Multi-objective topology optimization to reduce vibration of micro-satellite primary supporting structure

2.1. Establishment of finite element model of satellite base plate

As shown in Fig. 2, it is consist of 21029 nodes, 15183 body element in the FEM of the satellite base plate. 24 points of the central ring are used as the constraint boundary condition, and a lumped mass is used to simulate the optical camera load. The lumped mass is connected to the base plate by multi-point constraints. It makes use of 2A12 aluminum alloy, and the dynamic parameters [18] are shown in Table 1.

Fig. 2Finite element model of satellite base plate

a) Front view

b) Vertical view

2.2. Single-objective topology optimization

In operation, the maximization of stiffness is usually equivalent to the minimization of compliance; the compliance is defined by the strain energy. Therefore, the topology optimized mathematical model [16, 17], based on the solid isotropic material with penalization method for conditions of given loading and a single-stationary boundary, can be expressed using the formulation detailed below. It is constrained by the volume of the overall structure and makes minimizing compliance its objective:

where $C\left(\rho \right)$ is the compliance function, $V_k$ represents the material volume after optimization, ${\rho }_k$ is the design variable, and ${\rho }_{min}$ stands for the minimum value of the variable.

According to the mathematical model and practical requirements of the satellite, we optimize the design area on the basis of the stiffness topology optimization model, with compliance minimization set as its objective and volume fraction as its restriction. The result after iterative optimization is shown in Fig. 3.

The results show that to obtain a low weight, the optimized area turns into reinforced ribs. In addition, they take a central-divergent distribution, centering on the circle and distributed from the position of each camera to the border. According to the previous experience, central-divergent reinforced ribs can maximize the stiffness of the whole structure whilst maintaining a low weight. Hence, it is reasonable to take the result into consideration when designing for a low weight.

Fig. 3Result of single-objective optimization

Fig. 4Result of multi-objective optimization

2.3. Multi-objective topology optimization concerning stiffness and random response

By using the compromise programming approach, considering stiffness and random response, the multi-factorial MOTO function in the frequency domain restricted by volume can be generated as below [15, 17]:

where, $m$ is the total number of the loading case, $n$ is the total number of the secondary loads, $w_k$is the weight of the $k$th load, and $q$ is the penalty factor ($q\ge$2); $C_k\left(\rho \right)$ is the compliance function of the $k$th load, with two polar values respectively; ${\mathrm{\Lambda }}_k$($\rho$) is the random response function of the $k$th load, with two polar values respectively.

The boundary, restriction, and design area conditions are retaining as the same above. By iterative optimization, the following result is obtained, shown in Fig. 4.

To obtain a low weight, the optimized area turns into reinforced ribs. The distribution of the ribs is different to that shown in Fig. 4, and tends to be perpendicular to the border. Considering that the MOTO model adds a random response parameter, we make the assumption that the reinforced ribs that are perpendicular to the border help to reduce the RMS value of the random vibration response. To verify this assumption, a dynamic analysis was necessary to compare the two optimized results described above.

3.1. Construction of 3D model of base plate

In light of the results presented above, the symmetry of the satellite, and the requirements of the manufacturing technique, we present two designs of the base plate. For plan A, based on the SOTO result, the base plate is designed with divergent reinforced ribs as shown in Fig. 5. For plan B, based on the MOTO result, the base plate is designed with divergent reinforced ribs that are perpendicular to the border as shown in Fig. 6. However, there is not only the optical camera installing on the base plate, but also one Fiber Optic Gyroscope and two Star Trackers beside the camera, and one TT&C Antenna, two GPS Antenna installing in the reverse side beside the docking ring. So, for plan C, based on plan B and to meet the requirement of installing place and structural interfaces, improved design is worked out, as shown in Fig. 7. During the design process, the size of the plate, border, central circle, and reinforced ribs maintained the same volume to minimize the mass difference of three models to less than 5 %.

To simulate the dynamic conditions, and to reflect the dynamic characteristics of the base plate of the satellite, it is not only necessary to investigate the random response characteristics of the base plate but also significant to take into consideration the random response characteristics on the top of the cylinder where the antenna is located.

Fig. 5Base plate models of central-divergent distribution reinforced ribs (Plan A)

Fig. 6Base plate models showing distribution of reinforcing ribs (Plan B)

Fig. 7Improved design based on Plan B (Plan C)

3.2. Satellite FEM building

Based on the three models above, the FEM is built respectively. For plan A, the satellite FEM contains 34164 nodes, 24398 elements, 494 RBE2, 47 CONM2. For plan B, the satellite FEM contains 34114 nodes, 24362 elements, 494 RBE2, 47 CONM2. For plan C, the satellite FEM contains 34386 nodes, 24516 elements, 494 RBE2, 47 CONM2. During the vibration test, the satellite is fixed with the vibration table through the mounting surface of the docking ring. Therefore, in the finite element analysis, a node is set at the center of the docking ring. And then, the node and mounting surface of the docking ring are connected with the free degree of seven pressing points with the RBE2. During the analysis on the three plans, five sample nodes for vibration data collection are setting on the docking ring, base plate and cylinder, the specific locations are shown as in Fig. 8.

3.3. Comparative analysis of dynamic characteristics of above models

The dynamic analysis of the supporting base plate consists of modal analysis [18], and random response analysis. The modal analysis focuses on the frequencies of the main formations, including the reciprocating frequency and torsional frequency of the first-order movement in the $X$, $Y$, and $Z$ directions. The random response analysis concentrates on the comparison of the RMS value of the random response vibration for the three plans [19, 20]. Besides the comparison among the plans, we also compare the optimized model with initial model in which the base plate is a stock structure without lightweight design.

Fig. 8Sample nodes in FEM of satellite

3.3.2. Random response analysis

Smart video satellites adopt a carry-on launching model, and they are located at the margin of the supporting cabin in fairing of launch vehicle. During the launching process, the satellite encounters severe dynamic conditions. The excitation for random vibration are shown in Table 3 and Fig. 9.

By using Patran/Nastran [19], twelve cases are studied, which consist of a random response analysis of the $X$, $Y$, and $Z$ directions of plan A, plan B and plan C and the initial model. Two sample points from the base plates and two sample points from the cylinder top are selected to generalize the analytic result. An excitatory input point is also selected for contrast. The analytic results for each direction are presented as in Figs. 10, 11, and 12.

Table 3The data of random vibration excitation of smart video satellite

	Frequency range (Hz)	Acceptance conditions
		Power Spectrum Density ($g^2$ / Hz)	Total RMS of acceleration (g)
Value	20-150	+3 dB/oct	7.17
	150-280	0.04
	280-320	0.15
	320-380	0.1
	380-850	0.05
	850-1000	0.02
	1000-2000	0.005
Direction	$X$, $Y$, $Z$
Time	1 min for each direction

Fig. 9Curve of random vibration excitation of smart video satellite

Fig. 10RMS value of random vibration response analysis in X direction

a) Plan A

b) Plan B

c) Plan C

Fig. 11RMS value of random vibration response on base plate sample points in Y direction

a) Plan A

b) Plan B

c) Plan C

Fig. 12RMS value of random vibration response on base plate sample points in Z direction

a) Plan A

b) Plan B

c) Plan C

In the finite element analysis, different nodes can take different response values, even on neighboring nodes. To make a rational quantification, we adopt a mean value (accurate to the second decimal place) to compare the result. According to Table 4, the average response value of each analysis and the SAD value and variation range between each plans are presented. By considering the comparing results, the conclusions are drawn as follows:

Firstly, among the optimized three plans, plan A sees the largest response in three directions. However, compared with the initial model, the cylinder in $Y$ direction of plan A sees the largest declining response value with 7.07 g and the largest declining rate with 38.34 %. We can draw a conclusion that the SOTO plan and MOTO plan are both effective in reducing vibration to a great extent.

Secondly, it shows that compared with plan A, the random response value of the base plate and cylinder decline noticeably in Plan B, and that the $Y$ direction see the largest declining rate. The base plate’s random response value declines by 31.78 % and the value of the cylinder declines by 23.36 %. The declining values are 3.15 g and 3.87 g, respectively. 33.16 % is the largest declining rate for the satellite, representing the random response of the cylinder in the Z direction. Therefore, it can see that plan B is superior to plan A in reducing vibration. That is to say, MOTO is more conducive to vibration reduction.

Finally, compared with plan B, the random response value of the base plate and cylinder raised slightly in plan C. The $Y$ direction see the largest raising rate in cylinder which increase value is 0.23 g and variation range is 1.2 %. The largest variation range is 2.17 % representing the base plate of $Y$ direction and the raising value is 0.14 g. These comparing results show that plan C is larger than plan A in vibration value, but the variation range is so little enough that there is rarely difference for the satellite’s devices in mechanical environment between plan B and plan C. So, we can conclusion that plan C remain the optimum mechanical property of plan B.

Taking results into consideration, plan C can both meet the stiffness requirement and optimizes the vibration of the structure and satellite. Therefore, it is adopted in the design of the base plate of the video satellite.

Table 4Random response results

	Base plate			Cylinder
Direction	$X$	$Y$	$Z$	$X$	$Y$	$Z$
Initial model	13.05	12.63	11.97	21.37	19.86	18.44
Plan A	8.54	9.61	7.61	16.81	16.57	11.37
Plan B	7.53	6.6	5.36	12.85	12.63	7.51
Plan C	7.6	6.66	5.43	13	12.7	7.62
Initial Model/Plan A	4.51	3.02	4.36	4.56	3.29	7.07
SAD value
Initial Model/Plan A	0.3456	0.2391	0.3642	0.2134	0.1657	0.3834
Variation range
Plan A/Plan B	1.01	3.15	2.25	3.96	3.87	3.77
SAD value
Plan A / Plan B	0.1183	0.3178	0.2957	0.2356	0.2336	0.3316
Variation range
Plan B/ Plan C	0.07	0.06	0.07	0.15	0.07	0.11
SAD value
Plan B/ Plan C	0.0093	0.0091	0.0131	0.012	0.0055	0.0146
Variation range

3.4. Strength checking

3.4.2. Analysis of satellite overload

Overload analysis in 3 directions of the satellite is carried out with the conditions listed above. The Stress contour is shown in Fig. 13.

Analysis results show that 34.9 MPa with the $X$ maximum stress, 40.1 MPa with the $Y$ maximum stress and 69.8 MPa with the $Z$ maximum stress. The maximum stresses of the 3 directions are all happened to elements of the base plate where fixed with the docking ring.

The material of the base plate and docking ring is aluminum alloy which failure stress is 420 MPa. The structural safety margin is calculated taking into account the safety coefficient 1.8, shown in Table 5.

Table 5Calculation of structural safety margin

Operating stress (MPa)	Safety coefficient	Design load (MPa)	Failure stress (MPa)	Safety margin
69.8	1.8	125.64	420	2.34

Fig. 13Analysis of the satellite overload

a) Stress contour of $X$ direction

b) Stress contour of $Y$ direction

c) Stress contour of $Z$ direction

Based on the static and dynamic analysis above, two following conclusions will be gotten:

Fist, dynamic analysis shows that Plan C is able both to meet the requirements of stiffness, and has a great effect on the vibration damping of the satellite structure.

Second, static analysis shows that Plan C can satisfy the requirement of the satellite structure strength. According to the above two conclusions, Plan C is chosen for use in the design of the satellite’s base plate. The test structural sample is manufactured based on Plan C, shown as Fig. 14.