Du C., Xie L. Modeling and Control of Vibration in Mechanical Systems

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74 Modeling and Control of Vibration in Mechanical Systems

0 500 1000 1500 2000 2500 3000 3500 4000

0.05

0.1

Frequency(Hz)

FFT(RRO)(Vrms)

1−disk

2−disk

FIGURE 4.7

Comparison of single-disk and dual-disk RRO power spectrum at 8400 RPM (41%

improvement of σ value).

Classical Vibration Control 75

0 500 1000 1500 2000 2500 3000 3500 4000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Frequency(Hz)

FFT(NRRO)(Vrms)

1−disk

2−disk

S2(S3)

FIGURE 4.8

Comparison of single-disk and dual-disk NRRO power sp ectrum at 8400 RPM (28%

reduction of σ value).

76 Modeling and Control of Vibration in Mechanical Systems

FIGURE 4.9

Comparison of sing le-disk and dual-disk PES in time domain at 8400 RPM (38%

reduction of σ value).

Classical Vibration Control 77

C. Positioning accuracy improvement at 10200 RPM

A higher rotational speed of 10200 RPM is performed on both the dual-disk stack

and single disk cases. Figure 4.10 shows the obtained disk axial vibration by LDV.

The modes T1-T6 are quite obviou s with the amplitude reduced and the frequencies

shifted higher. Figure 4.11 and Figure 4.12 show the power spectrum of RRO and

NRRO. The 3rd and 5th harmonics are signiﬁcantly reduced as seen in Figure 4.11.

Figur e 4.12 reﬂects the corresponding disk vibrat ion modes T1-T6 in Figure 4.10.

As f or the slider related vibrations, S1 is redu ced signiﬁcantly again and S2 and S3

are shif ted, while S4 cannot be seen at all i n Figure 4.12 . As a result, the σ value of

RRO is improved b y 33%, NRRO b y 27%, and PES by 32%, compared with that of

the single disk approach. Figure 4.13 shows the comparison of PES i n time domain,

and it is observed that the amplitude of PES in the case of the dual-disk stack is

decreased.

0 500 1000 1500 2000 2500 3000

−9

−8.5

−8

−7.5

−7

−6.5

−6

−5.5

−5

−4.5

Frequency(Hz)

Dsplacement amplitude(log10(m))

2−disk

1−disk

FIGURE 4.10

Comparison of single-di sk and dual-disk axial vibration s measured via LDV at

10200 RPM.

The reduction of disk vibration amplitude is evaluated from Figures 4.2, 4.6 and

4.10 and tabulated in Table 4.1 for different rotational speeds. The improvements of

78 Modeling and Control of Vibration in Mechanical Systems

0 500 1000 1500 2000 2500 3000 3500 4000

0.05

0.1

0.15

0.2

Frequency(Hz)

FFT(RRO)(Vrms)

1−disk

2−disk

FIGURE 4.11

Comparison of single-disk and dual-disk R RO power spectrum at 10200 RPM (the

3rd and 5th harmonics reduced signiﬁcantly, 33% improvement of σ value).

Classical Vibration Control 79

0 500 1000 1500 2000 2500 3000 3500 4000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Frequency(Hz)

FFT(NRRO)(Vrms)

1−disk

2−disk

FIGURE 4.12

Comparison o f single-disk and d ual-disk NRRO power spectrum at 10200 RPM

(27% reduction of σ value).

80 Modeling and Control of Vibration in Mechanical Systems

FIGURE 4.13

Comparison of single-disk and dual-disk PES in time domain at 10200 RPM (32%

reduction of σ value).

Classical Vibration Control 81

TABLE 4.1

% reduction of σ values of PES, RRO and NRRO and di sk

vibration amplitude with stacked disks compared with single disk

Speed(RPM) RRO NRRO PES Disk vib ration amplitude

7200 23 18 21 24

8400 41 28 38 45

10200 33 27 32 35

the σ values of RRO, NRRO and PES are also summarized i n Table 4.1. Remarkable

improvement is achieved for every rotati onal speed. It can be seen clearly from the

table that the best improvement is obtained when operating at 8400 RPM. The second

best result obt ai ned is performed at 10200 RPM, where the dual-d isk stack appro ach

leads to an improvement of 32% in p ositioning accuracy.

For al l t he cases studied, we have observed a remarkably reduced amplitude of the

disk vibration and its reﬂection in positi oning accuracy improvement via the dual-

disk stack. This indi cates that the stacked disk is a simple way of reducing the disk

vibrations in spin st and tests, particularly when a single disk surface is accessed.

In Figures 4.2, 4 .6, and 4.10, we have also observed the fr equency shifting of disk

vibration modes.

Theoretically the maximum amplitude of disk vibration is given by [22]

V ∝

ρR

, (4.8)

where R is the disk outer radius, ρ is t he gas density, ω is the angular speed, t is

the disk thickness, E is the Young’s modulus of the di sk su bstrate, and d is the

disk substrate damping. Equat ion (4.8) indi cates that the disk vibration is inversely

proportional to t

. In the dual-d isk case, the disk stack th ickness

t = 2t. Hence

assuming that the change of E is negligible, the amplitude of disk vibration

V ∝

ρR

, (4.9)

which means that

V < V when the damping

d >

d. The experimental results in

Figur es 4.2, 4.6 and 4.10 veriﬁed the amplitud e reduct ion o f the disk vi bration.

On the other hand, it is known that the natural frequency of disk vibration mode is

deﬁned by [23]

2πR



12γ(1 − ν

)



1/2

, (4.10)

where f

is the natural frequency of the (m, n) mode, γ is t he mass per un it area of

the disk, ν is the Poisson’s ratio, and λ

is the dimensionless frequency parameter

which is generally a function of the boundary conditions on the plate, the ratio of the

82 Modeling and Control of Vibration in Mechanical Systems

inner to out er diameter, and Poi sson’s ratio. Denoting Ω as the rotational speed of

the disk, the frequency of the disk vibrati on mod e i s given by [23]

f = f

±nΩ. (4.11)

Assuming that the change of ν is also negligible, and considering that

t = 2t,

= 2

√

2πR



12γ(1 − ν

)



1/2

, (4.12 )

which implies some shifting of the disk vibration modes, as can be seen in the ex-

perimental results in Figures 4.2, 4.6 and 4.10.

In additio n to the doubled thickness, the mechanical properti es such as d and λ

[22][23] are certain to change correspondingly after stacking two disks. Further

research for details on how the stacked disks change these properties would be sig-

niﬁcant.

To summarize, t he proposed p rocess is indeed an approach to reduce th e disk

vibration on both disk surfaces for reading and writing, such as multi- disk servo

track w riters [25]. Thus such a disk vibration reduction approach can be used as an

alternative or complementary technique to the air shroud design [24] and the active

control approaches to fur ther improve positioning accuracy.

4.3 Self-adapting systems

In certain applications, system dynamic parameters vary with time. For example,

the mass of a system like a car, airplane or helicopter decreases as it consumes fuel.

Distinct ﬂight conditions such as ﬂight level, maneuvers, or landing produce different

types of excitatio n. When t he initial characteristics of the system no longer meet the

requirements of the system’s working conditions, the sy stem characteristics therefore

need to adapt t o th e parameter variatio ns of the excitat ion and the system itself. Two

ways are possible: one is to change the stiffness of the structure; the other is to add

moving masses so that their inertial effects counteract t he effects of the excitatio n

variation. Some self-adapting systems in use are bri eﬂy introduced as fo llows [8].

Self-tuning suspension

Self-tuning suspension systems are equipped suitably for slow variation of the sy s-

tem characteristics. One method, for example, used in self-tuning systems involves

analyzing the system’s vibration frequency and its timewise variation. The tuning

is usually performed by an actuator. After the system status is measured, the tuning

system will modif y the feature of the system, such as stiffness, damping and position

of a mass.

The self-tuning sy stem is tuned by a designed control algorithm. Simulations

using the control algorithm are required to identify the setting parameters of the

Classical Vibration Control 83

algorithm and t o verify the reliability of the system. Subsequently, the algorithm

must be validated by experimental tests on the real structure. The car suspension

is an example of self-tuning suspensions. It is able to adapt to various operating

conﬁgurations depending on the type of driving, the dr iving conditions, or the desired

comfort. One method to achieve this is t o modify the suspension stiffness or its

damping u sing various techniques.

Self-adjusti ng absorbers

The mechanical resonator, which is basically a mass-spring system, is used to con-

trol certain v ibration sources. It is placed where the vibrations are to be reduced. The

principle is that the resonance fr equ ency of the mass-spring system as a resonator is

adjusted with the excitation f requency in operation, and as the mass vibrates, the

vibratory level is decreased.

In certain structures, the excitation frequency evolves with time gradually. These

changes take place over a long period, compared to t he excitation frequency. As a

result, it is necessary to create a system whose resonance frequency would adjust

automatically to the variations of the excitation frequency. For a 20 Hz excitation

frequency, this variation usually takes place between 1 and 3 seconds. This kin d of

system actually works as the case of self-adjusting absorbers.

Self-adapting resonator

Self-adapting resonators are capable of adjusting to changes in the excitation fre-

quency. The hu b resonator used in a helicopter is an example. It consists of a ﬂapping

mass that v ibrates in a plane perpendicular to the rotation axis of the rotor. The mo-

bile ﬂapping mass is supported by three ﬂexible element s indexed at 120

◦

, and it

slides along the rotor axis. The stiffness of the three elements is designed so that

the resulting anti-resonance corresponds to the excitation frequency. The required

position of the mobile mass is determined by control algorithms through an actuator

and the mobile mass is moved by th e actuator. The z position of the mass makes the

variation of the inertia of the assembly and thus the anti-resonance frequency varies

accordingly.

4.4 Active vibration control

The self-adapti ng systems presented previously will not be sufﬁcient in the cases

where the source characteristics vary too fast for the involved algor ithms or the re-

quired level of performance is too high. Active methods should be used to decrease

the vibrations.

A vibration control system is called active if it uses external power to p erform

its function. It is comprised of a servomechanism with an actuator, a sensor, and a

microprocessor-based system. The actuator applies a force to the mass whose vi-

bration is to be reduced. The sensor measures t he motion of t he mass in terms of

displacement, velocity, or acceleration, depending on th e application. The micropro-