Sage Journals: Discover world-class research

Abstract

In this study, a passive vibration control method termed as translational root mounting method is proposed and investigated, for suppressing the low frequency vibrations of spacecraft solar panels. The translational root mounting method employs dampers installed at the roots of solar panels to dissipate the dynamic energy. The key of translational root mounting method is that translational root mounting method modifies the fundamental mode shape of the solar panel, by releasing the translation freedoms at its root. The root of the solar panel then has translational freedom instead of rotational freedom. As a result of mode shape modification, the translational root mounting method enables the root dampers to have sufficient working stroke, however without obviously reducing the solar panel’s fundamental frequency. Finite element model of a satellite solar panel with the length of 7 $m$ is established. Applying the finite element model, the limitation of a conventional passive vibration control method based on rotational root mounting is illustrated. To verify and illustrate the principle of translational root mounting method, the free vibration of the 7 $m$ -length satellite solar panel controlled by translational root mounting method is analyzed by finite element method. Finite element analysis (FEA) results indicate that the effect of a conventional passive control method is not obvious unless dampers with quite large damping coefficients are employed. In contrast, applying dampers with much smaller damping coefficients, the effect of translational root mounting method is fantastic comparing with the conventional passive method. To optimize the translational root mounting method and theoretically compare it to the conventional passive vibration control method, the solar panel is simplified into a non-linear one-degree of freedom system, and the analytical solution of its low frequency vibration is derived. Applying the analytical solution, it is found that damping ratio of the whole solar panel system does not increase with the damper’s damping coefficient, instead there are optimum values of damping coefficient. Applying the translational root mounting method, for the discussed 7 $m$ -length satellite solar panel, the damping ratio of which can be elevated from 0.0027 to 0.326.

Graphical abstract

Keywords

Spacecraft solar panel low frequency vibration vibration control passive method damping ratio

Highlights

Research highlight 1: This research proposes an efficient passive method (termed as translational root mounting method) based on root springs and dampers to suppress the low frequency vibration of spacecraft solar panel. The proposed translational root mounting method obviously increases the working strokes of root dampers by modifying the mode shape of the solar panel, and consequently the requirement on the damper’s damping coefficient is dramatically reduced.

Research highlight 2: Applying finite element analysis, the limitation of conventional passive methods using rotational root spring-damper system is illustrated, and the proposed translational root mounting method is analyzed and verified. Finite element analysis results approve that the proposed passive method outperforms the conventional passive method based on rotational elastic-damper interface, by modifying the mode shape of the solar panel.

Research highlight 3: An equivalent non-linear one-degree of freedom theoretical model of a satellite solar panel is established, to analytically optimize the proposed translational root mounting method. Applying the translational root mounting method, for a 7-length satellite solar panel, the damping ratio of which can be elevated from 0.0027 to 0.326.

Introduction

Solar panels are one of the most important appendages for spacecraft and provide power for whole system in on-orbit flight. In order to meet the limited launch weight and the increasing power requirement, spacecraft solar panels are becoming lager and more flexible, and they usually have very low damping ratios, high dimensional order, low modal frequencies, and parameter uncertainties in dynamics.^1,2 For a large and flexible solar panel, many disturbances such as orbit maneuver, thermal shock, debris impact, will result in low frequency vibration of solar panel. Since the spacecraft floats in the space, the entire spacecraft will response to the low frequency vibration of solar panel with motion or rotation. Thermally induced vibration of spacecraft appendages is a recurrent problem for many years. The vibrations typically occur on very low frequency booms or solar arrays due to sudden temperature changes at day-night or night-day transitions in the orbit. Sudden heating changes on a surface of an appendage may induce temperature gradients that generate time-dependent bending moments that deform the structure.^3,4 In 1990, the two solar panels of Hubble space telescope were found still “jittering” more than 6 months after launch, when they pass between day and night in orbit.^5,6 The vibrations of the two solar panels have significant influence on the stability needed for precise pointing, and NASA had no choice but replaced them by two damped solar panels in 2002. The in-orbit data of JAXA’s earth observation satellite “GOSAT” have demonstrated that when a satellite in Low Earth Orbit goes through an eclipse, the sudden thermal changes induce a difference in temperature between the sun side and the opposite side of the satellite’s solar array paddle. This temperature difference causes bending and then vibration of the solar array paddle.⁷ The Upper Atmosphere Research Satellite (UARS) is also a prominent example of a satellite that experiences thermal snap disturbances during eclipse transitions. The UARS satellite experiences attitude disturbances during orbital eclipse transitions due to rapid thermal bending of its rigid panel solar array. The attitude disturbances are of sufficient magnitude to violate the stability requirements for some of the science instruments on the satellite.^8,9

To date, the thermally induced vibrations of solar panels and the influences on the stability of a satellite have been well investigated. Johnston and Thornton⁸ in his research presented analytical and experimental investigations of solar panel thermal snap disturbances. Through the thermally induced satellite dynamics analysis and the solar panel thermal-structural experiments, it is revealed that the critical parameters for the thermal response of rigid panel solar arrays (solar panels) are the through-the-thickness temperature difference and its time derivatives. In past few decades, a number of failures caused by thermal vibrations of solar arrays have been investigated.^5,6,10–12

The issue on modeling and vibration control of flexible solar panels is rather difficult and challenging. To eliminate the harmful effects of vibrations of solar panels, the passive and active methods have been proposed and demonstrated. A successful example of solar panel vibration suppression in a passive way is Hubble space telescope. In 2002 passive dampers manufactured by CSA Engineering were installed between the solar panels and the main structure of Hubble. The solar panel’s damping ratio of the first mode was elevated form 0.012 to 0.035 by the dampers.⁵ Sales¹³ investigated the passive vibration control of flexible spacecraft using shunted piezoelectric transducers. The dynamic energy of the solar panel is converted into heat by the shunted piezoelectric transducer. The effectiveness of this passive method is quite limited unless massive of piezoelectric transducers are applied.¹³ Kong and Huang¹⁴ investigated a passive vibration suppression of a solar panel by a root damper, and found that a root damper with damping coefficient of $10^{5}$ order of magnitude is needed, and the damper nearly has no effect if the vibration of solar panel is quite small.

The passive methods are quite simple and consume no energy, and these two advantages are highly valued in aerospace applications. However, due to the low frequency of vibration and the limited strain variation, it seems quite difficult to further improve the effectiveness of passive methods which rely on dampers or damping materials. Thus, in the last decades, the active methods for vibration control of solar panel attracted many attentions. Among the researches on active control method, the smart materials such as piezoelectric materials and patches are widely used as actuators.^15–18 The active electric-magnetic devices are also valid actuators for the active control of solar panel vibration.¹⁹ In addition to the structural design, many researchers conduct researches on the control algorithms. To effectively suppress the vibrations of flexible solar panels, the fuzzy logic control with piezoelectric smart structure is studied by Xu et al.²⁰ A fuzzy logic controller which uses universal fuzzy sets is designed, and the effectiveness of the controller is demonstrated by finite element analysis. Azimi et al.²¹ investigated the vibration suppression of a flexible spacecraft during a large angle attitude maneuver. The solar panel appendages with surface bounded piezoelectric patches are investigated. Modified Sliding Mode and Strain Rate Feedback control theories have been used for attitude and vibration control simultaneously. Liu et al.²² established the Coupled Rigid-Flexible Dynamic model of a flexible solar sail. The vibration equations are derived considering the geometric nonlinearity of the sail structure subjected to the forces generated by the control vanes, solar radiation pressure, and sliding masses. The linear quadratic regulator based and optimal proportional-integral based controllers are designed for the coupled attitude/vibration models with constant disturbance torques caused by the center-of-mass and center-of-pressure offset, respectively. Although various active control methods are reported to be effective, the drawbacks are also obvious. Firstly, an active control system consumes extra energy and often needs special driving equipment, such as a power amplifier. Secondly, the smart materials used for actuating usually are limited in the actuating force and displacement. Other mechanical or electrical actuators are often unacceptable in size and mass. Finally, with many additional sensors, actuators, and controllers, the reliability of the active control system is also an issue.

Although many active control methods have been proposed, few of them have been applied in the launched spacecrafts, due to the aforementioned drawbacks. In reverse, an effective passive control method is more likely to be chosen in the practical design of a spacecraft. The passive control method outperforms the active control method in many aspects such as simplicity, energy consuming, and reliability. In this study, aimed to suppress the low frequency vibrations of spacecraft solar panels, a novel passive control method termed as translational root mounting method is proposed and verified, by finite element analysis and an equivalent nonlinear one-degree of freedom model. By installing a lateral spring-damper system at the root of the solar panel, the proposed passive method modifies the fundamental mode shape of the solar panel. The stroke lengths of the lateral dampers are then obviously increased as a result of the mode shape modification. Consequently, the dynamic energy of the solar panel can be efficiently dissipated by the lateral dampers.

In the next section, the finite element model of a representative spacecraft solar panel is established. The fundamental dynamics of the solar panel is illustrated by finite element analysis. In the third section, the limitation of a conventional passive vibration control method for spacecraft solar panel is discussed. In the fourth section, the principle of the proposed Translational Root Mounting Method for the control of low frequency vibration is illustrated and is verified. In the fifth section, an equivalent nonlinear one-degree of freedom model is established to theoretically explain the proposed Translational Root Mounting Method, and the solution of low frequency vibration of the solar panel is derived analytically. Based on the analytical solution, the influences of the lateral spring-damper system on the damping ratio of the solar panel are predicted and discussed. In the last section, the work of this study is summarized.

Low frequency flexible vibration of a satellite solar panel

A representative satellite solar panel and its finite element model are presented in Figure 1. The solar panel composes of four pieces of sandwich composite panels and a composite frame. In this study, the commercial finite code ABAQUS is employed to conduct finite element analysis. The solar panel are modeled by four node shell elements S4R, the frame is modeled by two node beam elements B31. The material properties and structural sets are listed in Table 1. In the finite element model, the solar cells are modeled as non-structural mass, and the satellite is modeled as a rigid body. The total mass of solar cells on panels is 10 $Kg$ . The total mass of the solar panel structure is 43.37 $Kg$ . The root of solar panel is rigidly connected to the side plate of satellite by displacement coupling restraint provided by ABAQUS. Restraining the displacement and rotation of the satellite, the mode analysis results of the solar panel are presented in Figure 2, the fundamental frequency of the solar panel is $0.477 Hz$ .

Figure 1.

Finite element model of a representative satellite solar panel.

Table 1.

Material properties and structural design of the solar panel.

Material properties	CFRP	$E_{11} = 145 GPa$
		$E_{22} = 9.75 GPa$
		$G_{12} = 5.69 GPa$
		$μ_{12} = 0.312$
		$ρ = 1650 Kg / m^{3}$
	Honeycomb core	$E_{11} = 1.57 \times 10^{- 4} GPa$
		$E_{22} = 1.57 \times 10^{- 4} GPa$
		$G_{12} = 2.945 \times 10^{- 5} GPa$
		$μ_{12} = 0.33$
		$ρ = 42 Kg / m^{3}$
Structural design	Beam	Material: CFRP
		Lay-up [45/−45/−45/45/45/−45/−45/45]
		Wall thickness = 1 $mm$
	solar panel	Material: CFRP, honeycomb core
		Lay-up of CFRP
		$[45 / - 45 / - 45 / 45 /$ ${core}_{29.2 mm} / 45 / - 45 / - 45 / 45]$

Figure 2.

Mode analysis results of solar panel.

In the space, the low frequency flexible vibration of a solar panel usually takes a long time to decay, for the reason that the dynamic energy of the solar panel can only be dissipated by the internal damping of material. To model the material damping in the finite element model, the commonly used Rayleigh damping is applied on the CFRP and honeycomb. It is well known that Rayleigh damping is the linear combination of mass matrix and stiffness matrix, that is $α M + β K$ . Where $M$ denotes mass matrix and $K$ denotes stiffness matrix. The mass damping coefficient $α$ and the stiffness damping coefficient $β$ in practice should be computed according to the dynamic responses.²³ In this study, as the interest is the passive method to suppress the low frequency vibration of the solar panel, we assume the damping coefficients $α$ and $β$ directly. In orbit, the low frequency vibration of solar panel lasts for a long time due to the lack of efficient damping mechanism. Thus, small Rayleigh damping coefficients $α$ and $β$ are assumed, that is $α = 0.01$ and $β = 0.0005$ . The free vibration of the solar panel is analyzed by finite element model. In the first “Static” analysis step, a static vertical displacement constraint of 300 $mm$ is applied on the free end of the solar panel. The displacement constraint is released in a followed “Implicit Dynamic” step, and then the free vibration of the solar panel is simulated. The predicted vertical displacement and acceleration time histories and spectral densities of the middle point of the solar panel free end are presented in Figure 3.

Figure 3.

Predicted vertical displacement and acceleration time histories and spectral densities at the middle point of the solar panel free end: (a) vertical acceleration and (b) vertical displacement.

The time domain curves and spectral densities presented in Figure 3 indicate that the vibration of the solar panel is initially a superposition of low frequency and high frequency modes (mainly mode 1 and mode 4). The high frequency vibration decays in a much shorter time due to the stiffness proportional damping $β K$ . After about $20 s$ , only the low frequency vibration corresponding to the fundamental mode remains. The damping ratio of the solar panel can be calculated according to the displacement time history, which is only $0.0027$ . Due to the orthogonality of modes, the dynamic energy of the fundamental mode cannot spontaneously transform to higher order modes and be dissipated. As the mass proportional damping $α M$ is small, the low frequency vibration decays quite slowly. Therefore, to reduce the decay time of the low frequency vibration, the most direct method is increasing the material damping coefficient $α$ . Although there are many studies on increasing the damping property of aerospace materials,^24–27 the quite high damping aerospace material is still an open challenge. To passively suppress the low frequency vibration of solar panels, installing dampers at the root of solar panel is a method easily comes to mind. Although the effectiveness of root dampers has been verified for instance on Hubble space telescope,⁵ in the next section we will discuss the limitation of root dampers for suppressing the low frequency vibration of a spacecraft solar panel.

Conventional method for low frequency vibration control of spacecraft solar panel based on root springs and dampers

Figure 4 exhibits the equivalence principle of a conventional passive method using root dampers. A spring-damper interface is installed between the spacecraft and the solar panel root. The vibration of solar panel leads to periodic moment on the spring-damper interface. The springs and dampers are then compressed or stretched due to the moment at the solar panel root, and the dynamic energy can be dissipated by the dampers. The most important feature of the conventional method is that the root of solar panel rotates during the vibration. In this study, the conventional passive method is termed as rotational root mounting method. A rotational spring-damper interface is established in the finite element model of the solar panel presented in Figure 1. The springs and dampers are modeled by the spring element and dash-pot element, as illustrated in Figure 5. The influence of spring stiffness on the solar panel fundamental frequency is analyzed by finite element model. In the meanwhile, by applying a vertical displacement of 300 $mm$ at the free end of the solar panel, the amount of compression(stretching) of the spring is also predicted. The predicted results are presented in Figures 6 and 7. FEA results indicate that using springs with quite large stiffness benefits the fundamental frequency of the solar panel, however the compression (working stroke) of the damper decreases with the spring stiffness. In contrast, if soft root springs are employed, the fundamental frequency of the solar panel will be lowered. As a result, the vibration velocity is lowered and the viscous damping force is also lowered. With either small working stroke or small damping force, it is not possible for a viscous damper to efficiently dissipate the dynamic energy of the solar panel.

Figure 4.

Equivalence principle of rotational root mounting method based on root springs and dampers.

Figure 5.

Finite element model of the satellite solar panel with a root spring-damper interface.

Figure 6.

Fundamental frequency–spring stiffness curve of the satellite solar panel with root spring-damper interface.

Figure 7.

Variation of root spring compression with its stiffness when a vertical displacement constraint of $300 mm$ is applied on the free end of the satellite solar panel.

Applying different damping coefficients $C$ and spring stiffness $K$ , free vibration responses of the solar panel are predicted by the finite element model. An initial vertical displacement of 300 $mm$ are applied on the free end of solar panel. The vertical displacement time histories of the solar panel free end are presented in Figure 8. When the damper with $C = 100 Ns / m$ is employed, the spring stiffness has no obvious influence on the descending speed of the vibration amplitude. After 40 $s$ , the vertical displacement amplitude only decreases from 300 to 230 $mm$ . When the damping coefficient $C$ further increases to 1000 and 10,000 $Ns / m$ , the influence of damper on the vibration is still quite limit if the spring stiffness $K = 10^{6} N / m$ is applied. In contrast, if spring with smaller stiffness $K = 10^{5} N / m$ is applied, increasing the damping coefficient $C$ is an effective method to reduce the decay time of free vibration. The overall damping ratios of the solar panel with root spring-damper interface are calculated for different $C$ and $K$ , and are listed in Table 2. To achieve a larger damping ratio, a root spring-damper system with small spring stiffness and large damping coefficient should be employed. The soft root spring ensures the working stroke of the dampers, and large damping coefficient ensures the viscous damping force. However, as the vibration frequency is lowered, even the damping coefficient increases to such a large value of $C = 10000 Ns / m$ , damping ratio of the solar panel is merely 0.058. Meanwhile, it is difficult to achieve a damper with such large damping coefficient if there are volume and mass restrictions. Therefore, because of the fundamental contradiction between working stroke and vibration frequency (which determines the viscous damping force), and also the limit damping coefficient $C$ , it is difficult to further improve the effect of conventional passive method illustrated in Figure 4.

Figure 8.

Predicted vertical displacement time histories of the solar panel free end after installing a rotational root spring-damper interface: (a) C = 100 Ns/m, (b) C = 1000 Ns/m, and (c) C = 10,000 Ns/m .

Table 2.

Damping ratio of the solar panel with root spring-damper system.

Damping ratio $δ$	$C = 100 Ns / m$	$C = 1000 Ns / m$	$C = 100000 Ns / m$
$K = 10^{5} N / m$	0.0042	0.009	0.058
$K = 10^{6} N / m$	0.0027	0.00285	0.0044

Passive vibration control of spacecraft solar panel based on translational root mounting method

Theoretically, to increase the energy dissipation of a damper in a certain amount of time, the working stroke and vibration frequency of the damper should be increased simultaneously. However, discussion in the last section indicates that the conflict between the working stroke of the damper and the vibration frequency is inevitable. The acceleration time history presented in Figure 3 indicate that the high frequency vibration decays in a relative short time. As a result, for suppressing the low frequency vibration via a spring-damper system, the valid vibration frequency of the damper cannot exceed the fundamental frequency of the solar panel with fixed root, which is 0.477 $Hz$ in this study. As there is a upper limit on working frequency of the damper, increasing the working stroke of the damper is the most direct method to improve the energy dissipating ability of the passive vibration control method. Thus, to suppress the low frequency vibration of a solar panel passively and effectively, the key is increasing the damper’s working stroke, and avoid the decreasing of solar panel’s fundamental frequency.

In this study, a passive method termed as translational root mounting method is proposed. The principle of translational root mounting method is illustrated in Figure 9. The translational root mounting method also applies spring-damper system for the vibration control. Different with the conventional method illustrated in Figure 4, the lateral translation stiffness of the solar panel root is reduced instead of the rotational stiffness. The solar panel root is connected to the spacecraft main structure by lateral spring-damper system, while the rotation of the solar panel root is restrained.

Figure 9.

Equivalence principle of translational root mounting method for low frequency vibration control of solar panel.

The finite element model of solar panel controlled by translational root mounting method is established, as illustrated in Figure 10. The root of the solar panel is connected to the satellite’s side plate in $y$ -direction by spring and dash-pot elements. A coupling restraint is established between the side plate and solar panel root, and the solar panel is fully restrained at the root except the translation freedom in $y$ direction. Assuming that the total stiffness of lateral springs $K = 1000 N / m$ , the mode analysis is conducted, and the fundamental mode shape of the solar panel is presented in Figure 11. The fundamental mode shape indicates that the root of solar panel has obvious lateral translation in the low frequency vibration, if a low stiffness lateral spring is applied. The lateral translation at the solar panel root enables large working stroke for the lateral damper in the vibration. Although the spring stiffness is only $1000 N / m$ , the fundamental frequency of the solar panel is only reduced from $0.477 Hz$ to $0.420 Hz$ . The influence of lateral spring stiffness on the fundamental frequency of the solar panel is presented in Figure 12. It indicates that the fundamental frequency of the solar panel do decrease with the spring stiffness, nevertheless, the decreasing of which is quite limited comparing with the conventional passive method.

Figure 10.

Finite element model of the satellite solar panel controlled by translational root mounting method.

Figure 11.

Modified fundamental mode shape of the satellite solar panel (lateral spring stiffness $K = 1000 N / m$ ).

Figure 12.

Fundamental frequency of the satellite solar panel versus lateral spring stiffness.

Applying the finite element model, the effectiveness of translational root mounting method is verified by predicting the free vibration responses of the solar panel. In FEA, a vertical displacement constraint of $300 mm$ is applied on the free end of the solar panel in a “Static” step, and then the displacement constraint is released in a “Implicit dynamic” step. Assuming the damper’s damping coefficient $C = 100 Ns / m$ , and assuming the total lateral spring stiffness $K = 1000 N / m$ and $K = 2000 N / m$ , the free vibration responses of the solar panel are presented in Figures 13 and 14. In Figure 13, the lateral displacement time histories of the solar panel free end demonstrate that the translational root mounting method is quite efficient in suppressing the low frequency vibration of solar panel. With smaller lateral spring stiffness $K = 1000 N / m$ , the low frequency vibration of solar panel decays much faster. In Figure 14, the displacement time histories of the solar panel root show that the working stroke of the lateral damper reaches centimeter-level, which is much larger than that of the conventional passive method presented in Figure 7. The results presented in Figures 12 to 14 illustrate that applying the translational root mounting method, the working frequency of the damper is not significantly lowered, while the working stroke of which is dramatically increased.

Figure 13.

Predicted vertical displacement time histories of the solar panel free end after installing the lateral spring-damper system (damping coefficient $C = 100 Ns / m$ ).

Figure 14.

Predicted lateral displacement time histories of the solar panel root after installing the lateral spring-damper system (damping coefficient $C = 100 Ns / m$ ).

To investigate the influence of damping coefficient $C$ on the damping ratio of the solar panel controlled by translational root mounting method, the free vibration responses of the solar panel with different $C$ are predicted by FEA. The vertical displacement time histories of the solar panel free end are presented in Figure 15. Based on the displacement histories, the damping ratios corresponding to different $C$ are calculated, and are presented in Figure 18. FEA results indicate that using translational root mounting method, there is an optimum $C$ for achieving the largest damping ratio. When the lateral spring stiffness is $1000 N / m$ , the optimum $C$ is $410 Ns / m$ , and the largest damping ratio of the whole solar panel system is 0.075. In comparison, damping ratio of the solar panel with fixed root is only about 0.0027. In order to further elevate the damping ratio of the whole solar panel system and to theoretically illustrate the difference between RRMM and translational root mounting method, in the next section the approximate analytical solution of the low frequency solar panel vibration is derived for the RRMM and translational root mounting method, and the translational root mounting method is discussed and optimized.

Figure 15.

Comparison of vertical displacement time histories of the solar panel free end with different damping coefficients (lateral spring stiffness $K = 1000 N / m$ ).

Theoretical analysis and discussion

As the fundamental mode of solar panel dominates the flexible vibration, the solar panel can be simplified to a cantilever beam with concentrated mass, and the whole solar panel system can be simplified to nonlinear one-degree of freedom systems, according to the mounting method at the root, as illustrated in Figure 16. In the equivalent nonlinear one-degree of freedom system, the mass $m$ is the effective mass of the fundamental mode in the vertical direction when the solar panel is fixed at its root. $K_{panel}$ denotes the equivalent stiffness of the solar panel, and can be calculate according to the fundamental frequency $f$ , and $K_{panel} = m (2 π f)^{2}$ . $K$ and $C$ are respectively the stiffness and damping coefficient of the root spring-damper system. $x_{1}$ is the displacement at the top of root springs, and $x_{2}$ represents the vertical displacement of the effective mass. For the solar panel with rotational root mounting, $l$ denotes the distance between the root springs and $L$ denotes the distance from the mass central of the solar panel to its root.

Figure 16.

Equivalent nonlinear one-degree of freedom model of the solar panel controlled by translational root mounting method and rotational root mounting method: (a) translational root mounting method and (b) rotational root mounting method.

The force equilibrium equations of the equivalent nonlinear one-degree of freedom system with rotational root mounting take the form:

\begin{matrix} \begin{matrix} (K \frac{l}{L} + K_{panel} \frac{L}{0.5 l}) x_{1} [t] - K_{panel} x_{2} [t] + {Cx}_{1}' [t] \frac{l}{L} = 0 \\ - K_{panel} \frac{L}{0.5 l} x_{1} [t] + K_{panel} x_{2} [t] + {mx}_{2}^{″} [t] = 0 \end{matrix} \end{matrix}

(1)

and the force equilibrium equations of the equivalent nonlinear one-degree of freedom system with translational root mounting take the form:

\begin{matrix} \begin{matrix} (K + K_{panel}) x_{1} [t] - K_{panel} x_{2} [t] + {Cx}_{1}' [t] = 0 \\ - K_{panel} x_{1} [t] + K_{panel} x_{2} [t] + {mx}_{2}^{″} [t] = 0 \end{matrix} \end{matrix}

(2)

By eliminating $x_{1} [t]$ , a third-order differential equation of $x_{2} [t]$ for the rotational root mounting is obtained:

\begin{array}{l} \frac{0.5 l^{2}}{L^{2}} C \frac{m x_{2}^{(3)} [t]}{K_{p a n e l}} + (\frac{\frac{0.5 l^{2}}{L^{2}} K}{K_{p a n e l}} + 1) m x_{2}^{"} [t] \\ + \frac{0.5 l^{2}}{L^{2}} C x_{2^{'}} [t] + \frac{0.5 l^{2}}{L^{2}} K x_{2} [t] = 0 \end{array}

(3)

The third-order differential equation of $x_{2} [t]$ for the rotational root mounting has similar form:

\begin{matrix} \begin{matrix} C \frac{{mx}_{2}^{(3)} [t]}{K_{panel}} + (\frac{K}{K_{panel}} + 1) {mx}_{2}^{″} [t] + {Cx}_{2}' [t] + K x_{2} [t] = 0 \end{matrix} \end{matrix}

(4)

Comparing equation (3) with equation (4), the only difference between the two dynamic equations is the coefficient $\frac{0.5 l^{2}}{L^{2}}$ for spring stiffness $K$ and damping coefficient $C$ . The effects of spring stiffness $K$ and damping coefficient $C$ of the rotational root mounting method is dramatically scaled down, comparing with the rotational modified root mounting method, as the coefficient $\frac{0.5 l^{2}}{L^{2}}$ is usually very small. Therefore, to accomplish identical stiffness and damping response for the overall solar panel system, the spring stiffness and damping coefficient of the rotational root mounting method should be much larger than the rotational root mounting method.

Assuming $x_{2} [t]$ :

\begin{matrix} \begin{matrix} x_{2} [t] = A e^{st} \end{matrix} \end{matrix}

(5)

and substituting equation (5) in to equation (4), a cubic characteristic equation can be derived:

\begin{matrix} a s^{3} + b s^{2} + cs + d = 0 \\ a = \frac{Cm}{K_{panel}} \\ b = \frac{(K + K_{panel}) m}{K_{panel}} \\ c = C \\ d = K \end{matrix}

(6)

Equation (6) can be rewritten as:

\begin{matrix} y^{3} + py + q = 0 \\ y = \frac{s + b}{3 a} \\ p = \frac{3 C^{2} K_{panel} - {(K + K_{panel})}^{2} m}{3 C^{2} m} \\ q = \frac{9 C^{2} (2 K - K_{panel}) K_{panel} + 2 (K + K_{panel})^{3} m}{27 C^{3} m} \end{matrix}

(7)

Equation (7) can be solved using the Cardano method, and three solutions are derived:

\begin{matrix} Δ = {(\frac{q}{2})}^{2} + {(\frac{p}{3})}^{3} \\ y_{1} = \sqrt[3]{\frac{- q}{2} + \sqrt{Δ}} + \sqrt[3]{\frac{- q}{2} - \sqrt{Δ}} \\ y_{2} = \frac{- 1 + \sqrt{3} i}{2} \sqrt[3]{\frac{- q}{2} + \sqrt{Δ}} + \frac{- 1 - \sqrt{3} i}{2} \sqrt[3]{\frac{- q}{2} - \sqrt{Δ}} \\ y_{3} = \frac{- 1 - \sqrt{3} i}{2} \sqrt[3]{\frac{- q}{2} + \sqrt{Δ}} + \frac{- 1 + \sqrt{3} i}{2} \sqrt[3]{\frac{- q}{2} - \sqrt{Δ}} \end{matrix}

(8)

Substitute equation (8) into equation (7), three solutions of $s$ are then derived. In the following discussion, by substituting the values of $K$ , $K_{panel}$ , $C$ , and $m$ into equation (8), it is found that $Δ > 0$ . Therefore, $s_{1}$ is a real solution and $s_{2}$ and $s_{3}$ are complex conjugate solutions:

\begin{matrix} s_{1} = \sqrt[3]{\frac{- q}{2} + \sqrt{Δ}} + \sqrt[3]{\frac{- q}{2} - \sqrt{Δ}} - \frac{K + K_{panel}}{3 C} \\ s_{2} = \frac{- 1 + \sqrt{3} i}{2} \sqrt[3]{\frac{- q}{2} + \sqrt{Δ}} + \frac{- 1 - \sqrt{3} i}{2} \\ \sqrt[3]{\frac{- q}{2} - \sqrt{Δ}} - \frac{K + K_{panel}}{3 C} \\ s_{3} = \frac{- 1 - \sqrt{3} i}{2} \sqrt[3]{\frac{- q}{2} + \sqrt{Δ}} + \frac{- 1 + \sqrt{3} i}{2} \\ \sqrt[3]{\frac{- q}{2} - \sqrt{Δ}} - \frac{K + K_{panel}}{3 C} \end{matrix}

(9)

Substitute equation (9) into equation (5), the expression of $x_{2} [t]$ can be derived:

\begin{matrix} x_{2} [t] = A_{1} e^{(M + N - \frac{K + K_{panel}}{3 C}) t} \\ + A_{2} e^{- (\frac{K + K_{panel}}{3 C} + \frac{M}{2} + \frac{N}{2}) t} Sin [\frac{\sqrt{3}}{2} (M - N) t + α] \\ M = \sqrt[3]{\frac{- q}{2} + \sqrt{Δ}} \\ N = \sqrt[3]{\frac{- q}{2} - \sqrt{Δ}} \end{matrix}

(10)

In equation (10), $A_{1}$ , $A_{2}$ , and $α$ are unknowns, and can be determined by the initial conditions $x_{2} [0]$ , $x_{2}' [0]$ , and $x_{2} [0] + \frac{m}{K_{panel}} x_{2}^{″} [0] = x_{1} [0] = x_{2} [0] \frac{K_{panel}}{K + K_{panel}}$ .

FEA indicates that if the solar panel presented in Figure 1 is fixed at its root, the vertical effective mass of which is $m = 35.834 Kg$ . As the fundamental frequency $f = 0.477 Hz$ , the equivalent stiffness $K_{panel} = m (2 π f)^{2} = 324.32 N / m$ . Assuming $K = 1000 N / m$ and $C = 400 Ns / m$ , applying the boundary condition $x_{2} [0] = 0.3$ , $x_{2}' [0] = 0$ , and $x_{2} [0] + \frac{m}{K_{panel}} x_{2}^{″} [0] = x_{2} [0] \frac{K_{panel}}{K + K_{panel}}$ , the analytical solution of $x_{2} [t]$ is derived:

\begin{matrix} \begin{matrix} x_{2} [t] \approx 0.02 e^{- 2.89 t} + 0.2831 e^{- 0.211 t} Sin [1.7193 - 2.7906 t] \end{matrix} \end{matrix}

(11)

Applying identical lateral spring stiffness and damping coefficient, the analytical solution equation (11) is compared with FEA results in Figure 17. Note, the FEA result is the vertical displacement time history of the solar panel free end. The initial value $x_{2} [0]$ only has influence on the amplitude of vibration, and the decay of the vibration is unaffected. Therefore, identical initial amplitude of 300 $mm$ is applied for both analytical solution and FEA, for better comparison. In Figure 17, the analytical solution of the equivalent two-degree of freedom model coincides well with the finite element model, which indicates that the equivalent two-degree of freedom model is sufficiently accurate to predict the low frequency vibration of the solar panel controlled by translational root mounting method.

Figure 17.

Comparison of analytical and FEA results on free vibration of the satellite solar panel controlled by translational root mounting method (lateral spring stiffness $K = 1000 N / m$ , lateral damper’s damping coefficient $C = 400 Ns / m$ ).

Applying the analytical solution of equivalent nonlinear one-degree of freedom model, that is equation (11), the damping ratio of the solar panel controlled by translational root mounting method can further be derived. As $x_{2} [t]$ is dominated by the $A_{2} e^{- (\frac{K + K_{panel}}{3 C} + \frac{1}{2} M + \frac{1}{2} N) t} Sin [\frac{\sqrt{3}}{2} (M - N) t + α]$ term in equation (10), the damping ratio of the solar panel takes the following form as:

\begin{matrix} \begin{matrix} δ = \frac{1}{2 π} \ln \frac{e^{- (\frac{K + K_{panel}}{3 C} + \frac{1}{2} M + \frac{1}{2} N) t_{1}} Sin [\frac{\sqrt{3}}{2} (M - N) t_{1} + α]}{e^{- (\frac{K + K_{panel}}{3 C} + \frac{1}{2} M + \frac{1}{2} N) t_{2}} Sin [\frac{\sqrt{3}}{2} (M - N) t_{2} + α]} \end{matrix} \end{matrix}

(12)

In equation (12), $t_{2} - t_{1} = \frac{2 π}{\frac{\sqrt{3}}{2} (M - N)}$ and equation (12) can be further simplified:

\begin{matrix} \begin{matrix} δ = \frac{(\frac{K + K_{panel}}{3 C} + \frac{1}{2} M + \frac{1}{2} N)}{\frac{\sqrt{3}}{2} (M - N)} \end{matrix} \end{matrix}

(13)

Note, equations (5)–(13) are derived for the rotational root mounting method. For the rotational root mounting method, similar analytical solution can be derived, simply by substituting the parameters $K$ and $C$ respectively by $\frac{0.5 l^{2}}{L^{2}} K$ and $\frac{0.5 l^{2}}{L^{2}} C$ . For the modified root mounting method, substituting $K = 1000 N / m$ , $K_{panel} = 324.32 N / m$ , and $m = 35.834 Kg$ into equation (13), the overall damping ratios corresponding to different damping coefficients are calculated. In Figure 18, the analytical results of damping ratios are compared with the FEA results. The FEA results of damping ratios are calculated by the time histories presented in Figure 15. Both analytical solution and FEA results illustrate that the damping ratio of the solar panel maximizes when $C = 410 Ns / m$ . Figure 18 indicates that the equivalent two-degree of freedom model is also accurate in predicting the damping ratio of the solar panel controlled by translational root mounting method.

Figure 18.

Variation of damping ratio of the whole satellite solar panel with lateral damper’s damping coefficient $C$ ( $K = 1000 N / m$ ).

Applying equation (13), the optimum damping coefficients and the maximum damping ratios of the whole solar panel system corresponding to different spring stiffnesses ( $K$ ) are predicted, and are presented in Figure 19. The optimum damping coefficient of the damper linearly increases with the spring stiffness $K$ , while the maximum damping ratio non-linearly decreases with $K$ . If the spring stiffness increases to $2000 N / m$ , the damping ratio of the whole solar is very close to the value when the root of the solar panel is fixed. Although reducing the spring stiffness $K$ benefits the damping ratio of the solar panel, the fundamental frequency of which will be lowered. Thus, in the practical design of a spacecraft solar panel, the spring stiffness $K$ should be chosen according to the requirement of the fundamental frequency. For the discussed solar panel in this study, if the lateral spring stiffness $K = 400 N / m$ , the corresponding optimum damping coefficient of the damper is $140 Ns / m$ . The fundamental frequency of the whole solar panel system is predicted to be $0.361 Hz$ , and the damping ratio of which is elevated to 0.326. Applying $K = 400 N / m$ , $C = 140 Ns / m$ , the free vibration time histories of the solar panel, predicted by FEA and analytical solution, are compared in Figure 20. With initial amplitude of $300 mm$ , the decay time of free vibration is less than $10 s$ .

Figure 19.

Variation of the maximum damping ratio of the satellite solar panel controlled by translational root mounting method with lateral spring stiffness $K$ .

Figure 20.

Comparison of analytical and FEA results on free vibration of the satellite solar panel controlled by translational root mounting method ( $K = 400 N / m$ , $C = 140 Ns / m$ ).

The translational root mounting method is further verified by predicting the dynamic response of the solar panel which experience the thermal snap disturbances during eclipse transitions. The solar panel is installed on a cube satellite of $2 \times 2 \times 2 m^{3}$ , as illustrated in Figure 1. The total mass of the cube satellite is 2000 kg, and the mass distribution is assumed to be even. Johnston and Thornton⁸ in his study predicted the temperature variation of a solar panel in the low attitude circular orbit, and he pointed out that there is a through-the-thickness temperature difference of $11 K$ on the solar panel. The through-the-thickness temperature difference will induce vibrations, which consist of a quasi-static deformation with superimposed oscillations.⁸ Applying the through-the-thickness temperature curve in Figure 3 of reference,⁸ the dynamic response of the cube satellite-solar panel system is predicted by FEA. In the finite element model, the cube satellite is modeled as a rigid body. Two root mounting methods are simulated, including the rotational root mounting and the proposed translational root mounting method. For the rotational root mounting method, the spring stiffness $K = 10^{6} N / m$ and the damper coefficient $C = 140 Ns / m$ . For the translational root mounting method, $K = 400 N / m$ and $C = 140 Ns / m$ are applied. The predicted vertical displacement curves of the solar panel free end and the attitude rotation angle curves of the cube satellite are compared in Figure 21. Applying the translational root mounting method, with two dampers of $140 Ns / m$ , the solar panel hardly vibrates during eclipse transition, and the attitude of the satellite is quite stable. In contrast, the solar panel has constant vibration during the eclipse transition if rotational root mounting is applied, and the attitude of the satellite is constantly influenced by the solar panel vibration.

Figure 21.

Predicted dynamic responses of the cube satellite-solar panel system experiencing eclipse transition: (a) vertical displacement time history of solar panel free end and (b) attitude rotation time history of the cube satellite.

If the rotational root mounting method is applied, to accomplish identical response for this solar panel system, the corresponding root spring stiffness $K = 400 N / m \times \frac{L^{2}}{0.5 l^{2}} = 149422 N / m$ , and the damping coefficient $C = 140 Ns / m \times \frac{L^{2}}{0.5 l^{2}} = 52297 Ns / m$ (the distance between the mass center of the solar panel to its root $L = 4.1 m$ , and the distance of the springs at the root $l = 0.3 m$ ). The damping coefficient needed for the rotational root mounting method is much larger than the translational root mounting method. Based on the same principle, if the spring and damper used in the rotational root mounting method are replaced by rotational type, similar nonlinear dynamic equations can by derived:

\begin{array}{l} \frac{1}{L^{2}} C \frac{m x_{2}^{(3)} [t]}{K_{p a n e l}} + (\frac{\frac{1}{L^{2}} K}{K_{p a n e l}} + 1) m x_{2}^{"} [t] \\ + \frac{1}{L^{2}} C x_{2^{'}} [t] + \frac{1}{L^{2}} K x_{2} [t] = 0 \end{array}

(14)

in which $K$ denotes the rotational stiffness and $C$ donates the rotational damping coefficient. Comparing equation (14) with equation (4), it illustrates that if rotational root springs and dampers are applied at the root of the solar panel, the effects of rotational stiffness $K$ and damping $C$ are both scared down by a coefficient $\frac{1}{L^{2}}$ . Thus, if rotational type root springs and dampers are applied, to accomplish similar response with the modified root mounting method, the root rotational stiffness $K = 400 N / m \times L^{2} = 7056 Nm / rad$ , and the damping coefficient $C = 140 Ns / m \times L^{2} = 2353 Nsm / rad$ . Equations (3), (4), and (14) illustrate that the rotational and translational root mounting methods theoretically are of the same principle in essence, excepting that the effect of stiffness $K$ and damping coefficient $c$ are simultaneously scaled down for the rotational root mounting method. For the discussed solar panel in this study, applying the translational root mounting method the requirement of the damping coefficient is significantly lowered, which makes it easier to be accomplished in practical engineering applications.

Conclusion

In a conventional passive method for suppressing the low frequency vibration of spacecraft solar panel, a rotational spring-damper interface is usually installed between the spacecraft and the solar panel. The conventional method inevitably meets the conflict between the working stroke and working frequency of the damper, and a damper with quite large damping coefficient is required. In this study, a novel passive method for suppressing the low frequency vibration of spacecraft solar panel is proposed. The novelty of the proposed method is that the mode shape of the solar panel is modified, by releasing the lateral translation freedom of the solar panel at its root. The mode shape modification enables the solar panel root have obvious translation if the solar panel vibrates. After connecting the solar panel root to the spacecraft by a lateral spring-damper system, large working stoke of damper can be achieved due to the mode shape modification of the solar panel. As a result, the dynamic energy can be sufficiently dissipated by the damper.

In this study, finite element model of a satellite solar panel with the length of $7 m$ is established. Applying finite element method, effectiveness of the proposed Translational Root Mounting Method is verified. FEA results indicate that applying a conventional passive control method, a damper with quite large damping coefficient is needed, and the fundamental frequency of the solar panel is dramatically lowered. In contrast, applying the proposed Translational Root Mounting Method, using damper with much lower damping coefficient, the low frequency vibration of the solar panel can be well suppressed in a much shorter time. For optimizing and theoretically explaining the principle of the modified root mounting method, the solar panel is simplified into a non-linear one-degree of freedom system, and analytical solution of the low frequency vibration is derived. Both analytical solution and FEA results illustrate that there is an optimum damping coefficient to achieve the largest damping ratio for the whole solar panel system. For the Translational Root Mounting Method, smaller spring stiffness requires smaller damping coefficient. For the 7 $m$ -length satellite solar panel in this study, with a lateral spring-damper passive control system of $K = 400 N / m$ and $C = 140 Ns / m$ , the fundamental frequency of the whole solar panel system is lowered from $0.477 Hz$ to $0.361 Hz$ , however the damping ratio of the whole solar panel system can be elevated from 0.0027 to 0.326. The non-linear dynamic equations of the solar panel corresponding to the rotational root mounting method are also derived, which illustrate that the effect of dampers are significantly scared down comparing with that of the modified translational root mounting method. Consequently, to accomplish similar damping effect as the modified root mounting method, much larger damping coefficient is needed for the rotational root mounting method.

Footnotes

Handling Editor: Chenhui Liang

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: Xiaohong Wang sincerely acknowledges the financial support from Innovation Team Project of Heilongjiang Institute of Technology, Grant No. 2018CX10, Doctoral Foundation Project of Heilongjiang Institute of Technology, Grant No. 2016BJ03, and Leading Talent Incubation Program of Heilongjiang Institute of Technology, Heilongjiang Province, China, Grant No. 2020LJ02.

ORCID iD

Hao Li

References

Qiu

. Acceleration sensors based modal identification and active vibration control of flexible smart cantilever plate. Aerosp Sci Technol 2009; 13: 277–290.

Maganti

Singh

. Simplified adaptive control of an orbiting flexible spacecraft. Acta Astronaut 2007; 61: 575–589.

Thornton

Kim

. Thermally induced bending vibrations of a flexible rolled-up solar array. J Spacecr Rockets 1993; 30: 438–448.

Zhang

Xiang

Liu

, et al. Stability of thermally induced vibration of a beam subjected to solar heating. AIAA J 2014; 52: 660–665.

Anandakrishnan

Connor

Lee

, et al. Hubble space telescope solar array damper for improving control system stability. In: IEEE aerospace conference proceedings, Big Sky, MT, USA, vol. 4, pp.261–276.

Foster

Tinker

Nurre

, et al. Solar-array-induced disturbance of the hubble space telescope pointing system. J Spacecr Rockets 1995; 32: 634–644.

Oda

Honda

Suzuki

, et al. Vibration of satellite solar array paddle caused by thermal shock when a satellite goes through the eclipse. In: Advances in vibration engineering and structural dynamics. London: IntechOpen, 2012 [Online]. https://www.intechopen.com/chapters/40863

Johnston

Thornton

. Thermally induced dynamics of satellite solar panels. J Spacecr Rockets 2000; 37: 604–613.

Lambertson

Underwood

Woodruff

, et al. Upper atmosphere research satellite attitude disturbances during shadow entry and exit. NASA STI/Recon Technical Report A 95, 1993, pp.1003–1017.

10.

Yan

. Thermally induced vibration of composite solar array with honeycomb panels in low earth orbit. Appl Therm Eng 2014; 71: 419–432.

11.

Liu

Pan

. Rigid-flexible-thermal coupling dynamic formulation for satellite and plate multibody system. Aerosp Sci Technol 2016; 52: 102–114.

12.

Liu

Cao

Huang

, et al. Thermal-structural analysis for an attitude maneuvering flexible spacecraft under solar radiation. Int J Mech Sci 2017; 126: 161–170.

13.

Sales

Rade

de Souza

. Passive vibration control of flexible spacecraft using shunted piezoelectric transducers. Aerosp Sci Technol 2013; 29: 403–412.

14.

Kong

Huang

. Design and experiment of a passive damping device for the multi-panel solar array. Adv Mech Eng 2017; 9: 1–10.

15.

Jiang

Liu

, et al. A new mechanism for the vibration control of large flexible space structures with embedded smart devices. IEEE/ASME Trans Mechatron 2015; 20: 1653–1659.

16.

Jiang

. An online learning-based fuzzy control method for vibration control of smart solar panel. J Intell Mater Syst Struct 2015; 26: 2547–2555.

17.

Tang

Chen

. Vibration reduction of flexible solar array during orbital maneuver. Aircr Eng Aerosp Technol 2014; 86: 155–164.

18.

Azadi

Fazelzadeh

Azadi

. Thermally induced vibrations of smart solar panel in a low-orbit satellite. Adv Space Res 2017; 59: 1502–1513.

19.

Sun

, et al. Design of an innovative active hinge for self-deploying/folding and vibration control of solar panels. Sens?Actuators?A Phys 2018; 281: 196–208.

20.

Jiang

. Fuzzy vibration control of flexible solar panel using piezoelectric stack. Appl Mech Mater 2011; 50–51: 214–218.

21.

Azimi

Shahravi

Fard

. Modeling and vibration suppression of flexible spacecraft using higher-order sandwich panel theory. Int J Acoust Vib 2017; 22: 143–150.

22.

Liu

Rong

Shen

, et al. Dynamics and control of a flexible solar sail. Math Probl Eng 2014; 2014: 1–25.

23.

Yang

Zhang

. Computation of rayleigh damping coefficients in seismic time-history analysis of spatial structures. J Int Assoc Shell Spatial Struct 2010; 51: 125–135.

24.

Khan

Siddiqui

, et al. Vibration damping characteristics of carbon fiber-reinforced composites containing multi-walled carbon nanotubes. Compos Sci Technol 2011; 71: 1486–1494.

25.

Wen

, et al. High damping and high stiffness cfrp composites with aramid non-woven fabric interlayers. Compos Sci Technol 2015; 117: 92–99.

26.

Ouyang

Wang

Yao

, et al. Improved damping and mechanical properties of carbon fibrous laminates with tailored carbon nanotube/polyurethane hybrid membranes. Polym Polym Compos 2021; 29: 1240–1250.

27.

Tang

Yan

. A review on the damping properties of fiber reinforced polymer composites. J Ind Text 2020; 49: 693–721.

Improvement on the passive method based on dampers for the vibration control of spacecraft solar panels

Abstract

Keywords

Highlights

Introduction

Low frequency flexible vibration of a satellite solar panel

Conventional method for low frequency vibration control of spacecraft solar panel based on root springs and dampers

Passive vibration control of spacecraft solar panel based on translational root mounting method

Theoretical analysis and discussion

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References