Modeling the Effect of a Temperature Shock on the Rotational Motion of a Small Spacecraft, Considering the Possible Loss of Large Elastic Elements Stability

The paper investigates the issues of a small spacecraft large elastic elements temperature shock influence within the framework of a two-dimensional model of thermal conductivity, taking into account the possible loss of stability during the temperature shock. The loss of stability causes additional movements of large elastic elements sections. The work explores this additional movement. The results were obtained using the Sophie Germain equation for thin plates. The thermoelasticity problem is solved, in which the parameters of the relative motion of large elastic elements, including the loss of stability, are estimated. The finite element method was used to solve the problem. Additional microaccelerations resulting from the loss of stability are estimated. Recommendations are given to avoid the loss of stability during the temperature shock. The results obtained can be used in the design of small technological spacecraft. In particular, when selecting the design parameters of large elastic elements. The selection of these design parameters, taking into account the results obtained in the work, will help to avoid the stability loss of elastic elements.


Introduction
The temperature shock of large elastic elements can affect the rotational motion of spacecraft, creating additional disturbances. This can be a serious problem when implementing gravity-sensitive technological processes on board a spacecraft (Porter et al. 2021;Lyubimova et al. 2019;Belousov and Sedelnikov 2013). Studies show that the problem of temperature shock is relevant only for small spacecraft, where the mass fraction of elastic elements in the total mass is the most significant (Shen et al. 2013;Taneeva et al. 2021;Sedelnikov and Orlov 2020). This is also confirmed by experimental data for the middle class technological spacecraft (Perminov et al. 2022;Shevtsova 2021;Sedelnikov 2014). When implementing gravity-sensitive processes, there is no effect of temperature shock on the level of microaccelerations of the internal environment (Perminov et al. 2021;Sharifulin and Lyubimova 2021;Anshakov et al. 2018).
A slightly overestimated evaluation of microaccelerations from the temperature shock is presented in (Sedelnikov and Orlov 2021;. It is based on the analysis of the most dangerous situation when the elastic element is in a flat state when leaving the Earth's shadow, and the normal to its undeformed surface is strictly perpendicular to the direction to the Sun. In (Sedelnikov et al. 2022), the degree of this overestimation is analyzed and it is noted that it depends on the initial deflection of the elastic element at the time of the temperature shock onset. The deflection can be caused by various factors, and the most significant are the natural oscillations of large elastic elements. If this deflection is significant, then the one-dimensional model of thermal conductivity may incorrectly describe the process of temperature shock (Sedelnikov et al. , 2022. In this case, the two-dimensionality of the thermal conductivity problem should be taken into account. On the other hand, a significant value of initial deflection and subsequent deformations due to the temperature shock, as noted in Sedelnikov et al. (2022), can lead to a loss of stability of the elastic element. This determines the presence of additional relative movements of the elastic element parts. The appearance of additional microaccelerations is a consequence of such movements. This issue requires a separate detailed consideration both in the aspect of assessing additional microaccelerations from the loss of stability of the elastic element, and in the aspect of developing recommendations to prevent the loss of stability in the design of the small technological spacecraft (Taneeva 2021).
It should be noted that the problem under consideration is broader than the evaluation of microaccelerations. Thus, when solving various tasks in space, small spacecraft with a single elastic element are used more frequently. A striking example of such spacecraft is the small Starlink spacecraft (McDowell 2020).
After the end of the active life of a small spacecraft, it turns into space debris, which is subject to cleaning from orbit (Aslanov and Ledkov 2022;Trushlyakov and Yudintsev 2020). Space debris is cleaned in different ways (Priyant and Surekha 2019). In some cases, it is proposed to use cable systems with various ways of fixing the cable (Botta et al. 2017). In this case, the temperature shock of large elastic elements can affect the rotational motion of the transported small spacecraft. The disturbing moments that arise in this case are quite capable of disrupting the connection between the cable and the small spacecraft being transported.

Mathematical Model of Thermoelasticity Considering Possible Loss of Stability
Let us consider the temperature shock when a small spacecraft exits the Earth's shadow, considering the elastic element as a homogeneous plate (Fig. 1).
To do this, we will set the third initial boundary value problem of thermal conductivity in the form of Sedelnikov et al. (2022): The system of Eq. (1) assumes the constancy of the thermophysical properties of the elastic element (the coefficient of thermal conductivity λ, the coefficient of temperature conductivity a, the degree of blackness ε, the length of the elastic element l in the coordinate system associated with the middle surface of the elastic element in the undeformed state). The presence of a temperature gradient in the direction of the z axis ( Fig. 1) is ensured due to fundamental differences in the boundary conditions of the third kind (Eqs. (2) and (3) of system 1). The temperature gradient in the x -axis direction is created due to the uneven heating of the surface layer of the elastic element with its significant deflection, despite the constancy of the solar heat flux Q 0 . The initial deflection u z = u z (x, t = 0) is associated with the natural oscillations of elastic elements and is considered to depend only on the longitudinal coordinate x and time t.
As it was shown in Sedelnikov et al. (2022), the presence of the initial deflection reduces the intensity of the elastic element heating. However, there are additional stresses associated with bending. In combination with thermal stresses, this can lead to the loss of stability of the elastic element and the appearance of additional microaccelerations associated with the movement of parts of the elastic element when losing the stability. Since the loss of stability will occur along the z axis, this phenomenon will mainly affect the rotational motion of the small spacecraft.
Let us write down the Sophie Germain equation, however, unlike (Sedelnikov et al. 2022), we additionally take into account the inertial term (Kawano 2013;Volmir 2022): where D is bending cylindrical stiffness of the platte; σ x , σ y , τ -normal and tangential stresses acting on the elementary part of the elastic element along the corresponding axes shown in Fig. 1.
Let us consider the case when u z = u z (x, t) (Sedelnikov et al. 2022). In this case (2) is converted to the form: In (Sedelnikov et al. 2022), σ x was understood as thermal stresses, and here the stresses from the initial deflection are also added. Let the initial deflection for each point of the plate be represented by a displacement vector � ⃗ u = u x , u y , u z . At the same time, we simplistically assume that at the initial deflection u x = u y = 0 . It is worth noting here that, due to the natural oscillations of elastic elements, deflections can be significant. Therefore, this simplification is not always fair. We assume that u z = u z (x, 0) . In this formulation, the components of the strain tensor: Assuming that all deformations (3) are elastic, we apply Hooke's law and find the corresponding stresses: where E is the Young's modulus, and μ is the Poisson's ratio.
Stresses xz contribute to the plate bending in the direction of the z axis (initial deflection) and stretching-compression in the direction of the x axis (Fig. 1).
The critical stresses for the plate stability loss are estimated by the formula (Yu et al. 2022): where K is a coefficient depending on the boundary conditions of fastening; b is the width of the plate. (4) The boundary conditions for the thermoelasticity problem, considering the introduced simplifications, will have the form: In (Sedelnikov et al. 2022), an evaluation of thermal stresses was obtained: where α is the coefficient of linear expansion.
The total voltages x are defined as: The minus sign in (8) emphasizes that these stresses are compressive. If x > cr , additional microaccelerations arising from stability loss should be taken into account.

Numerical Modeling
Let us consider a «Vozvrat-MKA» spacecraft as a simulated object, which was used to simulate a temperature shock in the works (Sedelnikov andOrlov 2020, 2021; Sedelnikov z, 0)) .  Table 1 ( Sedelnikov et al. 2022). Rigid fulfillment of the first boundary condition (7) over the entire cross-sectional area of the plate in the seal (Fig. 2) leads to the loss of its stability because of the temperature shock (Fig. 3).
Let us ease the fulfillment of the first boundary condition (7) (Fig. 4).
Then, under the same conditions of temperature shock, there is no loss of stability (Fig. 5).
The difference between the fixings in Figs. 2b and 4b is that in the case of rigid sealing (Fig. 2b), all components of the displacement vector are zero both at the initial deflection and at the temperature shock. In the second case (Fig. 4b), it is assumed that the zone of attachment of the elastic element to the spacecraft body may also be subject to temperature expansion. In this case, only the u x = u x (0, t) component is reset, and the rest take place during the temperature expansion. Such relaxation leads to the fact that the force arising in the sealing decreases (Sedelnikov et al. 2022):  It is maximal when xx = yy = 0 , i.e. in the case of rigid sealing. A decrease in force (10) causes a decrease in thermal stresses (8) and total stresses (9). Therefore, in Fig. 6, there is no loss of stability of the plate.
For the simulated situation, the assessment of critical stresses (6) gives: We will assume that the stresses from the initial deflection do not depend on the method of fastening the elastic element (Figs. 2b or 4b). Then, using the approximation of the initial deflection (Sedelnikov et al. 2022): u z (0, t) = −0.1 x 2 , from (9) we have: Expression (12) gives an evaluation of the maximum stresses in the temperature stress zone x ∈ 0, l∕2 . In the works (Sedelnikov andOrlov 2020, 2021), it was believed that free expansion of the elastic element occurs in the region x ∈ l∕2, l , and there are no longitudinal temperature stresses x .
The comparison of (11) and (12) shows that the stability reserve of the elastic element will be exhausted even without temperature stresses when the amplitude of natural oscillations exceeds approximately 25 0 .
To estimate the temperature stresses, we use the evaluation of the elastic element temperature field within the framework of the two-dimensional model of thermal conductivity presented in Sedelnikov et al. (2022) and expression (8). At the same time, due to the simplifying assumptions introduced earlier, we consider the dynamics of temperature changes in the middle layer of the plate for Fig. 3 The loss of stability under temperature shock, corresponding to the fastening shown in Fig. 2   Fig. 4 View of the plate with the initial deflection in the ANSYS software (a) and the scheme of fixing the plate layers in the sealing section (b) Fig. 5 No loss of stability under the temperature shock, corresponding to the fastening shown in Fig. 5 x = l/2. The dependence of temperature stresses on time is shown in Fig. 6.
Adding up with the stresses from the initial deflection in accordance with expression (9), they exceed the critical values (11). This leads to the loss of stability shown in Fig. 3.
Physically, the loss of stability means buckling of the elastic element sections from its surface. This causes additional movements of the elastic element points, leading to additional microaccelerations. In the framework of a two-dimensional model of thermal conductivity, we estimate additional deflections according to the formulas for rods (Volmir 2022): where f is the maximum value of deflection from loss of stability; n is the number of observed half-waves of loss of stability; F Then an approximate evaluation of the maximum microaccelerations from the loss of stability can be obtained by differentiating (14) twice in time: Then, using expression (9) for σ x , it is possible to obtain an evaluation of the maximum microaccelerations, taking into account the solution of the thermoelasticity problem. For the simulated case, an evaluation of the total microaccelerations from the temperature shock, taking into account the loss of stability, is presented in Fig. 7. Fig. 6 Dependence of the temperature stresses arising from the temperature shock on time: 1-for the case of fixing the elastic element corresponding to Fig. 2b; 2-for the case of fixing the elastic element corresponding to Fig. 3b   Fig. 7 Maximum microaccelerations from the temperature shock: 1-Within the framework of a one-dimensional model of thermal conductivity (Sedelnikov et al. 2022); 2-Within the framework of a twodimensional model of thermal conductivity without the loss of stability (fixing corresponds to Fig. 4b); 3-Within the framework of a two-dimensional model of thermal conductivity, taking into account the loss of stability (fixing corresponds to Fig. 2b) As can be seen from Fig. 7, at the moment of stability loss, microaccelerations increase, exceeding the values for the one-dimensional model of thermal conductivity. Then they decrease to values lower than without the loss of stability. This is due to the fact that the angle between the normal and the direction of the solar flux changes. At the same time, the intensity of heating of the elastic element in the zone of stability loss decreases.

Conclusion
The conducted studies have shown the danger of stability loss in terms of the additional microaccelerations occurrence. They significantly affect the overall picture of microaccelerations arising from the temperature shock.
It is shown that the loss of stability occurs when the elastic element is rigidly attached to the spacecraft body. Reducing the rigidity of the attachment helps to avoid the loss of stability and associated additional microaccelerations. This should be born in mind when using constructive methods to reduce microaccelerations.
When designing elastic elements for small technological spacecraft, it is necessary to choose such length and width ratios at which the value of the coefficient K in formula (11) is the maximum. This will help to avoid loss of stability and the appearance of additional micro-accelerations.
In the future works on the design of space technology for specialized technological purposes, it is proposed to develop an additional design restriction on the ratio of the elastic elements length and width based on the results of this work. At the same time, the optimization of design parameters should be carried out taking into account this limitation. Funding Not applicable.

Declarations
Ethics Approval Not applicable.

Consent to Participate
All authors agree to participate in the publication.

Consent for Publication
All authors agree to the publication of article materials in the journal.

Conflicts of Interest
The author declares that there is no competing financial interests or personal relationships that could have appeared to have influenced the work reported in this paper.