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## Assembly variation prediction for compliant aeronautical structures using fuzzy interval analysis

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Abstract

Variation sources’ distributions were described as discrete fuzzy numbers under the condition that only the ranges of variation sources were known. An assembly variation model for compliant aeronautical structures was built based on the theory of elastic mechanics and finite element analysis (FEA). The variation sources described by discrete fuzzy numbers were treated as the inputs of the assembly variation model, the fuzzy distribution of assembly variation was obtained with the aid of fuzzy interval analysis, and the effectiveness of proposed assembly variation model and prediction method were verified with the case of simulated wing-box skeleton assembly. The experimental results show that the predicted intervals based on the proposed assembly variation prediction method contains the measured assembly variation, which can estimate the assembly variation with poor information of the variation sources. Meanwhile, the proposed method provides the assembly variation intervals under different confidence levels. This method is a complement to the traditional assembly variation analysis method base on the Monte Carlo simulation (MCS).

Keywords： assembly variation prediction ; compliant aeronautical structures ; discrete fuzzy number ; variation source estimation ; influence coefficient method ; fuzzy interval analysis

DU Li, MEI Biao, ZHU Wei-dong, KE Zhen-zheng. Assembly variation prediction for compliant aeronautical structures using fuzzy interval analysis. Journal of Zhejiang University(Engineering Science)[J], 2019, 53(9): 1647-1655 doi:10.3785/j.issn.1008-973X.2019.09.002

## 1. 模糊区间分析表征的装配偏差模型

$u=f\left( {{v_1},{v_2},\cdots,{v_n}} \right).$

$A=\sum\limits_{i=1}^k {\frac{{A\left( {{x_i}} \right)}}{{{x_i}}}} =\frac{{A\left( {{x_1}} \right)}}{{{x_1}}} + \frac{{A\left( {{x_2}} \right)}}{{{x_2}}} + \cdots + \frac{{A\left( {{x_k}} \right)}}{{{x_k}}}.$

$\begin{split} \hat A=& \sum\limits_{i=1}^k {\left\{ {{{A\left( {\left[ {\overline {{x_i}} ,\underline {{x_i}} } \right]} \right)} \Big/ {\left[ {\overline {{x_i}} ,\underline {{x_i}} } \right]}}} \right\}}= \\ & \frac{{A\left( {\left[ {\overline {{x_1}} ,\underline {{x_1}} } \right]} \right)}}{{\left[ {\overline {{x_1}} ,\underline {{x_1}} } \right]}} + \frac{{A\left( {\left[ {\overline {{x_2}} ,\underline {{x_2}} } \right]} \right)}}{{\left[ {\overline {{x_2}} ,\underline {{x_2}} } \right]}} + \cdots + \frac{{A\left( {\left[ {\overline {{x_k}} ,\underline {{x_k}} } \right]} \right)}}{{\left[ {\overline {{x_k}} ,\underline {{x_k}} } \right]}}. \end{split}$

$\hat U=f\left( {{{\hat V}_1},{{\hat V}_2},\cdots,{{\hat V}_n}} \right).$

$\begin{split} \hat U=& \sum\limits_{i=1}^t {\left\{ {{{U\left( {\left[ {\overline {{u_i}} ,\underline {{u_i}} } \right]} \right)} \Big/ {\left[ {\overline {{u_i}} ,\underline {{u_i}} } \right]}}} \right\}} = \\ & \frac{{U\left( {\left[ {\overline {{u_1}} ,\underline {{u_1}} } \right]} \right)}}{{\left[ {\overline {{u_1}} ,\underline {{u_1}} } \right]}} + \frac{{U\left( {\left[ {\overline {{u_2}} ,\underline {{u_2}} } \right]} \right)}}{{\left[ {\overline {{u_2}} ,\underline {{u_2}} } \right]}} + \cdots + \frac{{U\left( {\left[ {\overline {{u_t}} ,\underline {{u_t}} } \right]} \right)}}{{\left[ {\overline {{u_t}} ,\underline {{u_t}} } \right]}}. \end{split}$

### 图 1

Fig.1   Variation analysis for assembly of compliant aeronautical structures

1）在定位阶段，零件被装夹到各自的夹具上，由于存在制造误差，零件在连接点处存在初始间隙 ${{v}}$

2）在夹紧阶段，通过施加夹紧力 ${{{f}}_{\rm{c}}}$将零件连接点夹持到理想位置，此时会引起观测点位移 ${{{u}}_{\rm{c}}}$

3）在连接阶段，通过铆钉或螺栓等紧固件对2个零件进行连接；

4）在释放阶段，移除夹具，释放夹紧力，零件变形回弹引起观测点位移 ${{{u}}_{\rm{r}}}$.

${{u}}={{{u}}_0} - {{{u}}_{\rm{c}}} + {{{u}}_{\rm{r}}}.$

${{{u}}_{\rm{c}}}={{{S}}_{\rm{c}}}{{v}}.$

${{{u}}_{\rm{r}}}={{{S}}_{\rm{r}}}{{v}}.$

${{u}}={{{u}}_0} - {{{S}}_{\rm{c}}}{{v}}+ {{{S}}_{\rm{r}}}{{v}}={{{u}}_0} + \left( {{{{S}}_{\rm{r}}} - {{{S}}_{\rm{c}}}} \right){{v}}={{{u}}_0} + {{S}}v.$

${{u}}={{{u}}_0} + {{S}}{{v}} + {{{u}}_{\rm{P}}} + {{{u}}_{\rm{T}}} + {{{u}}_{\rm{g}}}.$

### 图 5

Fig.5   Calculation process of fuzzy interval distribution of simulation wing-box skeleton

Tab.1  Comparison between measured and predicted assembly variation by fuzzy interval analysis and interval analysis based on assembly variation analysis methods

 观测点编号 ${U_{{\rm{fuz}}}}$/m ${U_{{\rm{int}}}}$/m ${U_{{\rm{mea}}}}$/m λ=0 λ=0.2 λ=0.4 λ=0.6 λ=0.8 λ=1 1 [−0.579 8，0.579 8] [−0.580 6，0.580 6] [−0.581 4，0.581 4] [−0.582 2，0.582 2] [−0.583 0，0.583 0] [−0.583 7，0.583 7] [−0.583 7，0.583 7] −0.544 2 [−0.554 4，0.554 4] [−0.554 4，0.554 4] [−0.554 5，0.554 5] [−0.554 5，0.554 5] [−0.554 6，0.554 6] [−0.554 6，0.554 6] [−0.554 6，0.554 6] −0.407 3 [−0.535 9，0.535 9] [−0.536 0，0.536 0] [−0.536 0，0.536 0] [−0.536 1，0.536 1] [−0.536 1，0.536 1] [−0.536 2，0.536 2] [−0.536 2，0.536 2] −0.269 4 [−0.518 4，0.518 4] [−0.518 5，0.518 5] [−0.518 6，0.518 6] [−0.518 7，0.518 7] [−0.518 8，0.518 8] [−0.518 9，0.518 9] [−0.518 9，0.518 9] −0.131 5 [−0.507 6，0.507 6] [−0.507 7，0.507 7] [−0.507 7，0.507 7] [−0.507 7，0.507 7] [−0.507 7，0.507 7] [−0.507 7，0.507 7] [−0.507 7，0.507 7] 0.007 6 [−0.581 2，0.581 2] [−0.581 9，0.581 9] [−0.582 7，0.582 7] [−0.583 5，0.583 5] [−0.584 2，0.584 2] [−0.585 0，0.585 0] [−0.585 0，0.585 0] 0.305 7 [−0.557 0，0.557 0] [−0.557 1，0.557 1] [−0.557 1，0.557 1] [−0.557 2，0.557 2] [−0.557 2，0.557 2] [−0.557 3，0.557 3] [−0.557 3，0.557 3] 0.259 8 [−0.539 1，0.539 1] [−0.539 1，0.539 1] [−0.539 1，0.539 1] [−0.539 1，0.539 1] [−0.539 1，0.539 1] [−0.539 1，0.539 1] [−0.539 1，0.539 1] 0.214 9 [−0.519 3，0.519 3] [−0.519 4，0.519 4] [−0.519 4，0.519 4] [−0.519 5，0.519 5] [−0.519 6，0.519 6] [−0.519 6，0.519 6] [−0.519 6，0.519 6] 0.168 10 [−0.500 8，0.500 8] [−0.500 8，0.500 8] [−0.500 8，0.500 8] [−0.500 8，0.500 8] [−0.500 8，0.500 8] [−0.500 8，0.500 8] [−0.500 8，0.500 8] 0.122 11 [−1.414 2，1.414 2] [−1.514 2，1.514 2] [−1.614 2，1.614 2] [−1.714 2，1.714 2] [−1.814 2，1.814 2] [−1.914 2，1.914 2] [−1.914 2，1.914 2] 0.657 12 [−1.051 9，1.051 9] [−1.151 9，1.151 9] [−1.251 9，1.251 9] [−1.351 9，1.351 9] [−1.451 9，1.451 9] [−1.551 9，1.551 9] [−1.551 9，1.551 9] 0.581 13 [−0.690 4，0.690 4] [−0.690 4，0.690 4] [−0.790 4，0.790 4] [−0.890 4，0.890 4] [−0.990 4，0.990 4] [−1.090 4，1.090 4] [−1.190 4，1.190 4] 0.505 14 [−0.810 3，0.810 3] [−0.814 0，0.814 0] [−0.817 6，0.817 6] [−0.821 3，0.821 3] [−0.825 0，0.825 0] [−0.828 6，0.828 6] [−0.828 6，0.828 6] 0.429 15 [−0.533 5，0.533 5] [−0.533 5，0.533 5] [−0.533 5，0.533 5] [−0.533 5，0.533 5] [−0.533 5，0.533 5] [−0.533 6，0.533 6] [−0.533 6，0.533 6] 0.353 16 [−1.409 8，1.409 8] [−1.509 8，1.509 8] [−1.609 8，1.609 8] [−1.709 8，1.709 8] [−1.809 8，1.809 8] [−1.909 8，1.909 8] [−1.909 8，1.909 8] −0.869 17 [−1.049 0，1.049 0] [−1.149 0，1.149 0] [−1.249 0，1.249 0] [−1.349 0，1.349 0] [−1.449 0，1.449 0] [−1.549 0，1.549 0] [−1.549 0，1.549 0] −0.728 18 [−0.686 7，0.686 7] [−0.786 7，0.786 7] [−0.886 7，0.886 7] [−0.986 7，0.986 7] [−1.086 7，1.086 7] [−1.186 7，1.186 7] [−1.186 7，1.186 7] −0.588 19 [−0.806 8，0.806 8] [−0.810 4，0.810 4] [−0.814 1，0.814 1] [−0.817 7，0.817 7] [−0.821 4，0.821 4] [−0.825 1，0.825 1] [−0.825 1，0.825 1] −0.447 20 [−0.536 7，0.536 7] [−0.536 7，0.536 7] [−0.536 7，0.536 7] [−0.536 7，0.536 7] [−0.536 7，0.536 7] [−0.536 7，0.536 7] [−0.536 7，0.536 7] −0.307

### 图 6

Fig.6   Platform for assembly variation measurement of simulant wing-box skeleton

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