be simplified as follows

\frac{\partial^{4} ω}{\partial x^{4}} + 2 \frac{\partial^{4} ω}{\partial x^{2} \partial y^{2}} + \frac{\partial^{4} ω}{\partial y^{4}} = \frac{q (x, y)}{D}

(7)

where $ω$ is the deflection at any point, $q (x, y)$ is the load applied at the sheet surface, D is the flexural rigidity of the sheet, $D = E h^{3} / 12 (1 - v^{2})$ , E is the elasticity modulus, v is Poisson’s ratio, and h is the sheet thickness.

For the rectangular sheet with four sides simply supported, the boundary conditions are as follows

{\begin{matrix} x = 0, x = a \to ω = 0, \frac{\partial^{2} ω}{\partial x^{2}} = 0 \\ y = 0, y = b \to ω = 0, \frac{\partial^{2} ω}{\partial y^{2}} = 0 \end{matrix}

(8)

where the parameter $ω$ is expressed in the form of double trigonometric series

ω (x, y) = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} A_{mn} \sin \frac{m π x}{a} \sin \frac{n π y}{b}

(9)

where $A_{mn}$ is the undetermined coefficient. The first 20 items of the double trigonometric series are used to calculate the approximate solution due to the low convergent rate of the series.

Obviously, the equation can satisfy the boundary conditions.²³ Substitute the formula into equation (7), then

A_{mn} = \frac{4}{π^{4} abD} \cdot \frac{\int_{0}^{a} \int_{0}^{b} q (x, y) \sin \frac{m π x}{a} \sin \frac{n π y}{b} dxdy}{{(\frac{m^{2}}{a} + \frac{n^{2}}{b^{2}})}^{2}}

(10)

So, the general solution of the flexural function of the four-sides simply supported sheet can be expressed as follows

\begin{array}{l} ω = \frac{4}{π^{4} a b D} \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} \frac{0 \int_{0}^{a} \int_{0}^{b} q (x, y) \sin \frac{m π x}{a} \sin \frac{n π y}{b} d x d y}{{(\frac{m^{2}}{a^{2}} + \frac{n^{2}}{b^{2}})}^{2}} \\ \sin \frac{m π x}{a} \sin \frac{n π y}{b} \end{array}

(11)

The solution of equation (11) is also known as the Navier solution.²⁴ In the X–Y plane, as shown in Figure 2, assume that $P_{1} (x_{1}, y_{1})$ represents the position of the drilling thrust force applied at the lower sheet. When the drill cutting bit acts on the lower sheet, the deflection at any point, $P (x, y)$ , on the lower sheet is as follows

ω_{1} = \frac{4 F_{d}}{π^{4} abD} \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} \frac{\sin \frac{m π x_{1}}{a} \sin \frac{n π y_{1}}{b}}{{(\frac{m^{2}}{a^{2}} + \frac{n^{2}}{b^{2}})}^{2}} \sin \frac{m π x}{a} \sin \frac{n π y}{b}

(12)

Assume that $P_{2} (x_{2}, y_{2})$ represents the coordinate of the clamping force acted at the upper sheet. The positional relation between the clamping force and the drilling thrust force can be expressed as $(x_{2}, y_{2}) = (x_{1} + R_{s} \cdot \cos θ, y_{1} + R_{s} \cdot \sin θ)$ , $θ \in [0, 2 π]$ , so when the clamping force is applied on the upper sheet, the deflection at any point, $P (x, y)$ , on the upper sheet is as follows

\begin{array}{l} ω_{2} = \frac{4 F_{p}}{π^{4} a b D} \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} \\ \frac{\int_{0}^{2 π} R_{s} \sin \frac{m π (x_{1} + R_{s} \cdot \cos θ)}{a} \sin \frac{n π (y_{1} + R_{s} \cdot \sin θ)}{b} d θ}{{(\frac{m^{2}}{a^{2}} + \frac{n^{2}}{b^{2}})}^{2}} \sin \frac{m π x}{a} \sin \frac{n π y}{b} \end{array}

(13)

Thus, the interlayer gap at any points, before the two sheets completely contact each other, is as follows

Δ ω = ω_{1} - ω_{2} + u

(14)

As can be seen from equations (12) and (13), the deflections of the upper and lower sheets at the same radius are different. But for upper or lower sheet, the deflections at the same circle are approximately equal. So, in this article, the average deflection of eight points located homogeneously at a circle of radius r is considered as the circle deflection.

Hence, when the drill thrust force acts at the lower sheet, the deflection at the circle of radius r is as follows

\begin{matrix} ω_{1} (r) = \frac{1}{8} \sum_{k = 0}^{7} \frac{4 F_{d}}{π^{4} abD} \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} \frac{\sin \frac{m π x_{1}}{a} \sin \frac{n π y_{1}}{b}}{{(\frac{m^{2}}{a^{2}} + \frac{n^{2}}{b^{2}})}^{2}} \\ \sin \frac{m π x_{1} + r \sin (k π / 4)}{a} \sin \frac{n π y + r \cos (k π / 4)}{b} \end{matrix}

(15)

where k represents the eight points located homogeneously at the circle.

When the clamping force is applied on the upper sheet, the deflection at the circle of radius r is as follows

{\begin{matrix} ω_{2} (r) = \frac{1}{8} \sum_{k = 1}^{7} \frac{4 F_{p}}{π^{4} abD} \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} \frac{\int_{0}^{2 π} R_{s} \sin \frac{m π (x_{1} + R_{s} \cos θ)}{a} \sin \frac{n π (y_{1} + R_{s} \sin θ)}{b} d θ}{{(\frac{m^{2}}{a^{2}} + \frac{n^{2}}{b^{2}})}^{2}} F (x, y) \\ F (x, y) = \sin \frac{m π x + r \sin (k π)}{a} \sin \frac{n π y + r \cos (k π)}{b} \end{matrix}

(16)

where k represents the eight points located homogeneously at the circle.

Then, by combining equations (15) and (16), the deflections caused by unit forces are as follows

{\begin{matrix} ω (O \to r) = ω_{1} (r) / F_{d} \\ ω (R \to r) = ω_{2} (r) / F_{p} \end{matrix}

(17)

In the investigation, the thickness of the sheet is $h = 2 mm$ , the length of the sheet is $a = 200 mm$ , and the width of the sheet is $b = 100 mm$ . The initial interlayer gap is $u = 0.4 mm$ .²⁵ The external diameter and the inner diameter of the clamping foot are $R_{e} = 22.5 mm$ and $R_{i} = 17.5 mm$ . So, the equivalent radius of the equivalent clamping force, $F_{p}$ , distribution linearly around the center of the hole are $R_{s} = (22.5 + 17.5) / 2 = 20 mm$ . The properties of the materials are displayed in Table 2. According to actual manufacturing condition, the drilling thrust force is $F_{d} = 130 N$ .²⁵ For $ω (O \to r)$ and $ω (R \to r)$ , the approximate calculation results are exhibited in Table 3.

Table 2.

Material properties of different components.

Component	Material	Elasticity modulus (GPa)	Poisson’s ratio
Sheet	2024-T3 aluminum alloy	72	0.34
Drill	Cemented carbide	–	0.3
Clamp foot	Stainless steel	200	0.3

Table 3.

The deflections of different radii under unit force (unit: 10⁻⁶ N/m).

R (mm)	$ω (O \to r)$	$ω (R \to r)$
0	2.991	2.425
2	2.984	2.422
4	2.972	2.417
6	2.927	2.399
8	2.882	2.379
10	2.828	2.354
12	2.769	2.323
14	2.706	2.289

The deflection data presented in Table 3 further prove the deformation, and the interlayer gap at drilling center is the largest. Substitute these values into equation (6), and the curve of the clamping force, $F_{p} (r)$ , is shown in Figure 4. The clamping force increases with the increment of the radius r. The calculation is consistent with the analysis above. The clamping force, $F_{p} (0) \approx 325.3 N$ , is considered as the optimal clamping force. The upper and lower sheets will entirely contact each other. When the clamping force continuously grows up, additional deformation would occur at the contact region.

Figure 4.

The clamping force at different radii.

Stacked drilling simulation

FE modeling

When the drill reaches the sheets’ surface with the one-side clamping force, the deflection of the sheets increases gradually and the interlayer gap varies simultaneously. In order to verify theoretical analysis, a three-dimensional FE model is established using ABAQUS 6.13. The model is built based on the research of W Tian et al.²⁰

The model consists of two aluminum alloy sheets and one clamping foot. The material properties are displayed in Table 2. The dimensions of the sheets are 200 × 100 × 2 mm. The clamping foot is a metal ring with external diameter of 22.5 mm and inner diameter of 17.5 mm. The sheets are built as deformable bodies, while the clamping foot is defined as rigid body. The components are meshed by C3D8R. To achieve the desired analytical accuracy with computational efficiency, the mesh density in the drilling and clamping regions is denser than other areas.²⁶ The mesh condition is shown in Figure 5.

Figure 5.

FE model mesh condition.

Multiple analytical steps with their specific conditions are sequentially introduced to the model. In initial step, the boundary conditions are set. The four sides of the sheets are constrained in Z-direction to make sure that the sheets are in the status of simply supporting. All freedoms of the clamping foot are restrained, except Z-direction. The initial interlayer gap is 0.4 mm. Contact properties are defined in step 1. The FE model includes contacts between clamping foot and upper sheet, and at the interface between the upper and lower sheets. A friction coefficient of 0.2 is specified for all of the contact surfaces.²⁰ The forces are loaded in step 2. The clamping force is applied on the clamping foot. The drilling force is loaded on the lower sheet through the hole of the upper sheet. Then, the deflections of the two sheets are calculated. The deflection of the each sheet is obtained from the average value of the eight points that locate homogeneously at the upper sheet exit or the lower sheet entrance. The interlayer gap is expressed as follows

Δ ω = \frac{1}{8} \sum_{i = 1}^{8} (ω_{1 i} - ω_{2 i} + u_{i})

(18)

where $ω_{1 i}$ is the point deflection at lower sheet, $ω_{2 i}$ is the point deflection at upper sheet, and $u_{i}$ is the initial interlayer gap.

Simulation results

During the simulation, the drilling thrust force is set as 130 N. The initial clamping force is 100 N. Then, the clamping force increases to 500 N gradually. The interlayer gap, $Δ ω$ , under different clamping forces is calculated and illustrated in Figure 6. When the clamping force rises from 0 to 325 N, the total interlayer gap between the sheets declines linearly. The interlayer gap becomes zero when the clamping force achieves 335 N. The interlayer gap between the sheets has been eliminated. At this point, the upper and lower sheets contact each other completely.

Figure 6.

Total interlayer gap under different clamping forces.

The Von Mises stress is used to evaluate the stress status.²⁷ When the clamping force reaches 335 N, the stress status and displacement situations are illustrated in Figures 7 and 8, respectively. Obviously, the stress and strain are distributed symmetrically around the drilling center. Compared with other cases under different clamping forces, the stress and strain conditions are the best. Therefore, the optimal clamping force is 335 N. The difference between the theoretical optimal clamping force and the simulation result is approximately 2.9%. The results agree with the theoretical analysis.

Figure 7.

Stress status of the stacked sheets.

Figure 8.

Displacement status of the stacked sheets.

Experiments

Experimental details

The 2024-T3 aluminum alloy sheet of 200 × 100 × 2 mm is chosen as the specimen. The cemented carbide drill used is of cylindrical shank, 5.1 mm diameter, 120° point angle, and 15° helix angle. The KISTLER9257B dynamometer is utilized to measure the drilling thrust force and the clamping force. The Zeus Axio CSM 700 confocal microscope is applied to measure the burr conditions at the upper sheet exit and at the lower sheet entrance. The drilling test is conducted using the robotic automatic drilling system developed by Zhejiang University, as shown in Figure 9.²⁸ The system consists of KUKA KR360-2 industrial robot, robot mobile platform, drilling end effector, testing bench, and stacked aluminum sheet specimen.

Figure 9.

Automatic robotic drilling system.

The drilling process is illustrated in Figure 10. The specimens are fixed on the fixture. The spindle speed is 6000 r/min. The feed rate is 200 mm/min.²⁸ The clamping force (loaded through clamping foot) is controlled by the air cylinder installed on the end effector. The clamping force increases from 100 to 500 N step by step. Because of the control accuracy of the air cylinder, the increment of each step is 50 N. At each force level, three holes are drilled.²⁰

Figure 10.

Local view of the drilling process.

Experiment results

The burr can be described by its burr cross section, as shown in Figure 11.²⁹ The burr heights measured under different clamping forces are exhibited in Tables 4 and 5 separately. The relationships of the average burr height and the clamping force are presented in Figure 12. Clearly, the burr heights drop significantly with the increment of the clamping force. When the clamping force is between 300 and 350 N, the slopes of both the curves turn small. Then, the fluctuations of both the curves are quite small when the clamping force is larger than 350 N. The burr heights tend to be stable at a value around 0.04 mm. It proves that the stacked sheets begin to contact to each other completely when the clamping force is larger than 300 N.

Figure 11.

Cross section and measured parameters of a burr.²⁹

Table 4.

The burr height at the upper sheet exit.

Clamping force (N)	Burr height at the upper sheet exit/mm
Clamping force (N)	Hole 1	Hole 2	Hole 3	Average
100	0.212	0.192	0.184	0.196
150	0.144	0.162	0.165	0.157
200	0.127	0.12	0.131	0.126
250	0.109	0.113	0.112	0.111
300	0.056	0.056	0.068	0.06
350	0.042	0.043	0.051	0.045
400	0.045	0.039	0.045	0.043
450	0.044	0.044	0.05	0.046
500	0.041	0.038	0.03	0.036

Table 5.

The burr height at the lower sheet entrance.

Clamping force (N)	Burr height at the lower sheet entrance/mm
Clamping force (N)	Hole 1	Hole 2	Hole 3	Average
100	0.122	0.133	0.123	0.126
150	0.094	0.097	0.094	0.095
200	0.066	0.063	0.071	0.067
250	0.064	0.063	0.062	0.064
300	0.056	0.06	0.064	0.06
350	0.042	0.043	0.043	0.043
400	0.041	0.039	0.035	0.038
450	0.038	0.034	0.04	0.037
500	0.041	0.038	0.04	0.04

Figure 12.

The curves of burr height and the clamping force.

The burrs at upper sheet exit and lower sheet entrance under 50-fold microscope are presented in Figures 13 and 14, respectively. As can be seen, the burrs at the upper and lower sheets are obvious when the clamping force is relatively small. When the clamping force achieves 300 N, the burrs are not so apparent. The burrs disappear when the clamping force continually rise to 350 N. The hole surfaces and hole edges of both sheets are smooth. The phenomenon shows that the optimal clamping force is between 300 and 350 N. The conclusion is identical to the theoretical calculation and FE simulations.

Figure 13.

Burrs at upper sheet exit.

Figure 14.

Burrs at lower sheet entrance.

Conclusion

In this study, a simple and clear relationship between the clamping force and the interlayer burr formation is presented. The smallest clamping force is considered as the optimal clamping force. Hence, the interlayer gap would be eliminated under the optimal clamping force. The shell theory is introduced to calculate the optimal clamping force. Numerical simulations and experimental verifications are conducted as well. The results further demonstrate the validity of the theoretical analysis.

Footnotes

Academic Editor: Xichun Luo

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The work described in this paper has been supported by grants from the Science Fund for Creative Research Groups of National Natural Science Foundation of China (no. 51521064) and special scientific research for civil aircraft (no. MJ-2015-G-081).

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