Vibration Isolation Platform with Multiple Tuned Mass Dampers for Reaction Wheel on Satellites

Abstract

Vibration isolation is a direct and effective approach to improve the ultraprecise pointing capability of an imaging satellite. To have a good trade-off between the resonance amplitude and the high frequency attenuation for the original vibration isolation platform, a novel vibration isolation system for reaction wheel (RW), including a multistrut vibration isolation platform and multiple tuned mass dampers, is proposed. The first step constructs the integrated satellite dynamic model including the RWs and the vibration isolation systems, while the static and dynamic imbalances of the rotor and base movements are considered in the modeling process. The transmissibility matrix of the vibration isolation system is then obtained, and its frequency domain characteristics are described. The third part presents the application of the vibration isolation system for RWs. The effective attenuation of RW disturbances by the new vibration isolation system is illustrated, and its safety performance is also verified. Finally, using the reasonable parameters of the vibration isolation system, its performance on the satellite is testified by numerical simulations. The study shows that the novel vibration isolation system presented cannot only be successfully applied to a satellite but also improve the attitude stability.

1. Introduction

Recently, spacecraft with high pointing accuracy and stability gains increasing importance in space missions. For example, the Hubble Space Telescope (HST) requires the pointing stability to be less than 0.007 arcsecond within periods up to 24 hours [1]. The James Webb Space Telescope (JWST) needs the line of sight motion to be 4 milliarcseconds [2]. The Terrestrial Planet Finder Coronagraph (TPF-C) must also maintain the pointing accuracy to 4 milli-arcseconds in order to meet the minimum science requirements [3].

For this class of spacecraft, attitude control torques are usually provided by reaction wheels (RWs) which produce continuous torques to realize high precision pointing control and perform large angle slewing maneuvers. However, Masterson et al. [4] found that the RW can also produce tonal disturbances, and broadband noises when the wheel spins, due to the imbalance of the rotor, the imperfections in the spin bearings, motor disturbances and motor driven errors, thus making it one of the largest disturbance sources onboard the spacecraft. To mitigate these effects on the spacecraft pointing control, the vibration isolation technology is often used for the RW. HST used viscous fluid dampers called the D-Struts to attenuate the axial disturbances of the RWs [5]. The Defense Satellite Communications System III Spacecraft applied four damped stainless steel spring isolators supporting an RW to provide six degrees of freedom wheel isolation [6]. The Chandra X-ray Observatory employed a hexapod isolator at each of its six RWs to achieve multidimensional vibration isolation [7]. This passive vibration isolation platform will be employed in JWST as the first isolation stage [8]. Moreover, Kamesh et al. [9] designed a flexible platform consisting of folded continuous beams arranged in three dimensions to act as a mount for each fly wheel.

However, in designing the above-mentioned vibration isolation systems of RWs, the researchers ignored the dynamic characteristics of the RW. The RW should constantly adjust its speed to realize attitude stabilization or attitude maneuver. Its disturbance force is caused by the static imbalance of the rotor, whose amplitude is proportional to the square of the wheel speed, and whose frequency is the same as the wheel speed. Consequently, a resonance condition will occur when the wheel speed and the corner frequencies of the above-mentioned vibration isolation systems come close. This resonance will cause more serious vibrations to the satellite, significantly influencing the image performance of the optical payload on the satellites. If the resonance amplitude is large enough, the vibration isolation system will lose the ability to transfer effective torques, causing rapid catastrophic attitude control system failure. A tuned mass damper (TMD) is a device mounted in structures to reduce the amplitude of mechanical vibrations, the application of which can prevent discomfort, damage, or outright structural failure. Therefore, it is frequently used in power transmission, automobiles, and buildings [10]. The achievements of the dynamic analysis and simulation method for the TMD are valuable. For example, Hoang et al. [11] have focused on optimal design of a TMD for a single degree of freedom structure and shown that for large mass ratios TMD becomes very effective in minimizing the primary structure response and robust against uncertainties in the parameters of the system. Li et al. [12] developed a single foot force model for the vertical component of walking-induced force and presented the optimal design parameters method for the multiple TMDs. For the dynamic analysis of space structures with multiple TMDs, Guo and Chen [13] presented the formulations of the reverberation matrix method. However, the previous dynamic models all ignored the base movements and the dynamic characteristics of the vibration source.

In addition, up to now, jitter performances of the spacecraft using the vibration isolation system for RWs have been predicted in some missions, such as the Space Interferometer Mission (SIM) [14], Terrestrial Planet Finder Coronagraph (TPF-C) [15], the GONES-N spacecraft [16], and Solar Dynamics observatory (SDO) [17]. In these literatures, the structures of spacecraft are all modeled by the finite-element method, with some using the vibration isolation system consisting of 6 decoupled second-order low-pass filters to approximate the effect of a passive fly wheel mount [18]. These accurate finite element models are necessary to predict the jitter performance of the spacecraft. However, the analysis using finite element models has some limitations. The spacecraft structure, for instance, should be well known, the dynamic coupling cannot be obtained, and the finite element model is not suitable for attitude controller designs based on the modern control theory.

This paper aims to design a new vibration isolation system for RW which cannot only satisfy the brace stiffness requirement but also reduce the resonance amplitude. The dynamic model of an integrated satellite system with RWs and the new vibration isolation system is constructed with Newton-Euler approach. The frequency domain characteristics of the vibration isolation system are described. Reasonable parameters of the vibration isolation system are employed to testify its performance on the satellite in numerical simulations.

2. Dynamics Model of the Integrated Satellite with RWs and Vibration Isolation System

2.1. Description of RW Disturbances

The present study only focuses on the rotor imbalance considered in the RW disturbances since the magnitudes of disturbances from motor errors and bearing imperfections are relatively small. The RW reference frames used in the modeling are shown in Figure 1 and are defined as follows:(1)quasi-geometric body frame of the rotor: 𝐟_𝑓(𝑜_𝑓𝑥_𝑓𝑦_𝑓𝑧_𝑓), whose origin is fixed at the geometric centre of the rotor, with its 𝑥-axis along the spin axis of rotor, and 𝑦 and 𝑧 axes in the rotating plane of the rotor perpendicular to 𝑥-axis. The axes do not rotate as the rotor does;(2)geometric body frame of the rotor: 𝐟_𝑤(𝑜_𝑤𝑥_𝑤𝑦_𝑤𝑧_𝑤), fixed to the rotor with its origin at the geometric centre, and its 𝑥-axis along the spin axis. In addition, 𝐟_𝑤 coincides with 𝐟_𝑓 in the initial state;(3)principal axes of the inertia frame of the rotor: 𝐟_𝐼(𝑜_𝐼𝑥_𝐼𝑦_𝐼𝑧_𝐼), fixed to the rotor with its origin at the centroid, its 𝑥-axis along the principal axis of the inertia in the spin axial direction. The vector 𝝆_𝑤 from 𝑜_𝑤 to 𝑜_𝐼 and the rotation angles 𝜇 and 𝜂 from 𝐟_𝑤 to 𝐟_𝐼 represent the static and dynamic imbalances of the rotor, respectively.

Figure 1 

Reference frames.

For the unification of notations, we define 𝐀_𝑛𝑚 as the coordinate transformation matrix from frame 𝐟_𝑚 to frame 𝐟_𝑛 and 𝐫_𝑛𝑚 the vector from 𝑜_𝑚 to 𝑜_𝑛. The static imbalance of the rotor can be expressed in frame 𝐟_𝑤 as 𝝆_𝑤=[𝛾𝜉𝜁]^𝑇, where 𝛾, 𝜉, and 𝜁 are three relatively small constants, showing the offset of the center of mass of the rotor from its spin axis. With high order small quantities ignored, the dynamic imbalance of the rotor can be expressed as 𝐀_𝐼𝑤=𝐀_𝑦𝜇𝐀_𝑧𝜂=≈,cos𝜇0−sin𝜇010sin𝜇0cos𝜇cos𝜂sin𝜂0−sin𝜂cos𝜂00011𝜂−𝜇−𝜂10𝜇01 (1) where 𝜇 and 𝜂 are small Euler angles from frame 𝐟_𝑤 to frame 𝐟_𝐼.

2.2. System Configuration and Reference Frames Definitions

In this research, the RW of a satellite is isolated separately, forming a combined system with the vibration isolation system. Figure 2 gives a schematic representation of the combined RW/vibration isolation system.

Figure 2 

Combined RW/vibration isolation system configuration.

As for the platform part, a passive multistrut vibration isolation platform based on the Stewart platform is adopted. Multiple tuned mass dampers are installed on the payload platform to attenuate the resonant amplitude. Each strut of the platform includes an upper part, a lower part, a spring, and a damper connecting the two parts. The struts are connected to the payload platform by spherical joints and the base platform by universal joints. The moments of inertia of the struts are taken into account in the modeling process. To build a general dynamic model of an integrated satellite with RWs, we assume that there are 𝐿 RWs, 𝑁 struts of each vibration isolation platform, and 𝐻 tuned mass dampers installed on the payload platform. Considering the convenience of the satellite modeling, this paper redivides the system into four parts as follows: each payload platform system, tuned mass dampers, struts, and the base platform system, where the payload platform system consists of a payload platform and a RW with rotor imbalances, and the base platform system includes all the base platforms and the satellite bus.

The reference frames used in the modeling are shown in Figure 3 and are defined as follows:(1)inertial frame: 𝐟_𝑒(𝑜_𝑒𝑥_𝑒𝑦_𝑒𝑧_𝑒), fixed in the inertial space;(2)body frame of the satellite: 𝐟_𝑏(𝑜_𝑏𝑥_𝑏𝑦_𝑏𝑧_𝑏), with its origin at the reference point of the satellite, and axes fixed in the satellite;(3)body frame of the payload platform: 𝐟_𝑝(𝑜_𝑝𝑥_𝑝𝑦_𝑝𝑧_𝑝), with its origin at the mass center of the payload platform, and axes fixed in the payload platform;(4)body frame of the base platform: 𝐟_𝑑(𝑜_𝑑𝑥_𝑑𝑦_𝑑𝑧_𝑑), with its origin at the mass center of the base platform, and axes fixed in the base platform;(5)body frame of the ℎth tuned mass damper: 𝐟_𝑡ℎ(𝑜_𝑡ℎ𝑥_𝑡ℎ𝑦_𝑡ℎ𝑧_𝑡ℎ), with its origin at the mass center of the ℎth tuned mass damper, and axes fixed in the ℎth tuned mass damper. It is the same orientation as 𝐟_𝑝(𝑜_𝑝𝑥_𝑝𝑦_𝑝𝑧_𝑝) in the initial state;(6)lower frame of the 𝑖th strut: 𝐟_𝑑𝑖(𝑜_𝑑𝑖𝑥_𝑑𝑖𝑦_𝑑𝑖𝑧_𝑑𝑖), attached to the lower part of the 𝑖th strut with its origin at the connection point to the base platform, 𝑥-axis along the strut, 𝑦-axis along the rotating axis (axis fixed to the strut) of the universal joint, 𝑧-axis perpendicular to the 𝑥 and 𝑦 axes according to the right hand rule;(7)upper frame of the 𝑖th strut: 𝐟_𝑢𝑖(𝑜_𝑢𝑖𝑥_𝑢𝑖𝑦_𝑢𝑖𝑧_𝑢𝑖), attached to the upper part of the 𝑖th strut with its origin at the connection point to the payload platform. It is the same orientation as 𝐟_𝑑𝑖(𝑜_𝑑𝑖𝑥_𝑑𝑖𝑦_𝑑𝑖𝑧_𝑑𝑖) in the initial state.

Figure 3 

Reference frames (vibration isolation system part).

In Figure 3, 𝐭 is the position vector from the origin of the inertial frame to the mass center of the payload platform, and 𝐝 is the position vector from the origin of the inertial frame to the mass center of the base platform. 𝐫_𝑡ℎ is the position vector from the origin of the inertial frame to the mass center of the ℎth tuned mass damper. 𝐩_𝑖 is the vector from the mass center of the payload platform to its connection point. 𝐪_𝑖 is the vector from the mass center of the base platform to its connection point. 𝐫_𝑑𝑏 is the vector from the origin of 𝐟_𝑏 to the origin of 𝐟_𝑑. 𝐒_𝑖 is the vector of the 𝑖th strut (vector from the origin of frame 𝐟_𝑑𝑖 to the origin of frame 𝐟_𝑢𝑖).

The modeling method adopted in this paper is based on the Newton-Euler approach employed by Dasgupta and Mruthyunjaya to derive the dynamic formulation of a Stewart platform [19]. Since the RW is introduced, the dynamic model of the payload platform system is similar to a spacecraft with a momentum exchange actuator. Therefore, the frame of the modeling process is as follows: firstly, establish the equations for the motion of the struts, the tuned mass dampers, each payload platform system, and the base platform system separately; then, combine those equations to form the dynamic formulation of the whole system. Particularly, it should be noticed that the motion of each strut is determined by the position and angular information of platforms, and that the forces and torques at joints are given by the dynamics of struts. In addition, the motion of each tuned mass damper is determined by the position and angular information of the payload platform, and that the forces and torques at joints are given by the dynamics of the tuned mass dampers.

2.3. Kinematics and Dynamics of Struts

Here and throughout, it is assumed that there are 𝑁 struts in the combined system. The 𝑖th strut vector 𝐒_𝑖 and its derivatives are defined as follows: 𝐒_𝑖=𝐭+𝐀_𝑒𝑝𝐩_𝑖−𝐝−𝐀_𝑒𝑏𝐪_𝑖,𝐒̇_𝑖=𝐭̇+𝐀_𝑒𝑝𝝎^×_𝑝𝐩_𝑖−𝐝̇−𝐀_𝑒𝑏𝝎^×_𝑏𝐪_𝑖,𝐒̈_𝑖=𝐭̈+𝐀_𝑒𝑝𝝎̇^×_𝑝𝐩_𝑖+𝐀_𝑒𝑝𝝎^×_𝑝𝝎^×_𝑝𝐩_𝑖−𝐝̈−𝐀_𝑒𝑑𝝎̇^×_𝑑𝐪_𝑖−𝐀_𝑒𝑑𝝎^×_𝑑𝝎^×_𝑑𝐪_𝑖, (2) where 𝝎_𝑝 and 𝝎_𝑑 are the angular velocities of the payload and the base platform, respectively. The superscript “×” denotes the cross product matrix.

The strut length and the unit vector along the strut are given by 𝑙_𝑖=𝐒_𝑖,𝐬_𝑖̂=𝐒_𝑖𝑙_𝑖. (3)

The angular velocity and angular acceleration of the 𝑖th strut are obtained by 𝝎_𝑙𝑖=𝝎_𝑙𝑖1+𝜔_𝑙𝑖2𝐬_𝑖̂,𝜺_𝑙𝑖=𝜺_𝑙𝑖1+𝜀_𝑙𝑖2𝐬_𝑖̂, (4) where 𝝎_𝑙𝑖1 and 𝜺_𝑙𝑖1 are the angular velocity and angular acceleration perpendicular to the strut and 𝜔_𝑙𝑖2𝐬_𝑖̂ and 𝜀_𝑙𝑖2𝐬_𝑖̂ are the components along the strut direction.

Consider 𝝎_𝑙𝑖1=𝐬^×_𝑖̂𝐒̇_𝑖𝑙_𝑖,𝜔_𝑙𝑖2𝝎=−^{𝑇𝑙𝑖1}𝐡_𝑖𝐬^𝑇_𝑖̂𝐡_𝑖,𝜺_𝑙𝑖1=1𝑙_𝑖𝐬^×_𝑖̂𝐒̈_𝑖−2𝑙_𝑖̇𝝎_𝑙𝑖+2𝑙_𝑖̇𝜔_𝑙𝑖2𝐬_𝑖̂−𝑙_𝑖𝜔_𝑙𝑖2𝐬^×_𝑖̂𝝎_𝑙𝑖1,𝜀_𝑙𝑖2=𝜺^𝑇_𝑙𝑖1𝐤_𝑖𝐤^𝑇_𝑖𝐬_𝑖̂+𝜔_𝑘𝑖𝜔_𝑦𝑖𝐤^×_𝑖𝐲_𝑖^𝑇𝐬_𝑖̂1−𝐤^𝑇_𝑖𝐬̂_𝑖². (5)

In the previous expressions, 𝜔_𝑘𝑖=𝝎^𝑇_𝑙𝑖𝐤_𝑖 and 𝜔_𝑦𝑖=𝝎^𝑇_𝑙𝑖1𝐲_𝑖. 𝐤_𝑖 is the unit vector along the fixed axis of the universal joint, which is known if the installation of the universal joint is determined. 𝐲_𝑖 is the unit vector along the moving axis of the universal joint. 𝐡_𝑖 is the normal vector of the plane formed by 𝐤_𝑖 and 𝐲_𝑖. Expressions of 𝐲_𝑖 and 𝐡_𝑖 are 𝐲_𝑖=𝐤^×_𝑖𝐬_𝑖̂𝐤^×_𝑖𝐬_𝑖̂,𝐡_𝑖=𝐲^×_𝑖𝐤_𝑖. (6)

Then, the sliding velocity and acceleration between the two parts of the strut are 𝑙_𝑖̇=𝐬^𝑇_𝑖̂𝐒̇_𝑖,𝑙_𝑖̈=𝐬^𝑇_𝑖̂𝐒̈_𝑖−𝐬^𝑇_𝑖̂𝝎^×_𝑙𝑖𝝎^×_𝑙𝑖𝐒_𝑖. (7) The linear accelerations of the centroids of the upper and lower parts of the strut can be expressed as 𝐚_𝑢𝑖=𝐚_𝑞𝑖+𝜺^×_𝑙𝑖𝐫_𝑢𝑖+𝝎^×_𝑙𝑖𝝎^×_𝑙𝑖𝐫_𝑢𝑖+𝑙_𝑖̈𝐬_𝑖̂+2𝝎^×_𝑙𝑖𝑙_𝑖̇𝐬_𝑖̂,𝐚_𝑑𝑖=𝐚_𝑞𝑖+𝜺^×_𝑙𝑖𝐫_𝑑𝑖+𝝎^×_𝑙𝑖𝝎^×_𝑙𝑖𝐫_𝑑𝑖, (8) where 𝐚_𝑞𝑖=𝐝̈+𝐀_𝑒𝑑𝝎̇^×_𝑑𝐪_𝑖+𝐀_𝑒𝑑𝝎^×_𝑑𝝎^×_𝑑𝐪_𝑖 is the linear acceleration of 𝑜_𝑑𝑖, 𝐫_𝑢𝑖 and 𝐫_𝑑𝑖 are the vectors from 𝑜_𝑑𝑖 to the centroids of the upper and the lower parts, respectively, and 𝑙_𝑖̇ and 𝑙_𝑖̈ are the sliding velocity and sliding acceleration between the two parts.

The details of one strut are shown in Figure 4. Considering the moments acting on the strut in the inertial frame, Euler’s equation for the whole strut can be obtained as follows: 𝐈_𝑢𝑖+𝐈_𝑑𝑖𝜺_𝑙𝑖+𝝎^×_𝑙𝑖𝐈_𝑢𝑖+𝐈_𝑑𝑖𝝎_𝑙𝑖=𝐒^×_𝑖𝐅_𝑠𝑖+𝑀_𝑢𝑖𝐡_𝑖−𝑐_𝑠𝑖𝝎_𝑙𝑖−𝐀_𝑒𝑝𝝎_𝑝−𝑐_𝑢𝑖𝝎_𝑙𝑖−𝐀_𝑒𝑑𝝎_𝑑−𝑚_𝑢𝑖𝐫_𝑢𝑖+𝑚_𝑑𝑖𝐫_𝑑𝑖^×𝐚_𝑞𝑖. (9)

Figure 4 

Details of one strut.

Newton’s equation for the whole strut is 𝑚_𝑢𝑖𝐚_𝑢𝑖+𝑚_𝑑𝑖𝐚_𝑑𝑖=𝐅_𝑠𝑖+𝐅_𝑢𝑖, (10) where 𝑚_𝑢𝑖 and 𝑚_𝑑𝑖 are the masses of the lower and upper parts of the 𝑖th strut, 𝐈_𝑢𝑖 and 𝐈_𝑑𝑖 are moments of inertia of the lower and upper parts of the 𝑖th strut with respect to point 𝑜_𝑑𝑖 (expressed in the inertial frame), 𝐅_𝑠𝑖 and 𝐅_𝑢𝑖 are the constrained forces at the spherical joint and the universal joint acting on the strut, respectively, 𝑀_𝑢𝑖𝐡_𝑖 is the constrained torque at the universal joint acting on the strut, and 𝑐_𝑠𝑖 and 𝑐_𝑢𝑖 are the coefficients of viscous friction in the spherical and universal joints, respectively.

Solving (9) and (10), we obtain 𝑀_𝑢𝑖=𝐬^𝑇_𝑖̂𝐂_𝑖𝐬^𝑇_𝑖̂𝐡_𝑖,𝐅_𝑠𝑖=𝑋_𝑖𝐬_𝑖̂−𝐬^×_𝑖̂𝐂_𝑖−𝐬^×_𝑖̂𝑀_𝑢𝑖𝐡_𝑖𝑙_𝑖,𝐅_𝑢𝑖=𝑚_𝑢𝑖𝐚_𝑢𝑖+𝑚_𝑑𝑖𝐚_𝑑𝑖−𝐅_𝑠𝑖, (11) where 𝐂_𝑖=𝐈_𝑢𝑖+𝐈_𝑑𝑖𝜺_𝑙𝑖+𝝎^×_𝑙𝑖𝐈_𝑢𝑖+𝐈_𝑑𝑖𝝎_𝑙𝑖+𝑐_𝑠𝑖𝝎_𝑙𝑖−𝐀_𝑒𝑝𝝎_𝑝+𝑐_𝑢𝑖𝝎_𝑙𝑖−𝐀_𝑒𝑑𝝎_𝑑+𝑚_𝑢𝑖𝐫_𝑢𝑖+𝑚_𝑑𝑖𝐫_𝑑𝑖^×𝐚_𝑞𝑖,𝑋_𝑖=𝐬^𝑇_𝑖̂𝐅_𝑠𝑖=𝑚_𝑢𝑖𝐬^𝑇_𝑖̂𝐚_𝑢𝑖+𝑘_𝑖𝑙_𝑖−𝑙_𝑖0+𝑐_𝑖𝑙_𝑖̇, (12) where 𝑘_𝑖 and 𝑐_𝑖 are the coefficients of stiffness and viscous damping of the 𝑖th strut, respectively, and 𝑙_𝑖0 is the nominal strut length.

2.4. Dynamic Equations for Tuned Mass Dampers

Here and throughout, it is assumed that there are 𝐻 tuned mass dampers in the vibration isolation system. Taking 𝑜_𝑡ℎ as the reference point, we can derive the dynamic equations of the ℎth tuned mass damper from theorems of moment and moment of momentum 𝑚_ℎ𝐀_𝑝𝑒𝐫̈_𝑡ℎ+𝐂_ℎ𝑡𝐀_𝑝𝑒𝐫̇_𝑡ℎ−𝐭̇−𝐀_𝑒𝑝𝝎^×_𝑝𝐭_ℎ+𝐊_ℎ𝑡𝐀_𝑝𝑒𝐫_𝑡ℎ−𝐭−𝐀_𝑒𝑝𝐭_ℎ=𝟎_3×1,𝐀_𝑝𝑡ℎ𝐈_ℎ𝝎̇_𝑡ℎ+𝝎^×_𝑡ℎ𝐈_ℎ𝝎_𝑡ℎ=−𝐊_ℎ𝑟𝜽_𝑡ℎ−𝜽_𝑝+𝐂_ℎ𝑟𝐀_𝑝𝑡ℎ𝝎_𝑡ℎ−𝝎_𝑝, (13) where 𝑚_ℎ and 𝐈_ℎ are the mass and moment of inertia of the ℎth tuned mass damper, respectively, and 𝐭_ℎ represents the vector from the origin of 𝐟_𝑝 to the origin of 𝐟_𝑡ℎ, with 𝝎_𝑡ℎ being the angular velocity of the ℎth tuned mass damper, 𝜽_𝑡ℎ and 𝜽_𝑝 being the attitude angles of the ℎth tuned mass damper and the payload platform, respectively. In this paper, the tuned mass damper is regarded as a 6-dof vibration system. Therefore, each tuned mass damper not only has translational stiffness and damping coefficient matrices 𝐊_ℎ𝑡 and 𝐂_ℎ𝑡 but also contains rotational stiffness and damping coefficient matrices 𝐊_ℎ𝑟 and 𝐂_ℎ𝑟.

According to (13), the constrained forces and the constrained torque at the joint between the ℎth tuned mass damper and the payload platform can be solved as 𝐅_𝑡ℎ=𝐂_ℎ𝑡𝐀_𝑝𝑒𝐫̇_𝑡ℎ−𝐭̇−𝐀_𝑒𝑝𝝎^×_𝑝𝐭_ℎ+𝐊_ℎ𝑡𝐀_𝑝𝑒𝐫_𝑡ℎ−𝐭−𝐀_𝑒𝑝𝐭_ℎ,𝐓_𝑡ℎ=𝐊_ℎ𝑟Δ𝜽_𝑡ℎ+𝐂_ℎ𝑟𝐀_𝑝𝑡ℎ𝝎_𝑡ℎ−𝝎_𝑝+𝐭^×_ℎ𝐂_ℎ𝑡𝐀_𝑝𝑒𝐫̇_𝑡ℎ−𝐭̇−𝐀_𝑒𝑝𝝎^×_𝑝𝐭_ℎ+𝐊_ℎ𝑡𝐀_𝑝𝑒𝐫_𝑡ℎ−𝐭−𝐀_𝑒𝑝𝐭_ℎ, (14) where Δ𝜽_𝑡ℎ=𝜽_𝑡ℎ−𝜽_𝑝.

2.5. Dynamic Equations for Each Payload Platform System

Because of the rotor imbalance, the position of the centroid of the payload platform system changes incessantly during the operation of the RW. For the convenience of modeling, we take 𝑜_𝑝 as the reference point and establish the absolute angular momentum of the system.

We assume that there are 𝐿 RWs, and that each RW has one matching vibration isolation system. In this section, the dynamic equations for the 𝑗th payload platform system are derived. For the simplification of these equations, we do not write the subscript 𝑗 in the later discussion, and the number of RWs can be selected by some space missions.

Omitting some derivation processes, here we present the absolute linear and angular momentums of the 𝑗th rotor directly as follows: 𝐩_𝑤=𝑚_𝑤𝐯_𝑝+𝝎^×_𝑝𝐫_𝑓𝑝−𝑚_𝑤𝐀_𝑝𝑤𝝆^×_𝑤𝐀_𝑤𝑝𝝎_𝑝+𝐀_𝑝𝑓𝛀,𝐡^𝑜𝑝_𝑤=𝑚_𝑤𝐫^×_𝑓𝑝𝐯_𝑝+𝝎^×_𝑝𝐫_𝑓𝑝+𝝎^×_𝑝𝐀_𝑝𝑤𝝆_𝑤+𝐀_𝑝𝑓𝛀^×𝐀_𝑓𝑝𝐀_𝑝𝑤𝝆_𝑤+𝑚_𝑤𝐀_𝑝𝑤𝝆^×_𝑤𝐀_𝑤𝑝𝐯_𝑝+𝝎^×_𝑝𝐫_𝑓𝑝+𝐀_𝑝𝑤𝐈^𝑜𝑓_𝑤𝑤𝐀_𝑤𝑝×𝝎_𝑝+𝐀_𝑝𝑓𝛀, (15) where 𝑚_𝑤 is the mass of the rotor and 𝐈^𝑜𝑓_𝑤𝑤=𝐀_𝑤𝐼𝐈^𝑜𝐼_𝑤𝐼𝐀_𝐼𝑤+𝑚_𝑤(𝝆^𝑇_𝑤𝝆_𝑤𝐄₃−𝝆_𝑤𝝆^𝑇_𝑤) is the moment of inertia of the rotor with respect to its geometric centre (in frame 𝐟_𝑤). 𝐈^𝑜𝐼_𝑤𝐼=diag(𝐈_𝑤𝐼𝑥𝐈_𝑤𝐼𝑦𝐈_𝑤𝐼𝑧) is the moment of inertia of the rotor with respect to its centroid (in frame 𝐟_𝐼); 𝛀_𝑗 represents the angular velocity of the rotor with respect to the payload platform, which has the same expressions in frames 𝐟_𝑓 and 𝐟_𝑤; 𝐯_𝑝 represents the absolute linear velocity of point 𝑜_𝑝.

The absolute linear and angular momentums of the payload platform are 𝐩_𝑝=𝑚_𝑝𝑝𝐯_𝑝,𝐡^𝑜𝑝_𝑝=𝐈^𝑜𝑝_𝑝𝑝𝝎_𝑝, (16) where 𝑚_𝑝𝑝 is the mass of the payload platform and 𝐈^𝑜𝑝_𝑝𝑝 is the moment of inertia of the payload platform (in frame 𝐟_𝑝).

Thus, we can get the absolute linear and angular momentums of the payload platform system as follows: 𝐩=𝐩_𝑝+𝐩_𝑤,𝐡^𝑜𝑝=𝐡^𝑜𝑝_𝑝+𝐡^𝑜𝑝_𝑤. (17)

By using the theorems of momentum and moment of momentum, the dynamic equations of the payload platform system can be written as 𝑚_𝑝𝑝𝐯̇_𝑝+𝝎^×_𝑝𝐯_𝑝=𝐅+𝐅_𝑎1+𝐅_𝑎2+𝐅_𝑑,𝐈^𝑜𝑝_𝑝𝑝𝝎̇_𝑝+𝝎^×_𝑝𝐈^𝑜𝑝_pp𝝎_𝑝=𝐓+𝐓_𝑎1+𝐓_𝑎2+𝐓_𝑐+𝐓_𝑑1+𝐓_𝑑2, (18) where 𝐅_𝑎1 is the coupling force acted on the payload platform by RWs, 𝐅_𝑎2 the additional coupling force due to the rotor static imbalance, and 𝐅_𝑑 the disturbance force generated by the rotor static imbalance, composed of Coriolis, tangential, and centrifugal forces due to the change of position of the centroid of the payload platform system. 𝐅 is the sum of forces at the spherical joints and tuned mass dampers joints acting on the payload platform. 𝐓_𝑎1 is the coupling torque acted on the payload platform by RWs. 𝐓_𝑎2 is the additional coupling torque due to the rotor static imbalance. 𝐓_𝑐 is the control torque generated by RWs. 𝐓_𝑑1 is the disturbance torque generated by the rotor static imbalance. 𝐓_𝑑2 is the disturbance torque generated by the rotor dynamic imbalance. 𝐓 is the sum of torques at the spherical joints and the tuned mass dampers joints acting on the payload platform. The expressions of these previous symbols are 𝐅_𝑎1=−𝑚_𝑤𝐯̇_𝑝+𝝎^×_𝑝𝐯_𝑝+𝝎̇^×_𝑝𝐫_𝑓𝑝+𝝎^×_𝑝𝝎^×_𝑝𝐫_𝑓𝑝,𝐅_𝑎2=−𝑚_𝑤𝝎̇^×_𝑝+𝝎^×_𝑝𝝎^×_𝑝𝐀_𝑝𝑤𝝆_𝑤,𝐅_𝑑=−2𝑚_𝑤𝝎^×_𝑝𝐀_𝑝𝑤𝛀^×𝝆_𝑤−𝑚_𝑤𝐀_𝑝𝑤𝛀̇^×+𝛀^×𝛀^×𝝆_𝑤,𝐅=−^𝑁_𝑖=1𝐀_𝑝𝑒𝐅_𝑠𝑖+^𝐻_ℎ=1𝐅_𝑡ℎ,𝐓_𝑎1=−𝐀_𝑝𝑤𝐈^𝑜𝑓_𝑤𝑤𝐀_𝑤𝑝−𝑚_𝑤𝐫^×_𝑓𝑝𝐫^×_𝑓𝑝𝝎̇_𝑝−𝝎^×_𝑝𝐀_𝑝𝑤𝐈^𝑜𝑓_𝑤𝑤𝐀_𝑤𝑝−𝑚_𝑤𝐫^×_𝑓𝑝𝐫^×_𝑓𝑝𝝎_𝑝−𝑚_𝑤𝐫^×_𝑓𝑝𝐯̇_𝑝+𝝎^×_𝑝𝐯_𝑝,𝐓_𝑎2=𝑚_𝑤𝐫^×_𝑓𝑝𝐀_𝑝𝑤𝝆^×_𝑤𝐀_𝑤𝑝+𝐫^×_𝑓𝑝𝐀_𝑝𝑤𝝆^×_𝑤𝐀_𝑤𝑝^𝑇𝝎̇_𝑝+𝑚_𝑤𝝎^×_𝑝𝐫^×_𝑓𝑝𝐀_𝑝𝑤𝝆^×_𝑤𝐀_𝑤𝑝+𝐫^×_𝑓𝑝𝐀_𝑝𝑤𝝆^×_𝑤𝐀_𝑤𝑝^𝑇𝝎_𝑝−𝑚_𝑤𝐀_𝑝𝑤𝝆^×_𝑤𝐀_𝑤𝑝𝐯̇_𝑝+𝝎^×_𝑝𝐯_𝑝,𝐓_𝑐=−𝐀_𝑝𝑤𝐈^𝑜𝑓_𝑤𝑤𝛀̇−𝝎^×_𝑝𝐀_𝑝𝑤𝐈^𝑜𝑓_𝑤𝑤𝐓𝛀,_𝑑1=𝐫^×_𝑓𝑝𝐅_𝑑,𝐓_𝑑2=𝐀_𝑝𝑤𝐈^𝑜𝑓_𝑤𝑤𝛀^×+𝐈^𝑜𝑓_𝑤𝑤𝛀^×𝑇𝐀_𝑤𝑝𝝎_𝑝+𝛀,𝐓=−^𝑁_𝑖=1𝐩^×_𝑖𝐀_𝑝𝑒𝐅_𝑠𝑖+^𝑁_𝑖=1𝑐_𝑠𝑖𝐀_𝑝𝑒𝝎_𝑙𝑖−𝝎_𝑝+^𝐻_ℎ=1𝐓_𝑡ℎ. (19)

2.6. Dynamic Equations for the Base Platform System

Taking 𝑜_𝑏 as the reference point, we can derive the dynamic equations of the base platform system from theorems of moment and moment of momentum 𝑚_𝑏𝐯̇_𝑏+𝝎^×_𝑏𝐯_𝑏+𝝎̇^×_𝑏𝝆_𝑏+𝝎^×_𝑏𝝎^×_𝑏𝝆_𝑏=𝐅^′+𝐅_ext,𝐈_𝑏𝝎̇_𝑏+𝝎^×_𝑏𝐈_𝑏𝝎_𝑏+𝑚_𝑏𝝆^×_𝑏𝐯̇_𝑏+𝝎^×_𝑏𝐯_𝑏=𝐓^′+𝐓_ext, (20) where 𝑚_𝑏 and 𝐈_𝑏 are the mass and the moment of inertia of the base platform system, respectively, and 𝐯_𝑏 represents the absolute velocity of point 𝑜_𝑏, with 𝝎_𝑏 being the angular velocity of the base platform system, 𝐅_ext and 𝐓_ext the external disturbing force and torque acting on the satellite being, respectively, 𝝆_𝑏 being the vector from 𝑜_𝑏 to the centroid of the base platform system, and 𝐅^′ and 𝐓^′ being the sum of forces and the sum of torques at the universal joints acting on the base platform system. They can be shown as follows: 𝐅^′=−^𝑁_𝑖=1𝐀_𝑏𝑒𝐅_𝑢𝑖,𝐓^′=−^𝑁_𝑖=1𝐀_𝑏𝑑𝐪_𝑖+𝐫_𝑑𝑏^×𝐀_𝑏𝑒𝐅_𝑢𝑖+^𝑁_𝑖=1𝑐_𝑢𝑖𝐀_𝑏𝑒𝝎_𝑙𝑖−𝝎_𝑏−^𝑁_𝑖=1𝐀_𝑏𝑒𝑀_𝑢𝑖𝐡_𝑖. (21)

Equation (21) only describes the sum of forces and the sum of torques on one combined system including the RW and the vibration isolation platform. If 𝐿 RWs are employed on a satellite, it means that there are 𝐿 combined systems. 𝐅^′ and 𝐓^′ can be rewritten as follows: 𝐅^′=−𝐀_𝑏𝑒^𝐿𝑁_𝑗=1𝑖=1𝐅_𝑢𝑗𝑖,𝐓^′=^𝐿_𝑗=1^𝑁_𝑖=1𝐀_𝑏𝑑𝑗𝐪_𝑖𝑗+𝐫_𝑑𝑗𝑏^×𝐀_𝑏𝑒𝐅_𝑢𝑗𝑖+^𝑁_𝑖=1𝑐_𝑢𝑗𝑖𝐀_𝑏𝑗𝑒𝝎_𝑙𝑗𝑖−𝝎_𝑏−^𝑁_𝑖=1𝐀_𝑏𝑒𝑀_𝑢𝑗𝑖𝐡_𝑗𝑖. (22)

In the previous equations, 𝑗 means the 𝑗th RW. The expression of each symbol including 𝑗 is the same as that in the only one combined system state. The transformation matrix 𝐀_𝑏𝑑𝑗, the vector from the mass center of the 𝑗th base platform to its connection point 𝐪_𝑖𝑗, and the vector from the origin of 𝐟_𝑏 to the origin of the 𝑗th body frame of the 𝑗th base platform 𝐫_𝑑𝑗𝑏 will be determined by the installation of the combined systems.

Hence, (13), (18), and (20) form the complete dynamic equations of the integrated satellite with the RWs and the new vibration isolation system.

As mentioned above, kinematics of the struts are all expressed by 𝐝 and 𝝎_𝑑; however, in the dynamic equations of the base platform system, 𝐯_𝑏 and 𝝎_𝑏 are used to describe the motion information of a satellite. In order to make the integrated satellite dynamic equations solvable, the relationship between the kinematics of the struts and the motion information of a satellite should be derived. As we know, 𝝎_𝑞=𝝎_𝑏. 𝐝 and its derivatives can be rewritten as 𝐝=𝐫_𝑏+𝐀_𝑒𝑏𝐫_𝑑𝑏,𝐝̇=𝐀_𝑒𝑏𝐯_𝑏+𝝎^×_𝑏𝐫_𝑑𝑏,𝐝̈=𝐀_𝑒𝑏𝐯̇_𝑏+𝝎^×_𝑏𝐯_𝑏+𝝎̇^×_𝑏𝐫_𝑑𝑏+𝝎^×_𝑏𝝎^×_𝑏𝐫_𝑑𝑏. (23)

3. Frequency Domain Characteristics Analysis of the New Vibration Isolation System

3.1. Derivation of the Transmissibility Matrix

Transmissibility is the amount of force or motion transferred across the isolation interface as a function of frequency. For the vibration isolation system presented in this paper, the disturbance acting on the payload platform has 6-DOF input, the satellite bus has 6-DOF output, and the transmissibility from the payload platform to the satellite bus is a 6 × 6 matrix, which is named as transmissibility matrix [20]. To derive the transmissibility matrix of the vibration isolation platform with multiple tuned mass dampers, the following assumptions are made.

The platform configuration does not change because the amplitude of vibration is small. The transformation matrix 𝐀_𝑏𝑑 from the satellite body frame to the base platform frame is an identity matrix according to the position relation between the satellite and the vibration isolation platform. The transformation matrices 𝐀_𝑒𝑝 and 𝐀_𝑒𝑏 from the payload and base platform frames, respectively, to the inertial frame can be seen as the identity matrices, and the square of angular velocities can be ignored because both the attitude angles and the angular velocities are small under the attitude stabilization control. Based on this small angle assumption, 𝝎_𝑡ℎ−𝝎_𝑝 can be rewritten as 𝜽̇_𝑡ℎ−𝜽̇_𝑝. Since the influence of strut’s moment of inertia and mass on the transfer function of the vibration isolation platform is small, they can be ignored. 𝑜_𝑏 is the same as the centroid of the base platform system; therefore 𝝆_𝑏 is a zero vector.

Based on the previous assumptions, the simplified dynamic equations of the integrated satellite are 𝑚_ℎ𝐫̈_𝑡ℎ+𝐂_ℎ𝑡𝐫̇_𝑡ℎ−𝐭̇+𝐭^×_ℎ𝜽_𝑝+𝐊_ℎ𝑡𝐫_𝑡ℎ−𝐭−𝐭_ℎ−𝜽^×_𝑝𝐭_ℎ=𝟎_3×1,𝐈_ℎ𝜽̈_𝑡ℎ=−𝐊_ℎ𝑟𝜽_𝑡ℎ−𝜽_𝑝+𝐂_ℎ𝑟𝜽̇_𝑡ℎ−𝜽̇_𝑝,𝑚_𝑝𝐭̈=−^𝑁_𝑖=1𝐅_𝑠𝑖+^𝐻_ℎ=1𝐅_𝑡ℎ+𝐅_𝑑,𝐈_𝑝𝜽̈_𝑝=−^𝑁_𝑖=1𝐩^×_𝑖𝐅_𝑠𝑖+^𝐻_ℎ=1𝐓_𝑡ℎ+𝐓_𝑑,𝑚_𝑏𝐛̈=−^𝑁_𝑖=1𝐅_𝑢𝑖,𝐈_𝑏𝜽̈_𝑏=−^𝑁_𝑖=1𝐪_𝑖+𝐫_𝑑𝑏^×𝐅_𝑢𝑖, (24)(25)(26) where 𝜽_𝑏 is the attitude angles of the satellite and 𝑚_𝑝 and 𝐈_𝑝 are the mass and moment of inertia of the payload platform system, respectively. 𝑚_𝑝=𝑚_𝑝𝑝+𝑚_𝑤, 𝐈_𝑝=𝐈^𝑜𝑝_𝑝𝑝+(𝐀_𝑝𝑤𝐈^𝑜𝑓_𝑤𝑤𝐀_𝑤𝑝−𝑚_𝑤𝐫^×_𝑓𝑝𝐫^×_𝑓𝑝). 𝐅_𝑑 and 𝐓_𝑑 are the sum of disturbance forces and sum of torques caused by RWs, respectively.

Denote 𝐱_ℎ=^{𝐫𝑡ℎ−𝐭ℎ}_𝜽𝑡ℎ, 𝐱_𝑝=^𝐭_𝜽𝑝, 𝐱_𝑑=^𝐛_𝜽𝑏, and (24) can be rewritten as 𝐌_ℎ𝐱̈_ℎ−𝐂_ℎ𝑝𝐱̇_𝑝+𝐂_ℎ𝐱̇_ℎ−𝐊_ℎ𝑝𝐱_𝑝+𝐊_ℎ𝐱_ℎ=𝟎_6×1, (27) where 𝐌_ℎ=𝑚_ℎ𝐄₃𝐈_ℎ,𝐂_ℎ𝑝=𝐂_ℎ𝑡−𝐂_ℎ𝑡𝐭^×_ℎ𝐂_ℎ𝑟,𝐂_ℎ=𝐂_ℎ𝑡𝐂_ℎ𝑟,𝐊_ℎ𝑝=𝐊_ℎ𝑡−𝐊_ℎ𝑡𝐭^×_ℎ𝐊_ℎ𝑟,𝐊_ℎ=𝐊_ℎ𝑡𝐊_ℎ𝑟. (28)

Based on the previous assumption of small attitude angles, the expression of 𝐅_𝑡ℎ and that of 𝐓_𝑡ℎ (see (14)) are presented as 𝐅_𝑡ℎ=−𝐂_ℎ𝑝𝑝𝐱̇_𝑝+𝐂^𝑇_ℎ𝑝𝐱̇_ℎ−𝐊_ℎ𝑝𝑝𝐱_𝑝+𝐊^𝑇_ℎ𝑝𝐱_ℎ, (29) where 𝐅_𝑡ℎ=𝐅_𝑡ℎ𝐌_𝑡ℎ,𝐂_ℎ𝑝𝑝=𝐂_ℎ𝑡−𝐂_ℎ𝑡𝐭^×_ℎ𝐭^×_ℎ𝐂_ℎ𝑡𝐂_ℎ𝑟−𝐭^×_ℎ𝐂_ℎ𝑡𝐭^×_ℎ,𝐊_ℎ𝑝𝑝=𝐊_ℎ𝑡−𝐊_ℎ𝑡𝐭^×_ℎ𝐭^×_ℎ𝐊_ℎ𝑡𝐊_ℎ𝑟−𝐭^×_ℎ𝐊_ℎ𝑡𝐭^×_ℎ. (30)

With the two-parameter isolator, the constraint force at the spherical joint acting on the upper part 𝐅_𝑠𝑖 is described as 𝐅_𝑠𝑖=𝑘_𝑖𝑠_𝑖̂𝑠^𝑇_𝑖̂𝐭−𝑘_𝑖𝑠_𝑖̂𝑠^𝑇_𝑖̂𝐩^×_𝑖𝜽_𝑝−𝑘_𝑖𝑠_𝑖̂𝑠^𝑇_𝑖̂𝐛+𝑘_𝑖𝑠_𝑖̂𝑠^𝑇_𝑖̂𝐪_𝑖+𝐫_𝑑𝑏^×𝜽_𝑏+𝑐_𝑖𝑠_𝑖̂𝑠^𝑇_𝑖̂𝐭̇−𝑐_𝑖𝑠_𝑖̂𝑠^𝑇_𝑖̂𝐩^×_𝑖𝜽̇_𝑝−𝑐_𝑖𝑠_𝑖̂𝑠^𝑇_𝑖̂𝐛̇+𝑐_𝑖𝑠_𝑖̂𝑠^𝑇_𝑖̂𝐪_𝑖+𝐫_𝑑𝑏^×𝜽̇_𝑏. (31)

Denote 𝐱=[𝐱^𝑇_𝑝𝐱^𝑇_𝑑𝐱^𝑇_ℎ]^𝑇, and 𝐉_𝑖=[𝑠_𝑖̂𝑠^𝑇_𝑖̂−𝑠_𝑖̂𝑠^𝑇_𝑖̂𝐩^×_𝑖−𝑠_𝑖̂𝑠^𝑇_𝑖̂𝑠_𝑖̂𝑠^𝑇_𝑖̂(𝐪_𝑖+𝐫_𝑑𝑏)^×𝟎_3×(ℎ×6)].

Equation (31) can be rewritten as 𝐅_𝑠𝑖=𝑘_𝑖𝐉_𝑖𝐱+𝑐_𝑖𝐉_𝑖𝐱̇. (32) Equations (24)–(26) can be rewritten as 𝐌𝐱̈+𝐂𝐱̇+𝐊𝐱=𝐔, (33) where 𝐌=diag𝑚_𝑝𝐄₃𝐈_𝑝𝑚_𝑏𝐄₃𝐈_𝑏𝐌_ℎ,𝐄𝐔=⁶𝟎_(1+ℎ)×6𝐅_𝑑𝐓_𝑑,𝐂=^𝑁_𝑖=1𝑐_𝑖𝐉_𝑖𝐩^×_𝑖𝐉_𝑖−𝐉_𝑖−𝐪_𝑖+𝐫_𝑑𝑏^×𝐉_𝑖𝟎_{(ℎ×6)×[12+(ℎ×6)]}+𝐂_1𝑝𝑝+⋯+𝐂_ℎ𝑝𝑝𝟎₆−𝐂^𝑇_ℎ𝑝𝟎₆𝟎₆−𝐂_ℎ𝑝𝐂_𝐡,𝐊=^𝑁_𝑖=1𝑘_𝑖𝐉_𝑖𝐩^×_𝑖𝐉_𝑖−𝐉_𝑖−𝐪_𝑖+𝐫_𝑑𝑏^×𝐉_𝑖𝟎_{(ℎ×6)×[12+(ℎ×6)]}+𝐊_1𝑝𝑝+⋯+𝐊_ℎ𝑝𝑝𝟎₆−𝐊^𝑇_ℎ𝑝𝟎₆𝟎₆−𝐊_ℎ𝑝𝐊₁, (34) where 𝐄_𝑛 is the 𝑛×𝑛 identity matrix and 𝟎_𝑛 is the 𝑛×𝑛 zero matrix.

The force and torque propagated into the satellite can be rewritten as 𝐅_𝑒=𝐂_𝑒𝐱̇+𝐊_𝑒𝐱, (35) where 𝐂_𝑒=^𝑁_𝑖=1𝑐_𝑖𝐉_𝑖𝐪_𝑖+𝐫_𝑑𝑏^×𝐉_𝑖𝟎_{[6+(ℎ×6)]×[12+(ℎ×6)]},𝐊_𝑒=^𝑁_𝑖=1𝑘_𝑖𝐉_𝑖𝐪_𝑖+𝐫_𝑑𝑏^×𝐉_𝑖𝟎_{[6+(ℎ×6)]×[12+(ℎ×6)]}. (36)

Denote 𝐗=[^𝐱_̇𝐱], and (33) and (35) can be described by the state equation as 𝐗̇=𝐀𝐗+𝐁𝐮,𝐘=𝐂_𝑧𝐗+𝐃𝐮, (37) where 𝐘=𝐅_𝑒𝐅,𝐮=^𝑑𝐓_𝑑𝟎,𝐀=^12+(ℎ×6)𝐄_12+(ℎ×6)−𝐌⁻¹𝐊−𝐌⁻¹𝐂,𝟎𝐁=^{[12+(ℎ×6)]×6}𝐌⁻¹𝐄₆𝟎_{[6+(ℎ×6)]×6},𝐂_𝑧=𝐊_𝑒𝐂_𝑒,𝐃=𝟎₆. (38)

Therefore, the transmissibility matrix of the vibration isolation platform with multiple tuned mass dampers can be obtained by 𝐆(𝑠)=𝐂(𝑠𝐄−𝐀)⁻¹𝐁+𝐃. (39)

3.2. Analysis of the Frequency Domain

To isolate the disturbances induced by the RW for six DOF, a matching vibration isolation system is adopted. Figure 5 gives a schematic representation of this combined system.

(a)

(b)

Figure 5 

Schematic representation of one combined system.

The disturbances of the RW are propagated through the vibration isolation system to the satellite bus. Therefore, for each combined system, the transmissibility matrix from the RW to the satellite bus can be obtained by (39). The frequency response curve of each transfer function in transmissibility matrix can be drawn, and all of these frequency response curves can be illustrated by a matrix form figure, as shown in Figure 6. It means that the six-DOF RW disturbances can be propagated through this transmissibility matrix to yield the predicted six-DOF base force and torque.

Figure 6 

6 by 6 frequency response curves of the transmissibility matrix for the vibration isolation system.

From Figure 6, it can be clearly seen that the 𝑧-translation or “bounce” model and the 𝑧-rotation or “torsional” model are independent due to the symmetry of the platform, while the motion in the 𝑥 direction is coupled with that in the 𝜃_𝑦 direction, and the motion in the 𝑦 direction is coupled with that in the 𝜃_𝑥 direction. The curve of the transmissibility frequency response function in the 𝑥-𝑥 direction is the same as that in the 𝑦-𝑦 direction. In addition, there is the same characteristic as mentioned above in the 𝜃_𝑥-𝜃_𝑥 direction and the 𝜃_𝑦-𝜃_𝑦 direction. These characteristics depend on the symmetry of the Stewart platform.

3.3. Application of the Vibration Isolation System

To realize the attitude stabilization control and pointing ultraprecisely, four RWs (the mass of each RW is 9 kg) are adopted. The effective torque which is below 2 Hz and caused by RWs to the satellite for completing the attitude stabilization control must not be influenced. Moreover, the disturbances must be attenuated by more than 90% at frequencies above 100 Hz, and the resonance amplitude should be as small as possible.

Based on the previous requirements, we should design four identical vibration isolation systems to be installed between each RW and the satellite bus. The diameter of each vibration isolation platform is 150 mm, the length of each strut is 183.7 mm, and the height of each vibration isolation platform is 106 mm. The mass and the moment of inertia of each strut are 𝑚_𝑢𝑖=0.1kg,𝑚_𝑑𝑖𝐈=0.2kg,_𝑢𝑖=1.7×10⁻⁴0004.885×10⁻³0004.885×10⁻³kg⋅m²,𝐈_𝑑𝑖=9.87×10⁻⁴0001.7449×10⁻²0001.7449×10⁻²kg⋅m². (40)

The stiffness coefficient and the damping coefficient of each strut are selected as 100000 N/m and 130 N·s/m, respectively. The corner frequencies obtained are within the interval 12.4 Hz~23.3 Hz.

To reduce the resonance amplitude of the vibration isolation platform, we use three tuned mass dampers installed on the payload platform. The parameters of the three tuned mass dampers can be obtained by “fminimax” function in MATLAB. The mass and the moment of inertia of three tuned mass dampers are as follows: 𝑚₁=0.15kg,𝑚₂𝑚=0.26kg,₃𝐈=0.31kg,₁=diag0.0220.01980.0297kg⋅m²,𝐈₂=diag0.02480.02230.0335kg⋅m²,𝐈_𝟑=diag0.03210.02890.0434kg⋅m². (41) The position vectors of the tuned mass dampers are 𝐭₁=0.1500^𝑇m,𝐭₂=−0.0750.075√30^𝑇𝐭m,₃=−0.075−0.075√30^𝑇m. (42) The translational and rotational stiffness coefficient matrices are 𝐊_1𝑡𝐊=diag1713.31713.33104.6N/m,_1𝑟𝐊=diag8.56638.566338.3328N/m,_2𝑡𝐊=diag6673667312092N/m,_2𝑟𝐊=diag33.365233.3652149.3043N/m,_3𝑡𝐊=diag160941609429165N/m,_3𝑟=diag53.647453.6474240.064N/m. (43) The translational and rotational damping coefficient matrices are 𝐂_1𝑡𝐂=diag35.6135.6147.94N⋅s/m,_1𝑟𝐂=diag1.78051.78053.7665N⋅s/m,_2𝑡𝐂=diag108.68108.68146.30N⋅s/m,_2𝑟𝐂=diag5.4345.43411.495N⋅s/m,_3𝑡𝐂=diag152.17152.17204.85N⋅s/m,_3𝑟=diag7.6097.60916.095N⋅s/m. (44)

With the substitution of the previous parameters into the transfer functions of the vibration isolation system, the frequency response curves can be drawn as shown in Figure 7, where the corner frequencies are within the interval [6.4022.5] Hz, and more than 24 dB (94%) disturbances are attenuated at frequencies above 100 Hz. The maximal resonance amplitude is 5.31 dB (184%), which can well meet the engineering requirements. Therefore, for the vibration isolation system with three tuned mass dampers, it is verified that the parameters selected are reasonable and this system can satisfy the previous requirements with the frequency domain method.

Figure 7 

Transmissibility curves of the vibration isolation platform with three tuned mass dampers.

Furthermore, to demonstrate the competitive advantages of this new vibration isolation system, we draw the frequency response curves of the vibration isolation platform without tuned mass dampers as shown in Figures 8, 9, 10, and 11. Because of the symmetry of the vibration isolation system, the curve of the frequency response in the 𝑥-𝑥 direction is the same as that in the 𝑦-𝑦 direction. Furthermore, there is the same characteristic as the previous one in the 𝜃_𝑥-𝜃_𝑥 direction and the 𝜃_𝑦-𝜃_𝑦 direction. Therefore, we only draw the transmissibility curves in the 𝑥-𝑥, 𝑧-𝑧, 𝜃_𝑥-𝜃_𝑥, and 𝜃_𝑧-𝜃_𝑧 directions.

Figure 8 

Transmissibility curves in the 𝑥-𝑥 direction.

Figure 9 

Transmissibility curves in the 𝑧-𝑧 direction.

Figure 10 

Transmissibility curves in the 𝜃𝑥-𝜃𝑥 direction.

Figure 11 

Transmissibility curves in the 𝜃𝑧-𝜃𝑧 direction.

To make the comparison clearer, the data of the resonance amplitude are displayed in Table 1.

Table 1 

Resonance amplitude comparison in different directions.

Table 1 reveals that the resonance amplitude is greatly attenuated by using the tuned mass dampers. In addition, when the three tuned mass dampers are employed on the vibration isolation platform, the corner frequencies and the high frequency attenuation of the original vibration isolation platform do not change, which means that the new vibration isolation system has a good trade-off between the resonance amplitude and the high frequency attenuation.

4. Numerical Simulations

According to the combined system (RW and new vibration isolation system) dynamic model, the effects of the vibration isolation system have been described by a numerical simulation. The static and dynamic imbalance parameters of the RW are presumed as about 1.36 g·cm and 17 g·cm², respectively. To verify the advantage of the vibration isolation system on the resonance frequency, we select the speed of the RW as 1.25𝜋𝑡, which can provide an effect torque of 0.125 Nm. When the stiffness coefficient and the damping coefficient of each strut are selected as 100000 N/m and 200 N·s/m, the disturbances attenuation curves with and without the vibration isolation system designed in this paper are shown in Figure 12.

(a)

(b)

Figure 12 

Disturbance attenuation curves in 𝑥-𝑥 and 𝜃𝑥-𝜃𝑥 directions.

Figure 12 clearly shows that the new vibration isolation system can attenuate the disturbances to a great extent. Then, we draw the comparison curves of disturbances attenuation with only the original vibration isolation platform and those with the new vibration isolation system as shown in Figure 13.

(a)

(b)

Figure 13 

Comparison curves of disturbances attenuation when the damping coefficient is 200 N·s/m.

In Figure 13, when the original vibration isolation platform is used, the disturbance torque and disturbance force can reach 0.05 Nm and 0.4 N at the resonance frequency, respectively. The amplitude of these disturbances can be attenuated by using the tuned mass dampers in comparison with the previous status.

In order to obtain larger high frequency attenuation for the vibration isolation platform, the damping coefficient of each strut should be selected as smaller than 200 N·s/m. To further verify the validity of the new vibration isolation system, we select the damping coefficient of each strut as 130 N·s/m and draw the comparison curves of disturbances attenuation as displayed in Figure 14.

(a)

(b)

Figure 14 

Comparison curves of disturbances attenuation when the damping coefficient is 130 N·s/m.

In Figure 14, when the speed of the RW crosses the resonance frequency, the vibration isolation platform with tuned mass dampers can work well, and the disturbance torque and disturbance force are 0.03 Nm and 0.25 N at resonance frequency, respectively. But for original vibration isolation platform, when the speed of the RW crosses the resonance frequency, the amplitudes of the disturbance force and disturbance torque always increase. These disturbances can cause rapid catastrophic failure. Therefore, it is evident that the tuned mass dampers cannot only attenuate the resonance amplitude but also ensure the security of the satellite system.

Based on the integrated satellite dynamic model with four combined systems, the attitude angular velocities of the integrated satellite can be simulated. From [21, 22], a PID controller is used to realize the three-axis stabilization. The parameters of PID controller 𝐾_𝑝, 𝐾_𝑖, and 𝐾_𝑑 can be obtained as 200, 0.1 and 2500, respectively. In this paper, the three initial attitude angles are all chosen as 1.5°. The four initial speeds of the RWs are all chosen as [300030003000−5196.2] rpm. The rotor motor, gimbal motor, attitude measurement, and the external disturbing torque acting on the satellite are ignored.

The mass and the moment of inertia of the satellite can be selected the same as those in [23], 𝑚_𝑏=1000kg,𝐈_𝑏=1100−20−10−20900−15−10−15800kg⋅m². (45)

It can be seen from Figure 15 that the attitude angular velocities can converge well to their equilibriums, and the response time is appropriate. The simulation results (see Figure 16) of the speeds of RWs show that the RWs work normally and can provide sufficient torques to realize the attitude stabilization control. All the simulation results indicate that the selected parameters of the vibration isolation system are reasonable, by which the satellite can realize attitude stabilization.

(a)

(b)

Figure 15 

Attitude angular velocities of the satellite with and without the vibration isolation system.

Figure 16 

History of speeds of RWs with the vibration isolation system.

When the satellite has no vibration isolation system for each RW, the maximal vibration amplitude of the attitude angular velocities is 6 × 10⁻⁶ rad/s, as shown in Figure 15(a). When the vibration isolation platform with tuned mass dampers is used, the maximal vibration amplitude is 5.2 × 10⁻⁷ rad/s, as shown in Figure 15(b). Under the disturbances of the RWs, the vibration isolation platform with tuned mass dampers is able to reduce the maximal vibration amplitude of the attitude angular velocities to 9% of that without any vibration isolation system. Therefore, the vibration isolation platform with tuned mass dampers can attenuate the disturbances to a considerable extent and can improve the attitude stability.

5. Conclusions

To isolate the vibration induced by the reaction wheel (RW), this study presented a new vibration isolation system which contains a multi-strut vibration isolation platform and multiple tuned mass dampers. This vibration isolation system can also be applied to isolating the vibration caused by other vibration sources, such as control moment gyroscopes and cryogenic cooler on satellites.

The dynamic model of the integrated satellite with this vibration isolation system and RWs by Newton-Euler Method has been derived. Compared to the dynamic models built by other researchers, this dynamic model not only takes the coupling effects between the RW and the vibration isolation platform into account but also considers the satellite bus movement and the dynamic characteristics of the RW. According to this dynamic model, jitter performance prediction of the spacecraft could be conveniently completed. Based on the assumption of small attitude angles, the transmissibility matrix of the vibration isolation platform with tuned mass dampers has been derived. It is indicated, from this transmissibility matrix, that the vibration isolation system presented in this paper has a good trade-off between the resonance amplitude and the high frequency attenuation. In addition, it is anticipated that the whole satellite dynamic model built and the calculation flow process of transmissibility matrix of this vibration isolation system will work for any other structure of vibration isolation systems. Finally, numerical simulations have been achieved, with results illustrating the effective attenuation of the RW disturbances by the vibration isolation system, and also validate the safety performance of it when the speed of the RW crosses the resonance frequency.

In further studies, for engineering realization of this new vibration isolation system, the influence of the installation position of this system, the offset of the center of mass of the vibration sources, and the dip angle of the payload platform on the performance of the attitude control of the spacecraft should be analyzed.

Acknowledgment

This work was supported by the Excellent Young Scholars Research Fund of Beijing Institute of Technology (Grant no. 2012YG0101).