2 CD-ROM Drive Base 2.1 Admittance Analysis

The mechanism of the piezoelectric shunt system can be divided into two part; energy transfer from mechanical system to electrical system, and dissipation of the transferred electrical energy in shunt circuit. Therefore, high performance of the piezoelectric shunt system can be achieved by cost effective energy transfer from mechanical system to electrical system.

Admittance of the piezoelectric structure is known as a representative parameter of electro- mechanical characteristic in the piezoelectric shunt system [12]. Figure 1 (a) shows the schematic diagram of the proposed CD-ROM drive base with piezoelectric shunt circuit.

When exciting frequency of the piezoelectric structure is much lower than natural frequency of piezoelectric material, equivalent electric model of the piezoelectric structure can be obtained as shown in Fig. 1 (b). In the equivalent electric model,

C

_p is the capacitance of piezoelectric material and L₁,C₁ and R₁ represent equivalent mass, spring and damping of CD-ROM drive base, respectively. The variable Z in Fig. 1 (b) represents electrical impedance, and subscripts

s, p

and

cir

represent structure, piezoelectric material and shunt circuit, respectively. If shunt circuit is assumed as serial resonant circuit like Fig. 1(b), the impedances of the equivalent electric model are expressed as follows.

Shunt Cir c uit Pie zoe le c tric Ma te r ial

i2

c2

R_cir

m2

L_cir

Schematic diagram

Shunt Circuit Piezo Material

_

Structure

1 1 1/k C

+

V is

Zs

1

1 c

R L1 m1

Zp

cir

s i

i / 2

1 k C_p

Zcir

icir

c2

R_cir

m2

L_cir

Equivalent electrical model

Figure 1. The proposed model consisting of the CD-ROM drive base with piezoelectric shunt circuit.

cir cir cir

cir cir

p p

s

R L j R s L s Z

C j s s k Z

C R L j

j s c

s k m s Z

Z Z

Z Z

) (

) 1 (

) 1 (

2

1 1 1

1 1 1

(1)

wheres represents Laplace variable. After some manipulations [4], the energy dissipated in the resistance of resonant shunt circuit,P_D, can be expressed as follows.

2 2 2

2 2

* 2

*

) 2Re(

1

) 2Re(

1

) 2Re(

) 1 ) 2(Re(

1 2

1

sp O cir p

p cir

O cir p

p cir

cir cir cir

cir cir cir

R cir D

Y Z V

Z Z Z

Z I Z Z Z

I Z I

I Z I

V P

˜ ˜

˜ ˜

˜

˜

˜

˜

˜

(2)

where,

V

_cir^R is the voltage applied at both ends of resonant shunt circuit,

I

_cir^* is complex conjugate of the current in the shunt circuit,

I

_O is the total current of the piezoelectric structure generated by external force,

V

_O is the applied voltage to measure the admittance, and

Y

_sp is the admittance of the piezoelectric structure in open circuit. It is observed from Eq.(2) that the dissipated energy is proportional to the electro-mechanical characteristic values (

I

_O,

V

_O,

Y

_sp) of the piezoelectric structure in open circuit. In most cases, the admittance of the piezoelectric structure in open circuit can be measured using impedance analyzer by applying constant voltage with corresponding frequency on the piezoelectric material mounted on the structure. Then, the dissipated energy is only a function of admittance of the piezoelectric structure in open circuit. This implies that the reduction of vibration in the piezoelectric shunt system is dependent on admittance of the piezoelectric structure and hence admittance can be a performance index in designing piezoelectric structure.

The dynamic response and admittance of the complicated CD-ROM drive base can be also obtained using commercial finite element codes such as ANSYS [13]. The equations of motion of the piezoelectric structure can be expressed as follows.

¿¾

½

¯®

­

¿¾

½

¯®

­

°¿

°¾

½

°¯

°® ­

¿¾

½

¯®

­

¿¾

½

¯® ­

¿¾

½

¯®

­

¿¾

½

¯®

­

] [

] [ ] [

] [ ] [ ] [

] [ ] [ ] [

] [ ] 0 [ ] 0 [

] 0 [ ] [ ] [

] [ ] 0 [ ] 0 [

] 0 [ ] [

Q F u

K K

K u K

u D M

t u

u

I I

I

_{I I}

I

(3)

where,

] [ , ]

[ F u

: vector of nodal structural forces and mechanical displacements

] [ , ] [ , ]

[ M D K

: structural mass, damping and stiffness matrix

] [ , ]

[ Q I

: vector of nodal electrical charge and potential

] [ , ]

[ K

_uI

K

_I : piezoelectric coupling and dielectric conductivity matrix

From the governing equation, modal frequencies and mode shapes can be obtained and the admittance of the piezoelectric structure can be determined as follow.

V

Y I ,

¦

i

Q

i

j

I Z

(4)

In the above, Vis the input voltage and

Q

_i is the point charge of the i-th node on the electrode.

(b) Mode 2 : 254.61 Hz

(c) Mode 3 : 285.5 Hz (a) Mode 1 : 218.9 Hz

Figure 2. Selected finite element modal analysis results of the drive base with piezoelectric patches The proposed CD-ROM drive base is a complex structure consisting of stiffened rib, boss and hole as shown in Fig. 1 (a). The length, width and height of the drive base proposed in this work are 180 mm, 140 mm and 40 mm, respectively. The drive base is made of ABS/PBT alloy. For the modal analysis four-node shell element is used and the total number of elements is 6797. On the other hand, the piezo patches are incorporated to the rear part of the drive base as shown in Fig. 1(a). The length, width and thickness of the piezo patch are 50 mm, 25 mm and 1 mm, respectively. Three representative mode shapes and corresponding natural frequencies are presented in Fig. 2. It is observed that the 1^st and 3^rd modes are major mode shapes of the rear part of the drive base. So, these modes are chosen as target mode to be reduced by the piezoelectric shunt damping. Next, admittance analysis is conducted to investigate electro-mechanical coupling effect of the piezoelectric system and to predict piezoelectric shunt performance of the drive base. To measure admittance in the exciting

frequency range, constant voltage (

V

_O

1 . 1 V

) is applied to the piezo patches and frequency is swept from 200 Hz to 600 Hz. The step size of sweeping frequency is 1 Hz. In numerical admittance analysis, charge for each node of electrode is obtained from harmonic analysis of the equations of motion (Eq. (3)) under the same excitation voltage and frequency range.

Then, admittance of the CD-ROM drive base with the piezo patches is calculated based on the charge of each electrode (Eq. (4)). Admittance consists of real and imaginary parts, which are called as conductance and susceptance. Figure 3 presents numerical result of admittance obtained from the finite element method. The frequencies and admittances at the peaks of conductance are determined as follows : 300 Hz – 1.47E-04, 368 Hz – 1.78E-04 and 572 Hz – 2.69E-04. Figure 4 presents measured admittance of the drive base. From the result, the frequencies and admittances are determined as follows : 316 Hz – 1.66E-04, 383 Hz – 2.00E- 04 and 533 Hz – 2,73E-04. The relative differences between experimental and numerical frequencies are 5 %, 3.9 % and 7.3 %, respectively. The differences of peak admittance values are 11%, 11% and 1.5 %, respectively. It is noted here that the error has been caused by bonding layer effect and dielectric loss effect which are not considered in the simulation model. From the admittance results, it is expected that the piezoelectric shunt will suppress vibration of the drive base at three peaks of admittance. In this work, the first and third modes are chosen as target modes to be controlled.

300 400 500 600

1.6

2.0 4.0 6.0 8.0 1.0 1.2 1.4 1.8

105

u

200

200 300 400 500 600

104

u

2.5 3.0

2.0

1.5

1.0

Figure 3. Admittance obtained from the finite element analysis

0.2 0.3 0.4 0.5 0.6 0.0

2.0 4.0 6.0 8.0 1.0 1.2 1.4 1.6

105

u

3.0

2.5

2.0

1.5

1.0

0.2 0.3 0.4 0.5 0.6

104

u

Figure 4. Admittance obtained from the experimental measurement

2.2 Shunt Performance

In order to demonstrate the effectiveness of the piezoelectric shunt circuit based on the admittance analysis, an experimental apparatus has been established. Resonant shunt circuit is connected to the piezo patches and tuned to suppress vibration of each target mode. A synthetic inductor consisting of OP amps and resistor is used in the resonant shunt circuit [17]. The piezoelectric actuator is attached to excite the CD-ROM drive base. Figures 5 and 6 present the measured performance of the piezoelectric shunt damping for two target frequencies in the frequency and time domains. It is clearly observed that the piezoelectric damping decreases the magnitude of frequency responses to 6dB at each mode. In the first mode, the magnitude of vibration in time domain is reduced from 26.2

P

m to 14.2

P

m after shunt circuit on. For the third mode, the magnitude is reduced from 17.1

P

m to 8.1

P

m.

One can find that 50 percent of amplitude reduction has been achieved. It is expected that vibration reduction by the piezoelectric shunt damping will give a significant improvement of performance of the CD-ROM drive.

200 300 400 500 600 -30

-20 -10 0 10 20 30 40

313.75Hz

Shunt Circuit OFF Shunt Circuit ON

0.2 0.4 0.6 0.8 1.0 1.2 1.4

-3.0 -2.0 -1.0 0.0 1.0 2.0 3.0

105

u

ON

Figure 5. Frequency and time responses of the piezoelectric shunt damping at mode 1

200 300 400 500 600

-30 -20 -10 10 20 30 40

381.2Hz

Shunt Circuit OFF Shunt Circuit ON

0.2 0.4 0.6 0.8 1.0 1.2 1.4

-2.0 -1.5 -1.0 -5.0 0.0 5.0 1.0 1.5 2.0 0

105

u

ON

Figure 6. Frequency and time responses of the piezoelectric shunt damping at mode 3

3 HDD Disk-Spindle System