chapter 4 piezoelectric material based vibration...

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CHAPTER 4

PIEZOELECTRIC MATERIAL BASED VIBRATION

CONTROL 4.1 INTRODUCTION

This chapter focuses on the development of smart structure using

piezoelectric patches in order to control the vibration actively. More advanced

technology and materials in industry lead to the implementation of lightweight

components for miniaturization and efficiency. Lightweight components and

certain materials, however, are susceptible to vibrations. (Autur KK 1997) The

flexible structures that make up these systems pose a great problem to vibration

control. Flexible structures are extensively used in many space applications, for

example, space-based radar antennae, space robotic systems, and space station,

etc. The flexibility of these space structures results in problems of structural

vibration and shape deformation, etc. Active control methods have to be

developed to suppress structural vibration and improve the performance of

these flexible structures.

In this current study, the main focus is to analyze the effect of Lead

Zirconium and Titanate (PZT) in vibration control over GFRP composite and

aluminium structures. Also, the position of PZT along with the length of the

structure is studied with various input voltage and control gains. The settling time

of the structures with the above parameters has also been studied.

4.2 PIEZOELECTRIC MATERIAL

Piezoelectric materials are active materials generally with high

bandwidths. The two properties that piezoelectric materials have are the direct effect

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and the converse effect. The direct effect of a piezoelectric material is an electric

polarization that occurs as the material is stressed by producing an electrical charge

at the surface of the material. The converse piezoelectric effect results in a strain in

the material when placed within an electric field. These properties make

piezoelectric materials the most popular smart materials. Lead Zirconium Titanate

(PZT) and Polyvinylidene Fluoride (PVDF) are two piezoelectric materials that are

most widely used in actuation and sensing. Differences in the composition of these

materials allow them to be used as actuators and sensors, respectively. PZT is

roughly 4 times as denser, 40 times stiffer and has a relative permittivity of 100

times greater as that of PVDF. The rigidity of PZT makes this material a perfect

candidate for actuators and, on the other hand, the flexibility and extreme sensitivity

of PVDF makes it a perfect candidate for sensing. In this current study, PZT has

been used for vibration control of cantilever structures. The specifications of PZT

are presented in Table 4.1.

Table 4.1 Specification of PZT patch

Length (mm) 76.2 Width (mm) 25.4 Thickness (mm) 2

modulus (GPa) 63 Density (kg/m3) 7500

0.28 Damping constants

Max. Input voltage (V) 270

4.3 MODELING OF THE STRUCTURE

In this study, the GFRP composite of (0°/0°/0°)s ply orientation and

aluminium cantilever beams have been taken for analysis of vibration control.

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The size of the beam is 500 mm x 50 mm x 2mm. The properties of the beams are

shown in Table 4.2.

Table 4.2 Properties of aluminium and GFRP

Property Aluminium GFRP Density (kg/m3) 2700 1800

Pa) 70 38.6 0.32 0.28

Figure 4.1 shows the cantilever beam modelled using ANSYS. The PZT

patches have been placed at various positions along the length of the beam such as

50 mm, 250 mm and 450 mm from the fixed end. The influences of these positions

over the settling time of the beams have been studied.

(a) 50 mm from the fixed end

(b) 250 mm from the fixed end

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(c) 450 mm from the fixed end

Figure 4.1 Position of PZT from the fixed end of the beam

The typical finite element used in the modeling and analysis of

piezoelectric crystal was (SOLID5), which has piezoelectric capacity in three

dimensional couple field problem. Like other structural solid elements, this

element has three displacement degrees of freedom per node. In addition to this

degree of freedom, the element has also potential degree for analysing of the

electromechanical coupling problems. Piezoelectric actuator inherently exhibits

anisotropic and yield three-dimensional spatial vibration in their response to the

piezoelectric actuation.

The models developed for the passive portion includes consistent

degree of freedom at the location where these elements interface.

For modeling the passive portion of the smart structure solid element used is

(SOLID45). The passive portion is made of aluminum and GFRP.

4.4 MODAL ANALYSIS AND DEVELOPMENT OF CONTROL

LAWS

Modal analysis was performed on both the aluminium and GFRP beam

to find out the natural frequency of the structure. The analysis was furthur carried

out for both passive and active structures. Table 4.3 presents the first four natural

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frequencies of aluminium and composite beams or structures. From this table, it

can be inferred that the addition of PZT patch increases both the mass and stiffness

of the system. But the increase was not proportional, causing the natural frequency

to increase. If they had proportionally increased, the natural frequency would have

remained constant. The natural frequency of the beams can be validated

analytically by using the Equation 4.1(Rao SS 2002).

Table 4.3 Natural frequencies of aluminium and GFRP beams

Modes

Natural frequency of aluminium (Hz)

Natural frequency of GFRP (Hz)

Passive Beam

Active Beam

Passive Beam

Active Beam

First Mode 6.65 6.63 5.982 5.76

Second Mode 41.18 41.12 37.456 36.09

Third Mode 115.5 114.4 112.56 111.3

Fourth Mode 226.28 225.2 225.4 224.64

, f = (4.1)

The harmonic response analysis was used to determine the steady

response of the linear structure under the harmonic loads. Under normal

circumstances, the PZT patches were actuated by a sine-wave power from the

power supply. This kind of PZT-structure coupled analysis accorded with the

conditions of the harmonic response analysis. Figure 4.2 shows the response of

harmonic analysis of the aluminium and composite beams. It can be noticed that

the peak occurs in the frequencies corresponding to the frequencies found by using

modal analysis.

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(a) Aluminium beam

(b) GFRP beam

Figure 4.2 Harmonic response of cantilever beam

From these figures, it can be inferred that only the vibration modes

corresponding to first, second and fourth modes have been obtained. This is due to

the fact that they correspond to the bending loads, since bending load is only

applied. Vibration modes corresponding to the third and fifth natural

frequencies would rise while applying the torsion loads. Only, when bending

loads are applied, their corresponding natural frequencies are validated.

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Closed loop simulation for active vibration control in smart structure

has been performed by using ANSYS. Control actions have been incorporated into

the finite element model by using APDL (ANSYS Parametric Design Language)

codes. Ks, Kc and Kv are the sensor, control and power amplification factors,

respectively. Ks and Kv are taken as 100 and Kc is changed in the analyses by

selecting the values starting from 10 with the step increase of 10. Only the

proportional control has been applied. The multiplication of Ks, Kc and Kv is the

proportional constant for the actuator voltage Va. Therefore, changing the values of

Ks, Kc and Kv and keeping the same multiplication do not seem to affect the results. In this control system, the controller used is that of a proportional

controller with a displacement feedback. The strain rate feedback control has also

been used for vibration control. From the literatures reviewed, displacement

feedback seems to enable better controlling action with higher actuation voltages

when compared to strain rate feedback. Modal analysis have been performed to find the undamped natural

as 1/(20fh), where fh is the highest frequency. In the transient analysis, the

0.0001

which taken in this study. The displacement has been calculated at the tip of the

beam and it is multiplied by Ks and then subtracted by zero. The zero value is the

reference input value. The difference between the input reference and the sensor

signal is called the error signal. The error value is multiplied by Kc and Kv to

determine Va at a time step. The part of the macro which enables the calculations for the closed loop

analysis for

ts=4 ! settling time

*DO,t,2*dt,ts,dt

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DDELE,N370,VOLT

Vmax=270

ref=0

*get,u1,node,346,u,z !tip displacement measurement

err=ref-ks1*u1

Va=kc*kv*err

D,N370,VOLT, Va

D,N359,VOLT,0

time,t

solve

*enddo

The actuation voltage to be applied for the piezoelectric actuator is

found by multiplying error signal by the gain Kc and Kv. The analysis continues

step by step for a specific duration after vibration amplitudes reach a steady-state.

From the results, calculated and

then used to determine the damping ratio for both aluminium and GFRP beams

they are used to compare the vibration characteristics of both aluminium and GFRP

beams.

To find out the influence of position of the PZT patch over the settling

time, the control gain of 10 and 20 has been considered as an input to the PZT. The

damping ratio and settling time for each position are found out. The position where

settling time is minimum has been selected as an optimum position of the PZT

patch.

After selecting the optimum position, the influence of input voltage to

PZT over settling time is determined by applying different control gain values.

The maximum input voltage of 270V for the selected PZT patch is taken into

account.

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4.5 DETERMINATION OF OPTIMUM POSITION OF PZT

The position of PZT over the aluminium and GFRP composite beam is

simulated by using finite element code to find out optimum location of PZT,

resulting in improved vibration control. Figures 4.3 to 4.5 show the settling time of aluminium beam for the

positions of piezoelectric 50 mm, 250 mm and 450 mm respectively. From these, it

can be clearly noticed that the settling of the beam is minimum when piezoelectric

patch is located at the distance of 50 mm from the fixed end of the beam.

Figure 4.3 Settling time of aluminium beam when PZT at 50 mm


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This is due to the position of piezoelectric in the high strain region

resulting in more controlled reduction in the tip displacement of the structure.

Similarly, the positions of piezoelectric at 250 mm and 450 mm have more settling

time. Settling time of aluminium structure for the various locations from the fixed

end of the beam is shown in Table 4.4.

Table 4.4 Settling time of aluminium beam with various positions of PZT

Distance of the PZT patch from the fixed end (mm) Settling time(s)

50 1.5

250 2

450 3.5

Similar to the aluminium cantilever beam, the GFRP beam has also

been simulated for the optimum position of PZT. Figures 4.6 to 4.8 show the

settling time of GFRP for the positions of PZT at 50 mm, 250 mm and 450 mm

respectively.

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Figure 4.6 Settling time of GFRP beam when PZT at 50 mm



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Table 4.5 again proves that the position of PZT at the distance of

50 mm from the fixed end has a lesser settling time when compared to other

positions via 250 mm and 450 mm respectively.

Table 4.5 Settling time of GFRP beam with various positions of PZT

4.6 DETERMINATION OF OPTIMUM CONTROL GAIN

Different gain values are given to the PZT patch in order to maximize

damping effect without exceeding the maximum voltage range that could be

applied to the PZT patch (270V). Since the optimum position of the PZT patch is

found to be at a distance of 50 mm from the fixed end, the need for the optimum

gain of the PZT patch arises. In order to obtain the optimum gain, values of 10, 20,

30 and 40 have been selected as an input to PZT patch and their corresponding

voltage graph is plotted for the aluminium structure from the Figures 4.9 to 4.12.

From these figures, it is evident that an increase in control gains increases the

voltage to PZT. When the control gain exceeds 20, the applied voltage to PZT

exceeds the maximum value of 270 V of the selected PZT.

Figure 4.9 Output voltage for the control gain of 10 Aluminium beam

Distance of the PZT patch from the fixed end (mm) Settling time(s)

50 2 250 2.4 450 3.7

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Figure 4.10 Output voltage for the control gain of 20-Aluminium beam



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From the results, it can be assured that the increase in control gain leads

to an increase in applied voltage as shown in Figure 4.13. The increase in applied

voltage results in the structure settles earlier due to the increased control action

offered by the PZT.

Figure 4.13 Actuator voltages for aluminium beam

When the PZT is located in the high strain region of the structure i.e

near the root of the structure, it provides better control action when compared to the

location at the free end. The same can be extended to GFRP structure as well.

Figures 4.14 to 4.16 present the settling time of the beam with applied voltage to

PZT.

Figure 4.14 Output voltage for the control gain of 10 - GFRP beam

050

100150200250300350400

0 10 20 30 40 50

Max

imum

vol

tage

of

actu

ator

(V)

Control gain

50



Similar to the aluminium beam, Figure 4.17 shows the actuator voltage

for the GFRP.

Figure 4.17 Actuator voltages for GFRP beam

0

100

200

300

400

500

0 10 20 30 40 50Max

imum

vol

tage

of

actu

ator

(V)

Control gain

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4.7 LOGARITHMIC DECREMENT

The logarithmic decrement is to be found in order to find the damping

ratio. Logarithmic decrement is found out for both the aluminium and GFRP beam

at their optimum position by using Equation 4.2 (Rao SS, 2002). Damping ratio

should not exceed more than 1 because it results in an over damped system.

= ln (x1/x2) (4.2)

where x1 = first maximum displacement, x2 = second maximum displacement

= logarithmic decrement.

4.8 DAMPING RATIO

Damping ratio is a dimensionless measure which describes how

oscillations in a system decay after a disturbance and is calculated by using

Equation 4.3 (Rao SS, 2002). It characterizes the frequency response of a second

order ordinary differential equation which is important in the study of system

damping and system control.

(4.3) Logarithmic decrement value is calculated both for aluminium and

GFRP beams with different control gains and position of PZT at 50 mm from the

fixed end. Table 4.6 shows the value of logarithmic decrements.

Table 4.6 Logarithmic decrement of aluminium and GFRP at 50 mm from the

fixed end

Control gain

Material 10 20

Aluminium 0.41 0.52 GFRP 0.35 0.4

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Damping ratio = = 0.16

The Damping ratio for Aluminium and GFRP beams with PZT patch at

a distance of 50 mm from the fixed end with various control gains are shown in

Table 4.7. From this, it can be inferred that the damping ratio of the aluminium

beam is higher than that of GFRP composite beam.

Table 4.7 Damping ratio of aluminium and GFRP at 50 mm from fixed end

Control gain Material 10 20

Aluminium 0.17 0.23 GFRP 0.14 0.16

Figure 4.18 depicts the damping ratio for aluminium and GFRP

composite structure. From this, it is clearly noticed that for the position of

PZT 50 mm from the fixed end and the control gain of 20, aluminium has a better

damping ratio leading to earlier settling of the structure. This is due to the fact that

GFRP structure has less stiffness when compared to aluminium.

Figure 4.18 Damping ratio for the PZT patch at a distance of 50 mm

from the fixed end

0

0.05

0.1

0.15

0.2

0.25

0 5 10 15 20 25

Dam

ping

rat

io

Control gain

Aluminium

GFRP

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4.9 CONCLUDING REMARKS Finite element modelling for the closed loop control system has been

developed by using ANSYS. Modal analysis and harmonic analysis have been

carried out to find the undamped natural frequencies of the system and it is

compared with the analytical results. With the modal frequencies as input, time step

is calculated for closed loop transient analysis. Transient analysis and closed loop

control laws are incorporated into the finite element models using APDL (ANSYS

Parametric Design Language) for different control gains and for different positions

of PZT patch on the flexible beam.

From the results, it can be noticed that attaching the piezoelectric patch

at a distance of 50 mm from the fixed end of the beam has a minimum settling time

and it is found to be the optimum position for placing the PZT patch on the beam

compared to the other positions. Similarly, an increase in input voltage tends to

have minimum settling time. For the control gain of 20, the maximum actuation

voltage falls within 270V, which is the maximum exciting voltage for PZT.

Also, from the analysis carried out for two different materials such as

GFRP and aluminium, it can be inferred that aluminium settles more quickly than

GFRP due to its high damping ratio.

chapter 4 piezoelectric material based vibration...

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