Clamping force model application on the aircraft structural assembly

The aeronautic manufacturing industry has been seeking to enhance competitiveness and product quality by applying the Industry 4.0’s technologies. Particularly, on the roadmap of the digital twin era, a way to achieve a reduction in manufacturing time and thus production cost is to obtain prediction models of the main elementary assembly operations and functions within aircraft manufacturing process, such as the clamping force applied by the temporary fasteners on the aircraft’s structural parts. Besides being a mandatory operation, it affects multiple tasks along the product’s assembly lifecycle. This work focuses on the role of the clamping force in the assembly process, establishing its functional model by means of an experimental approach based upon resources used on a real shop floor of a major aircraft manufacturer. To evince the main requirements that the clamping force tools can achieve, this work employs the Taguchi Design method, design of experiments, and process capability analysis. The model resulted from the aforementioned methods and tools allows the assembly behavior prediction and thus the control of the manufacturing process, ultimately yielding a better geometry quality.


Introduction
The tendency in the manufacturing systems for mass customization production whilst providing competitive companies with a reduction in costs, improvements in quality, increased flexibility, and product variety composes the main driving challenge of the Industry 4.0 revolution [1,2]. The main goal is then to optimize the value chain by implementing a dynamic and autonomously controlled production system based on real-time information and highly connected cyberphysical systems (CPS). From the operational perspective, the CPS reduces set-up times, labor and material costs, and processing times, resulting in higher productivity [3].
The key concept and important enablers for the cyberphysical systems (CPS) are the digital twins, which refers to the integration between simulation models and real-time data to provide a comprehensive physical and functional description of a component, product, or system, including all information that could be useful in all the lifecycle phases [4]. The technological realization of this concept is founded on the understanding of the different processes and variables that build up a given manufacturing system; modeling of the parameters of manufacturing processes is one of the first steps towards achieving this concept.

Aircraft structural assembly
More specifically, in the context of aeronautic manufacturing companies, it has been proven to be valuable to consider the aircraft structural assembly, which is labor-intensive and encompasses many steps that can represent as much as 50% of the total delivered part cost [5]. Figure 1 summarizes the main activities that are usually performed along the referred working process [5][6][7], and [8].
Considering the sub processes and operations depicted in Fig. 1, it is worth mentioning that multiple clamps are frequently used to serve the purpose of work-holding the intermediate product part in a fixture [9]. Henceforth, parts are pre-assembled, typically manually with temporary fasteners. The product needs to be stabilized to perform a hole and, if gaps are allowed, the presence of debris at the hole interface does appear. The debris may cause cracks over time. To avoid this, the parts are deburred after being disassembled and drilled [10]. The process in which the parts are drilled and fastened without disassembly is called one-up assembly [10,11], and [12]. Sealants are applied to the inner and outer surfaces of aircraft [13] and, to prevent leakage in the pressurized parts of the aircraft, the faying sealant is used with the controlled clamping force applied by the temporary fasteners during fastening operation.
Even when a robotic end-effector is used (Fig. 2) as it can deliver a force up to 90 kg over the workpiece during the automated drilling [8], the use of temporary fasteners to perform clamping force is essential to keep parts together and still, to avoid untighten parts during the assembly process.
To join all the parts to be fastened definitively, it is mandatory to pinning them in a way that minimizes the gaps between each other and does not allow improper movement or deformation during drilling and riveting.
Distortion of the structure naturally occurs because the parts being joined are susceptible to compliant forces. The compliant forces are imposed by the assembly jig and the clamping forces that are applied temporarily to keep the assembly in place [15]. If the resulting geometry variations are larger than the specified tolerance, it yields dimension and position errors in the final product [16]. These errors are noticeable as gaps and interferences between parts that create residual stresses in the structure when they are joined. Whether a gap exceeds the design tolerance, a shim or similar filler is used to assure a tolerance fit between the parts and minimize the impact of unexpected stresses in the structure [17]. Design    https:// www. infom oney. com. br/ merca dos/a-embra er-conse guiu-fazer-a-virada-dizpresi dente-da-empre sa/ amp/ The summarized overview of the aircraft assembly process posed so far evinces that the clamping force is a rather relevant variable that influences the manufacturing quality and costs. Therefore, a clamping force model aligned with an inspection in the process casts an initiative towards a digital twin of the assembly process, which shall support the manufacturing of aircraft, helping to minimize and control the primary part errors during its assembly. This concept can also be thought as a strategy of finding the optimal combination of individual parts to improve the geometrical quality of the product, known as the selective assembly technique [9].

Scope of the paper
This paper describes the methodological approach used to develop the representative model of a clamping force function to support the aircraft assembly process. To understand its behavior, besides evaluating the influence of some environment variables on the clamping force function response, a robustness analysis is further performed and the process capability is verified to assure accuracy resolution. The work is founded on experimental results obtained using tests performed with real coupons and tools.
To clarify how the proposed model contributes to the improvement of the aircraft assembly process, a simplified virtual simulation with different clamp force applications on a theoretical wing specimen has been carried out. Some important limitations are: • the geometrical deviation of primary parts was not considered; • use of only one material; • environment conditions remained constant; • as a boundary condition for the simulation, the ends of the spars are fixed emulating the condition that the assembly is on the tooling.
To evince the clamp force model as the first step towards a digital twin system, a framework has been proposed. Besides Sect. 1, this paper is organized as follows. Section 2 reviews the main attributes that impact the assembly process geometrical quality, emphasizing the relevance of the clamping force, and provides the background and literature review that positions the contribution of this work related to similar ones on aircraft assembly. Virtual Space through digital twin reviews how the automotive industry is tackling the clamp force topic towards the digital twin system. Section 3 details materials and the Taguchi method, design of experiment, and process capability approach used to develop the clamping force model. Results of the practical application of the proposed model, virtual simulation of different clamp force applications on an academic wing, and a digital twin framework proposal that describes the relevance of the clamping force model are shown in Sect. 4. The conclusions of this paper are the contents of Sect. 5.

Related work
Due to the manufacturing and structural assembly complexity, the geometrical quality of the parts depends upon many different parameters. Figure 3 shows the identified factors that affect their quality [5]. To achieve an effective geometry assembly quality control, all these parameters should be considered in a possible simulation model, which must, therefore, be able to representatively predict [4]: 1) the geometrical deviations on individual parts, 2) the variation propagation in an assembly, and 3) the effect of clamping and joining.

Physical space
The environmental condition may influence the assembly quality, as the aeronautical structure contains different materials with diverse proprieties jointed to shape the workpiece. High-temperature variation propagates thermal expansion that is related to changes in the material and geometric properties of the product and the tooling devices [15]. The part deviations due to variation in material properties and variation in previous manufacturing processes are also important [18], though they are usually assumed small compared to the part's dimensions. Despite they can cause significant product variability, during the assembly process, the aforementioned deviations can be absorbed by applying adequate assembly techniques with correct parameters.
The clamping force must be used to join and hold the parts to provide accuracy and position repeatability during the drilling and riveting processes, minimizing gaps between them [19,20]. The long-term structural jointed parts can be directly affected by the clamping force. Too low or too high clamping force might result in drawbacks: in the former, the part might not be placed to its nominal position, and in the latter, may lead to unwanted steering due to high friction between the parts.
Experimental results on carbon/epoxy laminates indicated that increasing clamping force on a joint improved static and fatigue life [21]. Higher clamp loads also reduce interlaminate burr and maintain the location of components during assembly processes [22] and [23]. The clamping force helps to inhibit parts movements, thus, preventing damage that could arise from the fixation without rigidity [24]. If it is not adequately tightened before the joint is drilled, gaps between parts are created, and then chips and debris get into the gap between the parts. As a result, this contamination can cause cracking.
The formation of drilling exit burr and the influence of interlayer gap on interlayer burr formation was analyzed in [25]. The conclusion was that an effective way to inhibit interlayer burr is to control the interlayer gap and the most common method is the preloading pressing force. To reduce burrs, several factors were also evaluated in [10], and, to minimize or eliminate the gap at the joint interface, the clamping force was considered the most important factor when compared to factors related to the drilling parameters (drill geometry, rotation speed, cutting feed, type of lubrication) and properties of the material to be drilled. This is due to the need to have a well-established workpiece that pursues stability to receive drilling holes [10].
An application study to predict the dimensional influence on a structural assembly used several factors [26]. In this work, the assembly sequence (positioning, clamping force, and fastening) had the greatest influence on the source of dimensional variation compared to the other factors analyzed such as types of tooling, product material specification, and types of fasteners applied [26]. Another study investigated the cutting parameters, clamping conditions, and drill geometry to optimize the process to meet the target quality. The results revealed the possible reduction of burr occurrence on both the entry and exit sides of the sheet, requiring no additional deburring. The overall conclusion from the investigation is that a properly constructed clamping system can significantly reduce the exit burr [27].
There exists typically three ways to provide clamping force to the workpiece. The first is associated with the primary parts riveting process applying clamping force on both sides, such as C-Frame [28][29][30]; Gantry [31,32]; and C-Frame end-effector with a positioning robot [33]. A second way is with a counterpoint process, where clamping is performed with a manual device, such as solutions developed to drill and fasten [34] and [35].
The third, though herein as the most important of the three ways to perform the clamping force, is the temporary fasteners. Temporary fasteners (TF) are any fasteners used to produce a temporary joint [36]. They can prevent both the relative motion between parts and the gaps between them during drilling by applying controlled clamping force [7]. They are used to pre-assemble the parts by positioning them into the pre-holes. Then, the pre-assembled parts can be moved to an automated drilling operation and the temporary fasteners are used to reference this automated process [7,37]. It is worth mentioning that, if the TF applies a non-controlled high clamping force, it can deform the parts generating contact between them [7]; consequently, it will not have control over the stress by tensioned assembly [38].
Since the focus of this work is on the clamping force through temporary fasteners, Table 1 presents the main difference among commercially available temporary fasteners.
Cleco™ apply a non-controlled and low clamping force (approximately 7 kgf) by a spring-pressed grip, joining the parts [10]. Tack rivets are blind rivets that the pop rivet gun pulls the mandrel into the rivet body, causing them to expand and grip the parts to be joined; once gripped, the mandrel snaps, holding the rivet in place. There is a patented Cleco™ temporary fastener type that improves the holding of the parts by hole-filling three-prong temporary fastener [39]. Another patent [40] is related to the one-up assembly process which defines the use of take-rivets on 5 up to 50% of the product holes. This process does not describe the clamping force or the definition to minimize gaps between layers. There is also a patented system that applies temporary clamping to the adjacent hole to be drilled. This system applies a force of approximately 900 kgf to eliminate the clearance before drilling, but it does not mention measurement and control of the gapor the tensioning the structure during its assembly [36].
Given the existence of manufacturing defects and the accumulation of assembly errors, non-compliant assembly appears among parts. In the engineering application, the clamping force is often used to eliminate the gaps between layers, but the improper clamping force may result in unwanted structure failure. The optimal clamping force can eliminate assembly gaps under the premise of reducing assembly stress [41].
About this subject, a finite-element (FE) stress analysis of aircraft structural double-lap bolted joints was performed to obtain the clamping pressure distribution and to estimate the stiffness of the joined plates within the clamped region [42]. Another research sought to understand the position error of the parts as assembly response, through the understanding of how forces and deformations change with the sequence of application of the clamping force to be performed [41].
To summarize this review, Table 2 presents the correlation between the main assembly functions and the clamping force performed with temporary fasteners.

Virtual space through digital twin in the automotive industry
Spot-welded sheet metal assemblies are the dominant types of assemblies in the automotive industry. The sheet metal parts are fixed in an assembly fixture. Then, the spot welding is performed and the assembly is released from the fixture. At this stage, springback can occur in the assembly because of implied stresses during the clamping and welding procedures. This springback and the geometrical quality of the assembly can be predicted by compliant variation simulation of the assembly procedure. Moreover, in welded sheet metal assemblies, springback occurs after releasing the clamps [43]. A novel method of design and optimization of compliant assembly fixtures was proposed [43]. The method defines the optimal type and location of holes, slots, and clamping units in compliant assembly fixtures. The presented method is not limited to any type of geometry and can be applied to complex 3D assemblies. Positioning and supporting the assembled parts [6,7,19,20,[22][23][24], Accessing by single side of the assembly [19,22] Removable and reusable [19,22] Reference position to CNC machine [7,19,22,37] Locator adjustment is a technique that increases the geometrical quality of assembly products by applying minor adjustments to locators of assembly fixtures. The goal of locator adjustments is to reduce the geometrical variations and deviations of the final assemblies by applying some minor adjustments to these locators and support points. Virtual trimming toolbox in a variation simulation program to optimize the amount of required adjustment for each locator in a fixture was proposed by [44]. In this method, the inspection data from the parts produced during the pre-production phase form the input of the procedure. Using these data and the variation simulation tool, the final geometry of the assemblies can be predicted by simulating the assembly process. Then, by applying an optimization algorithm, the optimal adjustments can be calculated so that the final deviations are minimal.
Another procedure to calculate the required adjustments was developed [45] and is referred to as Virtual Measurement Data Analysis (VMDA). Their method is based upon the statistical data measurements on the previously produced assemblies and computer-aided tolerance tools. If the need for adjustments arises, they can predict the outcome of different locator adjustments using the simulation tool. A method for adjusting locators by measuring the locator forces and using them in the control system before joining the parts was presented [46] and [47].
To make individualized adjustments, the geometry of each part should be known. Some techniques have been recently developed to calculate the deformed shape of each part by taking some pictures of it [48]. Hereafter, the deformed shape of each part can be provided before assembly as real-time production data to generate a digital twin for each physical assembly before starting the assembly process [49]. Thus, the assembly can be simulated for the digital twin using variation simulations. Then, using an optimization algorithm, the adjustments for the digital twin are obtained and applied to the physical assembly.
In the method proposed by [50], the real-time interaction scans data of each mating part of the physical assembly and directs the exact amount of adjustment to the fixture for clamping the parts before assembly. Data fusion is also performed to use data from different pictures to generate the deformed shape of the part. Accordingly, the presented system has digital twin features [51] and [52].
This section presents the importance of the clamping forces on structural aircraft assembly focusing on parts fixturing with controlled force to guarantee product robust manufacturing process to meet structural and geometrical quality requirements in a physical space. Virtual space through digital twin reviewed how the automotive industry is facing the clamp force subject towards the digital twin system. Based upon this review, it is possible to establish a correlation framework on aerospace structural assembly that evidence the link of the clamp force model as the first step of a digital twin system.
It is relevant to mention that it was not found a clamping force model in the literature. Such a model could complement the assembly process behavior prediction developed in the structural models of aeronautical riveted joints [15] and [53]. Considering this aspect, this work presents a contribution in this direction, presenting an experimental approach to build up a model of the comping force exerted with temporary fastener tools along with the aircraft part assembly.
The next section details the materials and methods used throughout the work, with a special emphasis on the statistical tools that lead to the aimed process model.

Materials and methods
The first step towards a complete model of the assembly process is to create a representative model of a function that most influences the variable's responses. Understanding the clamping force behavior can be the improvement needed to advance the geometric quality control on aircraft structural assembly since the importance of this function over the whole process has been made clear in the previous section. Figure 4 shows the methods applied in each step of this work to create the clamping force model.
Temperature variation is a variable that can influence the assembly geometrical quality, but one could argue if it is a relevant noise factor in the clamping force application. To answer this question, a signal-to-noise ratio in the Taguchi design [54] has been used. Then, it is necessary to set the clamping force application parameters that contribute to its results; to do so, a full factorial design experiment [55] was performed and analyzed to obtain the model. In the end, a process capability analysis was carried out to verify the clamping force accuracy resolution.

Taguchi method
Taguchi's robust method is an approach from quality engineering that seeks to increase the robustness of products by reducing the effects of process noise, as well as the appropriate choice of control factors, from the perspective of measuring high quality, low variability, and consistency of functional performance. It is unlikely or very expensive to perform an experiment that replicates all conditions in the real environment. Thus, Taguchi recommends a representative part, in which factors at two or most three levels are taken into account [54].
Taguchi's proposal is strongly based on statistical experimental techniques and focuses on minimizing the deviation from the target caused by uncontrollable factors, known as noise factors [54].
The parameter development by the Taguchi methodology is composed of two variables that are represented in a matrix shown in Fig. 5. The CFA (control factor) columns represent the control factors and each row has a specific parameter. NFA (noise factor) columns are noise factors with lines delineating different combinations of noise factor levels for each row of control factor parameters.
For each m row of the CFA factors, the n rows of the NFA provides at least n observations about the response of interest. These observations are used to calculate statistical performance measurements (PM), whose analysis provides the means to estimate and minimize the effect of noise factors on the performance of the analyzed process. The CFA factors are based on orthogonal array configuration that enables a reduced variance for the experiment with optimum settings of control parameters. Orthogonal arrays provide a set of balanced and minimum experiments to the desired output, serve as objective functions for optimization, and assist in data analysis and prediction of optimum results [56]. When there is a target value to be reached for the response, Taguchi makes use of the signal-to-noise ratio (S/N), configured in three types (S/N), shown in Eqs.  To evaluate if the temperature variation is a relevant noise factor in the clamping force application, it was set a Taguchi signal-to-noise ratio experiment with 3 factors in 3 levels having an L9 orthogonal array control factors, with two temperature variation ranges applied as noise factors. Figure 5 shows the clamping force Taguchi matrices, where 3 factors in 3 levels were defined, having the L9 orthogonal array configuration to the controlled factors, and the two levels of temperature, on the noise factors, were induced.

DOE full factorial
Many experiments involve the analysis of the effects of two or more factors. In general, factorial designs are most efficient for this type of experiment. Factorial design means that in each complete trial or replicate of the experiment, all possible combinations of the levels of the factors are investigated. The effect of a factor is defined to be the change in response produced by a change in the level of the factor [55]. A factorial design is necessary when interactions may be present to avoid misleading conclusions. Finally, factorial designs allow the effects of a factor to be estimated at several levels of the other factors, yielding conclusions that are valid over a range of experimental conditions [55].
The observations in a factorial experiment can be described by a model. The three-way effects model used for the analysis of the variance is described with Eq. 4: where µ is the overall mean effect, τ i is the effect of the ith level of the row factor A, β j is the effect of the jth level of column factor B, γ k is the effect of the kth level of column factor C, (τβ) ij is the effect of the interaction between τ i and β j , (τγ) ik the effect of the interaction between τ i and γ k , (βγ) jk is the effect of the interaction between β j and γ k , (τβγ) ijk is the effect of the interaction between τ i , β j , and γ k , and ε ijkl is a random error component.
The design of this experiment is of three factors with three levels; it means that it is necessary to have at least two replicates (n ≥ 2) to determine a sum of squares due to the error to include all possible interactions in the model [55].

Process capability
Process capability refers to the uniformity of the process. The variability of critical-to-quality characteristics in the process is a measure of the uniformity of output. A process capability analysis usually measures functional parameters. When the analyst can directly observe the process and can control or monitor the data-collection activity, the study is a true process capability study [57].
The main contributions of data analysis from a process capability perspective are as follows: predicting how well the process will hold the tolerances, assisting product developers/designers in selecting or modifying a process, assisting in establishing an interval between sampling for process monitoring, specifying performance requirements for new equipment, and reducing the variability in a process [57].
The histogram, as demonstrated in Fig. 6, can assist to estimate process capability. At least 100 or more observations should be available for the histogram to be moderately stable so that a reasonably reliable estimate of process capability may be obtained [57].
A simple and quantitative way to express process capability is through the process capability ratio Cp as modeled by Eq. 5. It can be more accurately defined as a process capability ratio that takes into account the process centering, the Cpk, presented in Eq. 6. where USL and LSL are the upper and lower specification limits, respectively; σ is the process standard deviation; Cpu is the process capability ratio Cp for the upper limit and Cpl for the lower limit. The higher Cpk value, the better the data, so is the tolerance, as shown in Table 3 and Fig. 6.
An important assumption to process capability and the ratios Cp and Cpk is that their usual interpretation is based on a normal distribution of process output. If the distribution is non-normal, then as previously cautioned, the statements about expected process fallout attributed to a particular value of Cp and Cpk may be in error. The tool used in the experiment is shown in Fig. 8. It is possible to configure the torque (Nm) and the speed rotation (RPM) applied to screw the temporary fastener. These variables influence the clamping force that the temporary fastener submits on the parts. The temporary fasteners (TF) were identified as shown in Fig. 9.

The experiment design
From the context of TF spindle-based tools, about three main factors were found to affect the clamping force application process: • Temporary fastener identification (TF): represents the same mechanical fastener features, but physically different. • Clamping force speed rotation application (SPE): is the rotation speed set on the tool to apply it to the temporary fastener measured in RPM.  To verify the tool performance application, three identical temporary fasteners have been picked up, so if the variance of the temporary fastener (TF) on the experiment is high, it means that the clamping force process is not robust and the TF is a representative factor that influences on the model.
The experimental coupons were defined in one scenario with: • Both plate material: aluminum alloy 2024 T3 • Both plate thickness: 2 mm • Hole diameter: 4.8 mm Table 4 shows the values defined on the three levels applied for each factor evaluated in the experiments.
The temperature variation range (TEMP) was verified with two different levels to understand the influence on the clamping force; it was set as a noise factor, due to it an environmental noise during the assembly process. It was observed by the sensor thermocouple and the clamping force display on the notebook screen as the calibrated load cell was pressed on each of the 3 temporary fasteners selected. It was induced and was observed in two ranges as the noise factor (low-and high-temperature range):

Threshold definition
Although there is no maximum allowable clamping force value in the literature for structural assembly, there are some references that can help in the value definition to be admitted for this work. Reasonably, there should be common forces performed on the part during assembly processes without generating residual stress on the joint to be drilled and subsequently fastened. These forces are associated with: • Applied force by tack rivets; • Axial force applied by the cutting tool during the drilling operation; • Clamping force applied to the drilling and fastening of robots and machines; • Clamping force applied by solid rivets.
Tack rivets with a diameter of 1/8″ (3.2 mm), which are used in the aircraft assembly, support and fix the assembly to be drilled. They are also the reference to the machine that locates the parts to be drilled, providing rigidity and tightening the parts to be drilled. Tack rivet brands have catalog clamping forces, as shown in Table 5.
The cutting tool axial feed force, during automated drilling, can reach 90 kgf, being much higher than the approximately 18 kgf exerted manually [8]. The variation in clamping force in a typically drilling cycle is approximately 81 kgf [58]. The clamping force performed by the robot's end-effector must be-necessarily-greater than the axial force of the cutting tool during the drilling process, to prevent the robot from moving out of the position where the hole is being drilled.
A typical process used by automatic riveters for panel fastening is riveting from the first hole drilled, in which the solid rivet is squeezed. The clamping force exerted by the squeezed rivet driven into a piece of aluminum alloy 2024-T3, by a 3.2-mm diameter rivet, is about 80kgf, with a squeezing force up to 14 kN [59].
In this research, it has been considered the clamping force limit application up to 90 kgf to respect the forces exerted on the product.

Results
This section presents the clamping force model creation selected by factors that aid the control of the manufacturing process attending structural and geometrical quality.

Taguchi method analysis
Following the L9 factorial design represented in Fig. 5, Table 6 presents the clamping force response yielded in the Taguchi experiments. Each parameter set in Table 6 was run 4 times, two at lower temperature and two at higher. While TEMP 1 has measured results above mean value on run 3 and 4 and bellow mean value on run 1, TEMP 2 has measured results above mean value on run 1 and 6 and bellow mean value on runs 2, 4, and 7. It evinces and ratifies that the temperature levels have a low influence on the clamping force results ( Table 7).
The Taguchi method made possible to understand that the temperature factor does not affect the clamping force behavior and to evidence that the ANOVA was run using TEMP as a factor to be analyzed. Table 8 shows the analysis of variance and the contribution of each factor to the model created. The temp has a low interference on the results, due to the p-value higher than 0.05 and the low contribution on the model (lower than 1%) demonstrating that the temperature does not affect clamping force accuracy.
To understand if the experimental results are based on a normal distribution of process output, the first analysis is verifying the residuals and their normal distribution. The residual analysis and model adequacy checking can be verified in Fig. 10, where it can be seen that the residuals vs fitted values (left) that indicate the error homogeneity are satisfactory, due to the red line behavior and due to the noise amplitude bellows 10% of the source. Value vs quantile normal graph (right) indicates that the experiment has a normal distribution and it is reinforced by the high p-value (0.6472) of the Shapiro-Wilk normality test analysis. Table 7 uses a specific target value as the best, so Eq. 3 was applied to determine the performance statistic of each parameter on the S/N ratio; with each minimum and maximum, it is possible to rank the parameter that has a higher influence on the clamping force application, as shown in Table 9. Both methods, Taguchi S/N ratio and ANOVA, identified that the SPE is the higher contribution, followed by TOR and TF, which has insignificant contribution (less than 5%).

DOE full factorial model
After running a fractional experiment to determine the relevance of environment behavior variables, it has been identified that the temperature is not relevant to the clamping force application, so a full factorial experiment was projected with a maximum possible resolution to fit the clamping force model. Table 10 shows the factors temporary fastener identification (TF), speed (SPE), torque (TOR), its levels, and the clamping force measured by the load cell. The runs were randomly performed.
The boxplot graphics shown in Fig. 11 make it clear that the speed (SPE) rotation and the torque (TOR) have a relevant influence on the clamping force, due to the high variance of the results. The variation of the temporary fastener (TF ID) demonstrated a much lower variance in the clamping force. Table 11 shows the analysis of variance and the contribution of each factor to the model created. The TF ID has low interferences on the results, due to the p-value higher than 0.05 and the low contribution to the model (lower than 1%) demonstrating that the clamping force applied by the tool has an accuracy that is repeatable independently of the temporary fastener used.
The TF's low contribution to the model (0.83%) and its high p-value (0.054) indicate that a temporary fastener does not affect the model, so applying clamp with temporary fasteners 1, 2, or 3 is considered the same solution. So, it is not considered as a variable of the clamping force model. Therefore, the final model residual analysis and model adequacy checking can be verified in Fig. 12, where it can be seen that the residuals vs fitted values (left) indicate that the error homogeneity is satisfactory, due to the red line behavior and due to the noise amplitude that is below 10% of the source. Value vs quantile normal graph (right) indicates that the experiment has a normal distribution and it is reinforced by the high p-value (0.6488) of the Shapiro-Wilk normality test analysis.
To create the final linear model, using R Studio software, it was identified the algebraic model as Eq. 7: The highest clamping force value measured used to create the linear regression model was 1675 N, and the residual error found on the model was 95.27 N, so the noise amplitude is below 6% of the source. The multiple R-squared represents the multiple correlations between the response variable and the two predictor variables, so 93.2% of the variation in the clamping force can be explained by the SPE and TOR (Table 12). Hereafter, the p-value confirmed that the sample results are consistent with a null hypothesis that is true.
The clamping force applied is controlled by the speed rotation and the torque can be depicted by the contour plot in Fig. 13 or by a three-dimensional plot in Fig. 14.

Process capability analysis
Even though the clamping force model was determined, to assure accuracy resolution for each parameter, it is necessary to take into account the process capability. To this end, two parameters were set: • Parameter 1: speed of 275 RPM and torque of 0.5 Nm To infer the capability processes, each parameter was run 100 times. Parameter 1 represents the lowest clamping force application and parameter 2 is the medium clamping force that respects the 100 kgf threshold. Table 13 shows the process capability analysis results for each clamping force level. Figure 15 shows the normality curve to parameter level 1, which has the Shapiro-Wilk test p-value of 0.137 and Liliefors test p-value of 0.574 demonstrating the normal    distribution data fit. Figure 16 shows the normality curve for parameter level 2, which has the Shapiro-Wilk test p-value of 0.299 and Liliefors test p-value of 0.944 demonstrating the normal distribution data fit.
The Cpk above 0.84 represents that around 1% of the clamping force application is out of tolerance for these two parameters as depicted in Fig. 15 and Fig. 16.

Clamping force model discussions and analysis
The Taguchi signal-noise ratio experiment was used to identify if variations in environmental conditions cause problems with the process's robustness. Taguchi's approach noted that the temperature variable did not cause variability in the important system response variables. It was also validated    that the temperature has a minor influence on the clamp force application due to the 0.04% contribution achieved on its ANOVA result. So, the experimental temperature conditions would not need to be controlled more tightly. While it has been needed 36 runs to perform analysis on the Taguchi method, the DOE full factorial required 54 runs to establish a clamp force response model. In total, it was run 90 experiments to create the model knowing that the temperature noise is not relevant to it. If a DOE full factorial would be run considering the temperature as a factor, it would need 243 runs (3 levels, 4 factors, and 3 replicate) to create the model considering all possible interactions included in the model.
Hence, the Taguchi method could not be used to create a complete model interaction as it is a crossed array experimental design, but it was important to determine the temperature low effect on the model and to reduce the number of runs.
Another relevant result was both methods' trends, as Taguchi and the full factorial specify that the SPE is the most relevant factor followed by TOR. The full factorial method identified that the TF has a low effect on the clamping force application model, as its p-value was higher than 0.05. It was demonstrated that there is a variation in the application (0.83%), but it is not relevant as it is the speed and torque factors. This confirms that the temporary fastener installation behavior is a robust process with low variability in operational conditions.
The capability analysis was used to have a larger tradespace data, so the data amplitude (difference between maximum and minimum results) would be higher than the previous methods. This improves the error result of the model. Table 14 summarizes the results comparing the two levels fully analyzed by the three methods applied.

Clamping force simulation
To demonstrate how the obtained clamping force model affects the geometries of the structural assembly, a simplified FEA  1) the clamping force applied in all reference holes with the low value; 2) the clamping force applied in all reference holes with the high value; 3) the first reference hole is clamped with high clamping force and the force decreases linearly up to the last hole that is clamped with the low force level. Figure 17 shows the geometrical deviation on the theoretical wing specimen applied with different clamping forces. A higher clamping force level induces higher deviation (over 0.18 mm) compared to that of a low clamping force level (around 0.05 mm). The changing clamping force (linear decrement) in the process creates a geometric distortion concentrated on the interface of the middle rib with the trailing edge spar.

Future development-digital twin framework
To clarify the interactions between the concept of Industry 4.0 and digital twins, that is, in which way the clamping force function model could be thought as a relevant part of a digital twin system, a framework has been depicted in Fig. 18. The following assumptions and simplifications were established to conceive the first draft of the Aircraft Structural Assembly Digital Twin Framework, namely:   • the Aircraft Structural Assembly working flow (Fig. 2) was simplified in the process simulation as clamping and joining; • the Geometric Assembly Quality Factors described in Fig. 3 were declared as simulation boundary conditions; • the inspection process provides current real data to the optimization algorithm; • the virtual simulation provides the simulated data, so the optimization algorithm can provide the clamping force demanded to be applied; • the clamping force model translates the clamping force value into real parameters to be applied to the physical aircraft structural assembly.  Further steps: to measure the product deformations with controlled clamping force on a full-scale aircraft assembly part. Creating correlations with the process parameters to a representative model of an assembly process behavior prediction of aeronautical assembly parts aiming at the start of the digital twin era.

Conclusion
This paper evidenced the relevance of clamping force on the entire assembly process, and its importance to assure geometrical quality. The methodology used to get the representative model of a clamping force function supports the assembly process control. To understand its behavior, besides evaluating the temperature influence on the function response, a robustness analysis was performed creating the algebraic model of Eq. 7, which reaches an R-squared of 0.932, and the process capability was verified to assure accuracy resolution. The work was founded on experimental results obtained using tests performed with real coupons and tools.
The method to create a clamping force model pictured in Fig. 4 can be scaled up to other functions, such as tooling stiffness or joining forces. It depends only on the way the functions and the responses are monitored during the assembly process. Data availability All data generated or analyzed during this study are included in this published article.

Code availability
The code generated to analyze the data and create the model was attached on the submission.

Declarations
Ethics approval Not applicable.

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Competing interests
The authors declare no competing interests.