Structural Design and Analysis of Telescope for Gravitational Wave Detection in TianQin Program

Abstract

Space gravitational wave detection, which could help humanity explore the mysteries of the universe, is a significant objective in the scientific world today, and several different countries and scientific groups have organized programs targeting its realization. The telescope for gravitational wave detection is a crucial component in the detection satellite, as it is the means of receiving and transmitting the interferometric laser beam; therefore, its structural design is very significant. This paper focuses on the telescope in the TianQin program. First, a structural design scheme is given based on a five-mirror optical system Then, some of the component’s parts are refined to improve its mechanical performance. Finally, a mechanical simulation analysis is performed to verify its reliability and feasibility during the rocket launch. The results reveal that the presented structural design scheme for the telescope is both safe and viable.

1. Introduction

Gravitational waves are ripples in the fabric of spacetime that were first predicted by Albert Einstein as a cornerstone of his General Theory of Relativity [1,2] when he found that linearized weak-field equations had wave solutions: transverse waves of space distortions that travel at the speed of light [3]. Thus, the origin of gravitational waves lies not in the force of gravity as Newton described it, but in Einstein’s revolutionary idea that mass and energy warp the four-dimensional continuum of spacetime. However, Einstein knew that gravitational waves would be so tiny that it would be almost impossible to capture or detect them. There was a long period of research silence on proving the existence of gravitational waves, until a significant discussion on the physical reality of gravitational waves at the Chapel Hill conference in 1957 aroused interest in the topic [4].

There are several challenges that must be overcome to achieve gravitational wave detection: first, there are several kinds of noise, such as displacement noise, optical noise, and electronical noise, which may ruin detection [5,6,7]; second, the signal strength of gravitational wave detection has reached the measurement limit of nature. This requires scientists to build the quietest and most stable ruler in human history to measure length changes even smaller than microscopic particles, and to distinguish the faint ripples of spacetime from the buzzing sound of the Earth.

The year of 2015, which should be embedded in the history of humankind, proved to be a remarkable era for astronomy and physics, with humans detecting the existence of gravitational waves via the laser interferometers of the LIGO program [8]. Each LIGO observatory consists of two 4 km-long perpendicular arms, where a powerful laser beam is split in two. Each beam travels down one of the arms, reflects off a mirror suspended at the end, and then travels back. Under normal conditions, when the beams return and recombine, they are carefully tuned to cancel each other out through a process called destructive interference, resulting in no light being detected at the photodetector. However, when a gravitational wave passes through the Earth, it minutely distorts spacetime itself, briefly stretching space in one direction and compressing it in the perpendicular direction. This infinitesimal change—far smaller than the width of a proton—alters the length of the arms, with one arm becoming slightly longer and the other slightly shorter. The detector uses this tiny change to decode the gravitational wave. Since the first detection in 2015, several exciting detections of gravitational waves have been reported [9,10].

In order to better detect gravitational waves, detectors should be put into space, where the background noise is reduced compared to that on the ground. LISA, a collaboration between ESA, NASA, and an international consortium of scientists and engineers, is a famous program that focuses on this work. Slated for launch in the mid-2030s, it will consist of three spacecraft flying in a triangular formation, 2.5 million km apart. By using laser beams to measure tiny distortions in spacetime with incredible precision, LISA will observe low-frequency gravitational waves inaccessible from Earth [11,12,13,14,15,16]. This mission will open a new window into the dark universe, revolutionizing our understanding of gravity, galaxy evolution, and the fundamental laws governing our cosmos. It promises to unveil the unseen history of the universe.

Chinese scientists and engineers have been working in this research hotspot for a number of years, founding two programs for space-based gravitational wave detection, Taiji [17] and TianQin [18], each comprising three detective satellites in space positioned at the vertices of a triangle.

Table 1. The main parameters of the Taiji and TianQin programs.

	Arm Length	Displacement Measurement Accuracy	Power of Laser	Aperture of Telescope	Residual Acceleration
Taiji	3 × 106 km	5–10 pm Hz−1/2	2 W	≤50 cm	3 × 10−15 m/s2/Hz1/2
TianQin	1.7 × 105 km	1 pm Hz−1/2	1–2 W	≤35 cm	1 × 10−15 m/s2/Hz1/2

lists the main parameters of these two programs [19,20,21,22].

The telescope is an essential component for detection, as it receives and emits the interferometric laser; thus, its layout and mechanical stability may have a significant effect on the quality and effectiveness of gravitational wave detection. The latest structural design of the telescope in the Taiji program was released in November 2024 by Shang Wang et al. [23]. According to their optical design, the structural layout of the telescope was to be an off-axis six-reflection configuration. In reality, it is an off-axis four-reflection structure with two reflective mirrors placed before and after the tertiary and quaternary mirror groups, respectively. Its primary mirror adopts a central single-point support configuration. The wavefront RMS of this telescope is designed to be within 1/20 λ (λ = 1064 nm). However, this paper concentrates more on the ground wavefront measurement and does not give a detailed structural analysis comprising different kinds of load cases. Apart from the structural design of the telescope in the Taiji program, an early structural layout of the telescope in the TianQin program was released in February 2023 by Shengyi Cao [24]. There are four reflective mirrors in this structural design, as well as a circular frame at the primary reflective mirror. Some structural elements like rods and support beams are adopted in this structural design scheme. This paper focuses on the optimization design of various supports within the telescope system. The support structures utilize carbon fiber design, but this paper does not address the issue of moisture absorption in carbon fiber structures and its impact on the system in space, nor does it discuss corresponding compensation measures. Additionally, the paper does not conduct various dynamic mechanical analyses such as sine vibration simulation and random vibration simulation.

Research performed on the telescope in the TianQin program to date has included elements such as stray light analysis, thermal stability analysis, and optical optimization [25,26,27]. The research emphasis of this paper is on the structural design and analysis of the telescope for the TianQin program.

2. Structural Design and Analysis Preparation

2.1. The Structural Design Requirements

The structural design of the telescope must meet several requirements, the most important of which are listed in

Table 2. The structural design requirements.

	Weight	Outline	Diameter of Primary Mirror	The First Natural Frequency	Maximal Deformation Under 1 g Load
Value	≤40 kg	≤600 mm × 600 mm × 900 mm	≤350 mm	≥200 Hz	≤10−5 m

. Moreover, the telescope must be safe under different load cases, which means that the stress must be lower than the mechanical limit. The details of the load cases are given in Section 3.

2.2. The Initial Structural Design

The initial structural design was based on optical design, resulting in a five-mirror system, as shown in Figure 1.

The five reflective mirrors, the connection plate, and the holding platform in the initial structural design scheme are all made of Zerodur. Zerodur is a special material with a very low coefficient of thermal expansion, making it ideal for optical instruments in space. The primary reflective mirror has a curvature radius of about 1300 mm and its surface is a paraboloid; the secondary reflective mirror has a curvature radius of about 50 mm and its surface is quadric; the surface of the third reflective mirror is a free-form surface; and the fourth and fifth reflective mirrors are both plane reflective mirrors, used to divert the light pathway.

The distance between the primary and secondary reflective mirrors is about 630 mm. A connection plate is used to connect the two, and, as the biggest part in the telescope, it must be carefully designed. This connection plate is like a cantilever beam, and, in general cases, it is made of metal due to its higher strength and stiffness. However, metal has a different coefficient of thermal expansion than glass, such as Zerodur; therefore, the connection plate is also made of Zerodur in this design. More details regarding the connection plate can be seen in Figure 2, which presents the connection plate in a transparent view, allowing for a clearer visualization of the reinforcing rib structure on the reverse side. The red surfaces in the figure represent bonding surfaces, including the bonding surfaces between the primary mirror and the connection plate, the secondary mirror and the connection plate, and the holding platform and the connection plate.

The initial connection plate has a varying section to ensure that the connection plate has varying bending stiffness, and some ribs are added in the bottom of the connection plate to enhance the stiffness more effectively. However, ribs may bring about new problems. One adverse trait is that the weight of the connection plate will also be larger, which may cause a bigger dynamic response. Therefore, a prudent tradeoff should be made.

There are three flexible supports in this initial scheme to absorb some of the excessive thermal stress and assembling stress. The bipod is considered an essential link in flexible supports, and, in this scheme, it is made of invar steel because of its low thermal expansion coefficient, good strength, sufficient stiffness, and excellent stability. The flexible support can be seen in Figure 3.

Table 3. The outline size and weight.

	Outline Size	Weight
Initial scheme	568 mm × 470 mm × 804 mm	37.138 kg

shows the outline size and weight of the initial structural design scheme. The properties of the structural part materials are given in

Table 4. The properties of the structural part materials.

	Density (kg/m3)	Young’s Modulus (GPa)	Poisson’s Ratio	Mechanical Limit	Coefficient of Thermal Expansion
Zerodur	2530	90.3	0.26	55 MPa (tensile strength)	0 ± 0.1 × 10−6
Invar steel	8130	147	0.24	280 MPa (yield strength)	1.2 × 10−6

.

2.3. The Fundamental Theory of Mechanical Analysis

In the subsequent sections of this paper, a mechanical analysis is performed for both the process of optimization and structural validation. ANSYS 16.0 was chosen as the software for the analysis. The fundamental theory for structural analysis is the Finite Element Method (FEM), which divides a whole structure into parts, then integrates the parts back into a whole. It discretizes a complex continuum structure into a finite collection of simple interconnected elements, and, through approximate analysis of each individual element, the governing equations for the entire structure are then assembled and solved.

Firstly, the structure is divided into several elements, each with a displacement function, strain function, and stress function, shown in Formulas (1)–(3), respectively, where $u(x,t)$ is the displacement function, $N(x)$ is the shape function matrix, $d^e(t)$ is the nodal displacement of the element, $B(x)$ is the strain-displacement matrix, and D is the elasticity matrix.

$$u(x,t)=N(x)d^e(t)$$

(1)

$$\epsilon (x,t)=B(x)d^e(t)$$

(2)

$$\sigma (x,t)=D\epsilon (x,t)=DB(x)d^e(t)$$

(3)

Based on the principle of minimum potential energy, the equilibrium equation of the element can be obtained by setting the variation of the total potential energy of an element to zero, as shown in Formula (4), where $M^e$ is the mass matrix of the element, $K^e$ is the stiffness matrix of the element, $C^e$ is the damping matrix of the element, and $f(t)$ is the nodal load matrix of the element. The damping matrix is usually given by model assumptions such as Rayleigh damping.${\Omega }^e$ represents an element domain.

$$M^e{\ddot{d}}^e(t)+C^e{\dot{d}}^e(t)+K^e{d}^e(t)=f^e(t)$$

(4)

$$M^e={\displaystyle {\int }_{{\Omega }^e} \rho N^{T}Nd\Omega }$$

(5)

$$K^e={\displaystyle {\int }_{{\Omega }^e}{B}^T}DBd\Omega$$

(6)

$$C^e=\alpha M^e+\beta K^e$$

(7)

Once the element equilibrium equation has been obtained, the global equilibrium equation (Formula (8)) can be assembled by all the element matrices and vectors according to the node numbers. This is the foremost equation in the FEM.

$$M\ddot{d}(t)+C\dot{d}(t)+Kd(t)=f(t)$$

(8)

In Formula (8), M is the global mass matrix, C is the global damping matrix, K is the global stiffness matrix, $d(t)$ is the global nodal displacement vector, and $f(t)$ is the global nodal load vector. In the following sections, modal analysis and several kinds of vibration analysis are performed; therefore, the FEM theory based on these analyses is briefly introduced in the following paragraphs.

FE analysis of modal analysis is based on the equilibrium equation of free vibration without damping, which can be seen in Formula (9). Formula (10) is the characteristic equation of Formula (9), and, when solved, it provides characteristic values and vectors. ${\lambda }_i$ is the characteristic value of the i order, ${\omega }_i$ is the angular frequency of the i order, and ${\varphi }_i$ is the vibration mode of the i order.

$$M\ddot{d}(t)+Kd(t)=0$$

(9)

$$(K-{\omega }^{2}M)\varphi =0$$

(10)

$${\lambda }_i ={{\omega }_i}^2$$

(11)

FE analysis of sine vibration typically involves solving the equations of motion in the frequency domain for a linearly elastic, harmonically excited system. For systems with proportional damping, the governing matrix equation is expressed by Formula (12).

$$(K-{\omega }^{2}M+i\omega C)d(\omega )=f(\omega )$$

(12)

$$\omega =2\pi f$$

(13)

$$H(\omega )={(-{\omega }^{2}M+i\omega C+K)}^{-1}$$

(14)

In Formula (12), M, C, and K represent the global mass, damping, and stiffness matrices, respectively; ω is the angular frequency, which is expressed in formula (13); d(ω) is the complex nodal displacement response vector; and f(ω) is the complex harmonic force vector. The frequency response function matrix, shown in Formula (14), can be solved from Formula (12). Based on this solution, the resonant frequencies and their corresponding amplification factors (Q-factors) at any nodal degree of freedom can be identified.

FE analysis of random vibration extends the deterministic framework into the probabilistic domain, where the input is characterized by a power spectral density (PSD) function. The core of the analysis involves computing the output response PSD matrix G_yy(ω), which is expressed in Formula (15), where G_aa(ω) is the input acceleration PSD matrix and H(ω) is the system’s frequency response function matrix. The root mean square (RMS) value of a response parameter y, such as stress, corresponding to one standard deviation (1σ), is obtained via Formula (16). This statistical output enables the estimation of probable peak responses via spectral methods.

$$G_{yy} (\omega )=H^{*}(\omega )[G_{aa} (\omega )]H^T(\omega )$$

(15)

$${\sigma }_{yy}=\sqrt{{\displaystyle {\int }_{f_1}^{f_2}{G}_{yy}(f)df}}$$

(16)

2.4. The Fundamental Theory of Hydroxide Catalysis Bonding

Because hydroxide catalysis bonding was selected in this study to connect glass parts, a brief introduction to this bonding theory is given here. Hydroxide catalysis bonding is a technique that can achieve direct atomic-level bonding of some materials, especially glass (e.g., Zerodur), through surface chemical reactions without introducing any adhesives or intermediate layers. The bonding mechanism is based on the dehydration and condensation of surface silanol (Si–OH) groups, forming a covalent siloxane (Si–O–Si) network identical in structure to the bulk material. There are four chemical steps in this bonding process: surface hydroxylation pretreatment, where the glass surface undergoes hydrolysis in an aqueous or alkaline environment to generate reactive silanol groups (Formula (17)); contact and dehydration condensation, where, after bringing the surfaces into close contact, adjacent silanol groups condense under mild heating, releasing water and forming covalent bonds (Formula (18)); interfacial network continuation, where the reaction propagates across the interface, extending the original silica network continuously (Formula (19)); and water diffusion and removal, where trace water generated during the reaction diffuses out of the interface, leaving no foreign residue (Formula (20)).

Si–O–Si (surfaces) + H₂O⟶2Si–OH

(17)

Si–OH + HO–Si⟶Si–O–Si + H₂O

(18)

[glass]–Si–OH + HO–Si–[glass]⟶[glass]–Si–O–Si–[glass] + H₂O

(19)

H₂O (interface)⟶H₂O (environment)

(20)

2.5. Improvement of the Initial Structural Design

Firstly, modal analysis was performed based on the initial structural design scheme. The first six natural frequencies and some main vibration modes are shown in

Table 5. The first 6 natural frequencies of the initial structural design scheme.

Order	1	2	3	4	5	6
Frequency (Hz)	268.70	326.74	391.58	450.53	518.01	684.58

and Figure 4.

Table 5 shows that the first natural frequency is 268.7 Hz, the second 326.74 Hz, the third 391.58 Hz, and the fourth 450.53 Hz. Figure 4 indicates that the connection plate dominates the lower-order vibrations, and increasing its stiffness should be a primary objective in the design.

Because of the cantilever beam structure, the free end where the secondary reflective mirror is situated may have a big dynamic response during the launch; therefore, some improvements and optimizations were performed to ameliorate this problem. Two structural design schemes are shown in Figure 5 and Figure 6.

Both structural design schemes for the connection plate add some ribs in the front section of the plate. Scheme A adopts an I-shaped beam in the front section, as I-shaped beams have a decent bending stiffness compared with other sections and are widely used in civil engineering and aerospace engineering. In addition, some tilted ribs are included in both flanks of the I-shaped beam in the front section of the connection plate. In contrast, scheme B uses a T-shaped beam as the main element in the front section, and some crossing ribs are added in the bottom of the connection plate. The weight of scheme A is 13.779 kg, and that of scheme B is 14.201 kg, though the weight difference between the two schemes is not obvious. Later, these two connection plates were added to the whole model. The weight of the whole telescope based on scheme A and scheme B was 38.001 kg and 38.423 kg, respectively. The modal analysis was based on two different models. A brief comparison of the initial scheme and the two optimized schemes can be seen in

Table 6. The natural frequencies of the initial structural design scheme and the two optimized schemes.

	Order	1	2	3	4	5	6
Frequency (Hz)	Initial scheme	268.70	326.74	391.58	450.53	518.01	684.58
	Scheme A	270.64	337.76	397.15	446.18	509.54	750.97
	Scheme B	275.74	349.76	403.40	457.93	517.00	735.01

.

The first four vibration modes can be seen in Figure 7 and Figure 8, revealing that the connection plate vibrates most in these modes. Figure 9 presents a more visual comparison, where the first six natural frequencies of the initial scheme are set as benchmarks; a bar is the deviated value between the new scheme and the initial one; a positive value means that the frequency ascends; and a negative value means that the frequency descends. As can be seen in this figure, the frequencies of the optimized schemes from the first order to the third order increase to some extent. Furthermore, scheme B has the upper hand over scheme A from the first order to the third order. When it comes to the fourth order, the frequency of scheme B rises, and that of scheme A slightly drops. At the fifth order, both frequencies decrease, but scheme A is worse than scheme B. At the sixth order, both frequencies escalate a lot. From this changeable tendency analysis, it can be concluded that scheme B is better; therefore, scheme B is adopted in the later analysis.

3. Structural Analysis of the Optimized Telescope

3.1. The Purpose of Structural Analysis

The purpose of structural analysis is to verify the credibility and viability of the structural design. In our case, the telescope for gravitational wave detection is designed to be sent to space via a rocket. In the rocket launch process, the telescope undergoes rigorous vibration. Therefore, different analyses based on various load cases have to be carefully performed to simulate the real load environment. In the following subsections, static analysis, thermal analysis, random vibration analysis, and sine vibration analysis are performed.

3.2. Some Information for Structural Analysis

The whole telescope is fixed to the satellite by three flexible supports; therefore, the bottom surfaces of the flexible supports were constrained in the structural analysis. Figure 10 shows the coordinate system and boundary condition. The following analyses are based on this model and coordinate system. Some more information regarding the FE analysis is shown in

Table 7. Some information for the FE analysis.

Constrained Situation	Types of Elements	Number of Elements	Number of Nodes	Damping Ration in Vibration Analyses
Six freedom degrees	Hexahedral and tetrahedral elements	69,304	142,926	0.05

.

For the subsequent analyses, solid elements—specifically, hexahedral and tetrahedral types—were used throughout the model. Considering the geometric dimensions and structural features, the finalized telescope model contained approximately 70,000 elements, and mesh refinement was applied in regions with significant geometric variations. Additionally, a minimum of three layers of elements were maintained through the thickness direction in areas such as stiffeners and ribs. In vibration analyses, the damping ratio was set as 0.05 according to our engineering experience. Parts made of Zerodur were assembled using hydroxide catalysis bonding, which directly joins two parts through chemical bonding without introducing foreign material at the interface. The bonding strength of Zerodur via this method is approximately 12 MPa. In the subsequent analyses, nodes at the bonded interface were initially connected by coupling their corresponding degrees of freedom. If the interfacial stress approached or exceeded 12 MPa under a given load case, further detailed simulation was performed for that specific condition using alternative contact and fracture models.

3.3. Static Analysis

Static analysis is a basic analysis for a variety of structures or apparatus. In this study, a static analysis under 1 g load was performed, where a gravitational load of 1 g was applied to the entire model as a gravitational field, providing an understanding of the displacement and stress in the structure. Generally speaking, 1 g load is a safe case; however, this analysis should not be excluded. Static load of 1 g was applied to the model in the +X, −Y, and −Z direction, respectively, as these three directions are representative of the telescope’s real orientations under Earth’s gravity during assembling, testing, and transport on the ground.

Table 8. The maximal displacement and stress in static analysis.

Direction	Maximal Displacement (×10−6 m)	Maximal Stress (MPa)
+X	3.032	2.821
−Y	7.135	5.069
−Z	3.092	1.992

and Figure 11, Figure 12 and Figure 13 show the result of the static analysis.

In the figures above, the maximal displacement is 7.135 × 10⁻⁶ m and the maximal stress is 5.069 MPa, both of which occurred when 1 g was applied in the −Y direction. The load case of 1 g in the −Y direction represents the most common situation because most optical measurements are performed under this load. The later wavefront RMS analysis was also based on this case. All values of displacement and stress were small in the X, −Y, and −Z directions. The static analysis, primarily conducted to verify that the telescope would not experience excessive deformation or stress during ground assembly, testing, and transportation, showed that the structural design scheme is viable. Furthermore, to ensure the overall wavefront aberration performance of the telescope, a brief analysis of wavefront RMS caused by mechanical deformation was performed. Taking into comprehensive consideration errors from the optical design, manufacturing, assembly, and wavefront measurement instruments, the overall wavefront aberration of the telescope system was expected to be better than 1/40 λ (λ = 1064 nm). This was ensured through careful and detailed optical design, mechanical manufacturing, and high-precision alignment. In particular, the optical design minimized the impact of external disturbances on wavefront aberration through optimized optical surface design. However, the relative positions of the primary and secondary mirrors are the most sensitive parts in the calculation of wavefront RMS. The maximal allowable relative position change of the primary and secondary mirrors is shown in

Table 9. The relative position change of maximal tolerance and the simulation result.

	Δx	Δy	Δz	Δα (Relative Pitch)	Δβ (Relative Yaw)
Maximal tolerance	8 × 10−6 m	8 × 10−6 m	2 × 10−5 m	4.8 × 10−5 rad	4.8 × 10−5 rad
Simulation result	≤10−10 m	4.736 × 10−6 m	5.183 × 10−7 m	3.423 × 10−5 rad	1.379 × 10−7 rad

, along with the simulation result of the relative position change under 1 g load in the −Y direction. The analysis indicated that the structural design meets the allowable wavefront RMS caused by mechanical deformation.

3.4. Thermal Analysis

Various measures are implemented to ensure the thermal stability of the telescope for gravitational wave detection in space, including the integration of sunshields, the deployment of active thermal control systems in the satellite cabin, and the installation of heating elements on the telescope mounting platform, enabling precise regulation of the on-orbit thermal environment. In the TianQin program, the temperature variation at critical locations of the telescope during orbital operation is designed to remain within ±5 mK. As the primary focus of this paper is structural design, detailed thermal control methodologies are not discussed. Nevertheless, to account for potential temperature fluctuations during ground tests or transporting phases, thermal analysis was conducted, with 22 °C as the reference temperature and structural responses evaluated under two thermal load cases, one where the temperature increases from 22 °C to 30 °C and one where the temperature decreases from 22 °C to 15 °C. The results are summarized in

Table 10. The maximal displacement and stress in the temperature fluctuation analysis.

	Maximal Displacement (×10−6 m)	Maximal Stress (MPa)
22 °C to 30 °C	1.377	4.775
22 °C to 15 °C	1.205	4.178

, showing that the maximal displacement and stress are 1.377 × 10⁻⁶ m and 4.775 MPa, respectively, in the temperature increase case. The cloud charts of displacement and stress in the temperature increasing case are presented in Figure 14. It is indicated that the telescope is safe under the ground temperature fluctuations.

3.5. Random Vibration Analysis

The real-world vibration environment that a satellite experiences during launch is inherently random, broadband, and unpredictable. The goal of random vibration analysis is not to predict the exact response, but rather to evaluate whether a satellite and its affiliated components can maintain structural integrity and functionality while being subjected to a sustained vibration that contains a wide spectrum of frequencies and intensities. In this kind of vibration, engineers are concerned about excessive stress. Because of its rigid mechanical environment, the telescope never participates in detection missions.

The input conditions of the random vibration analysis performed in this study are presented in

Table 11. The input conditions of random vibration analysis.

	20–100 Hz	100–600 Hz	600–2000 Hz	grms
Value	+3 dB/oct	0.05 g2 /Hz	−9 dB/oct	6

, where dB/oct describes the slope or rate of change of a spectrum curve (used by engineers to measure the decibel change in an amplitude (such as acceleration or power) when the frequency varies by one octave); dB is a logarithmic unit used to express a ratio calculated as 10 times the base-10 logarithm of the ratio of the two power spectral densities; oct denotes a doubling of frequency, for example, from 100 Hz to 200 Hz is one octave, and from 200 Hz to 400 Hz is also one octave; and g²/Hz is the acceleration power spectral density, the most critical quantity in random vibration analysis, describing the distribution density of vibration energy in the frequency domain. In the analysis, random vibration excitation was applied at the fixed constraint location, specifically, the bottom surfaces of the flexible supports. The grms (gravitational acceleration root mean square) input was 6. The grms represents the root mean square value of random vibration acceleration over the frequency history, which is equivalent to the standard deviation of the vibration signal.

There were two response points in this analysis: the first on the edge of the primary reflective mirror, and the second on the edge of the secondary reflective mirror. Figure 15 presents the location of the two response points.

The response curves are shown in Figure 16, Figure 17 and Figure 18. Figure 19 shows the stress (3σ stress) cloud charts of the telescope, revealing that the maximal values of stress have a 99.73% probability of not exceeding the values presented in the following results.

When random vibration load was applied in the X direction, the grms of response point 1 was 23.356 and the grms of response point 2 was 91.796, showing that the response was amplified significantly at point 2. The maximal stress was 187.11 MPa at one flexible support, which was less than 280 MPa, the yield limit of invar steel (from which the flexible supports were made). Except for at the flexible supports, the stress was less than 20 MPa, which was below the strength limit of Zerodur, indicating that all parts would be safe when load is applied in the X direction.

When random vibration load was applied in the Y direction, the grms of response point 1 was 15.610 and that of response point 2 was 92.118, showing that the response was also amplified significantly at point 2. The maximal stress was 254.24 MPa at one flexible support, which was less than 280 MPa, the yield limit of invar steel. Except for at the flexible supports, the stress was less than 28 MPa, which was below the strength limit of Zerodur, indicating that all parts would be safe when load is applied in the Y direction.

When random vibration load was applied in the Z direction, the grms of response point 1 was 17.616 and that of response point 2 was 28.635, neither of which was big. The maximal stress was 119.01 MPa at one flexible support, which was less than 280 MPa, the yield limit of invar steel. Except for at the flexible supports, the stress was less than 13 MPa, which was below the strength limit of Zerodur, indicating that all parts would be safe when load is applied in the Z direction.

In the random vibration analysis, response point 2 exhibited a big response, especially when random vibration load was applied in the X direction or Y direction. However, a highly dynamic response does not indicate a dangerous situation. The mass at the secondary reflective mirror was small, so the stress was not significant. The majority of the telescope parts are made of Zerodur and are connected by hydroxide catalysis bonding. The stress at these different bonding surfaces is more important; therefore, stress at these surfaces was extracted in particular. The different bonding surfaces were given different numbers: surface 1 was the surface between the primary reflective mirror and the connection plate; surface 2 was the surface between the secondary reflective mirror and the connection plate; surface 3 was the surface between the third reflective mirror and the holding platform; surface 4 was the surface between the fourth reflective mirror and the holding platform; surface 5 was the surface between the fifth reflective mirror and the primary reflective mirror; and surface 6 was the surface between the connection plate and the holding platform.

Table 12. Values of maximal stress at bonding surfaces.

	Maximal Stress (MPa)
	X	Y	Z
Surface 1	2.863	6.156	4.194
Surface 2	1.004	3.706	0.695
Surface 3	1.437	2.887	1.509
Surface 4	6.137	9.935	1.007
Surface 5	1.945	5.061	8.113
Surface 6	3.427	2.581	3.539

shows the maximal stress of the bonding surfaces, and Figure 20 presents a more visual comparison.

The maximal stress was 9.935 MPa at surface 4 when random vibration load was applied in the Y direction. However, all values of maximal stress were less than the bonding limit of hydroxide catalysis bonding, indicating that there would not be any crack disruptions at bonding surfaces during random vibration. Furthermore, the stress at surface 2 was very small, although the response was big at the secondary reflective mirror. According to the result of the random vibration analysis, this structural design scheme is safe and feasible.

3.6. Sine Vibration Analysis

Sine vibration analysis is a kind of harmonic response analysis. During launch, the engine of the rocket will produce significant thrust, and the impulse and vibration of the engine may make a significant difference to the telescope. This strong vibration is a kind of sine vibration in the low-frequency band, which is always lower than 100 Hz, and usually has large magnitude.

Table 13. The input conditions of the sine vibration analysis.

Frequency (Hz)	10–35	35–36	36–75	75–76	76–100
Value	32 m/s2	Transition	58.8 m/s2	Transition	45 m/s2

lists the input conditions of the sine vibration according to space engineering specifications. In the analysis, sine vibration excitation was applied at the fixed constraint locations, specifically, the bottom surfaces of the flexible supports.

Two response points were selected, as in the random vibration analysis. The response curves are shown in Figure 21, Figure 22 and Figure 23.

The amplification was slight when sine vibration load was applied in the X and Z directions. Although the case in the Y direction was slightly worse than that in the X and Z directions, its magnification was still very small. The maximal amplification factors were 1.13 and 1.34 at response point 1 and response point 2, respectively. To sum up, the amplification was not big because the first natural frequency was 275.74 Hz, which is far away from the stop frequency of the input vibration. This indicates that the telescope is safe under the load case of sine vibration.

4. Discussion

A detailed mechanical simulation comprising static analysis, thermal analysis, modal analysis, random vibration analysis, and sine vibration analysis was performed based on the optimized structural design scheme. All the values of deformation and stress were within acceptable ranges, showing the structural design scheme to be credible and viable, and the important parameters met the requirements of the structural design. A comparison of the parameters of the design and the requirements can be seen in

Table 14. The comparison between design requirements and design result.

	Weight	Outline	Diameter of Primary Mirror	The First Natural Frequency	Maximal Deformation Under 1 Gravity
Requirement values	≤40 kg	≤600 mm × 600 mm × 900 mm	≤350 mm	≥200 Hz	≤10−5 m
Design values	38.423 kg	568 mm × 470 mm × 804 mm	324 mm	275.74 Hz	7.135 × 10−6 m

.

5. Conclusions

This study focuses on the telescope for space gravitational wave detection in the TianQin program. First, an initial structural design scheme was devised. However, the modal analysis based on this initial structural design indicated that the connection plate should be improved to enhance the stiffness of the telescope. Therefore, some improvements and optimizations were performed to create a better connection plate. Finally, different kinds of mechanical analyses were conducted to verify the safety and credibility of the new structural design scheme, with the results revealing it to be safe and viable. A priority for future work is the practical implementation of the telescope, focusing on its manufacture and assembly.