Reliability improvement on assembly accuracy with maximum out-of-tolerance probability analysis and prior precise repair optimization

Abstract

For complex thin-walled structures, the accumulation of different error sources behaves a complex nonlinear relationship, causing the frequent occurrence of out-of-tolerance phenomenon, and retention ability of assembly accuracy is weak. Novel methods for improving accuracy reliability with probability analysis and prior precise repair compensation before practical assembly operations were studied. Firstly, considering the non-independent relationship among multi-dimensional error sources, their internal stress function with practical error items and the EOCDCF (Extreme Out-of-tolerance Caused by Deviation’s Coupling&Focusing Function) phenomenon were analyzed. Secondly, the probability method was adopted for evaluating and improving accuracy reliability. To be more specific, the occurrence probability model of EOCDCF event was established with Gartner-Ellis theorem in LDP (Large Deviation Principle), and the dynamic sliding window method was adopted to calculate its probability. Although feedback adjustment on reliability results could be proposed to avoid EOCDCF event considering online information, once it occurred, to control assembly accuracy and internal stress, the accurate repair simulation method based on IPSO (Improved Particle Swarm Optimization) algorithm was proposed considering the actual deviations, and detailed repair areas and quantity on parts would be designed in advance. Finally, a typical wing-box component was verified, and its assembly quality improvement rate was about 30%. This solution could enhance accuracy reliability and ensure assembly accuracy within the required range, and would also lay the precision foundation for product’s service performance.

Introduction

For complex and major products, such as aircraft and automobile, the main structures and other core bearing components are usually riveted/welded/screwed/glued together by a large number of thin-walled parts that having a low rigidity [1]. Since the assembled structures would endure complex alternating and impacting loads during their service period, for the numerical size and distribution status of the dimension accuracy and internal stress after assembly, they will affect the service indicators directly, such as product’s quality, fatigue and damage performance [2], [3]. For example, for aircraft products, (1) assembly accuracy of skin profile affects the aerodynamic parameters, (2) installation accuracy of movable wing components and manipulation mechanisms affect the maneuverability indexes, (3) step difference and butting gap between fixed skin and other movable parts affect the stealth performance, (4) positioning and joining quality, and the residual stress and damage generated during assembly process affect the mechanical performance, bearing strength and fatigue life of the assembled components and other structure integrity indicators (according to statistics, 80% of the fatigue damage of aircraft’s structure occurs at the joining area [4]), (5) geometric accuracy of outline profile and position of reference axis also affect the production interchangeability of the comprised large components.

As the final link of precision controlling in the development of light-weighted and thin-walled structures, assembly is relevant with other process links, such as product design, part manufacturing, assembly process planning, tooling assisted positioning, hole’s drilling&joining, measurement and quality evaluation, and it accounts for about 30%–60% of the workload as manufacturing complex products. As controlling assembly quality, it is faced with the following difficulties, such as: (1) huge number of parts with large size and weak rigidity, and tight restriction on aerodynamic shape (smooth streamline is often required), (2) assembly procedure is long, a quantity of error links making the error transmission path is extremely complex, (3) many types of dedicated assembly fixture/tooling/jigs are adopted [5], and their accuracy is usually 3–5 times higher than the accuracy of product parts, (4) the general Tolerance and Fit system is unsuitable to ensure the final assembly accuracy, and the support of complex and sound interchange/coordination theory is essential. For the current assembly theory based on geometric quantity controlling, it has the typical characteristics of “digital/empirical, passive repair and matching on geometric shape”. It is an open-loop assembly system without feedback improvements. Correspondingly, assembly accuracy is guaranteed mainly by compensation operations and worker’s personal ability [6], [7], making the structure’s performance cannot be guaranteed effectively and efficiently. In conclusion, the current assembly method lacks the ability to make the actual assembly accuracy meet design requirements, which means the reliability of assembly accuracy should be enhanced.

With the development of digital, automatic and intelligent technologies, design level and manufacturing consistency of major equipment have been improved significantly [8]. Assembly locates at trailing end of production, and the performance guarantee of assembled structures is gradually transferred from initial design&machining links to assembly links. As a result, assembly technology is highly concerned [9], [10], [11]. As a matter of fact, as assembling thin-walled structures, final accuracy is the superposition of initial geometric error and internal physical stress that generating in the entire assembly process. To guarantee structures’ performance, firstly, it is necessary to reduce the occurring probability of out-of-tolerance phenomenon with optimization and adjustment measures [12] in advance. To be more specific, considering the nonlinear accumulation of different error sources, it can be called as EOCDCF (Extreme Out-of-tolerance Caused by Deviation’s Coupling&Focusing Function) event. In fact, it’s difficult to improve accuracy reliability for thin-walled structures, reasons could be attributed to the following two aspects: (1) nonlinear coupling&focusing effect among multi-dimensional error sources would cause the assembly geometric accuracy difficult to meet design requirements, which means the EOCDCF event has a high occurring probability, (2) on practical engineering site, it is also often impossible to realize accurate repair compensation in advance, only manual trial and error method is adopted to ensure geometric accuracy, making the efficiency is low and structure’s internal stress often too large.

Enhancing the assembly accuracy, is also one of the eternal goals for product’s manufacturing. Considering the disturbance of multi-dimensional error sources, assembly error modeling based on rigid hypothesis is relatively mature, and it can represent the accurate motion deviation. However, as describing deviation disturbance caused by deformation of complex thin-walled structures, such as skin panel components, it is not suitable anymore. To enhance predicting accuracy of assembly quality, with actual deformation and geometric manufacturing error, Kang [13] modified and extended SDT (Small Displacement Tensor) expression of Jacobian-Tensor model. Then by updating matting constraints of actual flexible parts, Tlija [14] conducted finite element simulation with the distribution state of measured error, and Ballu [15] analyzed the nonlinear accumulation relationship with physical characteristics and optimization constraints (assembly loads, displacement boundary conditions). In addition, considering: (1) matting status of non-ideal surfaces, and (2) external and internal loads, methods such as linear complementary were also proposed, and it can be concluded that assembly tolerance analysis with SMS (Skin Model Shape) models could be taken as the objective function of a quadratic optimization problem. With DT (Digital Twin) solutions, Yi [16] calculated assembly error with updating and iterating mechanism. In addition, by fusing 3D geometric model and massive measurement data, Sun [17] and Masnaoui [18] proposed “semi physical assembly” technology, which could improve the success rate of one-time assembly. However, for the geometric-physical field coupling and error’s multi-scale transmission process, it may lead to the amplification and focusing effort, i.e. assembly deviation exceeds design expectation, then unstable and serious quality problems could occur. From another perspective, although the finite element simulation are adopted, it is still difficult to effectively use on-site measured data to reduce assembly uncertainty, making the accurate guidance on executing adjustment instructions cannot be fulfilled.

Probability modeling of assembly EOCDCF event, can quantify the accumulation results of different error items and reduce the influence of uncertainty factors, such as worker’s misoperation. Therefore, analysis on probability under the disturbance of multi-dimensional error sources, can prevent its occurrence. LDP (Large Deviation Principle) is an important portion in probability theory to describe the occurring of rare events [19], [20]. Then considering the assembly process of complex thin-walled structures, by comparing the probability value of current assembly state with defined probability threshold, the reliability of assembly accuracy can be quickly evaluated. Considering error sources in assembly process should be regarded as probability variables having an independent and identically distribution, LDP is applicable to estimate the probability of EOCDCF event, which means it can realize the application of probability method to improve accuracy reliability, as well as predicting serious quality problems. For parts with free-form surfaces, Corrado [21] investigated the influence of tail beams’ error on the gap among structure interfaces. Where the fitting condition and tolerance were modeled with probability density function. With rigid-flexible model, the tolerance zone and distribution of target assembly deviation could be obtained. Stricher [22] improved the influence coefficient method considering dynamic physical characteristics. Where the geometric nonlinearity caused by stiffness changes was considered, and the coupling result of joining defects, shape tolerances and rigidity changes on final assembly deviation was gained for beam component. With non-ideal SMS models and considering their contacting status among non-ideal surfaces, Zhang [23] modeled actual deformation with conjugate gradient and fast Fourier transform method, and the deep coupling effect of geometric error and contacting deformation were analyzed. It is mentioned for large-scale and thin-walled parts, due to assembly loads and initial geometric errors, forced clamping&joining actions are often adopted to eliminate clearance deviation. However, external internal stress will generate, and it has a harmful effect on service performance of assembled structures [24], [25]. For the above probability analysis on EOCDCF event, the internal assembly stress function with practical error items should be further considered, and compensation strategies as deviation cannot meet design requirements are also urgent.

Then to control assembly accuracy and avoid EOCDCF event in advance, with dividing error sensitivity into three indicator levels, Wu [26] proposed a quantitative analysis method with Monte Carlo simulation at features level, and its feasibility on multi-stations was also verified. However, this method cannot provide theoretical support for improving accuracy’s retaining ability. It’s known that non-rigid assembly simulation requires complex and difficult modeling tasks, as managing geometrical deviations during product’s entire lifecycle, Polini [27] introduced a DT tool and established a continuous and unambiguous flow of variation from part design to final assembly. Where available actual data sets from practical assembly site were allowed to develop new and accurate simulation models, such as part’s form deviations, fixture pressure and positioning error, bonding amount, assembly sequence, geometric shape and parts’ position. Then to enhance the reliability of assembly quality, Yoshizato [28] stated with accurate and timely adjustment strategies, unexpected assembly deformation and residual stress distribution could be predicted and reduced in advance. It is known that keeping geometric characteristics in a stable statistical state, is important to ensure the consistency between assembly error items of different structures. Guo [29] proposed optimization methods with ASFF (Assembly Station Flowing Fluctuation) analysis and feedback actions, and controlling measures based on historical data as reducing quality loss in aeronautical batch production was adopted. However, the SPC (Statistical Process Control) solution is off-line, it cannot realize real-time analysis. As assembling complex thin-walled structures, it is suggested that on-line feedback controlling mode needed be promoted both for small batch and large scale, i.e. it’s urgent to develop accuracy reliability analysis methods with strong applicability and good repeatability.

However, with above preventive actions, once the EOCDCF probability exceed defined threshold, compensation operations on key parts have to be carried out. Although grinding and shimming operations would restrict the assembly efficiency in batch production, the repair compensation method is often adopted at the final stage of practical engineering site, for controlling the geometric accuracy and internal stress [30]. This method refers to compensation on closed error links by repairing the size of certain error rings in assembly dimension chain. The comprised error links that to be repaired is defined as repair ring or repair area, and the repair size is called as repair quantity. In practical engineering, to predict the detailed repair scheme on the workpieces, is mainly based on personal experience of technicians and workers, and it don’t have a clear repair guidance. This situation often cause trial assembly and repeated disassembly. Taking actual physical factors into account, for tolerance analysis or gap estimation during shimming process, Falgarone [31] presented AnatoleFlex software to build a complete and realistic assembly tolerance simulation, which considered assembly sequence, joining defaults, composite material properties, form default, and contacting modeling. And the measured actual geometric deviation was taken as the input of simulation model. To minimize the repair workload, Heling [32] established tolerance-cost optimization model considering assembly dimension chain with statistical methods. Based on difficulty coefficient evaluating, Tlija [33] proposed tolerance allocation methods with Lagrange factor, which can evaluate assembly difficulty quantitatively and reduce repair probability. By investigating the efforts of repair compensation on deviation propagation, Wang [34] presented a tolerance analysis method for composites assembly based on T-Maps method, where the accumulation of anisotropic variations and propagation of clamping force modification and shimming operations were concerned. Considering the realistic models with geometrical surface deviations are essential for further functional analysis, Anwer [35], [36] stated SMS models could represent parts’ geometric deviations. Where measurement data and actual models can be used to compensate the accumulated error. In summary, to ensure the final dimensional error, current solutions for repair quantity are mainly based on the calculation of geometric dimension chain, the analysis result is the repair quantity range on the part, and it is difficult to determine the specific repair area and value size in advance. As a result, this situation results in a lack of quantitative and scientific guidance in practical engineering, and it has a limit effect on the reliability improvement on assembly accuracy.

Aiming at EOCDCF event and extensive repair operations, novel solutions for improving accuracy reliability with probability analysis and prior precise repair compensation before practical assembly operations were studied. Section 2 considered the internal stress function with practical multi-dimensional error items that having non-independent relationship, and probability model for accuracy reliability evaluation was modeled. Section 3 established the occurrence probability model of EOCDCF event with Gartner-Ellis theorem in LDP, and adopted the dynamic sliding window method to calculate its probability. Section 4 proposed the accurate repair simulation method based on IPSO algorithm with actual deviations, and designed detailed repair scheme in advance. Finally, wing-box component was taken to verify the effectiveness of improving accuracy reliability.