Determination of the transverse Young’s modulus (TYM) of wood by means of an input power technique

Construction and Building Materials 42 (2013) 11–21

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Construction and Building Materials

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Determination of the transverse Young’s modulus (TYM) of wood by meansof an input power technique

R.J. Alves, M.D.C. Magalhaes ⇑, E.V.M. CarrascoDepartment of Structural Engineering, Federal University of Minas Gerais (UFMG), Av. Antônio Carlos 6627, Campus Pampulha, Belo Horizonte CEP: 31270-901, Brazil

h i g h l i g h t s

" Transverse Young’s modulus (TYM) can be obtained using the power injection method." This alternative method allows for the tests to be made in an ordinary room (floor noise up to 40 dB)." It is seen that 86% of the TYM variation is attributable to the linear relationship between the dynamic and static TYM." The measured TYM values were 0.79 GPa and 1.27 GPa for the Ocotea porosa (Imbúia) and Tabebuia spp. (Ipê) respectively.

a r t i c l e i n f o

Article history:Received 17 August 2012Received in revised form 8 December 2012Accepted 19 December 2012Available online 9 February 2013

Keywords:Transverse Young’s modulusWood materialInput power technique

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⇑ Corresponding author. Tel.: +55 (31) 3409 1846;E-mail address: [email protected] (M.D.C. Maga

a b s t r a c t

The main goal of this study is to present an alternative non-destructive impact testing which can be usedfor the determination of the transverse Young’s modulus (TYM) of wood species. The method is based onthe steady-state power injection method, which is a non-destructive acoustic impact testing for the in situdetermination of dissipation and coupling loss factors of subsystems representing the physical system.The acoustic impact testing can be performed simply using a basic vibration test kit and an impedancetube, which is commonly used on the investigation of material acoustic impedance. The main advantagesof this approach for predicting the TYM are twofold: reliability and cost-effectiveness. The test can bemade simpler and in an ordinary quiet environment, as an adequate signal-to-noise ratio can be easilyachieved. Besides, this technique allows physical insight into the behavior of structural–acoustic systemsfound in wood buildings. Numerical and analytical work is presented in order to validate the experimen-tal tests. The minimum and maximum TYM values measured were 0.79 GPa and 1.27 GPa for the Imbuiaand Ipe species respectively. It is seen that 86% of the observed variation in the dynamic TYM is attrib-utable to the approximate linear relationship between the dynamic and static TYM, a very impressiveresult. In addition, the measured and calculated natural frequencies of the coupled system are presented.The relationship between them was significant. The species analyzed were as follows [1]: Amburanacearensis (Cerejeira), Cordia goeldiana Huber (Freijó), Ocotea porosa (Imbúia), Swietenia macrophylla(Mogno) and Tabebuia spp. (Ipê).

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1. Introduction

The use of Non-Destructive Testing (NDT) presents specificpeculiarities for each wood specimen. The NDT technique has beenwidely used to determine the dynamic mechanical properties ofwood and wood-based composites, such as dynamic modulus ofelasticity and damping ratio.

In the analysis of wood mechanical properties, many studieshave been considered on the use of ultrasonic frequencies [2–4].In Ref. [5], an ultrasonic system was used for ultrasonic tests usingdirect method in both parallel and perpendicular to grain

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directions. The time (t)–amplitude (A) signals were analyzed toevaluate the stress wave speed. The dynamic modulus of elasticitywas also evaluated, according to the theoretical relationship forhomogenous and isotropic elements. In another recent study [6],a nondestructive ultrasonic technique was developed for theevaluation of wooden rafter deterioration. Regression modelsdescribing the relationship between the artificial deterioration ofthe specimen and ultrasonic parameters were proposed. The meth-od was found to be reliable for evaluating wooden rafter deteriora-tion. The ultrasonic velocity in wood was found to be dependent onseveral factors such as moisture content, specific gravity, grain ori-entation, presence of natural or artificial defects/inclusions insidethe wood [7]. Correlation between ultrasonic velocity and staticmodulus of elasticity evaluated using ultrasonic and conventional

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12 R.J. Alves et al. / Construction and Building Materials 42 (2013) 11–21

method respectively were also investigated. Despite the fact thatultrasound techniques have many advantages, they depend onthe use of costly equipments.

In Ref. [8], three nondestructive methods based on stress wave,longitudinal vibration, and ultrasonic wave were examined andcompared. As expected, the longitudinal vibration method wasthe most precise and reliable technique to evaluate the mechanicalproperties of logs among the three acoustic-based nondestructivemethods.

More investigations are needed on the ability of using NDT inorder to accurately assess the modulus of elasticity of in situ tim-bers [9]. Weak correlations between the NDT tests performedand destructive tests were obtained. Even in laboratory conditions,when evaluating real-size old timber elements (and not usingmanufactured specimens) NDT evaluation is a very hard task thatshould be based on a wider range of NDT tests available. Low cor-relations between the results of the non-destructive tests andmodulus of elasticity obtained through mechanical evaluationwere obtained.

Vibration methods have been successfully used to estimate thephysical and mechanical properties of wood and wood-based com-posites for several decades in scientific research and forest prod-ucts industry [10]. The cantilever-beam vibration technique wasa reliable and cost-effective approach which provided a quicknon-destructive measurement of dynamic modulus of elasticityand damping ratio of wood-based composites.

In Ref. [11], the aim of the study was to estimate the longitudi-nal stress under static bending in wood beams made of the Japa-nese fast growing trees of the Melia azedarach species using theacoustoelastic technique. The technique is based on the relation-ship between the applied stress and the velocity of ultrasonic shearwaves propagating in wood normal to the stress direction. Theshear wave velocities decreased with an increase in compressivestress, and increased with an increase in tensile stress. The shearwave velocities variation changed depending on the applied stress.Further research on the influence of complicated stress state suchas combined stress may be needed before applying this technique.

Internal decay is a major structural defect of many tree species.The economic loss caused by internal decay is most significant forthe hardwood trees that are used to produce appearance-grade ve-neer and other high-value engineered wood products. Acoustictomography is an emerging nondestructive testing (NDT) technol-ogy for tree decay detection [12]. However, the method lacks thesensitivity to low-velocity features of decayed areas and thus haslimited capability in detecting early stages of decay in trees.

In Ref. [13], a non-invasive, non-destructive technique whichuses a low cost and portable data acquisition system was pre-sented for wood species identification. A square wood samplewas positioned on a polystyrene foam frame and had two of itsedges clamped. Next, a condenser microphone was positioned par-allel to the wood square sample and the sound pressure responserecorded as an impulse test was performed. For this test, a pendu-lum system (plastic ball attached to a string) was used to hit thewood sample in the middle. Although the authors declared thatthe tests were made in an ordinary quiet room, it is believed thatthis room should be a ‘quasi’ anechoic space in order to obtain agood signal-to-noise ratio for reasonable test reproducibility. Theresults presented confirmed this hypothesis as power spectrumdensity seemed to be very restrictive for the low sound pressurelevels considered which was below the audible threshold ofhearing.

The steady-state power injection method is a well-known sta-tistical energy method which is based on the generation of a sub-system energy matrix. The method has been used for thedetermination of statistical energy analysis loss factors for manyyears [14–17]. As energy cannot be directly measured, it must be

inferred from a set of measurements of dynamic variables at dis-crete points. The main assumption to this method is that theshort-time average input power and energies vary smoothly on atime scale much larger than the average period of the vibrationalprocess in operation. The input power must be quantified in orderto use this technique. In addition, the power injection method hasmainly been used for the determination of loss factors, not themodulus of elasticity.

Sound radiation is a significant subject with respect to struc-ture-borne sound. Experimental tests that have high precisionand wide application are required in order to provide practicalevaluation of sound radiation. In Ref. [18], sound radiation effi-ciency was measured when the specimen was impacted by a golfball. The specimen used in the experiment was a plywood boardof 12 mm thickness. The proposed method considered vibrationmeasurements and calculation of radiation impedance.

The interaction between an acoustic space and its flexibleboundaries is an important problem in the field of acoustics. Anal-ysis of this interaction has been of interest to many researchersduring the last half a century as reviewed by Pan [19] and Hongand Kim [20]. They provided solutions for coupled responses interms of the modal characteristics of the uncoupled structuraland acoustic systems.

According to the literature review presented above, there is still aneed among academic researchers and practitioners to develop acheaper and easier-to-implement non-destructive method for woodcharacterization. Therefore, this research was first undertaken as aresult of the need to develop an easy and reliable methodology forpredicting the acoustic response of impacted wood specimens inorder to obtain their transverse Young’s modulus (TYM). Thedetermination of the TYM was inferred using the power injectionmethod. Besides, it is seen that there is not many, if any, publishedpapers that have considered the analysis of structural–acousticsystems as a way of characterizing wood species. In summary, oneof the most important issues on wood construction design, i.e. thedetermination of the wood TYM, is addressed herein.

2. Materials and methods

Firstly, a FE model was considered in order to find the corresponding normalmodes for specific circular wood samples made of common species found in Brazil.A frequency range of 20–2000 Hz was considered on the numerical simulations. Theanalyses were performed as follow: the commercial FE software, namely ABAQUS[21], was used in order to obtain the mode shapes (and their corresponding naturalfrequencies) for the simply-supported circular wood plates. The simulations wereperformed and the corresponding sets of normal modes /p(y, z) extracted andstored. Then, a fluid–structure model was developed and implemented in MATLABfor the evaluation of the fluid–structure coupled natural frequencies.

Some of the physical and mechanical properties used on the simulations weremeasured, others obtained from the literature. For instance, the total loss factor gwas measured. It is known that the total loss factor of a partition depends on theboundary condition of the specimen. In other words, the total loss factor is equalto the sum of the internal loss factor of the material, the coupling loss factor tothe adjacent structures and the radiation loss factor to the surrounding media[25]. Therefore, a particular technique was used on the wood panel. It is an in situapproach. The duration of the response (vibration) due to impact excitation, whichis characterized by the reverberation time, was measured for the calculation of thestructural damping. The reverberation time is defined as the time interval in whichthe vibration energy level decays by 60 dB.

The wood panel was sealed around its periphery with Super Bond adhesive. Thisdoes not provide a clamped edge, as it is dependent on the stiffness and strength ofthe adhesive. However, it does provide a reasonable constraint on the flexural dis-placement but not slope.

As mentioned before, the values taken when the wood panel was tested in situin the impedance tube was used in the FE simulations, which were performed tocompare with the experimental results.

The experimental procedure for determination of decay times is described asfollows. On impacting a ‘simply-supported’ wood plate sample by a small plasticball, the analyzer was triggered and started to record the response signal at thereceiving point, where a microphone was attached and connected to the acquisitionequipment (a National Instruments data acquisition module type NI-9233). The in-put signal was filtered by conveniently configuring the channel parameters.

Table 1Mechanical properties of wood species [24,28].

Wood species qap (kg/m3) EL (GPa) ET (GPa)

Amburana cearensis (Cerejeira) 600 10.7 0.54Cordia goeldiana (Freijó) 590 11.1 0.60Ocotea porosa (Imbuia) 650 7.7 0.39Tabebuia spp (Ipê) 1010 18.1 0.91Swietenia macrophylla (Mogno) 630 10.7 0.54

R.J. Alves et al. / Construction and Building Materials 42 (2013) 11–21 13

The signal s(t) received at the receiving point is given by [26,27]:

sðtÞ ¼Z t

�1f ðsÞhðt � sÞds ð1Þ

where f(t) is input excitation force and h(t) is the impulse response of the system. Foran impulse at time ti one has

f ðsÞ ¼ Fodðs� tiÞ; then sðtÞ ¼ hðt � tiÞ ð2Þ

The Hilbert transform H of a function x(t) is used to produce the envelope of thesignal and subsequently allows determination of the decay rate of the signal. It isgiven by [27]:

Fig. 1. Photographs of the wood species used on the experimental te

Fig. 2. Set-up of the apparatus used on the experimental tests; (a) Pho

HðxðtÞÞ ¼ � 1p

Z þ1

�1xðsÞ 1

t � s

� �ds ¼ 1

pxðtÞ � 1

t

� �ð3Þ

where � means convolution.Hence, the amplitude decay curves were obtained by taking the Hilbert trans-

form of the received signal and then converting its absolute value to a logarithmicamplitude scale as follows:

CðtÞ ¼ 20log10jHðxðtÞÞj in dB ð4Þ

The values of damping found in this way are sometimes termed structuraldamping, to identify that the damping is dependent on both the damping inherentin the material and that which comes from other mechanisms including dissipationlosses at the boundary which might be significant.

To determine damping, the reverberation time T60, was first obtained using theleast square method [26] for fitting the best straight line to the data. The use of abest straight line fit to calculate T60 assumes diffuse field condition. Nevertheless,for non-diffuse field condition, it is recommended that an ensemble-average esti-mate of decay rates over a range of different excitation and receiver positions be ob-tained [18].

In other words, it is clear that at low frequencies a best straight line fit is notparticularly appropriate, partly because the field is not diffuse and few modes con-tribute and hence it is not normal to use Eq. (5). Nevertheless, as values were nec-essary in the FE model, the slope of the lines was used to give estimates, albeit

sts. (A) Cerejeira; (B) Freijó; (C) Imbúia; (D) Mogno and (E) Ipê.

tograph of the experimental set-up; (b) Schematic representation.

200 400 600 800 1000 1200 1400−20

−15

−10

−5

0

5

10

15

20

Z [d

B re

1N

.s/m

]

Frequency [Hz]

Fig. 4. Impedance of a 1 mm thick wood plate; —————Experimentally obtainedreal part of impedance ____ Input impedance of an infinite plate (Zinf ¼ 8

ffiffiffiffiffiffiffimBp

).

0 0.5 1 1.5 2 2.5 3 3.5 4-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time (s)

Out

put S

igna

l (V)

0 0.5 1 1.5 2 2.5 3 3.5 4-70

-60

-50

-40

-30

-20

-10

0

10

20

30

Time [s]

20lo

g 10 (|

H(x

(t))|)

[dB]

(a)

(b)

Fig. 3. Measurement of structural damping for a wood panel. (a) Typical response time history; (b) Hilbert Transform of the impulse response (decay curve).

Table 2The natural frequencies of the enclosed fluid in a 1 m length fixed-opened end ductand of a circular wood plate (species: Amburana cearensis) obtained via FE modelsimulation.

Modenumber

Uncoupled FN (Hz) acousticmodes

Uncoupled FN (Hz) structuralmodes

1 85.8 360.22 257.3 1020.43 428.8 1020.64 600.3 1885.65 771.8 1888.86 943.3 2176.3


possibly very approximate, and these have been used in the simulations. The lossfactor g, which is the ratio of energy lost to the reversible mechanical energy duringone cycle of vibration, is related to T60 by [26]:

g ¼ lnð106Þ2pfBT60

ð5Þ

On the other hand, the longitudinal Young’s modulus EL used on the simulationswere obtained from the literature [24,28] and presented on Table 1 below. Thetransverse modulus ET (TYM) was calculated by the relationship [24]: ET = EL/20.

Next, additional experimental tests were made, using a non-destructive acous-tic impact testing named input power technique. Power Spectrum Density (PSD)curves were obtained for 17 specimen of each wood species backed by the open-ended duct. Then, the peaks corresponding to the coupled natural frequencies wereidentified on each PSD curve. The measured and simulated natural frequencies ofthe coupled system were then compared by plotting the calculated values (usingthe analytical model) against the measured ones. A curve fitting using linear inter-polation was used.

The fluid–structure model system was composed of two subsystems: a volumeof fluid enclosed by a rigid-walled fixed-opened duct and a wood plate model(structure).

In general, the sound field in a circular duct can be expressed as the sum oftransverse and axial modal components. The cross-sectional modal modes repre-sent the circumferential and radial modes. The cut-on frequency for the first acous-tic cross-sectional modal mode fc is given by [22]:

fc ¼ 1:84co=ðpDÞ ð6Þ

where co is the sound speed in air and D is the diameter of the tube. In other words, fc

is the frequency cut-off for plane wave propagation in circular ducts. According toEq. (6), the cut-on frequency was found to be approximately 1,913 Hz. As a result,the three-dimensional eigenfunctions Xn, which neglect the cross-sectional elasticmodes can be written as [22]:


Xn ¼ cosnxp:x2Lx

� �for nx ¼ 1;3;5;7; . . . ð7Þ

where nx is the mode number, Lx is the tube length (=1 m).In summary, the eigenproblem analysis was restricted to hard-walled ducts.

The static cross-sectional mode (0,0) was the only transverse acoustic mode in-cluded on the calculations. In addition, no reflections from the duct open-end wereconsidered.

The response (normal displacement to the plate surface) of the plate to a har-monic point force excitation Fo at position (zo, yo) and at frequency x is given by:

wðz; y;xÞ ¼XP

p¼1

wp/p ð8Þ

where P is the total number of structural modes considered. Then, Eq. (8) may be anexpression for the wood plate displacement in terms of a summation of its assumed-modes. The modal response (displacement) of the wood plate is represented by wp.

Considering the fluid–structure interaction, the coupled system can be repre-sented as [22]:

�x2Un1 þ jxbn1Un1 þx2

n1Un1 ¼

c2o S

Kn1

� �XP

p¼1

ðjxwpCn1 pÞ ð9Þ

�x2wp þ jxbpwp þx2pwp ¼ �

qoSKp

� �XN1

n1¼1

ðjxUn1 Cn1 pÞ þFp

Kpð10Þ

where Un1 is the modal acoustic velocity potential amplitude; S is the wood platearea; x is the excitation frequency in radian/s and Fp is the generalized force appliedon the wood. On the left-hand side of Eq. (9) the additional term jxbn1

Un1 , in termsof the velocity potential, is inserted in order to include viscous damping in subsys-tem 1. Likewise, the term jxbpwp is added on the left-hand side of Eq. (10) in orderto represent the damping of the flexible wood plate. bn1

and bp) are the generalizedmodal damping coefficients, which are given by gn1

xn1 and gpxp for the tube andwood plate respectively; Kp is the modal mass of the structural mode p; xn1 andxp represent the natural frequencies of the duct and wood plate respectively.

The velocity potential function allows one to obtain all other acoustic parame-ters through the relationship for the pressure

p ¼ �qojxU ð11Þ

The spatial structural–acoustic coupling coefficient Cnp was evaluated as:

Cnp ¼1S

ZS

Xn/pdS ð12Þ

Eqs. (9) and (10) were used for the prediction of the wood modal displacementwp.

The acoustic modes considered on the formulation were calculated using Eq.(7). The structural modes used were obtained from the FE simulations. The forceexcitation and receiving point were located at opposite ends of the tube.

The weight of the ball W equals the mass times gravity times sine of the angle ofpendulum from vertical. It is given by:

W ¼ mg sinðhÞ ð13Þ

where m is the ball mass (m = 97 � 10�3 kg) and g is the gravity acceleration.

0 0.1 0.2 0.3 0.4 0.5-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Nor

mal

ized

Mod

al A

mpl

itude

Rigid-Walled Duct L

Fig. 5. Set of sound pressure modes of a duct with length equal to 1 m.

The potential energy is, of course, entirely from gravity. The only thing thatmatters is that the pendulum is slightly higher than at the bottom of the swing,and the potential energy is equal to the weight of the pendulum multiplied bythe change in height. Thus, the change in height was L(1 � cos (h)), where L wasthe pendulum length (L = 16 cm) and h was the release angle (h = 36o).

The ball was released freely onto the test object from a height of 3 cm, once persecond. Thus, the impact frequency was fs = 1 Hz. The momentum is I = mvo, wherethe velocity of the pendulum mass at impact is vo ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2gLð1� cosðhÞ

p. Thus, the

mean squared force F21 in the frequency band Df is given by [26]:

F21 ¼ 4f sI

2Df ð14Þ

For the most common case where the bandwidth of interest is an octave withcentre frequency f, the bandwidth is given by Df ¼ f=

ffiffiffi2p

.The eigenvalue problem was implemented in MATLAB. The matrix implementa-

tion for the calculation of the system frequency response is shown. Basically, aneigenvalue problem is described in its standard form as [23]:

½A� kI�½y� ¼ 0 ð15Þ

where A is the dynamic matrix, I is the identity matrix and y is the response function.The solution of the eigenvalue problem, for the fluid–structure interaction case de-scribed previously led to a system of a standard second order differential equationsin the form

½M�½€y� þ ½K�½y� ¼ 0 ð16Þ

where [M] and [K] are the mass and stiffness matrices respectively and [y] is the col-umn vector representing the generalized coordinates of the system.

For the experimental tests, five different species were considered (see Fig. 1 be-low): (a) Cerejeira (Amburana cearensis (Allemão) A.C. Sm.); (b) Freijó (Cordia goel-diana Huber); (c) Imbúia (Ocotea porosa (Nees e C. Mart.)); (d) Mogno (Swieteniamacrophylla King.) and (e) Ipê-champagne (Tabebuia spp.). A total of 17 specimenswere tested for each species.

The tests were made at the Wood and Advanced Materials Research Laboratory(WAMRL). Each specimen had the cross-section area equal to 150 mm � 170 mm.

Fig. 2 shows the experimental set-up. Each specimen was attached to the steelduct using an adhesive. The thickness of each wood sample was 1 mm. The basis ofthe criterion for ‘thinness’ adopted was kbhp < 1; where kb is the free structural wavenumber and hp is the plate thickness. The duct had its internal diameter equal to105 mm and length equal to 1000 mm.

The data acquisition system used was composed of a condenser microphone, aNI (National Instruments) data acquisition system (24 bits, 50 k samples/s) and aportable computer. A small ball with mass equal to 94.7 g was raised 3 cm heightand released as a pendulum providing an impulsive force on the wood specimen.The impulse response signal (sound pressure) was capture by the microphone posi-tioned at the opened-end of the tube. The recorded data were then analyzed andprocessed.

The linear equation which relates the radiated sound pressure with the force atthe driving point is given by:

pðx; y; z; f Þ ¼ HtFðf ÞF1ðf Þ ð17Þ

where HtF(f) represents the frequency response function describing the sound trans-fer for force and F1(f) the applied simple harmonic excitation force.

In many practical situations instead of linear equations, equations in meansquare quantities can be used. Thus, in the next section the results are presentedin terms of power spectrum density.

0.6 0.7 0.8 0.9 1

ength [m]

mode 1mode 2mode 3mode 4mode 5mode 6

They represent the first six fixed-free axial acoustic normal modes.

U, Magnitude

+0.000e+00+8.333e−02+1.667e−01+2.500e−01+3.333e−01+4.167e−01+5.000e−01+5.833e−01+6.667e−01+7.500e−01+8.333e−01+9.167e−01+1.000e+00

Step: Step−1Mode 1: Value = 5.12170E+06 Freq = 360.19 (cycles/time)Primary Var: U, MagnitudeDeformed Var: U Deformation Scale Factor: +1.050e−02

ODB: cerej_model.odb Abaqus/Standard 6.10−1 Thu Apr 05 19:31:05 E. South America Standard Time 2012

X

Y

Z

U, Magnitude

+0.000e+00+8.333e−02+1.667e−01+2.500e−01+3.333e−01+4.167e−01+5.000e−01+5.833e−01+6.667e−01+7.500e−01+8.333e−01+9.167e−01+1.000e+00



X

Y

Z

U, Magnitude

+0.000e+00+8.333e−02+1.667e−01+2.500e−01+3.333e−01+4.167e−01+5.000e−01+5.833e−01+6.667e−01+7.500e−01+8.333e−01+9.167e−01+1.000e+00



X

Y

Z

U, Magnitude

+0.000e+00+8.333e−02+1.667e−01+2.500e−01+3.333e−01+4.167e−01+5.000e−01+5.833e−01+6.667e−01+7.500e−01+8.333e−01+9.167e−01+1.000e+00



X

Y

Z

U, Magnitude

+0.000e+00+8.333e−02+1.667e−01+2.500e−01+3.333e−01+4.167e−01+5.000e−01+5.833e−01+6.667e−01+7.500e−01+8.333e−01+9.167e−01+1.000e+00



X

Y

Z

U, Magnitude

+0.000e+00+8.333e−02+1.667e−01+2.500e−01+3.333e−01+4.167e−01+5.000e−01+5.833e−01+6.667e−01+7.500e−01+8.333e−01+9.167e−01+1.000e+00



X

Y

Z

(a) F1=360.2 Hz (b) F2=1020.4 Hz

(c) F3=1020.6 Hz (d) F4=1885.6 Hz

(e) F5=1888.8 Hz (f) F6=2176.3 Hz

Fig. 6. Set of structural displacement modes for a circular sample of wood (Amburana cearensis). It is shown the first six simply-supported modes.


The input power technique considers the generation of an acoustic reverberantfield at single frequencies within a particular frequency range. The total power radi-ated can in fact be calculated directly using an approximation [22]:

Prad ¼ xgradEp ð18Þ

where Prad is the total power radiated, Ep is the spatial average time averaged soundenergy of the acoustic field generated in the tube and grad is the radiation loss factor.Thus, Ep and grad can be obtained according to the expressions [22]:

Ep ¼h�p2

i iVi

qoc2o

� �ð19Þ

grad ¼ qocoSr=ðxMÞ ð20Þ

where Vi is the volume of the tube, and h�p2i i is the spatial averaged mean square pres-

sure measured; r is the plate radiation efficiency [30], S is the panel area (m2), qoco isthe air characteristic impedance (=415 rayls), x is the angular frequency (rad/s) andM is the panel mass (kg).

The power injected by a point source into the wood plate, Pinp, is given at a sin-gle frequency by:

Pinp ¼ F21Reð1=ZpÞ ð21Þ

It was presumed that the exciting force acted over a surface with dimensionsmuch smaller than the bending wavelength, but big enough to prevent shear defor-mations at the excitation. This assumption is satisfied in most practical cases. As

Table 3The natural frequencies corresponded to the first six modes of the coupled system.The measured values were obtained from the mean value of 17 wood specimens(species: Amburana cearensis) backed by the duct acoustic volume).

Acousticmode number

Coupled FN (Hz)(calculated)

Coupled FN (Hz)(measured)

Correlation(%)

Error(%)

1 85.7 85.5 99.8 0.232 257.1 170.0 66.1 51.23 360.0 242.1 67.3 63.04 428.8 263.0 61.3 41.15 600.1 425.2 70.9 33.96 770.0 575.0 74.7 29.2

0 100 200 300 400 500 600 700 8000

100

200

300

400

500

600

700

800

Natural Frequency [Hz] - Calculated

Nat

ural

Fre

quen

cy [H

z] -

Mea

sure

d Discrete ValuesLeast-Square Curve Fitting

Fig. 7. Comparison between the measured and calculated natural frequencies of thecoupled system considering the wood species Amburana cearensis. (r2 = 0.98).


frequency increases, the plate will ‘appear to the force’ to be unbounded and reso-nant and anti-resonant behavior became insignificant (see Fig. 4 below). Eqs. (9)and (10) were used for the prediction of the wood modal velocity _wp .

Assuming that the input power is equal to the sound radiation from the plate,and also that the plate finite impedance Zp may be substituted by the impedanceof the corresponding infinitely extended system (see Fig. 4 in the next section),the phase speed of a quasi-longitudinal wave CL on the wood plate is given by [25]:

CL ¼ 1=ð14:4h2tFgradfq2

ph3pSpÞ ð22Þ

where h2tF ¼ Prad=ðF2

1qocoSrÞ, f is the frequency, qp is the wood plate density, hp and Sp

are the plate thickness and area respectively. Therefore, the transverse Young’s mod-ulus ET for each wood species was obtained using the expression [22]:

ET ¼ C2Lfqpð1� t2Þg ð23Þ

where t is the Poisson’ ratio.

3. Results and discussion

The determination of the mean squared force F21 was the first

step for the calculation of the dynamic TYM. As mentioned previ-ously, a small polymeric ball was released freely onto the test ob-ject from a height of 3 cm, and the impact frequency was fs = 1 Hz.The velocity of the pendulum mass at impact was equal to 0.8 m/s.

Next, for the wood specimen attached to the impedance tube,the experimental value of the loss factor, using the decay timetechnique (see Fig. 3), was hence found to be approximatelyg = 0.07. This value is an averaged-frequency value over the wholefrequency range. Its measurement was the first step for the deter-mination of the dynamic TYM. It might be higher than the one forthe wood specimen material, and it is suspected that the origin ofthis is most likely to be the high damping and losses at the edgeswhere the adhesive was situated.

As the specimens were highly damped (g = 0.068), due to theenergy material and boundary the power generated by the excita-tion, below the critical frequency fc (fc = 14,652 Hz), is independentof frequency and plate stiffness and inversely proportional to thesquare of the mass per unit area. It is seen in Fig. 4 that the plateimpedance reached a minimum at plate resonance and tended toa constant value Zinf as frequency increases. It is given by

Zinf ¼ 8ffiffiffiffiffiffiffimBp

ð24Þ

where B is the plate bending stiffness and m is the specimen massper unit area.

As expected, a maximum impedance difference of about 2 dBbetween the infinite and finite system impedance was found inthe whole frequency range, except at the 1/3 octave frequency cen-tre equal to 400 Hz, where a difference of about 6 dB was obtained.This was due to the panel first resonance frequency which occurredat approximately 360 Hz. As the excitation was applied on the cen-tre of the specimens, only the first natural frequency of the woodpanel was excited in the frequency range 20–2000 Hz (see Fig. 6).

In spite of all this complexity on measuring input impedanceson finite systems, according to Cremer [26], the power suppliedto a finite plate can be determined by the driving point impedanceof an infinite plate that has the same thickness and material prop-erties as the finite plate. This might be explained by the fact thatthe various modes act as independent energy reservoirs (due tothe spatial average of the square of the velocity) once the systemis excited by a broadband force. In other words, all modes will havethe same energy content on the average.

The first five natural frequencies FN corresponding to the acous-tic axial hard-walled duct modes (see Eq. (2)) and the structuralmodes /p (obtained via FEM) are presented in Table 2 below. Theacoustic and structural uncoupled natural frequencies presentedon Table 2 refer to the values obtained via analytical (Eq. (7))and numerical (i.e. using a FE model) simulation respectively usingphysical and mechanical properties of species Amburana cearensisfrom the literature.

Fig. 5 below shows the spatial variation of the sound pressuremodes in the duct. It is seen that at x = 0, maximum sound pres-sure occurs for all modes due to the rigid end. On the otherhand, at the open-end (x = 1 m), the sound pressure amplitudesare null.

Likewise, Fig. 6 below shows the structural modes obtained viaFEM for a circular thin plate made of wood (e.g. the Amburana cear-ensis). The structural normal displacement modes are presented forthe first six modes. As mentioned previously, it was assumed thatthe wood plate boundaries were simply-supported.

The results in Table 3 are consistent with earlier findings (seeTable 2) suggesting that the coupling between the acoustic andstructural subsystems was not strong enough to significantly alterthe system characteristics in terms of its eigenvalues.

Table 3 below presents the calculated coupled natural frequen-cies of a system composed of a wood plate backed by an acousticfixed-opened end duct. The coupled acoustic natural frequencieswere calculated as an eigenproblem. It is showed the measuredand calculated natural frequencies for the coupled system. Themeasured natural frequencies were obtained by taking the averageof the results for 17 tests (i.e. 17 wood specimens of speciesAmburana cearensis were tested). The calculated values wereobtained using both coupled equations (Eqs. (9) and (10)) andthe species properties measured (e.g. damping and density) andselected from the literature (stiffness).

The results presented in Table 3 indicate fair agreement be-tween the estimates of the measured and calculated natural fre-quencies of the coupled system. The relationship betweenmeasured and calculated values presented significant correlations

100 101 102 10320

30

40

50

60

70

80SP

L [d

B/H

z]

Frequency [Hz]100 101 102 103

20

30

40

50

60

70

80

SPL

[dB/

Hz]

Frequency [Hz]

100 101 102 10320

30

40

50

60

70

80

SPL

[dB/

Hz]

Frequency [Hz]100 101 102 103

20

30

40

50

60

70

80

SPL

[dB/

Hz]

Frequency [Hz]

100 101 102 10320

30

40

50

60

70

80

SPL

[dB/

Hz]

Frequency [Hz]

(a) (b)

(c) (d)

(e)

Fig. 8. The variation of SPL with frequency (1–2000 Hz) due to non-destructive dynamic impacting ball test using 17 specimens for each wood species; (a) Cerejeira; (b)Freijó; (c) Imbúia; (d) Mogno and (e) Ipê.


with values ranging from 61.3% to 99.8%. The difference betweenthe calculated (assuming a simply-supported boundary conditions)measured values possibly indicates that a mixed boundary condi-tion between free and simply-supported might have been realizedexperimentally due to the adhesive and fixture. In addition, somemeasurement errors actually occurred during the experimentaltest for measuring the total loss factor.

Fig. 7 shows the variation of the measured with the calculatednatural frequencies for the coupled system. It appears that the

highest error occurred at around 400 Hz. It is likely that this isdue to the coupling of the first structural mode to the axial acousticmodes as the other structural modes were excited at their nodes(see Fig. 6). It is seen that the coefficient of determination ‘r2’was 0.98. Therefore, there exists strong linear correlation betweenthe two quantities and the experimental test is believed to beacceptable. Therefore, the results obtained are fairly similar, andconsequently provides the validation of the FE model against itsanalytical counterpart.


Fig. 8 shows the variation of SPL (dB/Hz) with narrow frequencyfor five wood species. A number of 17 specimens were tested foreach wood species. The results were obtained using the dynamictest described previously. For each species, a total of seventeenspecimens were tested and the resulting noise measured. The SPLwas found to be significant at most frequency range. Althoughthese results are in substantial agreement with those of Rojaset al. [20], it is evident that the signal-to-noise ratio values hereinare significantly higher. These may be explained by the fact thatthe impedance tube amplified over twice as much the sound pres-sure level as the ones obtained in Ref. [20]. At low frequencies, i.e.below 400 Hz the tube natural frequencies can be identified as themodal density is also low. The number of peaks in the spectrum ismainly concentrated at frequencies above 400 Hz. This is due tothe dynamic and geometrical properties of the tube and woodspecimen. Fig. 8 also shows that the SPL mean values may be un-ique signature for each specimen. These results can be explained

1/3 Octave band frequency center (Hz)

TYM

(GPa

)

1600

1250

100080

063

050

040

031

525

020

016

012

510

08063504031,525201612

,5

1/3 Octave band frequency center (Hz)16

0012

5010

00800

630

500

400

315

250

200

160

125

1008063504031

,525201612,5

1/3 Octave b63504031

,525201612,5

1,75

1,50

1,25

1,00

0,75

0,50

Cerejeira

TYM

(GPa

)

1,75

1,50

1,25

1,00

0,75

0,50

Imbuia

TYM

(GPa

)

2,5

2,0

1,5

1,0

0,5

(a) (

(c) (

(e)

Fig. 9. The variation of the TYM (GPa) with 1/3 octave-band frequency centre. The stockconsidered for each species). (a) Amburana cearensis (Cerejeira); (b) Cordia goeldianaTabebuia spp (Ipê).

by the unique dynamic properties of a particular wood which sug-gests that each specimen was a vibrating structure responsible forthe duct acoustic field excitation. One acknowledges that otherlaboratories and/or methodologies may produce different results.However, this technique has promise as a valuable tool in the eval-uation of wood species identification.

The transverse Young’s modulus (TYM) of a particular woodspecimen is between 1/20 and 1/10 of the longitudinal one. Inother words, the speed of sound across the grain is only 20–30%of the longitudinal value [24]. It means that the TYM varies withgrain direction. In general, the wood sound speed decreases withtemperature and/or moisture. In addition, the TYM decreasesslightly as frequency increases.

Fig. 9 presents the variation of the TYM with one-third octavefrequency centre considering stress-waves propagation normal tothe wood fibers. The results were obtained by measuring the soundpressure levels at the receiving point and processing the data using

and frequency center (Hz)16

0012

5010

00800

630

500

400

315

250

200

160

125

10080


0012

5010

00800

630

500

400

315

250

200

160

125

1008063504031

,525201612,5


0012

5010

00800

630

500

400

315

250

200

160

125

1008063504031

,525201612,5

TYM

(GPa

)

1,75

1,50

1,25

1,00

0,75

0,50

Freijó

TYM

(GPa

)

1,50

1,25

1,00

0,75

0,50

Mogno

Ipê

b)

d)

charts show the mean and standard deviation of TYM values (17 specimens were(Freijó); (c) Ocotea porosa (Imbuia); (d) Swietenia macrophylla (Mogno) and (e)


the Eqs. (18)–(23). It is seen that that the values of TYM depend tosome degree on the frequency band at which it is evaluated. It isalso seen that this dependency is non-linear. According to Ref.[22], this phenomenon is due to the viscoelastic properties of thewood.

Fig. 10 presents a comparison between the TYMs of all speciesconsidered herein. Although the phase speed is directly related tothe modulus of elasticity (Eq. (23)), it is seen that the TYMs areroughly independent for most of the wood species (Imbúia, Freijó,Mogno and Cerejeira). For the Ipê, the TYM values obtained showsa significant discrepancy in comparison to the other species.

Table 4 shows statistical parameter values for the linear corre-lation between the dynamic and static TYM values for five differentspecies. The dynamic TYM values that were used on the correlationare frequency-averaged values over all 1/3 octave frequency bandsconsidered herein. The dynamic and static values were obtained

102 1030

2

4

6

8

10

12

14

16

18x 108

1/3 octave band frequency centre [Hz]

Tran

sver

se M

odul

us o

f Ela

stic

ity [P

a] ImbúiaIpêFreijóMognoCerejeira

Fig. 10. Comparison between the TYM mean values obtained for the five speciesconsidered in the frequency range 25–1200 Hz.

Table 4Measures of center and variability of the transverse Young’s modulus of the woodspecies.

Woodspecies

Average(Gpa)

Standarddeviation(Gpa)

Coefficientof variation

Minimum(Gpa)

Maximum(Gpa)

Cerejeira 0.82 0.253 0.31 0.46 1.66Freijó 0.89 0.267 0.30 0.51 1.64Imbuia 0.79 0.258 0.32 0.45 1.61Mogno 0.73 0.184 0.25 0.38 1.43Ipê 1.27 0.390 0.31 0.75 2.69

Fig. 11. Linear correlation between the dynamic and static TYM values for fivedifferent wood species.

from indirect measurements (i.e. measuring the sound pressureand then using the input power technique given by Eqs. (18)–(23)) and from the literature [31] respectively.

Fig. 11 shows the scatter plot of the data with the least squaresline superimposed. It is seen that 86% of the observed variation inthe dynamic TYM is attributable to the approximate linear rela-tionship between the dynamic and static TYM, a very impressiveresult. Similar results were found in [29], where the correlationsobtained with the dynamic modulus of elasticity via ultrasonictesting were very good but required the knowledge of the density,which adds complexity to the non-destructive testing technique.

4. Conclusions

The study presented herein is an alternative for improving thequality, reliability and reproducibility of the results. Firstly, it isverified that the input power technique is a reliable approachwhich provides a rapid and practical non-destructive measure-ment of the dynamic transverse Young’s modulus. A coefficientof determination of 86% was obtained.

Secondly, the input power approach outlined in this studyshould be a cost-effective technique that can be used in place oftraditional ultrasound techniques which depend on the use ofcostly equipments. In addition, experimental tests can now bemade in a noisier environment (e.g. an ordinary room) where back-ground noise levels (measured in 1/3 octave band) up to 40 dB canbe tolerated.

Thirdly, the experimental tests were validated by the numericalanalytical simulations using a modal model. The relationship be-tween the measured and calculated natural frequencies of the cou-pled system was significant. Comparison of the resonancefrequencies of different wood specimens requires some precau-tions. The resonance frequency measurements should always beconducted with the same apparatus and over the same frequencyrange. The results only indicate that this technique has promiseas a tool in evaluation of the dynamic TYM.

Finally, the acoustic-based technique might be alternatively ap-plied to track changes of mechanical properties along tree height.

Acknowledgments

The authors gratefully acknowledge the financial support pro-vided by the Federal University of Minas Gerais (UFMG) and theMinas Gerais State Research Foundation (FAPEMIG).

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Determination of the transverse Young’s modulus (TYM) of wood by means of an input power technique