Tolerance Analysis Method of Aircraft Door Sealing Structure Based on Linear Simplified Model

Abstract

As a moving component of the aircraft, the sealing state of the door will directly affect the aircraft's pressurization level and aerodynamic performance. The existing analysis methods of sealing structures mainly consider the compression deformation and contact stress of the seals. However, in practical engineering applications, it has been found that the sealing performance of cabin door depends on the interaction between the sealing structure and the seal. On the one hand, the tolerance fluctuation and elastic deformation of the sealing structure will cause the deviation between the actual compression ratio of the seal from the theoretical value; On the other hand, the deformation caused by the resilience of the seal will change the tolerance distribution of the sealing structure. In this paper, based on the local linear simplification of loading curve of the seal, the calculation efficiency of the large-size sealing structure’s deformation is improved, so that the compression ratio fluctuation of the seal is converted into the deformation tolerance which is included in the tolerance simulation of the sealing structures to obtain the distribution characteristics of step difference value on the aerodynamic surface as well as the different compression states of the seal. The validity of this analysis method is confirmed by comparing with the commercial software calculation and the practical measured data of the aircraft door.

1 Introduction

Aircraft doors are the accesses for personnel to enter and exit the cabin as well as for equipment maintenance. While the inadequate compression ratio of the seal or its separation from the sealing structure can lead to a decline in cabin sealing performance, potentially resulting in aircraft pressure loss and even posing serious flight hazards [1]. On the contrary, an excessively high compression ratio will lead to some problems such as increase in the operation force of the door, decrease in the service life of the seal and deterioration of aerodynamic surface flatness which is usually seen as a step difference between the door skin and the fuselage skin. Therefore, the seal properties and the relative deformation of the sealing structure have always been important factors in the design of the airframe structure. The existing structural sealing performance analysis is more concerned with the mechanical properties and compression state of the seal [2, 3], but in engineering applications, the state of the seal often has a coupling effect with the sealing structure. On the one hand, due to the elastic deformation of the sealing structure as well as the manufacturing and installation tolerance, the compression state of the seal will deviate from the theoretical design state in varying degrees. On the other hand, the seal under different compression states will cause the fluctuations in the deformation of the sealing structure. Therefore, the sealing performance of the door is determined by both.

The seal usually adopts a composite structure of fabric and rubber that is why traditional commercial software requires more calculation time for the deformation and contact analysis of the seal and sealing structure. In engineering applications, typical section analysis [2] or integral modelling analysis methods [4] are often used, while the former has a faster analysis speed, but its applicability is limited and not suitable for complex structures; The latter divides the structure and seal into 3D elements for calculation directly which has strong universality, but low computational efficiency and high hardware requirements. In order to balance calculation accuracy and analysis efficiency, a simplified analysis method applicable to large-size sealing structure of the door was proposed in reference [5]. While for the sealing structure, the tolerance not only leads to the compression ratio dispersion of the seal, but also affected by the seal will causes varying degrees of step difference problems between the door and the fuselage. Since Monte–Carlo method has been widely used in the analysis of tolerance characters of structural and mechanism [6, 7], this method can be used to analyse the relative relationship between parts tolerance and structural deformation at the door area.

In this paper, local linear simplification of the seal is firstly carried out based on the fitting of the loading curve, then the tolerance related to the compression of the seal is converted into the deformation tolerance of the structure. Furthermore, this deformation tolerance is incorporated into the step difference dimension chain formula between the door and the fuselage by Monte–Carlo method. The difference of the compression state of the seal is also analysed through the simulation of the example. Finally, the Bootstrap method is adopted to expand the measured data of the step difference values at different points of the door which can be used to analyse the distribution characteristics of practical case and verify the effectiveness of this method by comparing with the theoretical calculation results.

2 Linear Simplification of Seal and Deformation Analysis of Structure

2.1 The Relationship Between Mesh Size and Calculation Accuracy of the Seal

The computational efficiency and convergence will encounter many problems because of the non-linearity of materials and the contact algorithm when calculating the deformation of the sealing structure at large-sized level. Although using methods such as segmented modelling [4] can improve computational efficiency, the analysis strategy of 3D elements still restricts the selection of the minimum mesh size for the seal while the size affects the accuracy of the seal deformation and contact analysis. Taking an Ω-shape seal as an example, its section geometric dimension is shown in Fig. 1.a. which with a length of 140 mm. The uniaxial compression test data of its rubber material is shown in reference [1]. The Ogden model is used to describe the constitutive relationship of this rubber material, and the plane-strain model is used for analysis. The loading curves of the seal under different mesh sizes are calculated that are shown in Fig. 1.b.

Fig. 1

Calculating results of the seal (a model of compression; b loading curves)

As can be seen from Fig. 1b, with the increase of the compression value of the seal, the difference of the resilience calculated by different mesh sizes gradually increases, and the maximum deviation ratio increases to 32.63% at 9.5 mm. The compression ratio of aircraft door seals generally set in range 25–30% which means the corresponding resilience deviation ratio is 17.89–19.12% if adopting this Ω-shape seal. Therefore, when analysing the deformation of the sealing structure, no matter plan elements or 3d elements are used, the non-fine mesh will produce a large resilience deviation, while the excessively fine mesh has an extremely high requirement of the calculation step difference size during the contact process.

2.2 Linear Simplified Modeling Method for Sealing Structure

To solve the above problems, the simplified seal unit can be obtained by fitting the nonlinear loading curve shown in Fig. 1 [5]. Moreover, by locally fitting the estimated compression range of the seal, sufficient accuracy can be obtained even if it is just linear fitting.

The local linearization function of the loading curve of the seal with unit length is:

$$F = kx + c$$

(1)

where $x$ is the compression value of the seal, $F$ is the corresponding resilience, $k$ and $c$ are the coefficients of fitting function. The energy $U_{s}$ which is released by the seal before and after deform-ation in the door can be expressed as:

$$U_{s} = \int_{\Omega } {\int_{{x_{0} - \Delta u}}^{{x_{0} }} {(kx + c)} dxd\Omega }$$

(2)

$x_{0}$ and $\Delta u$ indicate the initial and the change of the compression value respectively, $\Omega$ is the area where the seal is deployed. Assuming that the seal and sealing structure remain contacting before and after deformation, which means there is no separation or penetration between them. After discretizing the sealing structure with finite element, the compression value change $\Delta u$ can be expressed by node displacement vector ${\mathbf{u}}^{e}$ and the shape function ${\mathbf{N}}$ of the sealing structure that is generally consist of holder component and compressor for the seal, using ${\mathbf{u}}_{A}^{e}$ indicate the node displace vector for the holder and ${\mathbf{u}}_{C}^{e}$ for the compressor, then the relationship between these vectors can be get:

$$\left\{ {\begin{array}{*{20}c} {{\mathbf{u}}^{e} = \left[ {\begin{array}{*{20}c} {{\mathbf{u}}_{C}^{e} } & {{\mathbf{u}}_{A}^{e} } \\ \end{array} } \right]^{T} } \\ {\Delta u = {\mathbf{Nu}}_{C}^{e} - {\mathbf{Nu}}_{A}^{e} = {\mathbf{N}}\left[ {\begin{array}{*{20}c} {\mathbf{I}} & {{\mathbf{ - I}}} \\ \end{array} } \right]{\mathbf{u}}^{e} } \\ \end{array} } \right.$$

(3)

where ${\mathbf{I}}$ is the unit matrix. Combinate Eq. (2) with (3) and using the variational calculus of ${\mathbf{u}}^{e}$, it can be seen:

$$\delta U_{s} = \delta {\mathbf{u}}^{{{\mathbf{eT}}}} \sum\limits_{ele} {\int_{{\Omega_{e} }} {\left( {kx_{0} + c} \right){\mathbf{N}}^{{\mathbf{T}}} \left[ {\begin{array}{*{20}c} {\mathbf{I}} & { - {\mathbf{I}}} \\ \end{array} } \right]^{T} - k{\mathbf{N}}^{{\mathbf{T}}} {\mathbf{N}}\left[ {\begin{array}{*{20}c} {\mathbf{I}} & { - {\mathbf{I}}} \\ { - {\mathbf{I}}} & {\mathbf{I}} \\ \end{array} } \right]{\mathbf{u}}^{e} d\Omega_{e} } }$$

(4)

Through equation above, the effect from the seal to the sealing structure is equivalent to adding a pair of generalized forces which have the same magnitude but opposite direction. The magnitude expression is:

$${\mathbf{F}}_{{\mathbf{s}}} = \sum\limits_{ele} {\left( {\int_{{\Omega^{e} }} {{\mathbf{N}}^{{\mathbf{T}}} (kx_{0} + c)d\Omega^{e} } - \int_{{\Omega^{e} }} {k{\mathbf{N}}^{{\mathbf{T}}} {\mathbf{N}}d\Omega^{e} \left( {{\mathbf{u}}_{C}^{e} - {\mathbf{u}}_{A}^{e} } \right)} } \right)}$$

(5)

where $\Omega_{e}$ is the elements that the seal is deployed and $ele$ is the number of elements.

3 Tolerance Analysis Method of Sealing Structure Considering Deformation

Taking the typical aircraft door structure which is shown in Fig. 2.a. as an example: Two hinge arms are hinged to the fuselage and when the door is closed in position, the lug is also hinged to the fuselage. The boundary size of the door is 180 × 180 mm, and all the parts are made of aluminium alloy except for the seal which is deployed around the quadrilateral of the door panel. The section and material are the same as the seal shown in Fig. 1.a. In this model, the preset seal’s compression state which the value is 4.3 mm is characterized by setting the interference between the door and the seal. The outer edge of the door panel that is also the compressor of the seal is 1.27 mm thick. The thickness of the holder is 2.03 mm, and the thickness of the fuselage skin is 1.52 mm. And the position relationship between the sealing structure and the seal is shown in Fig. 2b. The left side is the fuselage, and the right side is the door structure. The surface contour of the skin edges on both sides will directly affect the external step difference. At the same time, the positioning dimension tolerance of the seal’s holder component at the left side will affect the compression ratio of the seal which will change the deformation state of the cabin door.

Fig. 2

The schematic diagram of structure (a model of door; b section of sealing structure)

Due to the profile tolerance of a surface is the upper and lower limits of a series of spherical diameters that are tolerance values $t$ [8], combined with the linear simplified analysis method of sealing structure, the Monte-Carlo method can be used to simulate the deformation of sealing structure and the influence of skin contour deviations on the step difference as well as the relative distribution characteristic.

In the Fig. 2b. the profile of the fuselage skin edge and the door panel is $F_{1} = 0.00 \pm 0.80$ and $F_{2} = 0.00 \pm 0.70$ respectively, while the dimension requirement between holder and compressor is $F_{3} = 12.60 \pm 0.60$. Then the dimension chain calculation formula of the step difference between the two skin is:

$$F_{0} = G(F_{3} ) + F_{2} - F_{1}$$

(6)

The above equation $G_{{(F_{3} )}}$ represents the deformation tolerance of the door skin caused by the resilience of the seal which is a function of the installation size $F_{3}$. In order to obtain the distribution characteristics of the deformation tolerance, FEM analysis of the seal structure shown in Fig. 2 was carried out in which the seal was adopted the linear simplified model, and the analysis software was ABAQUS.

Except the hinge arms are modelled with C3D4 element, all other parts are modelled with 4-node/5-DOF shell element. Due to the independent interpolation of the translational and rotational DOF of the Mindlin shell element [9], the shape functions of shell element which is related to the normal displacement is relatively simple. Assuming the seal loads apply on the seal structure is line contact pressure, and incorporating the shape function into Eq. (5), it can be concluded that:

$${\mathbf{F}}_{s} = \sum {\left( {\left[ {\begin{array}{*{20}c} {\frac{1}{2}l^{e} q} \\ {\frac{1}{2}l^{e} q} \\ \end{array} } \right] - \frac{{kl^{e} }}{6}\left[ {\begin{array}{*{20}c} 2 & 1 \\ 1 & 2 \\ \end{array} } \right]\left( {\begin{array}{*{20}c} {\Delta {\text{u}}_{1} } \\ {\Delta {\text{u}}_{2} } \\ \end{array} } \right)} \right)}$$

(7)

where $\Delta {\text{u}}_{1} = u_{C1} - u_{A1}$ and $\Delta {\text{u}}_{2} = u_{C2} - u_{A2}$, the $l^{e}$ is the length of element, $\left( {u_{*1} ,u_{*2} } \right)^{T}$ indicate node displacement of the elements at both ends of the holder and the compressor. From Eq. (5), it can be seen that the loading effect of the seal can be regarded as two parts. First is the uniformly normal load corresponding to the initial compression value, and the other is the supports stiffness on the nodes. Furthermore, the second support stiffness on the right in Eq. (7) can be decomposed into:

$$\frac{{kl^{e} }}{6}\left[ {\begin{array}{*{20}c} 2 & 1 \\ 1 & 2 \\ \end{array} } \right]\left( {\begin{array}{*{20}c} {\Delta u_{1} } \\ {\Delta u_{2} } \\ \end{array} } \right) = \frac{{kl^{e} }}{6}\left[ {\begin{array}{*{20}c} 3 & 0 \\ 0 & 3 \\ \end{array} } \right]\left( {\begin{array}{*{20}c} {\Delta u_{1} } \\ {\Delta u_{2} } \\ \end{array} } \right) + \frac{{kl^{e} }}{6}\left( {\begin{array}{*{20}c} {\Delta u_{2} - \Delta u_{1} } \\ {\Delta u_{1} - \Delta u_{2} } \\ \end{array} } \right)$$

(8)

The first item on the right of Eq. (8) can be realized by applying a spring element at the corresponding node between the holder and compressor elements. While the second item is the coupling of $\Delta u_{1}$ and $\Delta u_{2}$ which can be realized through MPC element. However, for the purpose of simplifying the analysis model, this item can be ignored when the mesh size is small enough due to the magnitude of the difference between $\Delta u_{1}$ and $\Delta u_{2}$ is smaller compared with themselves.

The deformation analysis of this structure adopts a progressive local linearization method, which is first using linear fitting of the loading curve that the mesh size is 0.5 mm in Fig. 1 within a large interval. After preliminary analysis of the finite element model, the deformation interval of the seal is obtained, and then updated the linearization simplified element by fitting the deformation interval to obtain the final structural deformation result. In theoretical state (compression value is 4.3 mm) the parameters $k = 0.07\,{\text{N/mm}}$, $q = 0.28\,{\text{N/mm}}$. After these relevant parameters are brought into Eq. (8), the deformation of the door can be obtained as shown in Fig. 3. The maximum deformation of the door is 0.79 mm, and the deformation of the corresponding area of the fuselage skin is 0.19 mm which means the step difference in this area is 0.98 mm. Meanwhile, the minimum step difference in this model is 0.12 mm, both of which are the door panel protruding from the fuselage skin.

Fig. 3

The deformation contour plot of the door and fuselage

After the upper limit of 4.9 mm and lower limit of 3.7 mm compression of the seal are brought in, the minimum step difference is calculated as $G_{{(F_{3} )\,\min }} = 0.12_{ - 0.02}^{ + 0.03}$, and the maximum is $G_{{(F_{3} )\max }} = 0.98_{ - 0.14}^{ + 0.15}$. Usually, the positioning tolerance $F_{3}$ follows a normal distribution, while the absolute values of upper and lower limits of deformation tolerance $G_{{(F_{3} )}}$ are close to each other so the normal distribution can also be used for simulation. Relevant parameters are brought into Eq. (6) and Monte-Carlo method is used to simulate step difference for 3000 times. Then the maximum/minimum value of step difference as well as the probability distribution of step difference without seals are obtained, as shown in Fig. 4.

Fig. 4

Distribution of step difference

Due to the deformation of the sealing structure, the mean value of step difference distribution of the door is uneven around the edge, and the position tolerance of the holder $F_{3}$ is compensated by the deformation of the seal at the cost of 1.2 mm compression fluctuation of the seal. Combined with the deformation of the sealing structure, the actual compression ratio range of the seal is 2.76–4.53 mm which means the fluctuation of the seal compression ratio archives 9.83%. The corresponding contact stress of the seal at the minimum and maximum compression state is shown in Fig. 5. It shown that due to the deformation and positioning tolerance of the seal structure, the maximum contact stress of the seal is reduced from 4.59 MPa to 2.73 MPa, with a reduction ratio of 40.52%. By improving the stiffness of the seal structure and the installation and positioning accuracy, the contact stress loss can be reduced.

Fig. 5

Contact pressure of the seal

4 Comparison with Measured Data

The structural design and analysis of an airliner APU door [10] had been discussed in reference [11], but it did not discuss the rebound effect of seals on the edge of the composite door which was further analysed in the reference [5]. However, due to the manufacturing tolerance of the door components, there was a certain deviation between the analysis results and the physical data. At the same time, restricted by the number of samples, the validity of analysis results needs to be further discussed and confirmed. The configuration of the door is shown in Fig. 6.

Fig. 6

Configuration of the APU door

The composite material parameters are consistent with reference [11] which is carbon fibre and benzoxazine resin composites solidified at 185 °C. And the core material is the same as the in reference [5] which is GILLCORE HD 332O type aryl-paper honeycomb. Take the lower door as the analysis object in consideration of symmetry configuration. The analysis software is NASTRAN with CQUAD4 element used for the inner and outer skins as well as laminated plate. And CHEXA/CPENTA solid element used for the honeycomb core. The door is hinged to the fuselage through the titanium hinges and in the middle of these two doors, four shear pins fix the door to the fuselage when the door is closed.

As the seal holders are made of alloy material and directly connected to the frame of the fuselage, the stiffness is stronger than the composite panel, so only the deformation of the door is considered in the analysis. The local linear fitting equation of the seal and the calculation results of the deformation of the door are given in reference [5] in which the loading curve of the seal is obtained by experiment. Fitted within 0.6–5.0 mm, it can obtain that $F = 0.184x + 0.022$ while the initial compression $x_{0} = 4.60\,{\text{mm}}$. The results of the door deformation are shown in Fig. 7 in which large deformation occurs in the two-corner area, with the maximum deformation value of 3.75 mm in the right corner and 3.03 mm in the left corner.

Fig. 7

The deformation contour plot of the APU door

The sealing structure for this door is similarly with Fig. 2 [11], the profile of the fuselage skin edge and the door panel is $F_{1} = 0.00 \pm 1.00$ and $F_{2} = 0.00 \pm 0.75$ respectively, while the dimension requirement between holder and compressor is $F_{3} = 12.30 \pm 0.60$ through which the lower edge deformation of the door left corner is calculated as $G_{L} (F_{3} ) = 3.07_{ - 0.34}^{ + 0.55}$ and the right corner deformation is $G_{R} (F_{3} ) = 3.75_{ - 0.48}^{ + 0.54}$。

As $G_{L} (F_{3} )$ and $G_{R} (F_{3} )$ is presenting right skew distribution while the square root of the data is approximately normal distribution. Therefore, convert the Eq. (6) into:

$$F_{0} = \left( {\sqrt {G(F_{3} )} } \right)^{2} + F_{2} - F_{1}$$

(9)

Monte-Carlo method is used to simulate Eq. (9) by 5000 times, and the statistical information of the step difference distribution of two corner points are listed in Table 1. The probability density distribution of step difference at these two locations is shown in Fig. 8.

Table 1 Distribution information of step difference

Fig. 8

Distribution of the step difference at two corners

As a comparison, the actual measurement data of step difference of the left and right corner are given in Table 2.

Table 2 Measured data of two corners’ step difference (unit: mm)

Since the sample size of the measured data is relatively small, assuming that the step difference of the left and right corner follows the normal distribution then the parametric Bootstrap method [12,13,14] is used to resampling the step difference sample, and the confidence estimation of the distribution parameters is performed. The number of Bootstrap resample size at each corner is 2000 and the distribution parameters of the Bootstrap group of the step difference is shown in Table 3 which the confidence probability is 99%.

Table 3 Static information of step difference distribution

By comparing the data in Tables 1 and 3, the average value deviation of the average value and the theoretical analysis is 0.29 mm at left corner and 0.46 mm at right corner respectively, while the average value deviation of the standard deviation is 0.38 and 0.01 relatively. The static character data obtain by simulation are all in the confidence interval and the deviation of the average value is sufficiently accurate for estimating in engineering cases which shows there is a good agreement between theoretical analysis and actual measure data.

5 Conclusion

In this paper, the linear simplified method of the seal and Monte-Carlo method are used to analyse the factors affecting the deformation and step difference of the sealing structure for aircraft cabin door during which the finite element modelling strategy and calculation method are expounded. And the validity of this analysis method is verified by the actual data which static information is obtained by Bootstrap estimate method. The conclusions are as follows:

• (1) The loading curve of the seal is variated in the different analysis mesh sizes. And the linear simplified method of seal can used to analysis the deformation of large-size level sealing structure. Duo to the loading curve of the seal can be derived from planar structure simulation or experimental measurement, it won’t increase the calculation cost of the door seal structure compared to the traditional 3D finite element model.

• (2) The tolerance of the seal structure and the resilience of the seal are coupled interaction, on the one hand, the resilience of the seal lead to the uniform deformation of the seal structure, and on the other hand, the tolerance of the seal structure also makes the actual compression ratio of the seal varying degrees of change. The example shows the seal could compensate the installation tolerance of the seal’s holder part while the contact pressure is obviously decreased.

• (3) By comparing with the measured data, the seal structure deformation and tolerance analysis method which convert the installation tolerance of the holder parts into the deformation tolerance shows a good compliance. Therefore, the relevant content can be a reference for the optimization design of aircraft cabin door structures.