Dynamic Response and Optimal Design of Radio Telescope Structure under Wind Load Excitation

Abstract

The dynamic response of a radio telescope structure under wind load excitation significantly impacts the accuracy of signal reception. To address this issue, this study established a parametric finite element model of a radio telescope to simulate its dynamic response under wind load excitation. An improved Latin hypercube sampling method was applied in the design of experiments (DOEs) to optimize the structural dimensional parameters of various components of the radio telescope with the aim of reducing the dynamic response to wind load. A response surface model and multi-objective genetic algorithm (MOGA) were employed for multi-objective structural optimization of the radio telescope structure. The findings reveal that the thickness of the stiffening ribs, the length of the side of the square hollow pole, the thickness of the middle pole, and the inner diameter of the thin pole are the most influential structural parameters affecting the first-order frequency (F₁), second-order frequency (F₂), maximum deformation in the x-direction (D_X), and maximum deformation in the z-direction (D_Z) of the radio telescope, respectively. Optimizing the radio telescope results in a 40.00% improvement in F₁ and a 24.16% enhancement in F₂, while reducing D_X by 43.94% and D_Z by 64.25%. The study outcomes offer a comprehensive scheme for optimizing the structural dimensional parameters of various radio telescope components in regions characterized by multiple wind fields.

1. Introduction

To ensure high precision in the transmission and reception of radio telescope signals, the overall structure of the radio telescope is usually required to have minimal deformation and avoid structural resonance [1,2]. However, radio telescopes are usually situated in mountainous areas to accurately receive signals from outer space, which introduces a substantial dynamic response due to wind-induced loads. This dynamic response can significantly impact the precision of radio telescopes [3]. Therefore, optimizing the dimensional parameters of the radio telescope structure to minimize deformation and prevent resonance caused by wind-induced loads is imperative. Investigating the relationship between these dimensional parameters and the dynamic response to wind loads and subsequently optimizing them can greatly enhance the accuracy of radio telescopes in areas with varying wind conditions.

Several factors affecting the operating accuracy of radio telescopes have been studied, including wind loads [4,5,6,7], solar thermal load [8,9], and rail unevenness [10]. Various strategies have been proposed to mitigate the effects of these adverse factors on the operating accuracy of radio telescopes. For instance, Fu et al. [11] enhanced the pointing accuracy of the Tianma radio telescope by incorporating a novel inclinometer measurement system into the classical pointing model, resulting in a 65% improvement. To address issues with surface accuracy caused by the reflector’s large size and weight, Yan et al. [12] indirectly improved the radio telescope’s working accuracy by simplifying the reflector’s structure and reducing its weight. Sun et al. [13] employed a new algorithm to fine-tune large radio telescope panels with a main reflector, improving the accuracy of the antenna reflector profile from 0.28 mm to the current 0.19 mm. However, none of these studies have considered the structural vibrations resulting from wind load excitation. Furthermore, there is a scarcity of research dedicated to enhancing the working accuracy of radio telescopes by reducing the structural dynamic response.

Dynamic responses of structures under wind loads have been extensively investigated by scholars. For instance, Le and Caracoglia [14] investigated the stochastic structural dynamic response of a high-rise building subjected to transient unsteady winds employing a Wavelet–Galerkin analytical approach. Guo et al. [15] investigated the load transfer along the transmission path and the mechanical energy distribution in the dynamic response of a wind turbine under random wind loads, while also discussing the effects of the stiffness and motion of the support tower on the overall system response. Cao et al. [16] numerically studied the dynamic response of semi-rigid timber frames under wind load and analyzed the effect of uniform and non-uniform mass distribution on the acceleration at the top of the frame. Xie et al. [17] explored the characterization of the downwind and crosswind dynamic response of super high-rise buildings under the action of typhoons. Zhou et al. [18] delved into the effect of facade stiffness on the dynamic response of the cable network structure under wind load using ANSYS finite element software. However, few scholars have employed the DOE method based on the finite element model and Latin hypercube sampling method [19,20] to investigate the effects of the structural dimensional parameters of different parts in the radio telescope on the dynamic response under wind load excitation.

Once the relationship between the structural dimensional parameters of different parts in the radio telescope and its overall dynamic response is established, optimization algorithms can be utilized to refine these parameters, thereby minimizing the structure’s dynamic response to wind load excitation. Currently, multi-objective optimization algorithms are the most commonly used approach for optimizing structures with multiple parameters [21,22]. Scholars have employed this method to optimize the structural parameters of various systems, such as torsional flow heat exchangers [23,24,25], vacuum vessel sector multiple bottom support [26], antenna structure [27,28,29], wind turbine [30,31,32,33,34], and satellite [35], to achieve optimal performance. However, there is limited literature exploring the use of multi-objective optimization algorithms for optimally designing the structural dimensions of different parts of radio telescopes under wind load excitation.

Therefore, this study aims to identify the structural dimensional parameters of different components within a radio telescope structure to minimize its dynamic response to wind load excitation. The approach involves finite element analysis and a multi-objective optimization algorithm. The paper is organized as follows: Section 2 presents the development of a parametric finite element model of the radio telescope, along with modal test experimental design and wind load excitation random vibration simulation. The mesh convergence test and finite element model validity assessment are performed in Section 3. In Section 4, the structural dimensions of various radio telescope components are selected as design variables, and deformation and frequency are designated as optimization objectives and constraints. Section 5 focuses on the DOE design of the radio telescope’s structural dimension parameters using an improved Latin hypercube sampling method. It also verifies the validity of the response surface model and analyzes the relationship between each design variable and the objective values. Finally, the structural dimension parameters of the radio telescope are optimized through sensitivity analysis using the MOGA tool in ANSYS Workbench. The results of this study provide theoretical support for optimizing the structural dimensions of radio telescopes in areas with fluctuating wind conditions.

2. Structure and Numerical Model

2.1. Description of the Structure

The optimized design of the radio telescope with structural parameters is shown in Figure 1a. The total height of the radio telescope is 25 m, which includes the lightning rod. The reflector plane possesses a height of 20 m and a width of 15 m. Figure 1a provides an overview of the radio telescope’s components, primarily comprising the antenna back structure, reflector panels, azimuthal turnhead, and pedestal. During operation, wind loads exert a significant influence on the radio telescope, resulting in a notable dynamic response, characterized by vibration and deformation of the overall structure. Therefore, optimizing the dimensional parameters of different components within the radio telescope structure is expected to mitigate the dynamic response under wind load excitation. In this study, the radio telescope’s antenna back structure primarily consists of a combination of thin, medium, thick, and square hollow poles, while the base comprises three cylindrical segments with identical wall thicknesses, seamlessly joined.

2.2. FE Model and Boundary Condition

To investigate how various structural dimensions of the radio telescope affect its response under wind load excitation, the finite element method (FEM) [36] was adopted in this study to investigate the stochastic vibration analysis of the radio telescope under wind load excitation. Figure 1b showcases the developed finite element model. The poles used in the finite element model (thin, medium, thick, and square hollow pole) were modeled using beam elements, while the panels of the reflector surface, azimuth turnhead, and pedestal were modeled using shell elements. The material properties for this finite element model were based on structural steel, and the concrete parameters were taken according to the actual engineering materials. Subsequently, the entire radio telescope structure underwent hexahedral meshing using the mapping method, as depicted in Figure 1c. Following verification of mesh independence, the mesh size for the developed finite element model was determined to be 0.21 m. Additionally, in line with the radio telescope project’s installation process, the connections at the pole joints were fixed, and the bottom of the cylindrical support was fixed in all degrees of freedom.

2.3. Modal Test

To facilitate the subsequent analyses of the radio telescope’s dynamic response to wind load, a modal analysis of the entire structure was conducted. The modal test was carried out at a pitch angle of 15° since larger angles would subject the structure to higher loads that could compromise the radio telescope’s durability. The drive motor of the radio telescope was used as an excitation source for the modal test in this investigation (Figure 2a). The torque input and velocity output values of the motor were obtained during the modal test and converted into frequency domain signals. Then, the transfer function was calculated based on the converted frequency domain values. Finally, the amplitude-frequency and phase–frequency curves of the transfer function were converted to obtain the resonant frequency of the radio telescope. Notably, the first-order, second-order, and third-order frequency values of the radio telescope are selected as the validation values of the finite element model in this study.

2.4. Wind Load Testing

The random vibration method [37] was used in this study to apply loads to the overall structure of the radio telescope to reduce the computational amount of the finite element model and simulate more accurately the effect of random wind loads. Moreover, the random vibration loads were directly applied to the overall structure of the radio telescope based on the Davenport wind speed spectrum [38]. Furthermore, this study investigated the effects of different wind load directions on the simulation results and found that the wind loads in the x- and z-directions were the most unfavorable to the overall structure of the radio telescope. Therefore, the wind loads in the x- and z-directions were considered as the target load values for the subsequent structural dimensioning to minimize the negative effects of wind loads. The Davenport wind speed spectrum used in this study is shown in Figure 2b, and its power density function is

$$S_v(n)=\frac{4k{\overline{v}}_0^2}{n}·\frac{x^2}{{(1+x^2)}^{4/3}}$$

(1)

where x = 1200 n/${\overline{v}}_0^2$; ${\overline{v}}_0$ is the average wind speed at a height of 10 m, which is taken as 20.9 m/s in this study according to the Load Code for the Design of Building Structures GB50009-2012 [39]; k is a coefficient reflecting the roughness of the ground, which is taken as 0.00129 in this study according to reference [40]; and n is the frequency of the pulsating wind, with a value in the range of 10⁻³–10² Hz.

3. Finite Element Model Validation

The mesh convergence test of the finite element model based on the modal test was carried out in this study before evaluating the validity of the finite element model. The results of the test are presented in

Table 1. Modal analysis results for different mesh sizes.

Mesh Size	Model Mesh Diagrams	First-Order Modal Results	Second-Order Modal Results
0.15 m		1.245 Hz	1.739 Hz
0.21 m		1.245 Hz	1.777 Hz
0.25 m		1.251 Hz	1.786 Hz
0.4 m		1.271 Hz	1.818 Hz
0.8 m		1.284 Hz	1.892 Hz

. As can be seen in Table 1, the mesh change has only a slight effect on the first and second-order frequencies calculated by the finite element model. This indicates that the finite element model we have developed is convergent and valid. Consequently, the mesh size of 0.21 m was chosen in this study to reduce the computation time of the finite element model and to ensure the high accuracy of the computational results.

The results of the modal test results of the finite element model were compared with the experimental results in this study after determining the mesh size. The comparison results are presented in Figure 3. Figure 3a presents a comparison between the experimental results of the modal test and the finite element calculations for the first, second, and third-order frequencies. Specifically, for this study, the self-oscillation frequency of the radio telescope at a pitch angle of 15° was chosen for the comparison. As observed in Figure 3a, the results of the experimental modal test and finite element simulations were in close agreement, with errors of 4.58%, 1.72%, and 4.50%, respectively. These results affirm that the established finite element model can effectively replicate the real operational state of the radio telescope with reasonable accuracy. Consequently, this model can be used for simulating subsequent random vibration due to wind load and parametric structure design.

To further analyze the deformation characteristics of the overall structure of the radio telescope, the deformation cloud diagrams for the overall structure in the three types of frequencies are presented in Figure 3b–d. It can be observed that the maximum deformation in the first and second-order frequencies occurs at the top of the reflecting surface, while in the third-order frequency, it is observed at the front end of the receiver. Based on this, it is recommended to optimize the size of the rods in the region with the highest deformation mentioned above during the fabrication of radio telescopes.

4. Structure Parameter Design Optimization

To achieve optimal structural dimensions that minimize the dynamic response of the radio telescope under wind loading, thirteen structural dimensions were designated as design variables. These variables include thin pole wall thickness (T₁), thin pole inner diameter (R₁), middle pole wall thickness (T₂), middle pole inner diameter (R₂), thick pole wall thickness (T₃), thick pole inner diameter (R₃), square hollow pole cross-section width (L), square hollow pole wall thickness (T₄), reflector panels thickness (T₅), reflector panels stiffened rib height (H), reflector panels stiffened rib thickness (T₆), azimuth turnhead wall thickness (T₇), and pedestal wall thickness (T₈). Figure 4 illustrates various structural types used as design variables, along with the optimization process. To narrow the search domain for structural dimensions, practical engineering insights were combined with Equation (2) to further define the design variables:

$$\left\{\begin{array}{c}{\mathrm{T}}_1\in [1:8]\\ {\mathrm{R}}_1\in [12:45]\\ {\mathrm{T}}_2\in [2:15]\\ {\mathrm{R}}_2\in [30:130]\\ {\mathrm{T}}_3\in [3:20]\\ {\mathrm{R}}_3\in [150:200]\\ \mathrm{L}\in [160:350]\end{array}\right.\mathrm{,}\left\{\begin{array}{c}{\mathrm{T}}_4\in [5:45]\\ {\mathrm{T}}_5\in [1:12]\\ \mathrm{H}\in [90:200]\\ {\mathrm{T}}_6\in [8:32]\\ {\mathrm{T}}_7\in [10:50]\\ {\mathrm{T}}_8\in [12:36]\end{array}\right.$$

(2)

where the unit of length is mm. The first-order frequency (F₁), second-order frequency (F₂), maximum deformation in the x-direction (D_X), and maximum deformation in the z-direction (D_Z) of the overall radio telescope structure under wind load excitation are crucial for the optimization process. According to the wind speed spectrum in Figure 2, it can be found that the dynamic response of wind loading reaches a minimum and remains constant when the self-oscillation frequency of the structure exceeds 1 Hz. Moreover, the General code for steel structures (GB55006-2021) [41] and the stable operational requirements of the radio telescope stipulate that the maximum deformation of the structure must not exceed 210 mm. Consequently, based on the above analysis, this study imposed constraints on the values of the four target parameters, establishing the target equation as Equation (3).

$$\left\{\begin{array}{c}{\mathrm{F}}_1\ge 1.6\mathrm{Hz}\\ {\mathrm{F}}_2\ge 2.0\mathrm{Hz}\\ {\mathrm{D}}_{\mathrm{X}}\le 200.0\mathrm{mm}\\ {\mathrm{D}}_{\mathrm{Z}}\le 200.0\mathrm{mm}\end{array}\right.$$

(3)

In this study, once the design variables and objective functions were established, the structural parameters were globally optimized based on the DOE and MOGA functions in ANSYS Workbench. The specific optimization process is outlined in Figure 5 and comprises the following six steps:

(1)

Parametric finite element model construction: thirteen parametric variables, such as T₁, T₂, T₃, L, T₄, T₅, H, T₆, T₇, and T₈, were defined to carry out the evolution of structural dimensional parameters in different parts of the radio telescope. A parametric finite element model of the radio telescope, complete with fixed boundary conditions and loading specifications, was established in ANSYS.

(2)

Design of experiments (DOEs) execution: the design space, defined by Equation (2), was utilized to generate the dimensions for each structure. During this process, both the Latin hypercube sampling method [42] integrated into the ANSYS Workbench 2020R2 software and an improved Latin hypercube sampling method based on Ye et al. [43] were employed to execute DOE for the structural parameters. It is worth noting that the improved Latin hypercube sampling method is based on a principle of designing high-performance small-size base samples, determined by the maximum/minimum distance criterion from successive local enumeration, and then rapidly generating large-size test samples by “translating” the base samples. This approach effectively resolves the issues of time-consuming computations and low efficiency encountered in traditional optimal test design methods.

(3)

Finite element analysis: the radio telescope was analyzed in ANSYS, extracting key analyses like maximum deformation and self-oscillating frequency to evaluate performance at each design point.

(4)

Construction of response surface: a response surface model was built to correlate the 13 design parameters with the 4 target values reflecting the dynamic response of the radio telescope structure. This response surface was formed using the genetic aggregation algorithm integrated into the ANSYS Workbench software, considering the constraints provided in Equation (3). The genetic aggregation response surface, as illustrated in Equation (4), takes the form of an ensemble, represented as a weighted average of various metamodels.

$${\widehat{y}}_{ens}(x)={\displaystyle \sum _{i=1}^{N_M}{w}_i,}{\widehat{y}}_i(x)$$

(4)

where ${\widehat{y}}_{ens}$ = prediction of the ensemble, ${\widehat{y}}_i$ = prediction of the i-th response surface, $N_M$ = number of metamodels used, and $N_M\ge 1$, $w_i$ = weight factor of the i-th response surface. The weight factors satisfy:

$${\displaystyle \sum _{i=1}^{N_M}{w}_i=1}\mathrm{and}{w}_i\ge 0,1\le i\le N_M$$

(5)

(5)

MOGA parameter optimization: the optimization of the structural dimensional parameters of the radio telescope to minimize the dynamic response was carried out using the MOGA optimization function. The guidelines for this procedure were referenced from the DesignXplorer User’s Guide in ANSYS Workbench 2020R2 software [44].

(6)

Verification and output of optimal values: the optimization process was halted, and optimal values were recorded when the error between the predicted value and the actual calculated value fell below 5%. If the error exceeded this threshold, the process reverted to the second step for reevaluation.

5. Results and Discussion

5.1. Response Surface Validity Verification

In this section, the accuracy of the response surface models developed based on the thirteen design variables and the four objective functions was assessed. Based on the genetic algorithm, the response surface models of the design variables and the target values were established [45]. Two sampling methods, Latin hypercube sampling supplied with ANSYS Workbench 2020R2 software and improved Latin hypercube sampling, were compared to determine their impact on prediction accuracy. The results are summarized in Figure 6, where the prediction accuracies for different target values are presented. It was noted here that a total of 283 sample points was acquired using the Latin hypercube sampling supplied with the ANSYS Workbench software, whereas 105 sample points were collected using the improved Latin hypercube sampling method. To evaluate the accuracy of the prediction model, three coefficients were considered: the coefficient of determination (R²), root mean squared error (RMSE), and Pearson correlation coefficient (PCC). These coefficients were calculated by Equations (6)–(8).

$$R^2=1-\frac{{\displaystyle {\sum }_{i=1}^n{(y_i^{\prime }-y_i)}^2}}{{\displaystyle {\sum }_{i=1}^n{(y_i-\overline{y})}^2}}$$

(6)

$$RMSE=\sqrt{\frac{{\displaystyle {\sum }_{i=1}^n{(y_i^{\prime }-y_i)}^2}}{n}}$$

(7)

$$PCC=\frac{E(y_i^{\prime }-{\overline{y}}^{\prime })(y_i-\overline{y})}{\sqrt{{\displaystyle {\sum }_{i=1}^n{(y_i^{\prime }-{\overline{y}}^{\prime })}^2}}\sqrt{{\displaystyle {\sum }_{i=1}^n{(y_i-\overline{y})}^2}}}$$

(8)

where $y_i^{\prime }$, $y_i$, ${\overline{y}}^{\prime }$, and $\overline{y}$ are the predicted value, true value, average of predicted values, and average of true values, respectively.

It can be observed from Figure 6 that the computed values of the finite element model and the predicted values of the response surface model are closely clustered around the diagonal line, indicating a high degree of similarity between the two. Notably, the RMSEs of the predicted and observed values of the 4 target values of the dynamic response are close to 0, while the R² and PCC are above 0.99, indicating that the response surface model established based on the genetic aggregation algorithm has high accuracy. Therefore, the relationship between each design variable and the target value as well as the parameter sensitivity analysis can be analyzed based on the established response surface model. Furthermore, compared with Figure 6a–d and Figure 6e–h, it can be seen that the improved Latin hypercube sampling method proposed in this study is superior to the Latin hypercube sampling method in the ANSYS Workbench 2020R2 software, both in terms of sampling number and accuracy. Therefore, the improved Latin hypercube sampling method was selected for parameter optimization and sensitivity analysis in subsequent analyses.

5.2. Parameter Sensitivity Analysis

After determining the accuracy of the developed response surface model, the one-to-one correspondence between the 13 design variables and the 4 target values (F₁, F₂, D_X, and D_Z) was analyzed.

5.2.1. First-Order Frequency (F₁)

Figure 7 presents the relationship between the 13 design variables and F₁. It is evident from the figure that with the increase in T₁, R₁, T₅, H, T6, and T8, F1 shows a gradual decreasing trend. In contrast, an increase in T₂, R₂, T₃, R₃, L, T₄, and T₇ results in a synchronous increasing trend in F1. Therefore, to increase the F1 of the radio telescope under wind load excitation and to reduce the manufacturing cost, it is suggested to avoid taking too large values of T₂, R₂, T₃, R₃, L, T₄, and T₇ to decrease the values of T₁, R₁, T₅, H, T₆, and T₈. Based on the trend observed in Figure 7c–e, T₂ should be limited to around 7 mm, R₂ around 95 mm, and T₃ should be no more than 16 mm.

5.2.2. Second-Order Frequency (F₂)

Figure 8 illustrates the relationship between the 13 design variables and F₂. As can be observed from Figure 8, with the increase in T₁, R₁, T₂, R₂, T₅, H, T₆, and T₈, F₂ shows a gradual decrease. An increase in T₃, R₃, L, T₄, and T₇, results in a synchronous increase in F₂. Consequently, to ensure the increase in F₂ of the overall structure in the radio telescope under wind load excitation and to reduce the manufacturing cost of the radio telescope, a smaller value of T₁, R₁, T₅, H, T₆, and T₈ is recommended to be selected with the need to avoid taking too large values of T₃, R₃, L, T₄, and T₇ as well. In this case, based on the trends of T₁, R₁, T₃, and T₄ compared to F₂ in Figure 8a,b,e,h, the value of T₁ is not recommended to exceed 2 mm, R₁ should not exceed 16 mm, T₃ about 16 mm, and T4 about 30 mm.

5.2.3. The Maximum Deformation in the X-Direction (D_X)

The relationship between the 13 design variables and D_X is presented in Figure 9. It shows that with the increase in T₁, R₁, T₂, R₂, L, T_7, and T₈, D_X decreases initially and then increases. To minimize the maximum deformation in the x-direction, it is recommended to select T₁, R₁, T₂, R₂, L, T₇, and T₈ in the trough area of D_X while avoiding large values of T₃, R₃, T₄, T₅, H, and T₆. Based on the trends observed, T₁ should not exceed 2 mm, R₁ should be around 33 mm, T₂ around 14 mm, L around 225 mm, T₆ around 10 mm, T₇ around 40 mm, and T₈ around 18 mm.

5.2.4. The Maximum Deformation in the Z-Direction (D_Z)

The relationship between the 13 design variables and D_Z is presented in Figure 10. It indicates that with the increase in T₁, R₁, R₂, R₃, L, T₄, T₅, T₇, and T_8, the D_Z initially increases and then decreases. To reduce D_Z, it is recommended to avoid selecting T₁, R₁, R₂, R₃, L, T₄, T₅, T₇, and T₈ values that are in the peak D_Z region and to select smaller values. For T₂, T₃, H, and T₆, D_Z exhibits a multi-peak behavior. To minimize D_Z, T₂ should be about 15 mm, T₃ about 3 mm, H about 90 mm, and T₆ about 7 mm.

5.2.5. Sensitivity Analysis

Sensitivity analyses were conducted to quantify the effects of each design variable on the target values, and the sensitivity coefficients were further calculated. As can be clearly seen in Figure 11, the sensitivity coefficients of T₇ on the overall structure, F₁ and F₂, are almost 0, indicating that changes in T₇ do not significantly affect the self-oscillation frequency of the structure and a reduction in the changes in T₇ in the subsequent design of the structural dimensions is suggested.

Furthermore, based on the law that large sensitivity coefficients have larger impacts, as can be noted from Figure 11, T₁ and R₁ have the largest impact on D_Z and the smallest impact on F₂; T₂ and R₂ have the largest impact on D_X, and the smallest on F₂; T₃ and R₃ have the largest impact on D_Z and the smallest impact on F₁; L and T₄ have the largest impact on F₂, but L has the smallest impact on D_X, while T₄ has the smallest effect on D_Z; H has the largest effect on F₁ and D_X, and the smallest effect on D_Z; T₆ has the largest effect on D_X, and the smallest effect on D_Z; T₇ has the smallest effect on F₂ and the largest effect on D_Z; and T₈ has the largest effect on F₁, and the smallest effect on D_Z. Based on the above analysis, several measures are suggested as follows to control the F₁, F₂, D_X, and D_Z of the radio telescope, respectively, so as to increase F₁ and F₂ and reduce D_X and D_Z to achieve the optimization goal.

(1)

To increase the F₁ of the overall structure for the radio telescope to a greater extent by decreasing T₁, R₁, T₅, H, and T₆ and increasing T₂, R₂, L, T₄, and T₈.

(2)

To increase F₂ of the overall structure for the radio telescope to a greater extent by decreasing T₁, R₁, T₂, R₂, T₅, H, and T₆ and increasing T₃, R₃, L, T₄, and T₈.

(3)

To reduce x-direction deformation of the overall structure for the radio telescope by decreasing T₁, T₃, R₃, L, T₄, T₅, H, T₆, and T₇ and increasing R₁, T₂, R₂, and T₈.

(4)

To reduce z-direction deformation of the overall structure for the radio telescope to a greater extent by decreasing T₁, R₁, T₃, R₃, H, and T₇ and increasing R₂, T₂, L, T₄, and T₈.

5.3. Optimization Results

In this study, three optimization points were selected as candidates for the structural parameters of the radio telescope. Based on this, structural analyses were carried out for the initial and optimized nodes as well as the four target values (F₁, F₂, D_X, and D_Z). To compare with the initial design points, the values of design variables corresponding to the initial and optimized points are summarized in

Table 2. Values of design variables for initial and optimized design nodes.

Design Node	T1 (mm)	R1 (mm)	T2 (mm)	R2 (mm)	T3 (mm)	R3 (mm)	L (mm)	T4 (mm)	T5 (mm)	H (mm)	T6 (mm)	T7 (mm)	T8 (mm)
Initial	3	33	5	55.5	6	174	200	10	1.5	150	16	16	20
Optimized point 1	1.98	34.89	7.94	107.02	4.92	158.51	260.55	16.33	1.32	105.15	9.24	47.84	15.14
Optimized point 2	2.42	34.79	7.68	109.71	6.95	165.25	256.87	15.28	1.98	101.29	9.92	48.68	14.12
Optimized point 3	2.65	31.64	7.98	103.24	5.40	180.26	254.62	20.13	1.74	114.34	8.51	49.87	17.70

, respectively. To more clearly analyze variations in the objective values after optimization, the optimization results of four objective values are presented in Figure 12. As can be observed from Figure 12a,b, after optimization, F₁ and F₂ of the overall structure for the radio telescope can be improved by up to 40.00% and 24.16%, respectively. This means that, by using the proposed optimization scheme, the self-oscillation frequency of the structure can be improved greatly, as well as the resonance of the structure can be avoided under the excitation of wind loads. In Figure 12c, it can be seen that the D_X of the overall structure for the radio telescope is significantly reduced compared to the initial value, up to 43.94% in optimization point 1. The same result can also be observed in Figure 12d, i.e., the D_Z of the overall structure for the radio telescope is also significantly reduced after optimization, up to 64.25% in optimization point 3. This indicates that the proposed optimization scheme has a significant effect in reducing the x-direction deformation and z-direction deformation of the overall structure of the radio telescope.

Based on the above optimization results, it can be demonstrated that the wind-resistant performance of the radio telescope can be significantly improved through structural optimization. The proposed methodology, utilizing improved Latin hypercube sampling and optimization, contributes to the enhanced accuracy and efficiency of radio telescopes operating in multiple wind fields.

6. Conclusions

This study aimed to mitigate the dynamic response induced by wind loads on a radio telescope’s structure by conducting multi-objective optimization of its dimensional parameters across various components. The parametric finite element model of the radio telescope was established in ANSYS and verified by mode tests. Then, the 13 structural parameters of the radio telescope were taken as the input parameters of the design variables, the maximum deformation and self-oscillation frequency were taken as the objective functions, the improved Latin hypercube sampling method was used to perform DOE on the structural dimensional parameters, and MOGA was used for global optimization and sensitivity analysis. Based on the above analyses, the following main conclusions can be drawn.

• Errors between the finite element model simulation results and experimental test outcomes remained below 5% in each of the three modal tests. This attested to the precision of the simulation results, ensuring their reliability when subjected to subsequent wind-induced random vibrations.
• Employing the improved Latin hypercube sampling method in the DOE process across various structural components of radio telescopes of differing sizes not only enhanced the accuracy of the response surface model but also significantly economized computational resources associated with MOGA.
• Leveraging the response surface model, the study established a direct relationship between the 13 design variables and the respective objective values. Furthermore, the study derived the permissible parameter ranges for each design variable to achieve optimal objective values.
• Sensitivity analysis results unequivocally identified the structural parameters with the most substantial influence on the overall characteristics of the radio telescope, including F₁, F₂, D_X, and D_Z, pinpointing T₆, L, T₂, and R₁ as the most impactful variables, respectively.
• The optimization of the radio telescope can increase F₁ by 40.00% and F₂ by 24.16% and decrease D_X by 43.94% and D_Z by 64.25%.

These findings offer an effective solution for designing wind-resistant radio telescopes, particularly in regions characterized by multiple wind fields. Nevertheless, it is imperative to acknowledge that, due to experimental constraints, the study exclusively considered a single pitch angle and two wind directions. As a consequence, future investigations are encouraged to explore the optimization of wind-resistant design across varying working angles and wind directions, thereby minimizing the impact of wind loading on the radio telescope’s operational accuracy.