Benefit of Relaxation-Type Damping on the Performance of a Six-Degree-of-Freedom Microvibration Isolation Device for Control Moment Gyroscope

In order to provide an ultraquiet environment for spacecraft payload, a six-degree-of-freedommicrovibration isolation device for satellite control moment gyro (CMG) is proposed in this paper.-e dynamic characteristics of the microvibration isolation device are analyzed theoretically and experimentally.-e dynamic equations of the microvibration suppression device are established by using the Newton–Euler method. -e dynamic responses are numerically solved and the frequency-domain characteristics of the microvibration isolation device under base excitation are analyzed.-e analytical results are first verified numerically, and the two results are in good accordance. -e experimental apparatus is built, and the vibration isolation performance is investigated. -e acceleration transfer function is measured and the influence of the excitation amplitude on the vibration isolation performance is performed. It is shown that the amplification factor at the vicinity of the resonance frequency is within 10 dB, and the vibration isolation performance is significant at higher frequencies. -e vibration attenuation performance at the main frequency of the CMG (100Hz) is more than 30 dB. -e microvibration suppression device can effectively suppress the microvibration generated by CMG during orbital operation.


Introduction
Remote sensing or observation of the Earth is one of the most important tasks of satellite. In order to get an ultra-high-resolution image, many disturbances that affect the performance of the sensitive payload need to be controlled on board. e most annoying disturbance that degrades the performance of payload is thermal deflection [1,2] of satellite structure and microvibration ejected by moving parts such as cryocooler [3][4][5], flywheel [6,7], and control moment gyroscope (CMG) [8]. As a matter of fact, CMGs are widely used in fast maneuver satellites because of their significant angular momentum. Nevertheless, the high-speed rotors in CMGs also make them become one of the largest microvibration sources. In a recently published paper [9], the on-orbit microvibration measurement in a remotesensing satellite shows that flywheel and CMG are the main vibration source. erefore, it is urgent to isolate the microvibration generated by flywheel and CMGs in order to provide quiet environment for sensitive payload. Not surprisingly, many microvibration suppression methods including vibration source control and payload isolation [10] are employed to achieve the purpose. For example, the famous Hubble Telescope used the viscous damper developed by Honeywell to isolate flywheel vibration [11]. In Chandra X-ray Observatory [12], the Stewart-type passive vibration isolation device for flywheel is also considered, and the vibration isolation performance of which at the launch phase is also designed and tested. Li et al. [13] conducted research on the microvibration induced by flywheel and the effect on space camera; the experimental results on ground show that the camera can work normally after a vibration isolator is applied to the flywheel. Cobb et al. [14] proposed a passiveactive vibration isolation system to protect the optical payload from the unwanted microvibration on board. Spanos et al. [15] at JPL invented a flexible active vibration isolation and pointing system which was successfully applied in James Webb Space Telescope. Zhang et al. [16,17] proposed a new method to achieve better image quality of optical payload. In their research, both the control strategy and the passive vibration source isolation are implemented. Li et al. [18] also considered a similar system and the difference of which laying on the vibration isolation system is active. Zhang et al. [19] focuse on the design of internal vibration isolation structures for the CMG to reduce the vibration generated by the imbalance of the rotor. A fully passive two-stage isolation system based on viscoelastic materials has been developed in order to minimize the CMG microvibrations by Kawak [20]. is paper proposed a new type of vibration isolation device to isolate the CMG vibration transmitted to satellite. e device not only can attenuate vibrations at the main frequency of the CMG but also can suppress resonance amplification factor to a considerable low level. Some useful design guidelines and concrete engineering based experimental results can enrich the vibration isolation field for CMGs. is paper is organized as follows. Firstly, the dynamic model of the proposed vibration isolation device based on a relaxation-type damper is deduced using Euler-Newton method. e dynamic equation is then solved numerically. Secondly, the vibration isolation performances for all six degrees of freedom are analyzed and the coupling features in different directions are discussed. irdly, the experimental setup is built, and the vibration isolation performance is evaluated in three translational directions. Some valuable conclusions are drawn in the last section.

e Microvibration Isolation Device.
e Stewart-type structure is used very popularly to get vibration isolation performance in six degrees of freedom [21]. e schematic of the proposed microvibration isolation device for single CMG in this paper is shown in Figure 1, which is also based on a Stewart-type structure. e platform includes upper platform to connect with the CMG and the lower platform to be installed on the satellite. In between, six relaxation-type isolators are configured in a cubic pattern, which provides proper stiffness and damping to isolate the vibration generated by the CMG transmitting to the satellite. Figure 2 shows a simplified representation of the vibration isolation device in which some useful coordinates are defined. e local coordinates on the upper and lower platforms are Px p y p z p and Bx b y b z b , respectively. e global inertial coordinate system is Gxyz. e hinges of the upper and lower platforms with the isolator are defined as P i x pi y pi z pi and B i x bi y bi z bi . Oxyz is the inertial principal axis coordinate system of the CMG. e following symbols are used to build the dynamic equation of the system.

Modeling of the Strut and Upper
Platform. Different from many researchers who consider the single strut as a Kevin model or an active actuator, the strut used in this paper is a relaxation-type damper. e strut model and the load applied to the strut are shown in Figure 3.
As this vibration isolation device is a Stewart-type platform, the strut is symmetric along the center axis of the upper and lower platforms. e position vector l i of the i th strut in coordinate G { } can be expressed as T . e unit vector for the i th strut is given as e translational velocity and acceleration along the strut for the i th strut are where u 1i � ω p × (ω p × t pi ).  e acceleration of the mass center of the upper and lower legs can be expressed as Equations of moment of momentum for the upper and lower legs are given as Equation (6) can be simplified to express the force applied to the upper platform from one single strut, which is As shown in Figure 4, the force applied to the upper platform from one single strut is −F si . e gravity of the upper platform and the CMG is m 0 g, and the force and moment disturbance generated by the CMG are assumed as F mac and M mac .
By using Newton-Euler method, the dynamic equation for the upper platform is given as where q c ″ is the absolute acceleration of the upper platform in coordinate G { }, R is the position vector of the mass center of the upper platform in coordinate P { }, and I p is the inertia matrix of the upper platform in coordinate G { }.

Response Solution.
e force F i applied to the upper platform from each strut can be expressed as    where x di is the deflection of the joint of the damping and the parallel connected spring in the i th strut. Applying Laplace transform to equation (9), one can get the simplified equation where Combining equations (7)- (9), the dynamic equation of the vibration isolation device subjected to base excitation can be derived asw where Equation (11) gives the dynamic responses of the upper platform when the base is excited with harmonic vibration.

Performance of the Microvibration
Isolation Device e microvibration isolation device is designed to isolate vibration ejected by CMG and the parameters of which are given in Table 1.
e relaxation-type damper has been manufactured beforehand and the parameters of which are tested and recorded in Table 2. Meanwhile, the structure parameters of the microvibration isolation device are summarized in Table 3.

Natural Frequency Analysis.
First, the natural frequencies of the microvibration isolation in six directions are analyzed. By simplifying equation (11) and neglecting damping, one can get the free vibration equation of the isolation device without damping as In equation (12), K p x is the equivalent stiffness matrix and N i k i M p is the equivalent mass matrix. e first six-order natural frequencies of the vibration isolation system can be obtained by solving equation (12). e modal analysis is also conducted by using finite element software and the model of which is shown in Figure 5. e results of the natural frequency obtained from the finite element software are compared with the theoretical results solved from equation (12). e comparison results are given in Table 4 and the two results are in good accordance. Moreover, in order to understand the natural characteristics of this system thoroughly, the corresponding mode shapes are also summarized in Figure 6.
From the results of natural frequency and the mode shape, one can conclude that the third-order mode and the fifth-order mode are the Z-direction translational and Z-axis twist pure mode, respectively. e other four modes are coupled mode in X-direction and Y-direction.
3.2. Relaxation-Type Damper Performance. As mentioned above, the flexible strut used here is a relaxation-type damper. e advantage of this kind of damper over a Kevin model damper is that it can control the resonance amplification without losing capacity of attenuating higher-frequency vibration if the parameters of the damper are chosen carefully. To clarify this point more clearly, the transmissibility of the relaxation-type damper used in this paper is plotted in Figure 7. For the sake of comparison convenience, the transmissibility curve of a system with a Kevin model is also plotted in the same figure. One can see that although the two dampers exhibit the same resonance amplification factor, the vibration attenuation rate at the main frequency of CMG (100 Hz) has more than10 dB differences, which is the advantage of the relaxationtype damping.

Coupling Analysis.
e main reason that induced the coupling in different directions for the vibration isolation device considered here is that the mass center is out of the plane of the upper platform. Without loss of generality, the parameters h � 0 mm and h � 40 mm are chosen to illustrate the coupling characteristic of the device. e transmissibility of the two cases is plotted in Figures 8 and 9 for base excitation.
By observing the two figures, one can conclude that if the mass center of the CMG is in the plane of upper platform, there is no coupling for the device. When the mass center of the CMG is higher than the plane of upper platform, the X-direction translation, Y-direction rotation, Y-direction translation, and X-direction rotation are coupled. Two peaks in the transmissibility are observed. However, the movements in Z-direction translation and rotation are independent of the other direction in spite of the changing of the mass center of the CMG. is conclusion is very important when designing such kind of device to acquire better vibration isolation performance in certain direction.

Vibration Isolation Performance in Translational and
Rotational Direction. To evaluate the vibration isolation performance of the device, equation (13) is numerically solved by the Runge-Kutta method in MATLAB software. e base is applied with an excitation amplitude of 30 mg according to satellite experiment and the frequency range is from 2 to 40 Hz. e absolute transmissibility is used to represent the vibration performance, which is defined as the ratio of acceleration of upper and lower platforms. e translational transmissibility and rotational transmissibility are plotted in Figures 10(a) and 10(b). Meanwhile, the resonance amplification factor and the vibration attenuation rate at the main frequency of the CMG are summarized in Table 5. e transmissibilities in X-direction and Y-direction translation are basically the same because of the symmetrical feature of the device. In all six directions, the minimum natural frequency is 6.84 Hz (X-direction translational) and the maximum natural frequency is 28.65 Hz (Y-direction rotational). e vibration attenuation rate at 100 Hz is beyond 30 dB, which means that the vibration isolation efficiency is more than 94%. Moreover, the resonant amplification factor is below 3 times.

The Experiment
e Scheme. e experimental scheme is shown in Figure 11. It includes a microvibration excitation table, the vibration isolation device, accelerometers, LMS data acquisition system, and suspension system.
To cancel the gravity, a soft suspension system is built in Figure 12. e vibration table which simulates the excitation of sweep sine frequency applies the excitation to lower platform; both the accelerations on the vibration table and on the CMG are measured in order to calculate the absolute transmissibility.
e translational transmissibilities in Xdirection, Y-direction, and Z-direction are tested.   172 Shorter distance of hinges on upper platform l p (see Figure 2(b)) 28 Shorter distance of hinges on upper platform l b (see Figure 2 Figure 13, and the force-deflection curve is fitted by the LMS method, which is given in Figure 14. In the figure, one can calculate that the stiffness of the strip is about 0.245 N/mm, which yields a suspension frequency of about 0.3 Hz. e natural frequency of the suspension system is far lower than the first natural frequency of the vibration isolation system, which means that the suspension system can affect the experimental result a little. Moreover, in the vertical direction, the suspension length is more than 3 meters; therefore, the horizontal suspension frequency is also low enough.    that there is no coupling in this direction. e amplification factor at the resonance frequency is less than 2 times.

Frequency-Domain Results.
e transmissibilities in the three directions are shown in Figures 18 to 20. Both the experimental and theoretical results are plotted. Overall, the proposed vibration isolation device has excellent vibration attenuation performance. More than 20 dB attenuation is obtained at the main frequency of CMG, which is 100 Hz.
ere are slight deviations of the theoretical and experimental results in X-direction and Y-direction, though the resonance frequency is basically accurate. e reason for the deviation is possibly the coupling and the joint friction or assembly error. Nevertheless, the theoretical results can be used to predict such kind of vibration isolation device in engineering, since the trends are in good consistency and the deviations only occur at frequencies of small range.
In the Z-direction transmissibility, the theoretical and experimental results are in superb accordance. is is mainly because, as mentioned above, there is no coupling in this direction. In Figure 20, a transmissibility of a vibration isolation system with Kevin model is also plotted for comparison convenience. e curve is plotted by assuming a   same resonance amplification factor for both systems. It can be seen from the figure that the device proposed in this paper can provide better vibration isolation performance at the main frequency of the CMG. e device with relaxation-type dampers may isolate 10 dB more vibration than that with a Kevin model according to the experimental results. Moreover, the amplification factor at resonance frequency of this device is below 10 dB, which is an excellent index compared with traditional vibration isolation systems on market.

Effect of the Excitation Amplitude.
In fact, the model studied in this paper is a linear system. In other words, the amplitude of excitation is irrelevant to the vibration isolation performance. However, this feature is verified experimentally for the sake of completeness. e excitation amplitude    is varied as 10 mg, 20 mg, and 30 mg in Z-direction and the transmissibilities are plotted in Figure 21 for different excitation amplitudes. From the figure, one can easily distinguish that the amplitude of the excitation has rare effect on the vibration isolation performance.

Conclusions
A microvibration suppression device is proposed based on a relaxation-type damper and cubic configured Stewart platform in this paper. Both theoretical and experimental investigations are carried out. Some conclusions can be drawn as follows: (1) e height between the mass center of the CMG and the upper platform of the device is the main factor that induces coupling effect; this parameter should be chosen carefully when designing this kind of device. (2) e proposed device can provide more than 30 dB vibration attenuation at the main frequency of the CMG and less than 10 dB amplification at resonance, which is better than that of a traditional linear one. (3) e excitation amplitude has rare effect on the vibration isolation performance and the device can function in a wide amplitude range.
Data Availability e data are available upon request to the corresponding author.

Conflicts of Interest
e authors declare that they have no conflicts of interest.