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

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64 Modeling and Control of Vibration in Mechanical Systems

of a resilient member, i.e., stiffness, and an energy dissipator, i.e., damping. Typical

examples of passive isolators include metal springs, cork, pneumatic springs, and

rubber spr ings.

4.2.2 Absorbers

If a system is acted upon by a force whose excitation frequency nearly coincides with

its natural frequency, t he vibration of the syst em can b e reduced by using a dynamic

vibration absorber, which is simply a spring-mass syst em. We consider an auxiliary

mass m

attached to a machine o f mass m

through a spring of stiffness k

. The

motion equations of t he masses m

and m

are written as

¨x

+ k

−x

) = F

sinωt, (4.1)

¨x

+ k

− x

) = 0, (4.2)

where x

and x

represent displacement s of m

and m

, respectively. The steady-

state amplitudes of the mass m

and m

are given b y

−m

+ k

−m

)(k

−m

) −k

, (4.3)

+ k

−m

)(k

−m

) −k

. (4.4)

Equation (4.3) implies that if

the amplit ude X

of the machine m

will be zero. Consider the machine operat-

ing n ear its resonance ω

≃ k

before the addition of the dynamic vibration

absorber. If the absorb er is designed such that

, (4.5)

the vibration amplitu de of the machi ne, while operating at its original resonant fre-

quency, will be zero.

The above mentioned dynamic vibration absorber removes the original resonance

peak in the response of the machine, but introduces two n ew resonance peaks. If

it is necessary to reduce the amplitude of vibr at ion of the machine over a rang e of

frequencies, a damped dynamic vibration absorber can be used [5].

4.2.3 Resonators

Resonators are used in certain cases where it is easier and more efﬁcient to locate

anti-vibration systems near the vibration source. The main idea is to create another

source of vibrations which will cancel ou t the original vibration if it is correctl y

tuned. The p rinciple is to use the kinetic energy i n the resonant mass system, where

Classical Vibration Control 65

the mass is coupled to th e vibrating source either by kinematic coupling or via a very

ﬂexible link [8].

For example, in a rotating structure the stator is often excited by the rotor via

inertial effects such as imbalance or other effects such as the aerodynamic asymmetry

of the blades or ﬂaps. Positioning a resonato r on the rotor is to generate loads that

oppose the excitation loads. In the case of a helicopter, t he resonant mass is placed

on t he axi s of th e rotor hub. It is supported by three springs which allow it to vibrate

in a plane perpendicular to the axis of rotation. The motion of the counterweight is

mainly in-plane moti on so that the sy stem will generate in-plane loads to eliminate

the in-plane vibration .

4.2.4 Suspension

Suspension acts as the link between two structures that is designed in order to isolate

one of the structures from the other. This is a concept for structure isolation that is

achieved by canceling the variable loads transmitted to the support structure. The

characteristics of the l ink are determined by analyzing the dynamic behavior and vi-

brations without modifying its main function in st at ic characteristics. In p ractice, the

link characteristics can be modiﬁed by varying its stiffness or appropriate positioning

of t he system natural frequencies and its damping in terms of energy transfer. An ad-

ditional technique consists of i ntroducing a ﬂappi ng mass whose inertial effects will

neutralize the excitation inputs. Next we will brieﬂy introduce stiffness modiﬁcation

and damping modiﬁcation.

Stiffness modiﬁcation

The dynamic behavior of stru ct ures results from the exchange and dissi pation of

energy. Dynamic forces transfer their energy to the structure, which t hen responds

via several mechanisms, such as bending or extension. Dynamic behavior can be

modeled in several ways. The best known is Newton’s second law of motion. If

the external force is static or quasi-static, structural stiffness forces develop t o create

an equilibrium. External dynamic forces are balanced in a more complex way with

inertial and damping forces.

Stiffness, often schematically and conceptually represented by a spring , denotes

the capacity of a sy stem to store strain energy. The stiffness force follows Hook e’s

law, F

= k

∆x, where the stiffness constant k

is expressed in the unit o f force per

unit length. This is a linear model where the spring displacement is measured from

the rest length. More complicated laws exist, for example, the nonli near relation

F = k(x)∆x

, where the parameter a would depend on the particular material bein g

modeled, and the stiffness parameter k(x) is a fun ct ion o f how much the spring has

been elongated.

Dampi ng modiﬁcation

Damping deﬁnes the ability of a structure to dissipate energy. For an oscillatory

system, damping is a measure of how much energy is dissipat ed by the system dur ing

an oscillation cycle. For example, structural connections between components add

damping to a structu re.

66 Modeling and Control of Vibration in Mechanical Systems

Most systems possess damping to some extent, which is helpful in vibration con-

trol. If a system undergoes a forced vibration, its respon se or amplitude of vib ration

near resonance tends to become large if there is no damping. The presence of damp-

ing always limits the amplitude of vibration. Damping can be modiﬁed in the system

to control its response, by the u se of structural materials having hig h in ternal damp-

ing, such as laminated or sandwiched materials. An example is the use of viscoelastic

materials.

We know that th e response amplitude of a system at resonance ω = ω

under

harmonic excitation F (t) = F

iωt

is given b y

kη

αEη

, (4.6)

where η is the energy l oss factor, and the stiffness k is proportional to the Young’s

modulus, i.e., k = αE with a constant α.

When viscoelastic materials are used for vibration control, they are subjected to

shear or direct strains. A simplest arrangement is that a l ayer of the viscoelastic

material is attached to an elastic one. Anoth er arrangement is that a viscoelastic

layer is sandwiched between th e elastic layers. The material with the largest loss

factor w ill be subjected to the smallest stress, while the stress is proport ional to

the displacement. Hence viscoelastic materials having large loss factor are used to

provide internal damping for vibrat ion control. Damping tapes, consisting of thin

metal f oil covered with a vi scoelasti c adhesive, are used on vibrating stru ct ures.

Another application example is presented as follows with detailed discussion and

evaluation [14].

4.2.5 An application exa mple − Disk vibration reduction via stacked

disks

To suppo rt higher tr ack per inch (TPI) density hard disk drive, the positioning accu-

racy of test equipment such as spin stand and servo track writer has to be increased.

The Spin stand is commonly-used equipment for magnetic recording component test-

ing [10]. A servo track writ er (STW) [25] is used t o pattern the recording media with

servo i nformation for the HDD servo system. In both cases, spindle mot or and disk

vibrations [23] affect the positioning accuracy and limit how close adjacent tracks

can be p laced together, and thus restrict the TPI number that can be achievable. In

[11], damped laminated disks were used to reduce the amplitude of rocki ng modes

by sandwiching a viscoelastic layer in between two aluminum layers to increase disk

damping. Reference [24] investigated the effect of suppressing resonance amplitude

of disk vibrations by applying a squeeze ﬁlm damping. In addition to th e laminat ed

disks and the squeezed air bearing plate methods, in this section we discuss an ap-

proach with minimal mechanical alternation. That is t o stack more than one normal

recording media together for reading and writing on the disk surfaces.

The Guzik spin stand shown in Figu re 4.1 is used to evaluate the effectiveness of

dual-disk stack in positio n accuracy improvement. The disk vibrations in the axial

Classical Vibration Control 67

direction are also measured at the outer diameter (OD) region of the disk via LDV.

In the experiment, the spindle motor is spinning at 7200, 8400 and 10200 RPM

respectively, and the disk is an aluminum disk of 3 .5- inch in diameter and 0.8 mm in

thickness. Two such disks are stacked together and mounted on the spindle motor of

the spin stand. The experimental process is described as follows.

Step 1: Stack two disks together and mount on the spin stand;

Step 2: Spin the motor and write servo information on the top disk surface;

Step 3: Read back the wri tten information and obtain the PES.

To write servo information on the disk surface in between the two disks, the spin-

dle motor is stopped and the written surface of the disk is ﬂipped over, following

which, steps 1, 2 and 3 are repeated.

FIGURE 4.1

Disk and spindle mot or assembly of the spin stand.

The disk vib ration in the axial direction of the sing le disk and the dual-di sk stack

are measured via LDV. The position error signals are also col lected for b oth the

single disk and the d ual-disk stack cases. To evaluate the improvement of p osition

accuracy, 20 traces o f the position error signal are collected and denoted by P ES

68 Modeling and Control of Vibration in Mechanical Systems

(i = 1, ···, N ). N = 20. The repeatable runout (RRO) and the nonrepeatable

runout (NR RO) are given by

RRO =

i=1

P ES

, NRRO

= P ES

− RRO. (4.7)

A. Positioning accuracy improvement at 7200 RPM

Tested d isk vibration results of the dual-disk and the singl e d isk are sh own in Fig-

ure 4.2. Besides the harmonics with respect to the spin dle speed, the disk vibrat ion

modes can be seen clearly as denoted b y T1-T9. It is found that the frequencies of the

disk vibration modes T1-T9 are shifted and their amplitudes are apparently reduced.

The vibration modes are all shi fted by 40 Hz to higher f requencies. The power spec-

tra of the RRO and NRRO, computed from the collected PES data, are shown in

Figur e 4.3 and Figure 4.4. As seen in Figure 4.3, except for the 1st harmonic, other

dominant harmonics before the 11th harmonic are all reduced. In Figure 4.4, we

can observe that almost all the reduced disk vibration modes T1-T9 in Figur e 4.2

are reﬂected i n NRRO. The other peaks indi cated by S1-S4 can be identiﬁed to be

caused by slider vibrations. They are both lowered signiﬁcantly, which may be due

to better slider-disk interaction. It is not ed that the peak S2 for t he single d isk case is

shifted to S3 in the dual-disk case. This randomly happens and the writing process

at a d ifferent time may cause su ch a shifting. This phenomenon can be seen in other

testin g results which will be shown later. The standard deviations (or σ values) of

RRO, NRRO and PES are obtained. It turns out that the σ value of RRO is reduced

by 23%, NRRO by 18%, and the PES σ value is reduced by 21%. Figure 4.5 shows

the time sequences of PES i n both the cases and t he amplitude reduction is seen

apparently.

B. Positioning accuracy improvement at 8400 RPM

The disk or the dual-disk stack is rotating at 8400 RPM, and the d isk vibrations

in the axial direction are shown in Figure 4.6. It can be seen that the amplitudes of

the vibration modes are all reduced with shifts in frequencies, as indicated by T1-T6.

The power spectra of the RRO and NRRO are calculated from the tested PES and

shown in Figu re 4.7 and Figure 4.8. Fig ure 4.7 shows that almost all the ﬁrst seven

harmonics are decreased and as a result , the σ value of RRO is reduced by 41%.

In Figure 4.8, the correspond ing T1-T6 in Figure 4.6 can be f ound and they are all

suppressed in the case of the dual-disk stack. Notice that S1 is reduced signiﬁcantly,

but can not be traced in the tested disk vi brations in Figure 4.6. This further veriﬁes

that S1 is caused by the d isk-slider interaction. The corresponding S4 cannot be

found in this case, while T1 appears very close to S4. The resultant σ value of

NRRO is impr oved by 2 8%, and PES by 38%. Figur e 4.9 shows the comparison of

single-disk and dual-disk PES in time domain, and t he amplitude of PES in the case

of the dual-disk stack approach is much lower as compared to conventional single

disk approach.

Classical Vibration Control 69

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 4.2

Comparison of single-disk and dual-disk axial vibrations measured via LDV at 7200

RPM.

70 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 4.3

Comparison of single-disk and dual-disk RRO power spectrum at 7200 RPM (23%

improvement of σ value).

Classical Vibration Control 71

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)

2−disk

1−disk

FIGURE 4.4

Comparison of single-disk and dual-disk NRRO power sp ectrum at 7200 RPM (18%

reduction of σ value).

72 Modeling and Control of Vibration in Mechanical Systems

FIGURE 4.5

Comparison of sing le-disk and dual-disk PES in time domain at 7200 RPM (21%

reduction of σ value).

Classical Vibration Control 73

0 500 1000 1500 2000 2500 3000

−9

−8.5

−8

−7.5

−7

−6.5

−6

−5.5

−5

Frequency(Hz)

Dsplacement amplitude(log10(m))

2−disk

1−disk

FIGURE 4.6

Comparison of single-disk and dual-disk axial vibrations measured via LDV at 8400

RPM.