Experimental Measurement of Displacement

International Journal of Mechanical Engineering Education 38/3

Experimental measurement of displacement and vibration of piezoelectric smart structuresChandrashekhar Bendigeri (corresponding author),a Dattatraya Hegde,b K. Badari Narayanac and K. Ramachandrad

a Assistant Professor, Department of Mechanical Engineering, University Visvesvaraya College of Engineering, Bangalore University, Bangalore 560 001, IndiaE-mail: [email protected] Department of Mechanical Engineering, UVCE, Bangalore University, Bangalore-560 001, Indiac Goodrich Aerospace Service Pvt Ltd, White Field Road, Bangalore 560 048, Indiad Director (Ex), Gas Turbine Research Establishment, Bangalore 560 093, India

Abstract Recent advances in the fi eld of distributed sensing and actuation have led to the development of new ‘smart structures’. A smart (or intelligent) structure is able to sense and respond, and control its own characteristics and state. Piezoelectric smart materials have gained popularity as both sensors and actuators. In the present work a new experimental set-up is described that was designed to carry out static and dynamic experimentation on piezoelectric smart structures. Four types of experiment were carried out; results were also determined using analytical methods. The comparison of the analytical and experimental results showed good agreement.

Keywords piezoelectric; smart material; sensing; smart beam

Introduction

This paper presents a few experiments carried out to evaluate the static and dynamic responses of ‘smart structures’ embedded with piezoelectric materials. These exper-iments were carried out to assess the necessary instrumentation and testing proce-dures. The experiments were of four types:

(1) a smart beam made of aircraft-quality aluminum alloy with a built-in piezo-electric transducer (PZT), subjected to a known voltage to estimate the defl ec-tion in the beam;

(2) a smart beam made of steel with a built-in PZT, subjected to voltage;(3) a smart composite beam made of carbon/epoxy with a built-in PZT, subjected

to voltage to estimate the defl ection in the beam;(4) a smart beam made of aluminum with a built-in PZT, subjected to free vibration.

In this paper details of the instrumentation used, specimen preparation, conduct of the experiments and analysis of the results are presented.

Instrumentation

Table 1 lists the instrumentation used. Fig. 1 shows the different instrumentation and a smart beam used in the program.

186 C. Bendigeri et al.


TABLE 1 Instrumentation

Instrument Characteristics/use

1. Function generator Waveform: sine waveformFrequency range: 0.5 Hz to 3 MHz with digital displayOutput voltage: 30 V peak to peakUsed for generation of signal

2. Piezo sensing system Sensor: piezoelectric crystalOutput: sine waveFrequency range: 10 Hz to 1 kHzMaximum output voltage: 10 Volts.Used for sensing the voltage

3. Active band pass fi lter Maximum output voltage: 10 V peak to peak with unity gainFrequency range: 0.1–1000 HzOrder of fi lter: fourth orderUsed for fi ltering the waves

4. Vibration meter along with accelerometer

Frequency range 1–6000 Hz

Operating temperature of 125° CUsed for measuring of displacement/velocity/acceleration

5. Piezo actuation system Frequency range 10–400 HzInput: sine waveAC input voltage: +/− 10 V maximumDC voltage: 0–100 V Output voltage: 0–200 VUsed for generation of low-current and high-voltage source for driving the piezoelectric actuator

6. Dimmerstat with power transformer

Variable DC supply: 0–200 V and 1 AUsed to obtain variable DC supply

7. Linear variable differential transformer (LVDT)

±1 mm

Maximum output: 5 VNumber of digits: 5Excitation voltage: 1 V Frequency 4 kHzUsed for measuring the displacement

Fig. 1 (a) The instruments. (b) A smart beam and instrumentation.

(a) (b)

Piezoelectric smart structures 187


Certain qualities of piezoelectric materials needed to be taken account of in the experiments [1–5]:

• Aging tends to degrade piezoelectric performance with time, which is normally a decay of the poling process.

• High stress fi elds can depole piezoelectric material.• Breaching the Curie point depoles piezoelectric material.• Self-heating can occur at high stress levels.• Low fracture toughness and tensile strength results in brittle cracking.• Surface cracking can lead to electric arcing.• To take into account the hysteresis of piezoceramics (a property of producing

residual strains after the removal of the applied voltage), voltages were repeated in the reverse direction of the electric potential.

Also, the limitations of the equipment, such as the gain of the voltmeter being less at low frequencies, had to be considered. The mass of the accelerometer during measurement could produce some small variation in the results. The surrounding vibration sometimes could result in a small variation in the measured values, as could the cables. These factors should be borne in mind [6–8].

Aluminum smart beam

A thick beam made of aircraft-quality aluminum alloy was used. A piezoelectric patch was mounted on this beam and subjected to an electric fi eld. The block diagram of the experiment is shown in Fig. 2. The geometry of the smart beam is shown in Fig. 3. This case study involved the combination of isotropic and piezoelectric materials.

Specimen preparationThe aluminum beam was machined as per the dimensions shown in Fig. 3. Piezo-electric material made of PZT (manufactured by Sparkler Ceramics, Pune) was fi xed on the beam [9–12]. The procedure for fi xing the piezo was as follows:

Fig. 2 Block diagram of the experimental set-up for the aluminum smart beam.



Fig. 3 Geometry of the aluminum smart beam.

• An emery sheet of zero number was used to rub the beam where the piezoelectric material was to be fi xed.

• A pure cotton cloth and carbon tetrachloride were used to clean the surface.• The exact location for the PZT was marked.• Araldite (hardener and adhesive) was mixed thoroughly until white and uniform.• The araldite was smeared on the PZT and on the place on the beam marked

for the PZT.• The PZT was put on the beam and pressed down. Then for 24 hours it was kept

at room temperature for curing.• The PZT was supplied with leads known as pig tails from the manufacturer,

consisting of positive and negative terminals.• Another terminal, a small copper strip, was used to connect the pig tails to the

instrumentation.

Experimental procedureThe procedure for conducting the experiment was as follows:

• The aluminum beam with the piezoelectric material was clamped tightly at one side as a cantilever.

• The variable DC supply (dimmerstat with power transformer, variable DC supply 0–200 V and 1 A) was applied to the PZT on the beam.

• The voltage applied to the PZT on the beam caused it to defl ect.• The defl ection obtained at the tip of the beam was measured using a linear vari-

able differential transformer (LVDT) with specifi cation of ±1 mm, maximum output 5 V, number of digits 5, excitation voltage 1 V and frequency 4 kHz.



Analytical methodThe defl ection was also determined using an analytical method, from the following equation:

δ = ∗⎛⎝⎜

⎞⎠⎟ ∗m t

I Elb

b b2

where δ is the defl ectionm is the moment generated by the applied voltage, given by the equation

mV

t t

EI EI

EI EI= ∗ ∗ ∗ ∗

+6

3 431d

c b

b c

b c

Ib is the moment inertia of the beamIc is the moment inertia of the piezoelectric materiald31 is the piezoelectric constant (274*10−9 mm/V)V is the voltage applied (50–200 V)E is the modulus of the beam (70 MPa)l is the length of the beam (440 mm)tc is the thickness of the piezoelectric material (1 mm)tb is the thickness of the beam (5 mm)E is the modulus of the piezoelectric material (70 MPa)wb is the width of beam (50 mm)wc is the width of the piezoelectric material (25 mm)

and mmIw t

bb b= =

34

12520 833.

Iw t

cc c= =

34

122 0833. mm

Substituting the value of the moment of inertia of the beam and the piezoelectric materia and also substituting other values related to confi guration of the problem, one can get the fi nal displacement. For instance, for 50 volts,

δ = ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗

−6 274 10 50 70 10 520 83 70 10 2 083 5 440

1 5 3 70 1

9 3 3. .00 520 83 4 70 10 2 083 2 520 833 70 10

24

3 3 3∗ + ∗ ∗ ∗( )∗ ∗ ∗ ∗=

. . .

μm

Values obtained analytically are compared with those obtained from the experi-mental set-up in Table 2.

Steel smart beam

An active structural member made of steel with a PZT crystal (lead zirconium titanite) was used in this experiment. Fig. 4 shows the geometry of the cantilever beam with the piezoelectric actuator.

Experimental procedureAn LVDT was used to measure static defl ection of the tip of the beam at its free end. The signals from the LVDT were transmitted to a displacement indicator. The



0

0.02

0.04

0.06

0.08

0.1

0.12

0 100 200 300

Voltage (volts)

Def

lect

ion

(m

m)

Fig. 5 Static defl ection of the piezoelectric steel smart beam obtained from experiment.

Fig. 4 Dimensions of the steel beam with a piezoelectric material.

TABLE 2 Comparison of the experimental and analytical results for the aluminum smart beam

Voltage applied (V) Defl ection (μm)

Analytical Experiment

50 24 25100 48 51150 72 71200 96 95

block diagram for this experimental set-up would be similar to the one shown in Fig. 2.

The LVDT was placed at the tip of the beam and the initial defl ection was noted. Then voltage applied to the piezoelectric actuator varied from 50 to 200 V. For each voltage increment, the corresponding defl ection was measured. Fig. 5 shows the results.



Smart composite beam (carbon/epoxy and PZT)

In this experiment the base beam was made of carbon/epoxy composite. A piezo-electric patch was mounted on this beam and subjected to an electric fi eld. The geometry details and composite beam with PZT is shown in Fig. 6. This case study involves the combination of an orthotropic material such as the composite used and piezoelectric materials.

Specimen preparationThe carbon/epoxy composite beam was fabricated for this experiment. The resin LY 556 was heated at 60° C and hardener HT 972 was mixed with it. Acetone was added to the mixture of resin and hardener to achieve the required consistency. Carbon fi bers in fabric form were added (to 60% volume fraction) for reinforcement.

(a)

(b)

Fig. 6 Smart composite beam. (a) Details of geometry. (b) Beam with PZT actuator.



TABLE 3 Comparison of the experimental and analytical results for the composite smart beam

Voltage applied (V) Defl ection (μm)

Analytical Experiment

50 16 15.7100 31 30.6150 46 45.3200 61 60.4

Impregnation of the fabric was carried out on a heated plate at 60° C. Trapped air was removed using rollers. The curing was done for 3 hours at 120° C and a pressure of 7 bars; post-curing lasted 1 hour at 180° C. The beam was cut to shape using diamond-cutting equipment. The composite was a cross-ply lamination, with fi ber orientation angles in the alternate layers of 0°/90°/0°/90°.

The piezoelectric actuator was bonded on to the composite beam after cleaning the surface and marking the position. The leads of the pig tail provided with the piezoelectric materials were connected to the instruments via another terminal and the specimen was clamped to conduct the experiment.

Experimental procedureA block diagram of the experiment would be similar to one shown in Fig. 2. The procedure for conducting the experiment was as follows:

• The composite beam with the piezoelectric material was clamped tightly at one side as a cantilever.

• The variable DC supply (dimmerstat with power transformer, variable DC supply 0–200 V and 1 A) was applied to the PZT on the beam.

• The voltage applied to the PZT caused the beam to defl ect.• The defl ection at the tip of the beam was measured using an LVDT.

The measured results are compared with the analytically derived values in Table 3.

Aluminum smart beam subjected to free vibration

Free vibration analysis of smart beam made of aluminum alloy with a PZT patch was carried out. Fig. 7 show the geometry of the cantilever beam with the piezo-electric actuator. The block diagram of the experimental set-up is shown in Fig. 8.

Experimental procedure

• The function generator generated a sine wave at 10 V and this is passed on to the piezoelectric actuator drive amplifi er, where it was amplifi ed from 10 V to 145 V and connected to the actuator.



Fig. 7 Geometry of the aluminum beam with the piezoelectric material for vibration analysis.

Fig. 8 The block diagram for the free vibration test of the piezoelectric smart structure.

• The frequency from the signal generator was increased gradually and passed on to the actuator.

• An oscilloscope was attached to sensor and the high-amplitude reading on the coaxial resonator oscillator (CRO) corresponded to the natural frequency.

First and second natural frequencies were determined experimentally. The higher frequencies could not be measured due to the limitations of the instrumentation used.

A normal mode analysis was carried out to obtain the fundamental natural fre-quency for the smart beam with the PZT. The results obtained were compared with results obtained by analytical calculations (Table 4).



Analytical methodThe analytical equations pertaining to the fi rst fundamental frequency of the beam are presented here.

The circular frequency:

ω = ∗C a

Lt

2

where Ct has the value 3.52 for mode I, 22.4 for mode II, 61.7 for mode III and 121 for mode IV.

a E I A= ∗ ∗ρ

Moment of inertia:

Ibd=

3

12

Natural frequency:

FZ =∗ω

π2

The confi guration for the present experiment is as follows (see Fig. 7).

Length of beam: L = 200 mmWidth of the beam: b = 25 mmDepth of the beam: d = 3 mmArea of cross-section: A = 75 mm2

Substituting the values of present confi guration in the above equations we get the following values: moment of inertia I = 56.2 mm4; a = 4409585. Using this value of a we obtain the circular frequency ω = 388 rad/s and we then obtain fi rst natural frequency as 61 Hz. Similarly, the circular frequency for the second mode is obtained with the value of Ct as 22.4 and hence the natural frequency is obtained as 393 Hz. Table 4 compares the natural frequencies derived from this analytical method with those determined experimentally.

Although only the fi rst two natural frequencies could be determined from the experiment, the analytical and fi nite element approach could compute the natural

TABLE 4 Comparison of natural frequencies (Hz) of the smart beam obtained using analytical and

experimental methods

Analytical method Experiment

61 58393 380

1082 –2122 –



frequency of higher modes. As can be seen from Table 4, there was good agreement between the methods of estimation.

Conclusion

A set of experiments was carried out on the different smart beams with piezoelectric materials and related instruments attached. The static and dynamic behavior of the structural member was examined. The results of the experiments using piezoelectric materials in combination with isotropic and orthotropic materials were compared with the results obtained by analytical methods, and there was good agreement. The experimental results could similarly be used for the validation of fi nite element analysis.

References

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[2] Y. Matsuzaki and M. Toyama, ‘Vibration of a cantilevered beam decreasing development and retrieval analysis and experiment’, SPIE J, 2190 (1994), 705–710.

[3] J. Jin and R. C. Batra, ‘Effect of electromechanical coupling on static deformations and natural frequencies’, Transactions on Ultrasonic, Ferroelectrics, and Frequency Control, IEEE, 52(7) (2005).

[4] T. C. Manjunath and B. Bandyopadhyay, ‘Vibration suppression of timoshenko beams with embedded piezoelectrics using POF’, Int. J. Intelligent Technology, 1(1) (2006), 112–119.

[5] P. Bhattacharya, M. Rose and S. Homann, ‘Effects of piezoactuated damping on parametrically excited laminated composite plates’, J Reinforced Plastic Composites, 25(8) (2006), 801–813.

[6] J. Mathew, B. Sankar and L. Cattafesta, ‘Finite element modeling of piezoelectric actuators for active fl ow control applications’, AIAA J (2001), 891–898.

[7] G. L. C. M. de Abreu, J. F. Ribeiro and V. Steffen, ‘Finite element modeling of a plate with local-ized piezoelectric sensors and actuators’, J Brazilian Society Mechanical Science Engineering, 26(2) (2000), 214–232.

[8] F. J. Von Preissig and E. S. Kim, ‘Topics in fi nite element modeling of piezoelectric MEMS’, AIAA J (2002), 512–531.

[9] M. Hofer and R. Lerch, ‘Finite element calculation of wave propagation and excitation in periodic piezoelectric systems’, Fifth World Congress on Computational Mechanics (Vienna, Austria, 2002).

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Experimental Measurement of Displacement

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