Materials and Design 32 (2011) 2423–2428

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Materials and Design

journal homepage: www.elsevier .com/locate /matdes

Technical Report

Comparison of four test methods to measure damping properties of materialsby using piezoelectric transducers

Roberto Pereira a, Jorge P. Arenas a,⇑, Ernesto Zumelzu b

a Institute of Acoustics, Univ. Austral de Chile, PO Box 567, Valdivia, Chileb Institute of Materials and Thermo-Mechanic Processes, Univ. Austral de Chile, PO Box 567, Valdivia, Chile

a r t i c l e i n f o a b s t r a c t

Article history:Received 26 September 2010Accepted 27 November 2010Available online 4 December 2010

0261-3069/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.matdes.2010.11.070

⇑ Corresponding author. Tel.: +56 63 221012; fax: +E-mail address: jparenas@uach.cl (J.P. Arenas).

This article presents the experimental results of damping loss factor and Young’s modulus obtained forstiff and flexible materials through the use of four different methodologies: the Central ImpedanceMethod, the Modified Oberst Method, the Seismic Response Method, and the simply supported beammethod. The first three methods are based on the ASTM standard but using different experimental settingand different Frequency Response Functions. The fourth method corresponds to a non-resonant tech-nique used in the characterization of materials at very low frequencies. In this work, the results of damp-ing loss factor and Young’s modulus obtained through these four methods are compared, the variability ofresults is studied and the sensitivity of each technique when facing controlled temperature variations isverified.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

The study of structural properties in materials is becomingmore and more important in different disciplines of engineeringand mechanical design [1]. A number of investigations have beencarried out to modify the molecular structure of materials aimedat enhancing their internal damping without altering their otherphysical constants. These kinds of improvements involve develop-ing adequate methods to measure damping loss factor [2].

Stiffness and damping are some of the most important designcriteria for mechanical components and systems. Frequently,performance of a component or a structure is determined by com-bination of its stiffness and damping. This is particularly evidentwhen designing the dynamic characteristics of modern machinessince their increased speed and power, combined with lighterstructures, may result in intense resonances and in the develop-ment of self-excited vibrations [3].

In general, materials selection and component design are twoparallel streams followed when a mechanical component is de-signed. Firstly, a tentative material is chosen and data for it areassembled either from data sheets or from data books. In design,a choice of material can determine the price of a product and pro-duction paths. Later, a more detailed specification of the design andof the material is required. At this point it may be necessary to getdetailed material properties from possible suppliers or to conductexperimental tests [4].

ll rights reserved.

56 63 221013.

Damping loss factor is defined as the ratio between the energydissipated within the damping layer and the energy stored in thewhole structure, per cycle of vibration [1]. Use of constrainedand unconstrained damping material layers has been a helpful toolfor structural designers concerned with mitigating stress or dis-placement amplitude in vibrating systems. In addition, some re-search has been specifically aimed to optimize the damping ofthese layers [5].

The methodology established by ASTM [6] corresponds to astandardized test to measure loss factor and Young’s Modulus inmaterials. This test is based on the analysis of peaks in the Fre-quency Response Function (FRF) measured without interferingwith the system being analyzed. Consequently, this method im-plies the use of some specialized measurement instruments thatcould make the experimental setup highly expensive.

On the other hand, there is a variety of different contactingmeasuring approaches that can be employed for characterizingmaterials by resonance and non-resonance tests. Moreover, useof piezoelectric transducers is quite common in some of thesetests, where accelerometers and force sensors are by far the mosttraditional and widely used piezoelectric sensors employed inmodal testing [7]. Thus, carrying out tests using this type of trans-ducers becomes an alternative worthy to be analyzed.

Particular studies of contacting measuring approaches forcharacterizing materials are abundant in the technical literature,but comparative studies have not been reported. This work aimsto fill in this gap by presenting a comparison of four methodologiesto estimate the characteristics of damping and stiffness inmaterials. For the purpose of comparable results among different

Fig. 2. Central Impedance Method (CIM).

Fig. 3. Modified Oberst Method (MOM).

Fig. 4. Seismic Response Method (SRM).

2424 R. Pereira et al. / Materials and Design 32 (2011) 2423–2428

methods, all of the tests were performed under similar controlledconditions.

2. Theoretical review

2.1. ASTM methodology

In general, all resonant methods use the ASTM standard [6].This standard establishes a methodology based on the measure-ment of Frequency Response Functions (FRF) of clamped-free beam(uniform or composite) to determine the loss factor g, Young’smodulus E, shear loss factor gShear and shear modulus G of anabsorbing material under test.

For a uniform beam of length l, density q and thickness H, theloss factor g of a given mode n, at the resonance frequency fn canbe calculated by

g ¼ Dfn

fn; ð1Þ

where Dfn is the half power bandwidth of mode n. The system lossfactor is approximately equal to twice the critical damping ratio of avibrating system at resonance. The Young’s modulus is determinedfrom

E ¼ 12ql4f 2n

H2C2n

; ð2Þ

where Cn is a coefficient associated to mode n. In the case of anOberst beam (see Fig. 1) or a beam composed of a base beam anda layer of absorbing material of density q1 and thickness H1, theYoung’s modulus E1 of the absorbing material associated to modec of the Oberst beam, at resonance frequency fc, is obtained from

E1 ¼E

2T3 ða� bÞ þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiða� bÞ2 � 4T2ð1� aÞ

q� �; ð3Þ

where a = (fc/fn)2(1 + DT), b = 4 + 6T + 4T2, T = H1/H, and D = q1/q.The corresponding loss factor g1 of mode c is

g1 ¼ gc

1þMTð Þ 1þ 4MT þ 6MT2 þ 4MT3 þM2T4� �

ðMTÞ 3þ 6T þ 4T2 þ 2MT3 þM2T4� �

24

35; ð4Þ

where M = E1/E and gc is the loss factor of the composite beam com-puted from Eq. (1).

2.2. Resonant methods

The Central Impedance Method (CIM), the Modified OberstMethod (MOM) and the Seismic Response Method (SRM) are allresonant methods.

The CIM [8] uses contact transducers and is based in the analy-sis of the FRF defined as X/F, where X is the Fourier transform of thedisplacement signal x(t) measured at the beam center, produced bya force f(t) having a Fourier transform F, which is also applied at thebeam center. The principle is shown in Fig. 2.

MOM [9] uses the displacement z(t) at one of the bar’s end, pro-duced by a forced displacement y(t) at the bar center (see Fig. 3).Then, the FRF used in this method is Z/Y, where Z and Y correspondto the Fourier transform of z(t) and y(t), respectively.

Fig. 1. Oberst beam.

Different from previous methods, SRM [10] is based on the appli-cation of a forced motion s(t) at one end of the beam and measuringthe displacement response r(t) at the opposing end, as shown inFig. 4. Then, the FRF to be analyzed in this method is R/S, where Rand S are the Fourier transform of r(t) and s(t), respectively.

Both MOM and SRM require the use of a laser vibrometer orsome non-contact measuring method so as to avoid physical con-tact with the system.

2.3. Non-resonant method

One of the disadvantages of the resonant methods is that rela-tively large beams must be used to force resonant modes to appearin the low frequency region for the characterization of materials inlow frequency. This produces a series of practical difficulties,mainly when making the experimental setup. The methodologyproposed by Zaveri and Olesen [11], based on the theory statedby Timoshenko [12], and here referred to as SSM, requires a beamof longitude l, width b, thickness h, and density q, which is simplysupported at both ends. This beam is excited by a harmonic forcep(t) = P0sinxt at its center, which produces a displacement re-sponse h(t) = H0sin(xt + u), which is measured at the same pointwhere the force is applied, as shown in Fig. 5.

For a frequency f, below the first resonance frequency, the lossfactor g is calculated through the equation

g ¼ tan u1þ H0

P0

2p2 f 2 lbhqcos u

; ð5Þ

and Young’s modulus E is calculated through

E ¼ 2l3

Ip4

P0

H0cos uþ 2p2f 2lbhq

� �; ð6Þ

where I is the cross-sectional moment of inertia of the beam.As this method does not use resonances, ASTM standard is not

applicable in this case. In this way, according to what was stated

Fig. 5. Simply-supported beam Method (SSM).

Table 1Samples under study.

Method Sample type and length (cm) Number of samples

CIM and MOM Aluminum beams (32) 5Aluminum beams (25) 5Aluminum beams (18) 5ECCS–PET layers (32) 5ECCS–PET layers (25) 5ECCS–PET layers (18) 5

SRM Aluminum beams (16) 5Aluminum beams (12) 5Aluminum beams (9) 5ECCS–PET layers (16) 5ECCS–PET layers (12) 5ECCS–PET layers (9) 5

SSM Aluminum beams (32) 5Aluminum beams (25) 5ECCS–PET layers (32) 5ECCS–PET layers (25) 5

Table 2Piezoelectric transducers: FS = force sensor, Ac = accelerometer.

Method Excitation Response

CIM FS B&K 8230 Ac B&K 4518-003MOM Ac 4513-001 Ac B&K 4518-003SRM Ac 4513-001 Ac B&K 4518-003SSM FS B&K 8230 Ac B&K 4518-003

R. Pereira et al. / Materials and Design 32 (2011) 2423–2428 2425

in [11], the physical characteristics of a viscoelastic material arenot possible to be obtained starting from the known data of a basebeam.

3. Experimental procedure

3.1. Materials

The materials used in this study are an aluminum sample andan ECCS–PET, which is a thin and flexible polymer–metal com-monly used in the food industry for food conservation [13]. In thiscase, the aluminum was used as a base beam and the ECCS–PET

Fig. 6. General measurement setup. (A) B&K Pulse 3560-C, (B) B&K 2718 amplifier, (transducer, (F) personal computer. (C–E) and the location of the transducers depend on

was joined to this beam through a thin double adhesive. To obtainresonances in different frequencies, several aluminum beams andECCS–PET layers of different lengths were considered. These spec-imens are summarized in Table 1.

The width of all samples is 18.8 mm. The aluminum used has athickness of 2.5 mm and a density of 2927.402 kg/m3. The ECCS–PET used has a width of 18.8 mm, a thickness of 0.15 mm and adensity of 9447.083 kg/m3. Concerning the SSM, measurementswith different size beams were carried out with the purpose of ver-ifying if overlapping of curves obtained from beams having differ-ent length is produced.

3.2. Experimental setup

The different transducers used in this study are piezoelectric(see Table 2). Measurements were carried out inside a tempera-ture-controlled chamber at T1 = 24 �C and T2 = 40 �C. Sampleexcitation was done, for resonant methods, through an electro-dynamic shaker B&K 4810. For the SSM a shaker B&K 4809 wasused, due to its better performance at low frequencies. White noisewas used as excitation signal for all the experiments. The signalwas amplified by means of an amplifier B&K 2718. Acquisitionand further digital signal processing was done through a systemPulse B&K 3560-C. Hanning windows were used in all cases andFRF were calculated with a spectral resolution of 0.5 Hz. When nec-essary, the integration of the temporal signal was carried out withthe Pulse system. Fig. 6 shows a general diagram of the measure-ment setup used.

4. Results and discussion

4.1. Results of the base beam

As an example of the variability detected in the results obtainedthrough the resonant methods, Fig. 7 presents the results obtainedthrough CIM. This method presented the lowest statistical disper-sion in all trials.

With the aim of presenting the variability in the results ob-tained from each methodology, Fig. 8 presents the summary of re-sults obtained through the three resonant methods. Ends of errorbars correspond to the lowest and highest measured values,respectively, whereas the point where the graphics intersects theselines corresponds to the median of the group of measured values.Unlike the average, median of a data set is not too sensitive to ex-treme values; therefore it was chosen as a descriptive value.

Concerning aluminum loss factor g, results in the wider fre-quency range, the lowest variability and the highest correspon-dence with those presented in the literature [14] corresponded tothe results measured through CIM. Fig. 8 shows that the results ob-tained by MOM and SRM present a higher dispersion and highervalues, mainly at low frequencies.

C) B&K shaker, (D) response piezoelectric transducer, (E) excitation piezoelectriceach method.

0 500 1000 1500 2000 2500

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Frequency (Hz)

Loss

fact

or η

Fig. 7. Values of g obtained through CIM, T: 24 �C. �: l = 32 cm, m = 1; +: l = 25 cm,m = 1; s: l = 18 cm, m = 1; h: l = 32 cm, m = 3; �: l = 25 cm, m = 3; 4: l = 32 cm,m = 5; �: l = 18 cm, m = 3; }: l = 25 cm, m = 5 (m: excited mode).

2426 R. Pereira et al. / Materials and Design 32 (2011) 2423–2428

In contrast, regarding the results obtained for Young’s modulus,SRM values are the closest to those presented in [14]. CIM presentsa high variability and high values at low frequencies. Above500 Hz, these values become stable and are close to those reportedin the literature. MOM gives clear information only in the firstresonances present in the measured FRF, which implies a charac-terization covering a more limited frequency range. SSM givesdissimilar results depending on the length of the beam being stud-

0 500 1000 1500 2000 2500

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Frequency (Hz)

Loss

fact

or η

0 500 1000 1500 2000 25000

2

4

6

8

10

12

14

16

18

20x 1011

Frequency (Hz)

Youn

g´s

mod

ulus

E (P

a)

Fig. 8. Comparison of g and E values obtained through three resonant methodolog

ied. However, these results converge into physically right valueswhen the loss factor is computed and diverge for Young’s modulusas the frequency increases. In this case, limit values are acceptableand tend to coincide with those obtained through SRM.

The increase of temperature during measurements revealschanges in the values of the loss factor and Young’s modulus calcu-lated by means of each method, although a uniform tendency is notobserved. Concerning CIM, an increase in g values is observed ingeneral.

On the other hand, the results obtained by the non-resonanttest SSM are shown in Fig. 9. The results show the usual frequencydependences of the dynamic properties associated to typical metalor stiff structural materials, as presented in [15,16].

Here the tendency is clear and loss factor values slightly de-crease as temperature rises. Almost no variation in values is ob-served for Young’s modulus. These effects show the temperature-frequency dependence for a solid in which there is dense and reg-ular packing of molecules. It is a fact that an increase in frequencyhas an equivalent effect on the damping and stiffness as a decreasein temperature, and vice versa [1,17].

4.2. Results of the PET layer

Fig. 10 presents the summary of results for the PET layer ob-tained through the three resonant methods. We observe that mea-surements of loss factor carried out at 24 �C are quite similar whenusing MOM and SIM. Information is limited up to approximately330 Hz. The increase in temperature makes information given byFRF clearer and makes ASTM standard applicable [6]. In this way,information up to 850 Hz is obtained (see Fig. 10). Despite the

0 500 1000 1500 2000 2500

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Frequency (Hz)

Loss

fact

or η

0 500 1000 1500 2000 25000

2

4

6

8

10

12

14

16

18

20x 1011

Frequency (Hz)

Youn

g´s

mod

ulus

E (P

a)

ies. 5: SRM. �: MOM, h: CIM. Plots to the left: 24 �C; plots to the right: 40 �C.

20 40 60 80 100

0

0.2

0.4

0.6

0.8

1

1.2

Frequency (Hz)

Loss

fact

or η

20 40 60 80 100

0

1

2

3

4

5

6

7x 1010

Frequency (Hz)

Youn

g´s

mod

ulus

E (P

a)

Fig. 9. Comparison of g and E values obtained through the non-resonant method SSM. h: 24 �C. �: 40 �C. Dotted line: l = 32 cm. Continuous line: l = 25 cm.

0 100 200 300 400 500 600 700 800 9000

0.2

0.4

0.6

0.8

1

1.2

Frequency (Hz)

Loss

fact

or η

0 200 400 600 800 10000

0.2

0.4

0.6

0.8

1

1.2

Frequency (Hz)

Loss

fact

or η

0 100 200 300 400 500 600 700 800 900

0

1

2

3

4

5

6

7

8

9x 1012

Frequency (Hz)

Youn

g´s

mod

ulus

E (P

a)

0 100 200 300 400 500 600 700 800 900

0

1

2

3

4

5

6

7

8

9x 1012

Frequency (Hz)

Youn

g´s

mod

ulus

E (P

a)

Fig. 10. Comparison of g and E values for PET layer obtained through three resonant methodologies.5: SRM. �: MOM, h: CIM. Plots to the left: 24 �C; plots to the right: 40 �C.

R. Pereira et al. / Materials and Design 32 (2011) 2423–2428 2427

existing dispersion, similarities are observed among the obtainedcurves. In addition, for the three resonant methodologies we ob-serve that estimated loss factor values increase as temperaturerises. At high temperatures the material becomes soft and reachesa rubbery state. This fact has also been observed and discussed inthe technical literature [17].

On the other hand, there is no coincidence in the values obtainedfor Young’s modulus through the studied methods, so there is noclear tendency on its values. It is well known that some of the errorsin the estimated values are related to the ratio of layer thickness tobase beam thickness. The layer should be thick enough to cause a

difference between the bending stiffnesses of the layer and basebeam. This difference must be sufficiently large in order to measurethe bending stiffness of the layer with precision and repeatability,which is not the case studied here (0.15 mm of the PET layer com-pared to 2.5 mm of the base beam). These effects have been previ-ously discussed by Jones and Parin [18] and Pritz [19].

Nonetheless, the increase in temperature produced, in all cases,a decrease in the measured values. This fact is typical for highlypolymeric materials and it has been widely covered in the litera-ture [17,20]. Thus, considering the structural influence of the PETpolymer on the behavior of the ECCS–PET composite, the values

2428 R. Pereira et al. / Materials and Design 32 (2011) 2423–2428

obtained for both loss factor and Young’s modulus through thethree resonant methods are reasonable enough. On the contrary,SSM did not give realistic results; hence they were not includedin this section.

5. Conclusions

This work has been aimed in the comparison of the results ofloss factor and Young’s modulus obtained through the applicationof four methodologies using piezoelectric transducers.

The following conclusions can be drawn from the experiments:

� In general, the use of piezoelectric transducers in resonant con-tacting measuring methods produces a high variability in theresults of loss factor and Young’s modulus. Therefore, a largernumber of measurements will be necessary to reduce the statis-tical dispersion and thus render more accurate values.� Concerning SRM and MOM, forces and moments produced by

both the mass and location of the accelerometer used to mea-sure the system response to the applied force originate distor-tions in the FRF measured. Consequently, this fact produceserrors in the estimation of the calculated physical parameters.� Characterization of the aluminum loss factor may be trustwor-

thily done through CIM. The most accurate results of Young’smodulus are presented by SRM. In the same way, CIM presentsreasonable results of Young’s modulus at higher frequencies,which is complemented with those values obtained throughSSM.� Congruent results in a reduced frequency range have been pre-

sented by the characterization of ECCS–PET loss factor. Temper-ature increase allows characterization within a larger frequencyrange, although no clear tendencies have been observed for lossfactor.� On the other hand, it can be concluded that SSM provides accu-

rate results in the low frequency range as piezoelectric trans-ducers are used to characterize single layer beam samples. Onthe contrary, SSM does not provide realistic results for multi-layer beam samples. In this case, it is very difficult to excitepurely bending waves, on which the method is based.

Therefore, the measurement method must be carefully decidedif piezoelectric transducers and test beams are used. By the com-plementary use of the four methods, a material could be character-ized in different frequency ranges, considering the limitations ofeach method. Thus, these experimental methodologies can be usedto obtain more detailed material properties when designingmechanical components and systems. Measured values of dampingcan help in material selection for preventing or alleviating some

specific negative effects such as undesirable resonances, impactsbetween vibrating parts, accelerated wear, noise generation, andharmful vibrations transmitted to human operators.

Nonetheless, more experimental studies are needed for a rigor-ous validation of the methods.

Acknowledgment

This work has been supported by CONICYT–FONDECYT No.1070375, which is gratefully acknowledged.

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