momentum wheel assemblies

Journal of Sound and Vibration 331 (2012) 2984–3005

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Design and analysis of an intelligent vibration isolation platform for reaction/momentum wheel assemblies Wei-Yong Zhou n, Dong-Xu Li College of Aerospace and Material Engineering, National University of Defense Technology, No. 47, Yanwachi Street, ChangSha 410073, People’s Republic of China

a r t i c l e in f o

abstract

Article history: Received 4 July 2011 Received in revised form 8 February 2012 Accepted 13 February 2012 Handling Editor: K. Worden Available online 8 March 2012

This study focuses on design and analysis of an intelligent vibration isolation platform for reaction wheel assemblies (RWAs) and momentum wheel assemblies (MWAs). A passive platform consisting of four folded beams is designed and analysed for MWAs. A simple and effective mathematical model is developed for the system consisting of the platform and MWAs, and this model is used to investigate the passive vibration isolation performance. Further development is performed to produce an intelligent platform for RWAs, with piezoelectric sensors and actuators bonded to the vertical beams. The flywheel imbalance and impulse load are assumed to be input disturbances for the investigation of the active vibration isolation performance by the finite element method (FEM). The simulation results show that the passive vibration isolation platform is particularly effective for the suppression of a high frequency range vibration for MWAs, and the intelligent platform using velocity feedback control effectively attenuates the dynamic amplification of amplitude at resonance for RWAs. Thus, it is concluded that the passive platform can be used as a vibration isolation platform for MWAs and that the intelligent one can be used for RWAs. Crown Copyright & 2012 Published by Elsevier Ltd. All rights reserved.

1. Introduction The micro-vibrations generated by mechanical systems onboard spacecrafts (for example, reaction/momentum wheel assemblies (R/MWAs), cryo-coolers, thrusters, solar array drive mechanisms and mobile mirrors, etc.) can affect the performance of instruments with high pointing accuracy and stability [1–5]. Because these disturbances are significant in the 1 kHz frequency range [6] and they excite flexible modes of spacecraft, they cannot be controlled or reduced by attitude and orbit control systems. Minimising micro-vibrations becomes a significant problem for spacecrafts with high stability requirements, for example, the Hubble Space Telescope, GOES-N, GOCE and Hinode (Solar-B), among others [2,3,5,7–9]. Many vibration isolation systems have been developed to suppress the disturbances from these sources. However, because of the broad frequency range (0–1 kHz) of the disturbances [6], there is no single vibration suppression method that can meet all requirements. In general, passive vibration isolation systems are effective in the high frequency range, and active vibration isolation systems are effective in the low frequency range. Thus, a hybrid system of passive and active vibration isolation is usually designed and implemented to suppress the entire range of disturbance frequencies. Generally, RWAs and MWAs are considered to be the largest micro-vibration sources onboard spacecrafts [10–16]. Many passive and active vibration isolation systems have been designed and researched to attenuate the vibrations from

n

Corresponding author. Tel.: þ 86 731 84573186; fax: þ 86 731 84512301. E-mail addresses: [email protected] (W.-Y. Zhou), [email protected] (D.-X. Li).

0022-460X/$ - see front matter Crown Copyright & 2012 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jsv.2012.02.018

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Nomenclature Variables A centre of upper fixed plane of the platform A, B, C, D system matrix in stateh space form iT Ag gradient function, Ag ¼ @=@x @=@y @=@z Am, Bm system mode matrix in state space form aa , bb structure constant BA compliance matrix of the platform at point A Bd, Bdp, Bds, Bdf derivative operator matrix Bc control force assembling matrix b width of the beam Cc global damping matrix caused by active control Cs global damping matrix Cm damping and gyroscopic effect matrix in the mode coordinate Ceda element active damping matrix c elasticity matrix cp elasticity matrix of the piezoelectric material cs elasticity matrix of the host structure cb modulus of elasticity of the beam cp modulus of elasticity of the piezoelectric actuator/sensor csx,csy,csz,csa,csb,csg damping coefficient at S De electric displacement vector Ds disturbance force assembling matrix d piezoelectric constant matrix E electric field vector e piezoelectric matrix ex, ey unit vector of x, y in the body-fixed coordinate Fea applied electrical charges Fc(t) control force input in the finite element equation Fef applied mechanical force Fs(t) input disturbance force in the finite element equation Fes sensed electrical charge Fx(t) disturbance force in the x direction Fy(t) disturbance force in the y direction FCoM CoM of the R/MWAs Fr force generated by static imbalance Fx, Fy, Fz disturbance force in the x, y, z directions at node 1 Fx,in input force caused by Us Fj,out output force at O in the direction of j, j¼x,y,z F ji output force at node i in the direction of j, j ¼x,y,z Fx out, Fy out, Fz out disturbance force output at O f1 input force vector at node 1 f2 output force vector at O G gyroscopic force matrix G0 assembled gyroscopic matrix G modulus of rigidity of the beam Gc signal conditional gain Gp modulus of rigidity of the actuator/sensor H transfer matrix from input voltage to force in state space form hAS equivalent support distance from A to S

hAF

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vertical distance from A to the CoM of R/MWAs hAO vertical distance from A to O h1 vertical distance from S to the body coordinate of R/MWAs (or P1) h2 vertical distance from S to O i imaginary unit iðtÞ electrical current I identity matrix Ibx, Iby, Ibz moment of inertia of the base of the R/MWAs Ifx, Ify, Ifz moment of inertia of the flywheel K assembled stiffness matrix Ks diagonal stiffness matrix at S KA effective stiffness matrix of the platform at point A Kp1 stiffness matrix at P1 Km stiffness matrix in the mode coordinate Kfu mechanical–electrical coupling stiffness matrix Keuu element mechanical stiffness matrix Keuun Keuu þKeuf ½Keff 1 ½Keuf T Keuf element electrical–mechanical coupling stiffness matrix Kefu element mechanical–electrical coupling stiffness matrix Kefus element mechanical–electrical coupling stiffness matrix of the sensor Keff element dielectric stiffness matrix kx, ky, kz, ka, kb, kg linear and angle elastic modulus at S lh1, lh2, lh3, lh4 length of the horizontal beam of the platform lv1, lv2 length of the vertical beam of the platform M assembled mass matrix M1 mass matrix of R/MWAs in the body coordinate Mm mass matrix in the mode coordinate Mese mass matrix of the beam element Mepe mass matrix of the piezoelectric sensor and actuator element Meuu element mass matrix of the intelligent beam Mx,in input torque caused by Ud Mj,out output moment at O in the direction of j, j¼x,y,z M ji output moment at node i in the direction of j, j¼x,y,z mb mass of the base of the R/MWAs md dynamic imbalance mass mf mass of the flywheel ms static imbalance mass N shape function matrix Ns shape function matrix of host structure Np shape function matrix of piezoelectric patches Nw(x), Nqy(x) shape function matrix of the beam O centre of mounted plane of the platform P1 mass point q modal coordinate q, q1, q2 transverse rotation of the beam Rf  constant of the amplifier R1 R2 rotating speed range of MWAs

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S Tx(t) Ty(t) Tsx1 TAsx1 TAs1 Ts1 Ts2 Tr tb ta,ts u Ud Us V eo V ea V Ve Va Vea Ves w1,w2 X x1

xs x, y, z z

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decoupled elastic support point element disturbance torque in the x direction disturbance torque in the y direction displacement transfer matrix from x1 to xs displacement transfer matrix from A to S force transfer matrix from S to A force transfer matrix from S to the body coordinate of MWAs force transfer matrix from S to O torque generated by dynamic imbalance depth of the beam (actuator, sensor) depth of the actuator or sensor displacement vector dynamic imbalance of the flywheel static imbalance of the flywheel output voltage by the sensor input voltage on the actuator electric potential vector element electric potentials vector control input vector input electrical potential vector sensed electrical potential vector transverse displacement of the beam state variable in state space form displacement matrix of the body coordinate of MWAs (or P1) relative to the initial inertial coordinate relative displacement matrix at the elastic support S displacement in the direction of x, y, z effective distance of the intelligent beam

Greek variables

a, b, g

Euler angle from the transfer order of 1–2– 3(x–y–z)

generalised nodal displacement nodal displacement of the element nodal displacements of the actuator element nodal displacements of the sensor element e strain vector z damping ratio l complex eigenvalue rp mass density of host structure rs mass density of piezoelectric material r stress vector s real part of l modal matrix corresponding to the first U six modes fs phase of the static imbalance fd phase of the dynamic imbalance v dielectric constant matrix O rotating speed of the flywheel o frequency o1, o2, o3, o4, o5, o6 natural frequencies of the platform and R/MWAs od imaginary part of l [ ]T transpose of [ ] [ ]1 inverse of [ ] ½ U (@/@t)[ ] ½ UU (@2/@t2)[ ] A, B, C, Dy element numbers 1, 2, 3, 4y node numbers

d de dea des

Abbreviations CoM FEM LQR MWAs RWAs

centre of mass finite element method linear quadratic regulator optimal control momentum wheel assemblies reaction wheel assemblies

RWAs and MWAs on spacecrafts. Honeywell [17] developed a viscous-damped passive isolator for the Hubble Space Telescope (HST) to suppress the vibrations from RWAs. Vaillon and Philippe [18] designed and tested an elastomer-based passive vibration isolator. The isolator can be used to isolate the six degrees of freedom of a disturbance source with an attenuation performance exceeding 20 dB above 25 Hz and 40 dB above 50 Hz. Makihara et al. [19] studied a semi-active vibration isolator for MWAs. The isolator consists of a piezoelectric material and a switch-controlled passive circuit. Because no external energy is supplied to the system, it is stable even when a control malfunction occurs. Kamesha et al. [20] proposed a flexible low-frequency platform for RWAs consisting of four folded beams, and the finite element method (FEM) was used to analyse its passive and active vibration isolation performance. Further, Kamesha et al. [21] have tested the usefulness of their structure on a Kistler transducer; and their results suggest that their platform is useful for the vibration isolation; however, the rotating speeds of the RWAs in [21] have an interval of 600 rev/min, which may lose certain critical speeds because certain natural frequencies of the system would be changed by the gyroscopic effect. Although many vibration isolators have been developed for RWAs and MWAs, few studies have focused on the gyroscopic effect of flywheels during their design and analysis. However, the gyroscopic effect of a flywheel will change certain natural frequencies of the system and may generate dynamic interactions with the spacecraft elements, as reported by Narayan et al. [10]. In this paper, an intelligent vibration isolation platform for RWAs and MWAs is designed and analysed, and the gyroscopic effect of flywheel is considered. The platform, consisting of four folded beams, is designed and analysed to suppress the vibrations from MWAs, which mainly emit high-frequency disturbances; furthermore, the platform is also developed for RWAs, which emit disturbances over a broad frequency range. The adaptation to RWAs is achieved by bonding a piezoelectric sensor and actuators to the vertical beams in order to construct an intelligent structure that can attenuate the dynamic amplification of amplitude at resonance. This paper proceeds as follows: Section 2 reviews the micro-vibrations emitted by RWAs and MWAs. Section 3 first describes the platform, then develops the mathematical model and, at the end, analyses and discusses the passive vibration isolation performance of the platform. Section 4 applies the piezoelectric sensors and actuators on the platform

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to construct an intelligent structure and then analyses and discusses the active vibration isolation performance. Section 5 summarises the paper and presents the conclusion. 2. Disturbance produced by RWAs and MWAs 2.1. RWAs and MWAs RWAs and MWAs are widely used in high stability and accuracy spacecrafts to provide attitude control torque or maintain stability. A typical R/MWAs consists of a rotating flywheel mounted on a shaft, supported by bearings and driven by a brushless DC motor, with the whole assembly encased in a cover. In general, a RWAs has zero average speed; it rotates in either direction, generally up to 3000–4000 rev/min, to provide control torque in both directions on the speed axis. An MWAs usually rotates at a high mean speed (4000–10,000 rev/min) to provide momentum bias and stability to the spacecraft, as well as to provide control torque in both directions on the speed axis. 2.2. Disturbance from R/MWAs A number of studies have investigated the disturbances produced by R/MWAs [10–16]. The most common R/MWAs configurations are symmetrically supported by ball bearings, as shown in Fig. 1. Though R/MWAs are balanced during operation, the disturbance forces and torques that they produce can still degrade the performance of high-precision instruments. Disturbances from a spinning wheel assembly generally arise from four main sources: flywheel imbalance, mechanical bearing noise, motor drive errors and motor disturbances [12]. In general, flywheel imbalance (static imbalance and dynamic imbalance) is considered to be the largest disturbance source [14], which is at the fundamental harmonic. The secondary disturbance source is the bearing disturbance caused by bearing irregularities, which occur at both sub- and super-harmonics of the rotational speed but mainly at super-harmonics. In addition to the disturbances mentioned above, a disturbance is also caused by the flexibility of the R/MWAs. Because of the gyroscopic effect of the flywheel, certain natural frequencies of the R/MWAs, generally the rocking mode, change with the rotational speed. Forward whirl stiffens the rocking mode, while backward whirl softens it. In addition to the excitation of the flywheel imbalance by the forward whirl, other disturbances (for example, the higher harmonics) may also excite it. The backward whirls are mainly excited by other disturbance sources, for example, bearing lubricant and random noise. When the forward and backward whirl modes are active, dynamically amplified disturbance forces are generated [11]. In general, RWAs and MWAs operate below their critical speed. The flywheel imbalance, the frequency of which is equal to the rotational speed, would not excite the R/MWAs structural modes. However, to attenuate the vibrations from the R/MWAs, the natural frequencies of the first six degrees of freedom of the system that contains the platform and R/MWAs are usually far less than those of the R/MWAs. Therefore, the flywheel imbalance disturbance will excite the structural modes of the system. In this study, the flywheel imbalances are simulated to investigate the vibration isolation performance. 2.3. Flywheel imbalance disturbance The fundamental harmonic disturbances emitted by the R/MWAs are mainly caused by static and dynamic imbalances, which arise from the asymmetry of the flywheel with respect to its rotational axis. Static imbalance results from the offset of the CoM of the wheel from its spinning axis, which produces a radial centripetal force when the flywheel is rotating. Dynamic imbalance is caused by the misalignment of the wheel’s principal axis and its spinning axis, which produces a radial torque when the flywheel is rotating [12–14]. Representations of the static dynamic imbalances are shown in Fig. 2.

Fig. 1. Typical R/MWAs configuration and disturbance.

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Fig. 2. Static imbalance and dynamic imbalance.

The force generated by the static imbalance is given by the following expression: F r ¼ ms r O2 ¼ U s O2

(1)

where the static imbalance is reproduced by considering a static mass imbalance ms at a distance r from the spinning axis (note that the static imbalance Us ¼msr is a flywheel mass property). The force can be expressed as follows: Fx ðtÞ ¼ U s O2 cosðOt þ fs Þex Fy ðtÞ ¼ U s O2 sinðOt þ fs Þey

(2)

where fs is the phase of the force and ex and ey are the unit vectors in the body-fixed coordinate. Similarly, the torque generated by the dynamic imbalance is given by the following: T r ¼ md rdr O2 ¼ U d O2

(3)

where md is the dynamic imbalance mass and the dynamic imbalance Ud ¼mdrdr is a flywheel mass property. The torque caused by the dynamic imbalance can be written as follows: Tx ðtÞ ¼ U d O2 cosðOt þ fd Þex Ty ðtÞ ¼ U d O2 sinðOt þ fd Þey

(4)

where fd is the phase of the torque caused by the dynamic imbalance. Details of the mass imbalance disturbances can be found in [15]. The input disturbance force vector f1(t) caused by the static imbalance and dynamic imbalance is obtained as follows: h iT 2 2 2 2 (5) f 1 ðtÞ ¼ U s O cosðOt þ fs Þ U s O sinðOt þ fs Þ 0 U d O cosðOt þ fd Þ U d O sinðOt þ fd Þ 0

3. Passive vibration isolation platform design and analysis for the MWAs Because the RWAs and MWAs work at different rotating speeds, the relevant vibration isolation strategies are different. In general, the MWAs have a rotating speed of approximately 4000–10,000 rev/min, and its disturbance is mainly produced in the high-frequency range. Therefore, passive vibration isolation is effective for the MWAs. 3.1. Design criteria There are two basic design criteria for the vibration isolation platform for R/MWAs. The first criterion requires that the platform should isolate the disturbances from R/MWAs, which means that it should suppress the vibrations in all six degrees of freedom (two radial translation modes, two radial rocking modes, the axial translation mode and the axial rotation mode). A detailed description of the modes of the R/MWAs can be found in [12]. The second criterion requires that the attitude control torque should be transferred from the R/MWAs, which means that it cannot affect the attitude-control operation [19]. In this paper, the two criteria are met by constraining the natural frequencies of the system comprising the platform and the R/MWAs. The first criterion is achieved by maintaining the first six natural frequencies (both axial and radial) of the system below an upper limit frequencies, and the seventh or higher natural frequencies which may be excited the flywheel imbalance, above the operate speed range. Assuming that the rotational speed range of the MWAs is [R1 R2] rev/min, the upper limit pffiffiffi natural frequency of the first six natural frequencies of the passive vibration isolation platform should be lower than ð 2=120ÞR1 Hz, and the seventh natural frequency should be higher than (1/60)R2 Hz. In this paper, considering the gyroscopic stiffening effect, the upper limit frequency is designed to be 25 Hz. The second criterion is met by keeping the

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axial rotational natural frequency at values much larger than those of the working frequencies of the control system. In this paper, the lower limit of the axial rotational natural frequency is designed to be 8 Hz. Because the system containing the platform and the R/MWAs is axially symmetric, the two radial translation modes are similar, and the frequencies have the same value. Thus, the two modes are represented by a radial translational mode in this paper, as are the two radial rocking modes and frequencies.

3.2. Platform description Generally, the diameter of the platform is limited by the diameter of the R/MWAs. The detailed dimensions of each beam are determined by the natural frequencies of the system, to meet the requirements of criteria in Section 3.1. The concept of folded beam vibration isolator for RWAs was first proposed by Kamesha et al. [20]; however, in this paper, we propose another type of fold beam isolator for the RWAs and MWAs, which is as shown in Figs. 3 and 4. The platform consists of four single-folded beams that are orthogonal to each other. Each folded beam consists of three horizontal beams and two vertical beams. One end of the single-folded beam is attached to the upper fixed plane, and the other is connected to the base of the platform. Note that A is the centre of the upper fixed plane of the platform, O is the centre of the mounted plane of platform and hAO is the vertical distance from A to O.

Fig. 3. Finite elements of the platform [20].

Flywheel FCoM hAF

lh1

lh3

lh2 1

lh4 S

lv2

lv1

A

lh3

lh4

lh2

lh1

hAS hAO lv1

lv2 z

b

tb

O Base centre

Fig. 4. Sketch of the elastic support and R/MWAs.

y x

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Table 1 presents the dimensions and material properties of the folded beam and also lists the piezoelectric actuators and sensors used in Section 4. Table 2 gives the natural frequencies of the R/MWA. 3.3. Passive vibration isolation performance analysis 3.3.1. Method and model The method developed in Ref. [22] is first employed to analyse the stiffness matrix of the platform, and it is then used to develop the mathematical model. 3.3.1.1. Method. We consider a decoupled elastic support point element S, which can provide decoupled linear support over six degrees of freedom; i.e., the stiffness matrix Ks at point S is diagonal. One end of S is rigidly connected to a point mass at point P1 with a vertical distance of h1, and the other end is rigidly connected a fixed point P2, which is as shown in Fig. 5(a). The stiffness matrix Kp1 at point P1 produced by the elastic element S can be determined as follows. For a small displacement x1 related to the initial position of P1, the relative displacement at S can be expressed by xs ¼ Tsx1x1 (Tsx1 is the corresponding displacement transfer matrix from P1 to S). The elastic forces generated at point S can be determined to be Fs ¼Ksxs ¼KsTsx1x1. Finally, the force applied by Fs at P1 can be determined to be Fp1 ¼ Ts1Fs ¼Ts1KsTsx1x1 (Ts1 is the Table 1 Geometric, material and mass properties of the folded beam and R/MWAs. Geometric properties Horizontal beam lh1/lh2/lh3/lh4 Vertical beam lv1/lv2 Depth and width of the beam tb/b Depth of the actuator/sensor ta/ts Vertical distance from FCoM to A hAF Height of the platform hAO Equivalent distance h1 Equivalent distance h2

0.020/0.030/0.020/0.025 m 0.050/0.050 m 0.0015/0.010 m 0.00025/0.00025 m 0.02 m 0.055 m hAS þhAF hAO  hAS

Material properties Modulus of elasticity of the beam cb Modulus of rigidity of the beam G Modulus of elasticity of the piezoelectric actuator/ sensor cp Modulus of rigidity of the actuator/sensor Gp Mass properties Mass properties of the flywheel mf/Ifx/Ify/Ifz Mass properties of the base of the R/MWAs mb/Ibx/Iby/Ibz Rotating speed area MWAs RWAs

2.1  1011 Pa 8.0  1010 Pa 6.8  1010 Pa 2.6  1010 Pa 4.0 kg/0.021/0.021/ 0.040 kg m2 2.0 kg/0.014/0.014/ 0.020 kg m2 [3600, 6000] rev/min [  3600, 3600] rev/min

Table 2 Natural frequencies of the R/MWAs. Mode

Rocking

Axial translation

Radial translation

Frequency (Hz)

120

200

80

Fig. 5. Sketch for the method and simplified model: (a) sketch of the method and (b) sketch of the simplified model.

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corresponding force transfer matrix from P1 to S). Thus, the stiffness matrix caused by the elastic support at P1 is determined to be Kp1 ¼Ts1KsTsx1. Detailed expressions of Tsx1 and Ts1 can be found in the appendix. Note that during the deduction, the higher-order terms are ignored, and small angles are considered (i.e., sin a  a,cos a  1). Considering the platform in Fig. 4, the effective elastic matrix KA at point A can be obtained by the effective stiffness method. Applying a unit force vector at point A to obtain the compliance matrix BA,KA is obtained by KA ¼ B1 A . The stiffness matrix of the platform obtained by the effective stiffness method is as follows: 2

830:659

6 0 6 6 6 0 KA ¼ 6 6 0 6 6 4 117:576

3

0

0

0

117:576

0

830:659 0

0 1653:633

117:576 0

0 0

0 0

117:576

0

18:983

0

0

0

0

0

18:983

0

0

0

0

0

19:291

0

7 7 7 7 7 7 7 7 5

Using the method introduced above, it is found that KA can be deposed by the displacement and stiffness transfer matrix as follows: 2

kx 6 0 6 6 6 0 6 KA ¼ TAs1 Ks TAsx1 ¼ 6 0 6 6 6 h k AS x 4 0

0

0

0

hAS kx

ky 0

0 kz

hAS ky 0

0 0

hAS ky

0

ka þ hAS ky

0

2

0

3

7 7 7 7 7 07 7 7 07 5 kg 0 0

0

0

0

2 kb þ hAS kx

0

0

0

0

(6)

where hAS can be obtained by hAS ¼(KA[2,4]/KA[2,2]) ¼0.0223 m. Thus, the stiffness matrix at A can be represented by a diagonal stiffness matrix Ks at point S below A, with a vertical distance of hAS. K ¼Ks is determined to be the following: 2 6 6 6 6 46 Ks ¼ 1:0  10 6 6 6 4

4:2816 0

0 4:2816

0 0

0 0

0 0

0 0

0

0

0

0

4:6056

0

0

0

0

0:0112

0

0

0

0

0

0

0:0112

0

0

0

0

0

0

0:0208

3 7 7 7 7 7 7 7 7 5

3.3.1.2. Mathematical model. Using the method above and assuming that point P1 is the CoM of the MWAs and that the stiffness of the platform is represented by an elastic element S, the simplified model of the platform and the MWAs is as shown in Fig. 5(b). Note that h1 ¼hAS þhAF and h2 ¼hAO  hAS. The Newton–Euler method [23] is employed to develop the mathematical model of the system consisting of the platform and the MWAs. The equations of motion, in matrix form, are as follows: M1 x€ 1 þ Gx_ 1 þ Ts1 Cs Tsx1 x_ 1 þ Ts1 Ks Tsx1 x1 ¼ f 1 Ts2 Cs Tsx1 x_ 1 þTs2 Ks Tsx1 x1 ¼ f 2

(7)

where M1 is the mass matrix of the R/MWAs in body coordinates, x1 ¼ ½ x y z a b g T is the relative displacement vector, G is the gyroscopic force matrix, Cs is the damping mass matrix at S, f1 is the input disturbance force vector, and f1 is the output force vector at O, Tsx1 is the displacement transfer matrix, and Ts1 and Ts2 are the force transfer matrices. Detailed expressions for each term in Eq. (7) can be found in the appendix. Note that the viscous damping forces are also assumed to be decoupled at point S, and the damping force applied at FCoM (CoM of the R/MWAs) can be obtained to be Ts1 Cs Tsx1 x_ 1 . Table 3 shows the first six natural frequencies of the system obtained using both the mathematical model and the finite element method (FEM). The results obtained using the mathematical model compare well with those obtained using the FEM, although they are slightly higher because the mass of the folded beam is ignored in the effective stiffness method. The seventh frequency obtained by FEM is greater than 300 Hz; it corresponds to a local vibration mode that is far greater than the rotational speed. Thus, the effective stiffness matrix effectively represents the passive vibration isolation platform. Table 3 Natural frequencies of the system. Mode

1

2

3

4

5

6

7

FEM (Hz) Mathematical model (Hz)

8.72 8.74

8.72 8.74

9.36 9.37

13.80 13.94

14.39 14.49

14.39 14.49

330.23 N

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120

80 60 40

20 15 10 5

20 0

ω1 ω2 ω3 ω4 ω5 ω6 ω=Ω

25 Natural Frequency (Hz)

100 Natural Frequency (Hz)

30

ω1 ω2 ω3 ω4 ω5 ω6 ω=Ω

0

1000

2000 3000 4000 Rotating Speed (RPM)

5000

6000

0

0

200

400 600 800 Rotating Speed (RPM)

1000

1200

Fig. 6. Campbell diagram of the system: (a) Campbell diagram and (b) Campbell diagram (local view).

3.3.2. Campbell diagram In a rotating system, the eigenfrequencies often depend on the rotational speed as a result of the gyroscopic effect. The variation of the eigenfrequencies with the rotational speed is often plotted in a diagram called a Campbell diagram. The Campbell diagram of the system comprising the platform and MWAs is as shown in Fig. 6. Because there is a distance (h1a0) between the CoM of the MWAs and the decoupled elastic support S, the radial rocking mode and radial translation mode of the system are actually coupled with each other. Thus, there are two pairs of backward and forward whirl curves. One such pair is o1 and o2; o1 is the backward whirl curve that decreases with the speed of the flywheel, and o2 is the forward whirl curve that increases with the rotational speed. The other pair is o3 and o4. The backward whirl curve o3 decreases with the rotational speed, while the forward whirl curve o4 increases with the rotational speed of the flywheel, with an asymptote of o ¼(Ifz/(Ifx þIbx))O [24]. The axial translation natural frequency o5 and the axial rotational natural frequency o6 are independent of the rotational speed, because the axial equation is decoupled from the others in the mathematical model. The values of o5 and o6 are 13.9 Hz and 9.3 Hz respectively. Because the radial rocking mode and the radial translation mode are coupled with each other, it is impossible to separate them on the curves, and the modes can transform from one type to the other as a result of the gyroscopic effect. For example, o2 is most like a radial rocking mode at low rotational speed, and it increasingly tends to a radial translation mode with increasing rotational speed. As stated in [24], ‘‘The spin speeds at which one of the forcing functions has a frequency coinciding with one of the natural frequencies of the system are usually referred to as critical speed’’. In general, the forcing frequency of the flywheel imbalance is o ¼ O. Thus, there is a critical speed at which there is an intersection of the forward whirl curve with the fundamental harmonic curve, o ¼ O. At this rotational speed, the flywheel imbalance triggers dynamic amplification of amplitude, and the system experiences large amplitude vibrations. As shown in Fig. 6, although there are two forward whirl curves, there is only one critical speed, which is 650 rev/min. The selection of a single curve occurs because there is only one intersection of the o ¼ O curve with the forward whirl curve o2. The asymptote of the other forward whirl curve (o4) in this paper is (Ifz/(Ifx þIbx))O ¼ (0.040/0.035)O ¼1.143O; as a result, there is no intersection with o ¼ O. In general, synchronous disturbance, such as the flywheel imbalance, would not excite the backward whirl frequency [24]. However, the disturbances, such as impulse load, bearing noise, may excite the backward whirl frequency and cause dynamic amplification of amplitude [11]. These phenomena are generally not referred to as critical speeds because they are relatively small.

3.3.3. The gyroscopic force term The gyroscopic force term deserves more attentions during design and analysis of the vibration isolation platform, because it changes the critical speed of the rotating system. For example, if the gyroscopic term is ignored, the two pairs of backward and forward whirl curves in Fig. 6 would become two straight horizontal lines, and there would be two critical speeds, which are 449 rev/min and 969 rev/min respectively. However, there is only one critical speed if the gyroscopic term is considered, which is 650 rev/min (as shown in Fig. 6). Furthermore, neglect of the gyroscopic effect may cause failure of the vibration isolation platform at certain rotational speed. Though there is only one critical speed in this paper because the MWAs has the mass properties of (Ifz/(Ifx þIbx)) 41, if the mass properties of the MWAs have a mass characteristic of (Ifz/(Ifx þIbx)) o1, there would be another intersection between o4 and o ¼ O, then another critical speed would occur at a higher rotational speed. If the critical speed is in the working speed range of the MWAs, the platform should be redesigned to avoid such a coincidence. For example, assuming

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a MWAs had the mass properties of (Ifz/(Ifx þ Ibx))¼0.98, then there would be an intersection of o ¼ O with o4 at the rotational speed of 4600 rev/min (another critical speed), and the platform would cause large dynamic amplification when the MWAs rotates at 4600 rev/min, which would cause failure of the platform. If it happened, the stiffness of the platform should be redesigned to make the intersection located at a lower rotational speed. Thus, the gyroscopic effects of flywheels should be considered during the design and analysis of the vibration isolation platform. 3.3.4. Response to flywheel imbalance In general, the static imbalance Us and dynamic imbalance Ud are two parameters to represent the manufacture imperfect of the flywheel. Although they are applied simultaneously in the real system, they are introduced separately in the paper, to clarify their corresponding effects on the output force and moment. Though the static imbalance Us produces an input disturbance force at the CoM of the flywheel, it does not only introduce an output force; rather, it also introduces an output moment at O; as does the torque input caused by Ud, which also introduces an output force at O. If the Us and Ud were introduced simultaneously in the simulation, the output force and moment obtained at O would be an addition results caused by both, then it would be difficult to demonstrate each effects on the output force and moment. Hence, in this paper, they are introduced separately. However, in real system, if they are applied simultaneously, the output force and moment caused by them can be added considering their phase differences. Introducing the static imbalance Us and dynamic imbalance Ud to the equations of motion as an input force and a torque, respectively, we can obtain the output force and the moment at O. The comparison of the input and output forces and moments is shown in Fig. 7. Fig. 7 shows that when the R/MWAs is mounted on the platform, the disturbance output level remains low when the critical rotating speed is passed, although the input imbalance (static and dynamic) triggers resonance at the critical speed

0.5

8

Fxin Fxout

7

Mxout

0.45 0.4

6

0.35 Mx (Nm)

Fx (N)

5 4 3

0.3 0.25 0.2 0.15

2

0.1 1 0

0.05 0

1000

2000 3000 4000 5000 Rotating Speed (RPM)

0

6000

1.4

0.14

1

0.12 Mx (Nm)

Fx (N)

1.2

0.6

0.02 0 2000 3000 4000 Rotating Speed (RPM)

5000

6000

Mxin Mxout

0.06

0.2

1000

6000

0.08

0.04

0

2000 3000 4000 5000 Rotating Speed (RPM)

0.1

0.4

0

1000

0.16

Fxout

0.8

0

0

1000

2000 3000 4000 Rotating Speed (RPM)

5000

6000

Fig. 7. Force and moment output caused by flywheel imbalance: (a) force input and output at O caused by static imbalance (Us ¼ 1.8 g cm, Ud ¼ 0), (b) moment output at O caused by static imbalance (Us ¼1.8 g cm, Ud ¼ 0), (c) force output at O caused by dynamic imbalance (Us ¼0, Ud ¼ 3.6 g cm2), and (d) torque input and moment output at O caused by dynamic imbalance (Us ¼ 0, Ud ¼3.6 g cm2).

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(650 rev/min), which generates a dynamically amplified output disturbance. This result illustrates that, except for values near the critical speed (650 rev/min), the output disturbance force and moment caused by the flywheel imbalance are minimised by the platform. Hence, it can be deduced that the platform is able to isolate the high-frequency disturbance. It can be concluded that passive vibration isolation is particularly effective for the MWAs, which works at 3600–6000 rev/min. The force and moment output at O can also be calculated by the FEM as follows: 25 X

F j,out ¼

F ji

j ¼ x,y,z

i ¼ 22

Mx,out ¼

25 X

M xi þðF z23 F z25 Þðlh1 þlh2 þ lh3 ÞF yout hAO

i ¼ 22

M y,out ¼

25 X

M yi þðF z22 þ F z24 Þðlh1 þlh2 þlh3 Þ þ F xout hAO

i ¼ 22

M z,out ¼

25 X

Mzi þðF y22 F x23 F y24 þ F x25 Þðlh1 þ lh2 þlh3 Þ

(8)

i ¼ 22

where F ji and M ji are the force and moment at node i in the direction of j, respectively; F jout and M jout are the force and moment at O along the j direction, respectively; i¼ 22, 23, 24, 25; and j ¼x, y, z. The node numbers are as shown in Fig. 4. Fig. 8 shows the comparison of the force and moment at O calculated using the mathematical model and the FEM. The results that are obtained using the mathematical model are in good agreement with those obtained using the FEM, which confirms that the mathematical model is reasonably accurate. Thus, the mathematical model is useful for the

0.04

0.7

FEM Mathematical Model

0.6

0.03 Mxout (Nm)

Fxout (N)

0.5 0.4 0.3

0.025 0.02 0.015

0.2

0.01

0.1

0.005 0

0 0

1000

2000 3000 4000 5000 Rotating Speed (RPM)

0

6000

3.5

0.07

FEM Mathematical Model

0.06

2000 3000 4000 5000 Rotating Speed (RPM)

6000

x 10-3 FEM Mathematical Model

2.5 Mxout (Nm)

Fxout (N)

1000

3

0.05 0.04 0.03

2 1.5

0.02

1

0.01

0.5

0

FEM Mathematical Model

0.035

0 0

1000

2000 3000 4000 5000 Rotating Speed (RPM)

6000

0

1000

2000 3000 4000 5000 Rotating Speed (RPM)

6000

Fig. 8. Comparison of the force and moment obtained using the FEM and mathematical model: (a) force output at O caused by static imbalance (Us ¼ 1.8 g cm, Ud ¼ 0), (b) moment output at O caused by static imbalance (Us ¼ 1.8 g cm, Ud ¼ 0), (c) force output caused by dynamic imbalance (Us ¼ 0, Ud ¼ 3.6 g cm2), and (d) moment output caused by dynamic imbalance (Us ¼0, Ud ¼ 3.6 g cm2).

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2995

representation of the various design parameters (stiffness and support distance) that characterise the platform in further design. 4. Active vibration isolation platform design and analysis for RWAs 4.1. Design criterion Though the passive vibration isolation platform can be used to suppress high-frequency disturbances caused by MWAs (which usually work at high-rotational speeds), when the platform is used for RWAs, which work at 73600 rev/min, excessive oscillation occurs if the rotating speed is close to the critical speed, for example, 650 rev/min (which is the critical speed as shown in Fig. 8). Therefore, the passive vibration isolation technique is not adequate for RWAs. Although viscous damping can be added to suppress the amplitude at resonance, such damping degrades the performance of the vibration isolation platform in the high-frequency range. Under these circumstances, the active vibration isolation platform offers an alternative. Hence, the active vibration isolation technique is implemented on the platform to attenuate dynamic amplification of amplitude at resonance. The active vibration control platform is constructed by bonding piezoelectric sensors and actuators to both sides of the vertical beams, as shown in Fig. 9. Thus, the horizontal beams are unaltered, and the vertical beams are modified to be intelligent structures. In the finite element model, these structures are element I (nodes 6, 10), element J (nodes 7, 11), 11 (nodes 8, 12), element ? 12 (nodes 9, 13), element ? 17 (nodes 14, 18), element ? 18 (nodes 15, 19), element ? 19 (node 16, element ? 20 (nodes 17, 21), as shown in Fig. 3. The finite element model is developed using the Euler–Bernoulli 20), and element ? beam theory. 4.2. Modelling and formulation of the intelligent platform Fig. 9 shows the intelligent beam with piezoelectric sensors and actuators bonded to both sides, in local coordinates. It is assumed that the piezoelectric sensors and actuators are bonded perfectly, and the bond layers are thin enough such that their mass and stiffness contributions are negligible. However, the stiffness contributions of the piezoelectric sensors and actuators are considered. The applied voltage on the actuator is assumed to be uniform along each beam plane. 4.2.1. Constitutive equation The linear constitutive equations can be expressed by the direct and converse piezoelectric equations, and the (e,E)-type piezoelectric constitutive equations are used for finite element discretisation: De ¼ ee þ vE

r ¼ ce þeT E

(9)

where De, E, e and r are the electric displacement, electric field, strain, and stress vector, respectively, and c, e, and v are the elasticity, piezoelectric and dielectric constant matrices, respectively. The piezoelectric coefficient matrix can be written in matrix form as follows: e ¼ dc

(10)

where d is the piezoelectric constant.

Feedback voltage z

e

Va (t)

Actuator layer

ta Flexible beam

tb ts e Vo

Sensor layer

x

Control Law

Sensor output w2

w1 z

q1

l

q2

x

Fig. 9. Structure of an intelligent beam.

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The constitutive equation in the elastic field is

r ¼ cs e

(11)

where cs is the elasticity constant matrix of the beam. The constitutive equations for the piezoelectric beams, as shown in Fig. 9, can be written as follows: D1 ¼ e31 e1 e31 E3

s1 ¼ c11 e1 w33 E3 e31 ¼ d31 c11

(12)

4.2.2. Finite element formulation Using the standard discrimination technique produces the following [25]: u ¼ Nd

e ¼ Bd d E ¼ Ag V

(13) T

where u is the displacement, N is the shape function matrix, Ag is the gradient function, Ag ¼ ½ @=@x @=@y @=@z  , Bd is the derivative operator between the strain e and the generalised nodal displacements d, and V is the electric potential. The transverse displacement and transverse rotation of the beam element are denoted by w and y, respectively, as shown in Fig. 9. The local displacement is given by de ¼ ½ w1 q1 w2 q2 T . The transverse displacement and transverse rotation are expressed by shape functions as ½ w1 w2 T ¼ Nw ðxÞde and ½ q1 q2 T ¼ Nqy ðxÞde , respectively. Nw(x) and Nqy(x) are the shape functions. In this paper, Nw(x) is assumed to be a cubic polynomial function of x, and Nqy(x) is the derivative _ w ðxÞ. The operator Bd can also be determined from Nw(x) using the constitutive equations. The of Nw(x), ½Nqy ðxÞ ¼ ½N piezoelectric surface is assumed to be equal potential, and thus, the electric potential variable for the piezoelectric patch is constant. Using the virtual work principle, the finite element equations can be obtained as follows [25,26]: e

Meuu d€ þ Keuun de ¼ Fef Keuf ½Keff 1 Fec

(14)

Kefu de þ Keff Ve ¼ Fec

(15)

where Meuu ¼ Mese þ Mepe ¼ Keuu ¼ Kese þ Kepe ¼ Keuf ¼

Z Q

Z Z

Q se

Q se

NTs rs Ns dQ þ

Z

BTds cs Bds dQ þ

Q pe

NTp rp Np dQ

Z

Q pe

BTdp cp Bdp dQ

Z h iT BTdp eT Bdf dQ ¼ Kefu , Kff ¼ BTdf vT Bdf dQ Q

Eq. (14) is the mechanical equation, and Eq. (15) is the electrical equation. Meuu is the element mass matrix of the intelligent beam, Keuu is the element mechanical stiffness matrix, Keuf is the element electrical–mechanical coupling stiffness matrix, Kefu is the element mechanical–electrical coupling stiffness matrix, Kff is the element dielectric stiffness matrix, Fef is the applied mechanical force, Fes is the applied electrical charge, de is the nodal displacement and Ve is the electric potential. Substituting Eq. (15) into Eq. (14), we obtain the following expression: e Meuu d€ þKeuun de ¼ Fef Keuf ½Keff 1 Fea

where

Keuun

¼

(16)

Keuu þKeuf ½Keff 1 ½Keuf T :

4.2.3. Sensor equations The piezoelectric strip can be used as a sensor by measuring the electric charge on the electrodes. Assuming that the applied charge on the sensor is zero and that the converse piezoelectric effect is negligible, we obtain the sensed electrical potential as follows: Ves ¼ ½Keffs 1 Kefus des

(17)

Fes ¼ Kefus des

(18)

The sensed electrical charge is

W.-Y. Zhou, D.-X. Li / Journal of Sound and Vibration 331 (2012) 2984–3005

2997

Actuator layer ta tb

Flexible beam Sensor layer

ts Rf

_ +

e

Vo

Fig. 10. Current amplifier connections for the piezoelectric sensor.

In practice, the piezoelectric output voltage is amplified by either a charge amplifier or a current amplifier [27]. We use the current amplifier, which is as shown in Fig. 10; and determine the output voltage by the sensor to be e e Ves ðtÞ ¼ Rf F_ s ¼ Rf Kefus d_ s

(19)

where Rf is the constant of the amplifier and i(t) is the electrical current. In this paper, the sensor output voltage is obtained to be V eo ¼ Rf cp d31 zbðq_ 2 q_ 1 Þ

(20)

where z ¼ ððt b þt s Þ=2Þ is the effective distance. 4.2.4. Control law From the sensing signal Ves , the actuator voltage can be determined using different control laws, such as velocity feedback, displacement feedback, linear quadratic regulator (LQR) optimal control, and others. Generally, the LQR control method is more effective for the control of vibration; however, the LQR requires the full state of the system, which is very difficult to determine. Although a state observer or estimator can be designed to estimate all of the state values from the limited measurement signals, the risk of failure arises, and the scheme violates the general rule that devices for space applications require very high reliability. Velocity feedback control is relatively simple and provides higher reliability and stability. Thus, negative velocity feedback control is implemented to attenuate the amplitude at resonance. Using the current amplifier, the voltage applied to the actuator Vea is obtained as follows: e

Vea ðtÞ ¼ Gc Ves ðtÞ ¼ Gc Rf Kefus d_ s where Gc is the signal conditional gain. As the velocity amplitude

e d_ s

(21)

increases, the feedback voltage is amplified.

4.2.5. Actuator equations For the actuators, assuming that the electric charge induced by the displacement is negligible, the electrical charge can be calculated by the following: Fea ¼ Keffa Vea

(22)

Combining Eqs. (16), (21) and (22), we obtain the following actuating equation: e e Meuu d€ a þ Ceda d_ s þKeuun dea ¼ Fef

Ceda

where is the active damping matrix introduced by the active control force term and intelligent beam shown in Fig. 9, there is des ¼ dea ¼ de .

(23) Ceda

¼

Gc Rf Keufa Kefus .

For the

4.2.6. The state space equations of motion Assembling all of the finite elements of the platform, we obtain the corresponding global mass and stiffness matrices and mechanical and electrical load vectors. The system equations of motion are expressed as follows: Md€ þ ðCs þG0 Þd_ þKd ¼ Ds Fs ðtÞ þ Bc Fc ðtÞ dðt0 Þ ¼ 0 d_ ðt0 Þ ¼ 0 Fc ðtÞ ¼ Kuf Va

(24)

where G0 is the gyroscopic matrix; Cs is the damping matrix, which can be written in the form of Cs ¼ aaMþbbK; aa and bb are the structure constants; Ds and Bc are the disturbance force and control force assembling matrix, respectively; and Va is the control input vector.

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If the current amplifier is used, the control law used is velocity feedback control, and the equations of motion can be expressed as follows: Md€ þðCs þ Cc þ G0 Þd_ þ Kd ¼ Ds Fs ðtÞ

dðt0 Þ ¼ 0 d_ ðt0 Þ ¼ 0 Cc ¼ Bc Kuf Gc Rf cp d31 zb

(25)

The velocity feedback control can enhance the viscous damping, effectively suppressing the resonance amplitude [26]. Increasing the viscous damping also enhances the stability of a rotating system [24]. Assuming the state variable X ¼ ½ d d_ T , the system equations of motion in Eq. (24) can be written in the standard state space form as follows: _ ¼ AX þ BFc ðtÞ þ DFs X Y ¼ CX

(26)

where " A¼

½0 M

1

#

I K

M

1

ðCs þ G0 Þ

" ,

B¼

# ½0 , 1 M Bc

h C¼ 0

Rf Kfus

iT

" and

D¼

# ½0 : M Ds 1

The negative velocity feedback input can be expressed as follows: Fc ¼HY¼HCX; thus, the state equation can be written as _ ¼ ðAþ BHCÞX þ DFs X

(27)

The complex eigenvalue problem of Eq. (27) can be obtained as follows: ½lIðAþ BHCÞfUg ¼ 0

(28)

The complex eigenvalue is determined from l ¼ s þ odi. The damping ratio can be used to evaluate the effect of active vibration control and suppression for the intelligent structure. The damping ratio is defined as the negative of the normalised real part of the complex eigenvalue [28]:

s

z ¼  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

(29)

s2 þ o2d

4.3. Active vibration control performance analysis 4.3.1. Controllability analysis Without considering the gyroscopic effect, the first six natural frequencies of the system are below 30 Hz, and the following frequencies are all above 300 Hz. Hence, the equation of motion can be represented by modal equations using the first six modes as follows: Mm q€ þ Cm q_ þ Km q ¼ UT Ds Fs ðtÞ þ UT Bc Fc ðtÞ

(30)

where q¼ Ud is the modal coordinate, U consists of the eigenvectors corresponding to the first six modes, Mm ¼ UTMU is the mass matrix in the modal coordinate, Cm ¼ UT[Cs þG0]U is the damping and gyroscopic effect matrix in the modal coordinate, and Km ¼ UTKU is the stiffness matrix in the modal coordinate. Because G0 is not a symmetrical matrix, the damping matrix Cm cannot be decoupled by the modal coordinate. The state space form of the matrix can be obtained as follows: " # " # ½066 I66 ½0 (31) , Bm ¼ Am ¼ T M1 M1 M1 m U Bc m Km m C The controllability of the system can be determined by the order of the matrix as follows: rank½Bm ^Am Bm ^    An1 m Bm  ¼ 10 orank½Am  ¼ 12

(32)

Eq. (32) demonstrates that the system is not capable of controlling all six modes. From the pattern of the bonded piezoelectric sensors and actuators, it may be deduced that the axial rotational mode is the uncontrolled mode. Eliminating the eigenvector corresponding to the axial rotational mode in the matrix U allows the following to be obtained: rank½Bm2 ^Am2 Bm2 ^    An1 m2 Bm2  ¼ 10 ¼ rank½Am2 

(33)

Thus, the active control system can control the vibration in five degrees of freedom (two radial translations, two radial rocking, and one axial translation) of the RWAs, but not at the axial rotation mode.

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4.3.2. Damping ratio comparison Assuming that the rotating speed of the RWAs is 681 rev/min and that the coefficients of the damping matrix are aa ¼bb ¼0.0001, we can obtain the eigenvalues and damping ratios before and after the velocity feedback control, as given in Table 4. Note that the eigenvalues of the system change with the rotational speed because of the gyroscopic effect. Comparison of the damping ratio before and after the application of control in Table 4 shows that the active control system adds damping to the five modes, which means that the active control can effectively attenuate the vibration at resonance. The damping ratio of the axial translation mode is higher than the others, implying that the active control system more effectively attenuates the vibration of this mode. However, the damping ratio is not changed for the axial rotation mode, which confirms the conclusion in Section 4.3.1 that this mode cannot be controlled by the active control system. 4.3.3. Response at resonance Figs. 11 and 12 illustrate the response of the intelligent platform at the critical rotational speed to static imbalance disturbance and dynamic imbalance disturbance, respectively. Fig. 11 indicates the response of the intelligent Table 4 Damping ratio comparison. Mode

Resonance type

Eigenvalue (before control)

Damping ratio (before control)

Damping ratio (after control)

o1 o2 o3 o4 o5 o6

Backward whirl Forward whirl Backward whirl Forward whirl Axial translation Axial rotation

 0.01053 þ4.73353i  0.03196 þ 11.26559i  0.07831 þ 15.40333i  0.09754 þ 21.81197i  0.068670 þ 14.786481i  0.031682 þ 10.04091i

0.00223 0.00284 0.00508 0.00447 0.00465 0.00316

0.09558 0.02934 0.01099 0.02764 0.12093 0.00316

3

x 10-4

8 uncontrolled velocity feedback control

4 1

Fxout (N)

Displacement (m)

2

0 -1

2 0 -2 -4

-2 -3

2

-6 -8 0

5

10 15 20 Time (seconds)

25

30

0

x 10-3

10 15 20 Time (seconds)

25

30

uncontrolled velocity feedback control

0.4 0.3

1 Mxout (Nm)

0.2

0.5 0 -0.5

0.1 0 -0.1 -0.2

-1

-0.3

-1.5 -2

5

0.5 uncontrolled velocity feedback control

1.5

Angle (rad)

uncontrolled velocity feedback control

6

-0.4 -0.5 0

5

10 15 20 Time (seconds)

25

30

0

5

10 15 20 Time (seconds)

25

30

Fig. 11. Response to static imbalance at the critical rotational speed: (a) displacement x of node 1 (CoM of the RWAs), (b) Fx out at O, (c) angle displacement a of node 1, and (d) Mx out at O.

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W.-Y. Zhou, D.-X. Li / Journal of Sound and Vibration 331 (2012) 2984–3005

4

x 10-5

1.5 uncontrolled velocity feedback control

uncontrolled velocity feedback control

3

1 0.5

1

Fxout (N)

Displacement (m)

2

0 -1

0 -0.5

-2 -1

-3

-1.5

-4 0

4

5

10 15 20 Time (seconds)

25

0

30

x 10-4

10 15 20 Time (seconds)

25

30

0.08 uncontrolled velocity feedback control

3

uncontrolled velocity feedback control

0.06 0.04 Mxout (Nm)

2 Angle (rad)

5

1 0 -1

0.02 0 -0.02 -0.04

-2

-0.06 -0.08

-3 0

5

10 15 20 Time (seconds)

25

0

30

5

10 15 20 Time (seconds)

25

30

Fig. 12. Response to dynamic imbalance at the critical speed: (a) displacement x of node 1 (CoM of the RWAs), (b) Fx out at O, (c) angle displacement a of node 1, and (d) Mx out at O.

60 uncontrolled velocity feedback control

uncontrolled velocity feedback control

1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1

40

Fzout (Nm)

Displacement (m)

x 10-3

20 0 -20 -40 -60

0

1

2

3

4 5 6 7 Time (seconds)

8

9

10

0

1

2

3

4 5 6 7 Time (seconds)

8

9

10

Fig. 13. Response to harmonic resonance excitation at the axial translation mode: (a) displacement z of node 1 (CoM of the RWAs), (b) Fz out at O.

platform when the rotating speed is 681 rev/min (681 rev/min is the critical rotating speed without control). Note that the stiffness of the piezoelectric sensors and actuators increases the critical rotational speed in this case in comparison to its former value (650 rev/min). The static imbalance is assumed to be 1.8 g cm, and the dynamic imbalance is assumed to be zero.

W.-Y. Zhou, D.-X. Li / Journal of Sound and Vibration 331 (2012) 2984–3005

3001

In the stable state, the amplitude of the disturbances is reduced to less than one-third of the starting value after active control, as shown in Fig. 11(a–d). Thus, the velocity feedback control is effective in controlling the amplitude of the vibration at resonance, as shown in Fig. 11(a–c), and the intelligent platform is effective in attenuating the disturbance force and moment, as shown in Fig. 11(b,d). Fig. 12(a–d) represents the equivalent results when the dynamic imbalance is assumed to be 3.6 g cm2 and the static imbalance is assumed to be zero.

1

x 10-6

0.6

0.03

0.4

0.02

0.2

0.01

0 -0.2

-0.01 -0.02

-0.6

-0.03

-0.8

-0.04 -0.05

-1 0

0.5

1

1.5 2 2.5 3 3.5 Time (seconds)

4

4.5

0

5

x 10-6

1

1.5 2 2.5 3 3.5 Time (seconds)

4

4.5

5

uncontrolled velocity feedback control

0.04 0.03

0.4

0.02

0.2

0.01

Fyout (N)

0.6

0 -0.2

0 -0.01

-0.4

-0.02

-0.6

-0.03

-0.8

-0.04

-1

-0.05 0

0.5

1

1.5

0.4

2 2.5 3 3.5 Time (seconds)

4

4.5

5

0

10-2

Voltage of veritcal beam 1 Voltage of veritcal beam 2

0.3

10-3

0.2 0.1

Fxout (N)

Voltage (V)

0.5

0.05 uncontrolled velocity feedback control

0.8 Displacement (m)

0

-0.4

1

uncontrolled velocity feedback control

0.04

Fxout (N)

Displacement (m)

0.05

uncontrolled velocity feedback control

0.8

0 -0.1 -0.2

0.5

1

ω2

ω3

1.5

2 2.5 3 3.5 Time (seconds)

4

4.5

5

ω1

10-4

ω4

10-5 10-6

-0.3 -0.4

0

0.5

1

1.5

2 2.5 3 3.5 Time (seconds)

4

4.5

5

10-7

0

10 20 30 40 50 60 70 80 90 100 Frequency (Hz)

Fig. 14. Response to the radial impulse force: (a) displacement x of node 1 (CoM of the RWAs), (b) Fx out at O, (c) displacement y of node 1, (d) Fy out at O, 17), and (f) F (e) input voltage of vertical beam 1 (element I) and 2 (element ? x out frequency response.

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Fig. 13 illustrates the response of the intelligent platform to a harmonic resonance excitation of F z ¼ 0:5 sinð2po5 tÞ, where o5 is the axial translation frequency. The platform more effectively controls the amplitude of the axial translation disturbance force at resonance. 4.3.4. Response to impulse load An external radial translation impulse load of Fx ¼1 N is assumed to be applied to node 1 (CoM of the RWAs) at 0 s for a duration of 1 m s at a rotational speed of 3000 rev/min. Fig. 14 illustrates the response of the intelligent platform. The active control increases the decay rate of the vibration. A beat-like phenomenon appears in Fig. 14(a)–(d). Frequency analysis indicates that it is caused by two similar natural frequencies with relatively high amplitude (o2 and o3), as shown in Fig. 14(f). Fig. 14(f) also shows that there are four frequencies (o1, o2, o3, and o4) excited by the impulse load Fx. The impulse load in the x direction does not excite the axial modes (o5 and o6) because the axial translational mode and the rotational mode are decoupled from the others. Fig. 14(c) and (d) demonstrates that, as a result of the gyroscopic effect, the impulse load in the x direction also excites vibration in the y direction, which is similar to the vibration in the x direction, except for a phase difference. An external axial translation impulse load of Fz ¼1 N is also assumed to be applied to node 1 (CoM of the RWAs) at 0 s for a duration of 1 ms at a rotational speed of 3000 rev/min. Fig. 15 illustrates the time and frequency domain response of the intelligent platform. Only the axial translation mode (o5) is excited by the load, and the active control more effectively attenuates the axial translation vibration than the radial vibration. 4.3.5. Response to different gains Fig. 16 shows the stable state disturbance force amplitude and control voltage responses for different gains with the same static imbalance at a critical speed of 681 rev/min. The stable disturbance force amplitude Fx out decreases quickly

0.05

10-2

uncontrolled velocity feedback control

0.04

ω5

10-3

0.03 Fzout (N)

Fzout (Nm)

0.02 0.01 0 -0.01

10-4 10-5

-0.02 10-6

-0.03 -0.04

10-7

-0.05 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time (seconds)

0

10 20 30 40 50 60 70 80 90 100 Frequency (Hz)

Fig. 15. Response to axial translational impulse force: (a) Fz out for impulse force and (b) Fz out frequency response.

6

70

5.5 5

50 Voltage (V)

4.5 Fxout (N)

Voltage of veritcal beam 1 Voltage of veritcal beam 2

60

4 3.5 3 2.5

40 30 20

2 10

1.5 1 0

50 100 150 200 250 300 350 400 450 500 Gain

0

0

50 100 150 200 250 300 350 400 450 500 Gain

Fig. 16. Fx out and voltages versus gain: (a) Fx out versus gain and (b) voltage of vertical beam versus gain.

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Fig. 17. Effect of velocity feedback control: (a) ‘‘common’’ beam, (b) ‘‘intelligent’’ beam, and (c) ‘‘equivalent’’ beam.

with increasing gain at first and then slows down, nearly becoming a horizontal line. This trend implies that increasing the gain has little effect on suppression of the disturbance when the gain is larger than 200. When the gain is high enough (larger than 200), the high velocity feedback control effects maintain equal slopes at both ends of the beam. This result can be interpreted in terms of the example of a cantilever beam, as shown in Fig. 17(a). If a harmonic force F(t) is applied to the free end, there will be a harmonic slope angle q(t) on the free end for the ‘‘normal’’ beam. However, if the beam is an ‘‘intelligent’’ beam with a bonded piezoelectric sensor and actuator, the moment applied by the actuator will try to force the slope angle to zero, as shown in Fig. 17(b). As the gain increases, the feedback control voltage also increases, and the free-end boundary gradually becomes a sliding-roller boundary condition [29,30], as shown in Fig. 17(c). Fig. 17 shows that the eventual effect of an ‘‘intelligent’’ beam is the constraint of the y degree of freedom; as a result, it has a limited effect on the control of the vibration of the beam. The ‘‘intelligent’’ beam becomes ineffective if a particular force and torque are applied. For example, if a harmonic force F ¼ sin ot is applied to the free end of a cantilever beam and a harmonic torque T ¼  (Fl/2) is applied simultaneously, the slope of the free end is always zero, and the sensed voltage is also zero. In this case, the ‘‘intelligent’’ beam becomes a ‘‘common’’ one. However, the problem above can be solved by bonding segmented sensors and actuators to the beam and using a multiinput–multioutput control system. 5. Conclusion This paper presents the design and analysis of an intelligent vibration isolation platform for a reaction/momentum wheel assemblies considering the gyroscopic effect of the flywheel. A passive vibration isolation platform based on a folded beam is designed and analysed for a momentum wheel assemblies (MWAs). A mathematical model is developed for the system consisting of the platform and the MWAs. The passive vibration isolation platform is simplified to be a decoupled elastic support with a gap below the MWAs. The results confirm that the mathematical model can effectively represent the passive vibration isolation performance. The passive vibration isolation platform is particularly effective for the suppression of disturbances produced by the MWAs, the disturbance frequencies of which are mainly in the high-frequency range because MWAs generally operate at the high-speed range. However, passive vibration isolation is not adequate for a reaction wheel assemblies (RWAS) because excessive oscillation occurs if the rotating speed is close to the critical speed. To attenuate the vibration amplitude at the critical speed for RWAs, the passive vibration isolation platform is modified to be an intelligent one. In the simulation, piezoelectric sensors and actuators are bonded to the vertical beam of the platform to constitute an intelligent platform, and the velocity feedback control law is used to attenuate the vibration caused by static imbalance, dynamic imbalance and impulse load. The simulation results demonstrate that the intelligent vibration isolation system is effective for the suppression of dynamic amplification at resonance. Thus, it can be concluded that the intelligent platform is effective for the isolation of vibrations from RWAs. Appendix

2 6 6 6 6 6 M1 ¼ 6 6 6 6 4 2

mf þ mb

0

0

0

0

0

0

mf þmb

0

0

0

0

0 0

0 0

mf þ mb 0

0 If x þ Ibx

0 0

0 0

0

0

0

0

If y þIby

0

0

0

0

0

If z þ Ibz

3

0 2 6 6 6 6 6 Ks ¼ 6 6 6 6 4

0

1

0

0

0

h1

60 6 6 60 Tsx1 ¼ 6 60 6 6 40

1

0

h1

0

0

1

0

0

0

0

1

0

0

0

0

1

07 7 7 07 7, 07 7 7 05

0

0

0

0

0

1

3 7 7 7 7 7 7, 7 7 7 5 3

kx

0

0

0

0

0

0

ky

0

0

0

0

0

kz

0

0

0

0

0

ka

0

0

0

0

0

kb

0

0

0

0

0

07 7 7 07 7 7, 07 7 07 5 kg

3004

W.-Y. Zhou, D.-X. Li / Journal of Sound and Vibration 331 (2012) 2984–3005

2

0 60 6 6 60 G¼6 60 6 6 40 0 2 6 6 6 6 6 Cs ¼ 6 6 6 6 4

0

0

0

0

0

07 7 7 07 7, 07 7 7 05

0

0

0

0

0

0

0

0

0

0

OI f z

0

0

OIf z

0

0

0

0

0

csx

0

0

0

0

0

csy

0

0

0

0

0

csz

0

0

0

0

0

c sa

0

0

0

0

0

c sb

0

0

0

0

0

2

1 6 0 6 6 6 0 A Ts1 ¼ 6 6 0 6 6 4 hAS 0

3

0

0 3 0 0 7 7 7 0 7 7 7, 0 7 7 0 7 5 c sg

0

0

0

0

0

1

0

0

0

3

0

1

0

0

hAS

0

1

0

0

0

0

1

07 7 7 07 7, 07 7 7 05

0

0

0

0

1

2

1 6 0 6 6 6 0 Ts1 ¼ 6 6 0 6 6 4 h1 0 2

0

0

0

0

0

1

0

0

0 0

3

0

1

0

h1

0

1

0

0

0

0

1

07 7 7 07 7, 07 7 7 05

0

0

0

0

1 0

1

0

0

0

hAS

60 6 6 60 A Tsx1 ¼ 6 60 6 6 40

1

0

hAS

0

0

1

0

0

0

0

1

0

0

0

0

1

0

0

0

0

0

1 3

2

1 6 0 6 6 6 0 Ts2 ¼ 6 6 0 6 6 4 h2 0

3

07 7 7 07 7, 07 7 7 05

0

0

0

0

0

1

0

0

0 0

0

1

0

h2

0

1

0

0

0

0

1

07 7 7 07 7: 07 7 7 05

0

0

0

0

1

References [1] U. Roberto, M. Flavia, Artemis micro-vibration environment prediction. Spacecraft structures, Materials and mechanical testing, Proceedings of a European Conference held at Braunschweig, Germany, 4–6 November 1998, European Space Agency (ESA), Paris, ESA-SP, Vol. 428, 1999, pp. 489–495, ISBN: 9290927127. [2] T. Wacker, L. Weimer, K. Eckert, GOCE platform micro-vibration verification by test and analysis, Proceedings of the European Conference on Spacecraft Structures, Materials and Mechanical Testing, 2005 (ESA SP-581), 10–12 May 2005, Noordwijk, Netherlands. [3] T. Shimizu, S. Nagata, S. Tsuneta, T. Tarbell, C. Edwards, R. Shine, C. Hoffmann, E. Thomas, S. Sour, R. Rehse, O. Ito, Y. Kashiwagi, M. Tabata, K. Kodeki, M. Nagase, K. Matsuzaki, K. Kobayashi, K. Ichimoto, Y. Suematsu, Image stabilization system for Hinode (Solar-B) solar optical telescope, Solar Physics 249 (2008) 221–232. [4] N. Jedrich, D. Zimbelman, M. Turczyn, J. Sills, C. Voorhees, B. Clapp, Cryo cooler induced micro-vibration disturbances to the Hubble space telescope, Fifth Cranfield Space Dynamics Conference, July 2002. [5] C.L. Foster, M.L. Tinker, G.S. Nurre, W.A. Till, Solar-array-induced disturbance of the hubble space telescope pointing system, Journal of Spacecraft Rockets 32 (1995) 634–644. [6] G.S. Aglietti, R.S. Langley, E. Rogers, S.B. Gabriel, Model building and verification for active control of micro-vibrations with probabilistic assessment of the effects of uncertainties, Proceedings of the Institution of Mechanical Engineers Part C: Journal of Mechanical Engineering Science 218 (2004) 389–399. [7] N. Yoshida, O. Takahara, T. Kosugi, K. Ninomiya, T. Hashimoto, K. Minesugi, S. Tsuneta, K. Ichimoto, S. Shimada, Systematic approach to achieve fine pointing requirement of SOLAR-B, IFAC ACA2004, 16th Symposium on Automatic Control in Aerospace, June 14–18, 2004, St-Petersburg, Russia. [8] S. Miller, P. Kirchman, J. Sudey, Reaction wheel operational impacts on the GOES-N jitter environment, AIAA Guidance, Navigation and Control Conference and Exhibit, Hilton Head, South Carolina, 2007, pp. 1–12, 20–23, AIAA 2007-6736. [9] O. Takahara, N. Yoshida, K. Minesugi, T. Hashimoto, S. Nagata, M. Koike, S. Shimada, T. Nakaoji, Microvibration transmissibility test of Solar-B, The 24th ISTS (International Symposition on Space Technology and Science), May 30–June 6, 2004, Miyazaki, Japan. [10] S.S. Narayan, P.S. Nair, A. Ghosal, Dynamic interaction of rotating momentum wheels with spacecraft elements, Journal of Sound and Vibration 315 (2008) 970–984, doi:10.1016/j.jsv.2008.02.020. [11] K.C. Liu, P. Maghami, C. Blaurock, Reaction wheel disturbance modeling, jitter analysis, and validation tests for Solar Dynamics Observatory, AIAA Guidance, Navigation and Control Conference and Exhibit, Honolulu, Hawaii, 18–21 August, 2008, pp. 1–18, AIAA 2008-7232. [12] B. Bialke, A compilation of reaction wheel induced spacecraft disturbances, 20th Annual AAS Guidance and Control Conference, February 1997, AAS Paper 97-038. [13] R.A. Masterson, D.W. Miller, Development and Validation of Empirical and Analytical Reaction Wheel Disturbance Models, Master Thesis, Department of Mechanical Engineering, Massachusetts Institute of Technology, 1999. [14] R.A. Masterson, D.W. Miller, R.L. Grogan, Development and validation of reaction wheel disturbance models: empirical model, Journal of Sound and Vibration 249 (2002) 575–598. [15] L.M. Elias, D.W. Miller, A coupled disturbances analysis method using dynamic mass measurement techniques, AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, 22–25 April 2002, AIAA2002-1252. [16] S. Taniwaki, Y. Ohkami, Experimental and numerical analysis of reaction wheel disturbances, Japan Society of Mechanical Engineers International Journal 46 (2) (2003) 519–526. [17] L.P. Davis, J.F. Wilson, R.E. Jewell, J.J. Rodden, Hubble Space Telescope Reaction Wheel Assembly Vibration Isolation System, NASA Workshop on Structural Dynamics and Control Interaction of Flexible Structures, G.C. Marshall Space Flight Center, March 1986. [18] L. Vaillon, C. Philippe, Passive and active microvibration control for very high pointing accuracy space systems, Smart Materials and Structures 8 (1999) 719–728. [19] K. Makihara, J. Onoda, K. Minesugi, New approach to semi-active vibration isolation to improve the pointing performance of observation satellites, Intelligent Smart Materials and Structures 15 (2006) 342–350. [20] D. Kamesha, R. Pandiyan, Ashitava Ghosal, Modeling, design and analysis of low frequency platform for attenuating micro-vibration in spacecraft, Journal of Sound and Vibration 329 (2010) 3431–3450. [21] D. Kamesha, R. Pandiyan, Ashitava Ghosal, Passive vibration isolation of reaction wheel disturbances using a low frequency flexible space platform, Journal of Sound and Vibration 331 (2012) 1310–1330.

W.-Y. Zhou, D.-X. Li / Journal of Sound and Vibration 331 (2012) 2984–3005

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[22] W.Y. Zhou, G.S. Aglietti, Z. Zhang, Modelling and testing of a soft suspension design for a reaction/momentum wheel assembly, Journal of Sound and Vibration 330 (2011) 4596–4610. [23] H.C. Corben, Philip Stehle, Classical Mechanics, second ed., Wiley, New York, 1960, pp. 136–152. [24] G. Genta, Dynamics of Rotating Systems, Springer Science þ Business Media, Inc., Printed in the United Sates of America, 2005. [25] C.Q. Chen, W.M. Wang, Y.P. Shen, Finite element approach of vibration control using self-sensing piezoelectric actuators, Computers and Structures 60 (3) (1996) 505–512. [26] S.H. Chen, Z.D. Wang, X.H. Liu, Active vibration control and suppression for intelligent structures, Journal of Sound and Vibration 200 (2) (1997) 167–177. [27] N. Jalili, Piezoelectric-Based Vibration Control, Springer Scienceþ Business Media, Inc., Printed in the United Sates of America, 2010. [28] C.I. Tseng, Electromechanical Dynamics of a Coupled Piezoelectric/Mechanical System Applied to Vibration Control and Distributed Sensing, Ph.D. Thesis, University of Kentucky, 1981. [29] S. Narayanan, V. Balamurugan, Finite elment modelling of piezolaminated intelligent structures for active vibration control with distributed sensors and actuators, Journal of Sound and Vibration 262 (2003) 529–563. [30] H.S. Tzou, Piezoelectric Shells—Distributed Sensing and Control of Continua, Kluwer Academic, London, 1993.