Numerical simulation of ply angle deviation and thermal deformation behavior of CFRP reflector

As we know, the ply angle deviation of Carbon Fiber Reinforced Polymer(CFRP) reflector is difficult to obtain accurately in engineering applications, it will make thermal deformation of reflector unpredictable due to the discreteness of ply angle. In this paper, based on constitutive relationships between generalized internal force and strain of laminated composite, a novel ply angle deviation model based on normal distribution is proposed. Furthermore, an orthogonal test was designed and significance analysis were carried out by means of a 2.6m aperture reflector with polygonal back structure. Assumed that the ply angle deviation obeys normal distribution, a number of samples were further randomly generated, and the sample mean and standard deviation of the thermal deformation of reflector were calculated. According to χ2-test of goodness of fit, the thermal deformation of reflector still follows normal distribution. Finally, an example was employed to verify the validity and effectiveness of the proposed method. The results reach a good agreement with the experimental data, thus it is capable of providing a promising prospect for engineering practice.


Introduction
With the rapid development of high-throughput communication satellites (HTS), extremely high thermal stability requirements for antennas in space environment will be needed to achieve a high gain, low sidelobe and high C/I(Carrier to Interference ratio) goal [1][2].Because of simple structure, excellent weight, low cost and good structural adaptability to different aperture reflectors, the reflector with a polygonal carbon fiber square-tube back structure is being widely used in HTS [3][4][5].Unfortunately, a periodic temperature load change of the reflector in orbit environment will introduce thermal expansion and thermal stress.Surface distortion will be occurred in case of thermal mismatch between reflector components, which is an undesirable situation for the reflector performance [6].
Domestic scholars had studied reflector materials, reflector structures [6][7][8][9][10][11],and thermal deformation test methods [2,[12][13].We know the deviation of ply angle can cause the nonaxisymmetric deformation of the laminated plate, and the deformation will lead to a undesirable thermal deformation of the reflector.Unluckily, we have not carried out relevant work yet.In 1989, foreign scholar Robert [14] proposed a concept of statistical variables to analyze the thermal deformation of high-precision reflectors; William [15] obtained a sample mean value and standard deviation of mechanical property parameters through testing samples, and randomly generated parameter values of each element in the finite element model by the Monte Carlo method.Lang [16] established the displacement deformation gradient matrix based on the linear Gradient Based Superposition method, obtained the relationship between the random variable and the displacement deformation field of the reflector.Meanwhile, increasement of the number of plies could reduce the impact of ply angle deviation on thermal deformation, but it fulfilled at the cost of weight increasement [17].
For the above reasons, we introduce the concept of probability and statistics in this paper, develop the ply angle deviation model of laminated composite materials, and give the thermal deformation analysis method of the reflector based on normal distribution random samples.By using the orthogonal experimental design, the significance of the influence factors of thermal deformation was analyzed, and the better conditions were obtained.According to some random samples subject to normal distribution, the probability level and distribution function of reflector thermal deformation were predicted.This will provide a reference for reflector design and process control as well as offer reference for engineering application.

Reflector structure and material
The reflector projection aperture D is φ2600mm, the offset distance H is 500mm, and the focal length F is 4500mm, as shown in figure 1.The paraboloid equation in the reflector design coordinate system O m -X m Y m Z m is Z=(X 2 +Y 2 )/4F.The reflector is composed of a dish, a polygonal back structure, patches and a deployable arm.The dish is bonded with the back structure through some discretely distributed L-shaped carbon fiber patches, and the deployable arm is connected with the back structure by screws, as shown in figure 2. The reflector is a carbon fiber aluminum honeycomb sandwich structure, composed of front skin, back skin and aluminum honeycomb core.The back structure and the deployable arm are all composed of carbon fiber laminated rectangular square tubes.3. The dish, the polygon back structure, the patches and the deployable arm were meshed by Quad4 shell element.The patches was connected with the dish by four MPC/RBE2 units; The deployable arm was connected with the back structure by eight MPC/RBE2 units.

Load and boundary conditions
The deployable arm is fixed on a two-dimensional pointing mechanism, and the pointing mechanism installed on the satellite deck.Combined with the working conditions of reflector in orbit, we define the constraints as follows: the translational displacement of the four vertices of the deployable arm root segment in the three directions of Xm, Ym and Zm is zero.The temperature load is 120 ℃.

Thermal deformation calculation method
A thermal deformation example of polygonal back frame reflector at 120℃is shown in figure 4.
Assumed that the number of nodes on the dish is p, the initial coordinate is 20℃, and it becomes ( i x , i y , i z ) at 120℃.To eliminate the rigid displacement caused by the deployable arm, all nodes of the dish after thermal deformation will be done the best fitting than the p nodes before thermal deformation, and the fitting value

Structure size optimization
The size of back structure is as shown in figure 5, and the value of L1 and L2 in size-1 is 1.75m and 1.35m, the value of L1 and L2 in size-2 is 1.6m and 1.2m, the value of L1 and L2 in size-3 is 1.45m and 1.05m.According to the above result, the influence of the size of the back structure on the thermal deformation of the dish is obtained as shown in figure 6, the ply angles of layer-1 and layer-2 are [0°/±45°/90°/90°/±45°/0°]s and [50°/0°/-50°/0°/0°/-50°/0°/50°]s.It can be seen there is nonlinear stiffness coupling between the back frame and the dish.What's more, compared with the axial zero expansion design, the quasi-isotropic design for the back structure has a better thermostability.

Thermal deformation calculation model of reflector with ply angle deviation
A configuration of laminated composite structure is shown in figure 8.The total number of layers is n, and the thickness is h. the ply angle of each layer is θi (i=1, 2,..., n), the thickness of each layer hi=h/n.
x y  According to the constitutive relationship between generalized internal force and generalized strain of laminated structure [18], it satisfies equation ( 2 where   N 、   M are the internal force and internal moment per unit width of the laminated structure cross section, respectively;   Here,

AB BD
is the total stiffness matrix.  A is the tension-compression stiffness matrix,   B is the tension (compression) bending (torsion) coupling stiffness matrix,   D is the bending (torsion) stiffness matrix, the calculation method is shown in Equation ( 6)-( 8).(1  ) cos 4( )) 8 Take one sample of [1°/46°/-45°/89°/C15/90°/-44°/46°/-1°] (C15 represents the height of aluminum honeycomb core) as an example, substitute the values in Equations ( 9), (10) and table 2 into the finite element model and assign the value of the dish pavement.1), the RMS value of reflector thermal deformation are 0.027mm and 0.135mm, respectively.Obviously, it can be seen that there are the following changes after considering the ply angle deviation.

Thermal deformation analysis based on normal distribution random samples
Firstly, there is no thermal deformation coupling effect between the deployable arm and the back structure.Secondly, the structure size of patches is very small, and the influence of the discreteness of the ply angle deviation is relatively small.Therefore, when doing the thermal deformation analysis, the angle deviation between the dish and the back structure is mainly discussed.And it is assumed that: 1) The ply angle deviation of dish and back structure is subject to the normal distribution model, and each ply angle is independent of each other; The theoretical ply of dish is [0°/45°/-45°/90°/C15/90°/-45°/45°/0°].Meanwhile, the theoretical ply of six tubes of the back structure is [0°/45°/-45°/90°/90°/-45°/45°/45°/0°]s.Then, the random deviation of each ply angle of reflector is △ θi (i=1, 2, ..., 104) and satisfies equation (11).
2) According to the normal distribution and the required value of ply angle deviation ±δ (δ=0.5°,1°), it can be generated N sample values of deviation angle with M layers by equation (12) in @Matlab software.In terms of the μ test method [19], take N equal to 100.
3) On the basis of random samples of ply angle deviation, update each ply angle according to equation ( 13), analyze thermal deformation and obtain thermal deformation results εi (i=1, 2, ..., N).
4) Based on the maximum likelihood estimation method, the mean value ˆ  and standard deviation ˆ  of the reflector thermal deformation samples are shown in equations ( 14) and ( 15)

Othogonal experimental design
According to the orthogonal test design method [19], selected an orthogonal table L9(3 4 ), the orthogonal test results and the analysis of variance were shown in table 3 and table 4, respectively.The horizontal number factors contained A (ply angle deviation of dish) and B (ply angle deviation of back structure), where level 1 represents δ=0°.Level 2 represents δ = 0.5°, and level 3 represents δ = 1°.A×B is the interaction between factors A and B. R is the range, 2 i s is the sum of squares of changes in the column, i f is the degree of freedom in the column, e is the error, and T is the sum of all data.Table 3. Results of orthogonal test.The analysis results are as follows: a) An interaction on the thermal deformation is found between the dish and the back structure, that means the back structure has a coupling effect on the thermal deformation of the dish.
b) The interaction on the thermal deformation is the most significant, the dish takes the second place, and the significance of the back structure is the weakest.c) Because A×B interaction is significant, consider unbiased statistics in equation ( 16), so 11 23 32 ˆˆ(ab) =(ab) =(ab) is better.In fact, the angle deviation of paving layer is not zero, and factor A is significant, so the better case is A2B3.

Analysis and discussion
Based on the above result, simulation results of reflector thermal deformation are shown in figure 9

Verification and discussion
According to the corresponding material, ply design and ply angle deviation control, the 2.6m polygonal back structure reflector was manufactured.The thermal deformation test was carried out using the photogrammetric method [12][13], and the equipment was the MPS/FP single-camera industrial photogrammetric system.Test status in the thermal vacuum tank is shown in figure 10.
1) The test result is 0.075mmRMS according to equation (1), and is distributed in the sample interval [0.07, 0.08).The displacement deformation is shown in figure 11.The relative error is only 8.7% between the simulation result with a value 0.069mmRMS and the test.2) According to the same method in this paper, the thermal deformation analysis result of the dish is only 0.061mm RMS based on the ±0.5° ply angle deviation.Compared with 0.069mmRMS, the thermal coupling effect between the dish and the back structure is only 13.1% of the thermal deformation of the dish, as indicated in figure 12.It shows that the thermal coupling effect between the back structure and the dish can be improved by optimizing the ply angle deviation.At the same time, with the increase of the ply-angle deviation, the nonlinear of the dish thermal deformation value greatly increases.

Conclusion
In this paper, based on the ply angle deviation model and the idea of probability statistics, a method of reflector thermal deformation analysis is proposed.Because the ply angle deviation is difficult to obtain in the actual manufacturing process, so random samples subject to normal distribution is given as well as used as simulation samples.Meanwhile, χ2-test of goodness of fit is applied to verify the correctness of the model.Results show that simulation values are consistent with experimental values.So it can be applied for engineering applications in the future.

Figure 3 .
Figure 3. Finite element model of reflector.Figure 4. A thermal deformation of reflector.

Figure 4 .
Figure 3. Finite element model of reflector.Figure 4. A thermal deformation of reflector.
variables include rotation angle Rx around the Xm axis and rotation angle Ry around the Ym axis, because the twodimensional pointing mechanism has the ability to rotate along the Xm and Ym axes.So, the root mean square (RMS) value ε of reflector thermal deformation (unit: mm) is shown in equation (1).

Figure 5 .
Figure 5. Size of back structure.Figure 6. Influence of back structure size on thermal deformation of reflector.The front skin ply of the dish is [0°/±45°/90°], and the back skin ply is [90°/±45°/0°].As indicated in figure7below, the thermal compatibility with the stiffness of the back structure will be deteriorated with increasing magnitude of heigh of honeycomb core.

Figure 6 .
Figure 5. Size of back structure.Figure 6. Influence of back structure size on thermal deformation of reflector.The front skin ply of the dish is [0°/±45°/90°], and the back skin ply is [90°/±45°/0°].As indicated in figure7below, the thermal compatibility with the stiffness of the back structure will be deteriorated with increasing magnitude of heigh of honeycomb core.

Figure 7 .
Figure 7. Influence of height of core on thermal deformation of reflector.

Figure 8 .
Figure 8. Structural form of laminated composite.According to the constitutive relationship between generalized internal force and generalized strain of laminated structure[18], it satisfies equation (2).

M
are the internal force and internal moment caused by temperature change, respectively; When temperature load t  is applied on the structure, there is   N =  M =0.Substituting it into equation (2) obtains equation (3).Equations (4)-(5) provide the calculation method for the correlation matrix in Equation (3), there are cos l

•
The value of matrix   D changes;• The deterioration of thermal deformation caused by the ply angle deviation cannot be ignored.

Figure 10 .
Figure 10.Thermal deformation test status.Figure11.Thermal deformation test result.2) According to the same method in this paper, the thermal deformation analysis result of the dish is only 0.061mm RMS based on the ±0.5° ply angle deviation.Compared with 0.069mmRMS, the thermal coupling effect between the dish and the back structure is only 13.1% of the thermal deformation of the dish, as indicated in figure12.It shows that the thermal coupling effect between the back structure and the dish can be improved by optimizing the ply angle deviation.At the same time, with the increase of the ply-angle deviation, the nonlinear of the dish thermal deformation value greatly increases.

Figure 11 .
Figure 10.Thermal deformation test status.Figure11.Thermal deformation test result.2) According to the same method in this paper, the thermal deformation analysis result of the dish is only 0.061mm RMS based on the ±0.5° ply angle deviation.Compared with 0.069mmRMS, the thermal coupling effect between the dish and the back structure is only 13.1% of the thermal deformation of the dish, as indicated in figure12.It shows that the thermal coupling effect between the back structure and the dish can be improved by optimizing the ply angle deviation.At the same time, with the increase of the ply-angle deviation, the nonlinear of the dish thermal deformation value greatly increases.

Figure 12 .
Figure 12.Influence of ply angle deviation on thermal deformation of dish

Table 2 .
Change of coefficient of thermal expansion caused by angle deviation.

Table 4 .
Analysis of variance.