Computer Simulation of the Natural Vibrations of a Rigidly Fixed Plate Considering Temperature Shock

Abstract

This paper presents the results of a computational experiment on the natural vibrations of a homogeneous rigidly fixed plate after a temperature shock. Unlike in many well-known studies, in this work, the plate is not stationary at the moment of thermal shock. This formulation has wide practical applications. For example, as a result of the unfolding of solar panels, free vibrations are excited. The purpose of this work was to analyze the effect of temperature shock on the characteristics of the plate’s own vibrations. Specifying the parameters of natural vibrations and considering temperature shock make it possible to model the vibration process more adequately. The simulation parameters simulate the conditions of the space environment. Therefore, the results of this study can be applied to the study of thermal vibrations in solar panels and other large elastic elements of spacecraft.

1. Introduction

The problem of thermal vibration is relevant in space technology. The problem occurs when a small spacecraft orbits the Earth and periodically finds itself in the shadow of the Earth. When entering and exiting the shadow, the small spacecraft experiences temperature shock [1,2]. Studies have shown (for example, [3]) that thermal vibrations are excited by temperature shock. The widespread use of flexible and lightweight solar panels (for example, ROSA [4], UltraFlex [5], IT SET [6]) is hindered by a lack of research into the control of thermal vibrations. For example, during experiments with ROSA on board the International Space Station, the solar panel could not be folded due to intense thermal vibrations [7]. The first missions with ROSA solar panels have already been successfully conducted (for example, DART [8]). However, a sufficient number of studies have still not been carried out. A deep and comprehensive analysis of thermal vibrations is needed.

The intrinsic vibrations of solar panels reduce the efficiency of their operation. In [9], the authors present results showing a significant decrease in the overall efficiency of solar energy conversion when the panels operate under vibration conditions. During temperature shock, the parameters of the natural vibrations change. This should be considered when evaluating the electrical power generated by the solar panel [10]. In order to improve the performance of the panels, control algorithms are being developed [11]. Damping devices are used to reduce the intensity of the impact of vibrations on the spacecraft body [12]. New design solutions are used both in the manufacturing technology of the solar panels themselves [13] and in the methods of attaching the panels to the spacecraft body [14].

The intrinsic vibrations of the solar panels, along with other disturbance factors, lead to the need for the isolation of the target equipment from vibration [15]. Technological equipment for the implementation of gravity-sensitive processes is subject to mandatory vibration isolation [16,17]. Examples of such processes are the growth of single crystals [18] and the behavior of a liquid in microgravity [19,20]. The equipment used in Earth remote sensing spacecraft is increasingly being vibration-insulated [21,22], or image quality restoration methods are being used [23,24]. This is especially true for small spacecraft [25]. According to the authors in [26], the control of natural vibrations is an important aspect for the successful achievement of the tasks assigned to the spacecraft. Moreover, the vibration parameters affect the control law of the vibration-isolating device, thus impacting its effective operation [27,28].

In [29], along with the spacecraft’s movements into and out of the Earth’s shadow, the shading of elastic elements by the spacecraft’s body is considered. This also leads to temperature shock, although this situation is unlikely for small spacecraft due to their small body size [30]. Only partial shading of the elastic elements is possible.

In general, the correct dynamic characteristics of solar panels, considering thermal vibrations, are important for the design of a spacecraft and its control system [31,32].

Thus, the study of the effect of temperature shock on the characteristics of the natural vibrations of the solar panel of a spacecraft is important and relevant if the efficiency of space technology is to be improved.

2. Materials and Methods

The actual design of solar panels is quite complex [33]. They are unique and cannot be generalized for numerical modeling. Therefore, many authors simplify the design of the solar panels in their research. Sometimes, the Euler–Bernoulli beam is used as a model [34,35]. However, it is most often a homogeneous plate [36,37]. In this paper, the homogeneous plate model is considered. One edge of the plate is rigidly sealed. The other three edges are free. Thermal shock of the solar panel occurs when a small spacecraft enters the Earth’s shadow and then emerges from it [38,39]. At the same time, a solar radiation flux (approximately 1400 W/m² in Earth’s orbit) appears or disappears almost instantly (except for the penumbral area [40,41]). During the simulation, it was assumed that the plate had a flat shape at the time of the temperature change, and there were no internal stresses in it.

The numerical simulation was performed in the ANSYS environment. The design scheme is shown in Figure 1 (the upper left edge of the plate is rigidly sealed).

The following ANSYS elements were used for thermal analysis:

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SOLID226 is a solid 20-node element of coupled analysis with options for a structural and thermal solution;

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SURF152 is an element of the surface layer with one additional space node and options for setting properties through permanent elements (used to model the radiating surfaces of the plate).

The scheme of fixing one of the edges of the plate is shown in Figure 2.

During the simulation, a curved deformed state of the plate was observed realized for the first time (Figure 3). The temperature field was not recorded. Due to the initial deflection, free- plate vibrations occurred. When the plate reached a flat state, after several perfect vibrations, a heat flux of 1400 W/m² was instantly applied. At this point, the temperature field of the plate was considered uniform. At the same time, one boundary surface of the plate was exposed to heat flow, and it radiated heat into the surrounding space according to the Stefan–-Boltzmann law [42]. The other boundary surface only radiated. The initial data for the simulation areis presented in

Table 1. The main parameters of the simulated plate.

Parameter	Designation	Value	Dimension
Plate material	-	MA2 [43]	-
Density	$\rho$	1780	$\mathrm{k}\mathrm{g}/{\mathrm{m}}^3$
Young’s modulus	$E$	4 × 1010	$\mathrm{P}\mathrm{a}$
Shear modulus	$\mu$	1.6 × 1010	$\mathrm{P}\mathrm{a}$
Poisson’s ratio	$v$	0.3	-
Plate length	$l$	1	$\mathrm{m}$
Plate width	$b$	0.5	$\mathrm{m}$
Plate thickness	$h$	0.006	$\mathrm{m}$
Rayleigh stiffness-weighted damping constant	$\beta$	0.0001	s
Initial deflection of the free edge	$u_{y0}$	0.002	$\mathrm{m}$

Where $\beta ={\displaystyle \frac{2\cdot \xi }{p_1}}$ ($p_1=4.822$ Hz—the frequency of natural vibrations), $\xi =0.0004$—viscous damping ratio [44], $\xi ={\displaystyle \frac{\delta `}{\sqrt{{(2\pi )}^2+\delta {`}^2}}}$ ($\delta `$—logarithmic decrement).

.

3. Results of Numerical Simulation

The plate is undeformed in the plane position. During the first stage, the short edge of the plate (bottom right in Figure 1a) is evenly lowered by 2 mm. Then, this edge of the plate is released. In this case, free natural vibrations occur, which, considering the forces of internal friction of the plate, are damped (Figure 3).

To obtain the results, an implicit analysis with a time step of 15 points per vibration period was used. The initial deflection value was chosen in such a way that it exceeded the amplitude of thermal vibrations by an order of magnitude (for example, the experimental data in [45]). This corresponds to the actual vibration amplitude of the solar panels (for example, data from [46]). As can be seen from Figure 2, the vibrations are damped due to the consideration of internal friction in the plate material when modeling them [47,48]. Let us compare the first natural vibration frequency. It is known [49] that the first natural frequency of a cantilevered plate can be approximately calculated using the following formula:

$$p_1={\displaystyle \frac{3.52}{l^2}}\sqrt{{\displaystyle \frac{D}{\rho h}}}$$

(1)

where D is the cylindrical bending stiffness of the plate.

Calculations using Formula (1) show that the theoretical vibration frequency is approximately 4.822 Hz. Figure 4 shows the error $\delta$ of the calculated frequency relative to the theoretical one. As the number of elements increases, this error increases slightly due to an increase in the rounding error [50]. In general, it does not exceed 2.3%. This indicates good convergence.

The vibration frequency range (4.712–4.721 Hz) is small and depends on the choice of the mesh. It can be assumed that the vibration frequency is constant and corresponds to a value of 4882 ± 0.1 Hz. The amplitude decreases from the initial deflection value of 2 mm to about 0.005 mm in 20 s of vibration. This is due to the dissipation of vibration energy due to internal friction.

In practice, natural vibrations can occur with perturbations, which act during the deployment of solar panels [51,52]. In the problem considered in this paper, the amplitude of these vibrations significantly exceeds the thermal vibrations excited as a result of temperature shock. Studies show that the same pattern is observed during the operation of some spacecraft [53,54].

In another computational experiment, the plate was subjected to temperature shock at time $t^{*}=3 \mathrm{s}.$ (its parameters are presented in

Table 2. The main parameters of the thermal shock and the thermophysical parameters of the simulated plate.

Parameter	Designation	Value	Dimension
Thermal conductivity	$\lambda$	40	$\mathrm{W}/\left(\mathrm{m}\cdot \mathrm{K}\right)$
Stefan–Boltzmann coefficient	$\Theta$	5.67 × 10−8	$\mathrm{W}/\left({\mathrm{m}}^2\cdot {\mathrm{K}}^4\right)$
Heat flux	$Q$	1400	$\mathrm{W}/{\mathrm{m}}^2$
Ambient temperature	$T_c$	3	$\mathrm{K}$
Initial temperature of the plate	$T_0$	200	$\mathrm{K}$
Emissivity	$e$	0.8	-
Specific heat	$C$	1130.4	$\mathrm{J}/\left(\mathrm{k}\mathrm{g}\cdot \mathrm{K}\right)$
Coefficient of linear expansion	$\alpha$	26 × 10−6	$1/\mathrm{K}$
The moment of temperature shock	$t^{*}$	3	s

). Up until this point in time, it performed its own vibrations, as shown in Figure 3.

At the same time, the conditions of near-Earth space are modeled, corresponding to a heat flow value of 1400 W/m² [2]. The initial boundary value problem of thermal conductivity with boundary conditions of the third kind was posed in [2]. In [2], it was solved analytically without considering the mobility of the plate at the time of the temperature shock. However, if we consider the mobility of the plate during the temperature shock, it is extremely difficult to obtain an analytical solution. The heat flow incident on the surface of the plate will change, and its approximation will be approached and limited. It will not be universal. Therefore, in this paper, this problem was solved numerically in the ANSYS package, considering natural vibrations (Figure 3) at the time of the temperature shock. Figure 5 shows the temperature dependences of the upper layer of the plate exposed to temperature shock and the lower shadow layer of the plate.

It is evident from Figure 5 that the surface layer, onto which the flux falls, begins to heat up immediately after the onset of the temperature shock. During heating, radiation from the surface layer is considered according to the Stefan–Boltzmann law [41]. This is precisely why a decrease in the rate of heating is observed. This is expressed in the bend of curve 1 in Figure 5. Much more complex temperature dynamics are observed on the shadow surface layer of the plate. At first, it cools down due to the fact that the initial temperature of the layer is higher than the ambient temperature. The heat flux from the heated layer has not yet reached the shadow surface layer. Therefore, it cools down for some time after the onset of the temperature shock. Then, the sign of the temperature derivative changes due to the fact that, due to thermal conductivity, the heat flux from the heated layer reaches the shadow surface layer. The temperature of the layer begins to rise. At first, this growth is accelerated (this is expressed in the upward-bent branches of nonlinear dependence 2 in Figure 5 in the corresponding section). However, the heat flux from the heated surface layer weakens. This happens because the external heat flux is considered constant (1400 W/m²), and the radiated flux increases with increasing surface layer temperature. This leads to a change in the sign of the second derivative of the temperature. The growth becomes slow (the branches of dependence 2 in Figure 5 bend downwards). Over the 1000 s of heating considered, the plate is far from a state of thermodynamic equilibrium. However, the temperature gradient along the plate thickness levels out and remains almost constant for a long period of time.

The dynamics of the vibrations considering temperature shock are shown in Figure 6.

A comparative analysis of Figure 3 and Figure 6 shows the presence of significant changes in the parameters of the plate’s own vibrations, both with and without considering the impact of temperature.

4. Discussion

The results of the computational experiment showed that the temperature shock significantly affected the natural vibrations of the studied plate. Firstly, the natural vibrations without considering the temperature shock occurred near the static equilibrium position (the flat, undeformed shape of the plate). The temperature shock created a temperature gradient across the thickness of the plate [2]. This led to its deflection. Thus, the vibrations occurred near a constantly changing equilibrium position. This position was determined by the difference in the temperature of the surface layers of the plate (the dynamics of these temperatures are shown in Figure 5). An increase in this difference led to an increase in the deflection of the plate and a shift in the equilibrium position to the zone of negative values. This is clearly seen in Figure 6. At the moment of temperature shock ($t^{*}$ = 3 s), an abrupt deflection of the plate and a shift in the equilibrium position of the vibrations occurred. Thus, it can be argued that, when considering thermal vibrations, the equilibrium position of the plate is not static. This must be considered when modeling the natural vibrations of solar battery panels of spacecraft. Their deviation can cause the disorientation of small spacecraft at the time of filming. This will reduce the quality of the target task.

Secondly, the natural frequency of the vibrations changed. Due to the deflection of the plate, thermal stresses arose within it. These stresses reduced the frequency of the natural vibrations of the plate. The authors of Ref. [55] came to the same conclusion. This reduction in the conducted computational experiment was within 5%. Therefore, it can be argued that considering the thermal vibrations changes the amplitude–frequency characteristics of the vibrations themselves. This is important when assessing the disturbance factors caused by the natural vibrations of solar panels [1,2].

Thirdly, the natural vibrations and thermal vibrations continue until the thermodynamic equilibrium of the plate is established. This occurs when two conditions are simultaneously met.

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The heat flux incident on the heated surface layer is equal to the sum of the flux radiating into the surrounding space and the flux leaving the plate due to thermal conductivity.

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The heat flux at the shadow surface layer is equal to the flux radiating into the surrounding space. The fulfilment of these conditions ensures the constancy of the temperature of both surface layers. By neglecting the heat flows inside the plate, we can state the constancy of the temperature gradient across the plate thickness (the constant temperature difference between the heated and shadow surface layers of the plate). This means that there is a static distribution of internal thermal stresses within the plate. This stressed state (in the absence of other force effects) corresponds to a static equilibrium position (a curved equilibrium shape).

In the example under consideration, as already noted, the plate is far from a state of thermodynamic equilibrium (Figure 5). Therefore, thermal vibrations will be observed for a long period of time. However, the amplitude of these vibrations will be small after a certain period of time due to damping. This period is determined by the sensitivity of the target tasks to the vibrations of the solar panels. Upon reaching this time point, from a practical point of view (for the quality of the achieved target tasks), the vibrations can be ignored. For a number of target tasks performed by small spacecraft, this is of great importance. Thus, the analysis performed demonstrates the importance of considering temperature shock when modeling the natural vibrations of the solar panels of a spacecraft.

5. Conclusions

The paper examines and analyzes the changes in the parameters of the natural vibrations of solar panels, considering temperature shock.

First, a change in the equilibrium position for the vibration process during temperature shock was established. This change was associated with the deflection of the plate with one rigidly fixed and three free edges as a result of temperature shock.

Second, the appearance of thermal stresses as a result of plate deflection caused a decrease in the frequency of natural vibrations. A decrease of about 5% was observed.

Third, it was established that the onset of thermodynamic equilibrium is a fairly long process. However, natural vibrations, taking into account thermal vibrations, should not always be considered a disturbance factor. Due to their small size, the impact on the quality of the performance of target tasks is negligible.

Since temperature shock occurs regularly when a spacecraft moves into and out of the Earth’s shadow, it must be considered when modeling the dynamics of the natural vibrations of solar panels. This is especially true when performing tasks such as the remote sensing of the Earth or implementing gravity-sensitive processes. This means that static formulations of thermoelasticity problems (e.g., [1,2]) may not be sufficient to achieve the required accuracy of modeling the dynamics of an elastic spacecraft. Therefore, along with approximate analytical dependencies of the components of the vector of displacements of solar battery panel points, numerical estimates of the parameters of natural vibrations that consider temperature shock are also important. Together, they allow us to recreate a complete picture of the phenomenon and contribute to the development of effective laws for controlling the motion of elastic spacecraft.