PROCEEDINGS of the
23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Development of a quick and non-invasive measurement method forthe extraction of the dispersion relation in CLT plates for theevaluation of the elastic parameters
Arved THIES1; Federica MORANDI2; Luca BARBARESI2; Massimo GARAI2; Jörn HÜBELT1; Niko
KUMER31

Hochschule Mittweida, Germany
2

3

University of Bologna, Italy

Stora Enso WP Bad St. Leonhard GmbH, Austria

ABSTRACTThis contribution discusses a quick and non-invasive method for the extraction of dispersion relations in CLT
panels, with the aim of providing an estimate of the elastic parameters of the plates starting from the fittingof the experimental dispersion curve to the theoretical models. 
Measurements were performed at the Stora Enso CLT mill on panels that have different thickness andorthotropic ratio. An instrumented hammer was used to excite the plate and accelerometers were used toacquire the impulse responses. Thanks to the reciprocity principle, the measurement points were kept fixed
and the hitting point was moved; this allowed to reduce the measurement time and to decrease the number ofaccelerometers that must be attached.
The dispersion curves were evaluated using the phase shift method and the experimental results were fittedwith the theoretical models for acoustically thick plates; finally, the elastic parameters were estimated. 
The results give slightly low estimates of the Young’s modulus and the shear modulus in the stiffer direction,
while confirming the datasheet values in the less stiff direction, with deviations depending on the panel
thickness. The precision of the hits is also analysed, while a reproducibility study is left for further
developments.
Keywords: Cross Laminated Timber, Bending Wavenumbers, Structural Acoustics

1. INTRODUCTION
The CLT construction technique is widely used and can count on a distinct performance in building
physics and structural features (1-3). Among the most relevant acoustic parameters that must be known
for the characterization of a propagation medium, stand the velocities of longitudinal and bending
waves, which are directly dependent on the elastic parameters of the material. Although thesequantities can be accurately predicted for homogeneous materials, the anisotropic and inhomogeneous
nature of wood complicates the estimate. Methods for the extraction of the elastic parameters by either
vibrational testing (4) or by measurement of the dispersion relation (5) have been shown.
This paper presents a preliminary analysis of another quick and non-invasive method for thedetermination of the mechanical characteristics of CLT plates based upon the measurement of thedispersion relations. The work seeks a correlation between the vibro-acoustical characteristics of the
CLT panels and the internal composition of the plies, in terms of numbers and thickness of the planks.
To reach this goal, the velocities of longitudinal and bending waves were measured on different CLT
panels characterized by: different total thickness, different number and thickness of the plies, different
wood species and different surface qualities.
The dispersion relations are measured for eight panels in which all these parameters are varying
and, through a fitting with theoretical models, the elastic constants of the panels are estimated. The
final aim is to provide a comparison between the so-calculated values and the results of the structural
1 athies1@hs-mittweida.de; huebelt@hs-mittweida.de2federica.morandi6@unibo.it; luca.barbaresi@unibo.it; massimo.garai@unibo.it

3 niko.kumer@storaenso.com

5036

tests that are done for quality control purposes, and that imply a huge expense for laboratory tests.
Section 2 provides a brief presentation of the theoretical fo rmulations for the dispersion relationsof acoustically thick plates in contrast to the thin plate theory. Furthermore, the phase shift method
which was used for the evaluation of the dispersion images is described.
Section 3 describes the measurement setup, the panels tested and the process used to extract theelastic parameters from the dispersion relations. Section 4 discusses the results of the comparison interms of the Young’s modulus ‫ ܧ‬and the shear modulus ‫ ܩ‬and provides an estimate on theuncertainty of the results calculated with the proposed method. Finally, the conclusion discusses theadvantages and the limitations of the proposed approach and discusses potential improvements of themethod.

2. BACKGROUND
2.1 Acoustically thick plates
The dispersion relation for bending waves in acoustically thin plates is well known and given in
Eq. (1), where ‫ ܧ‬is the Young’s modulus, ‫ ܫ‬the second moment of area, Ɋ the Poisson ratio and ݉ԢԢ
the surface mass of the plate. This equation is based on the Kirchhoff model of the thin plate and thus
only holds under the assumption that shear forces are negligible.
ర‫ܫܧ‬ܿ஻ǡ௄ ൌ ඨξɘሺͳ െ Ɋଶ ሻ݉ᇱᇱ(1)

The most common model for thick plates is the Mindlin model (6), which takes shear forces into
account. The dispersion relation for the bending waves according to Mindlin is given in an implicit
formulation by Eq. (2). As a result of taking shearing forces into account, the maximum bending wave
velocity ܿௌ is limited by the non-dispersive shearing wave velocity which is given in Eq. (3), where‫ ܩ‬is the shear modulus and ɏ the density. The longitudinal wave velocity ܿ௅ is given in Eq. (4), while
the coefficient ܶ, which takes the non-uniformity of the shear stress into account, is given in Eq. (5).
The bending wavenumber ݇஻ǡ௄ which is used for the calculation, can be derived from Eq. (1).
ቆͳ െ

ଶ
ܿ஻ǡெ
ܿ௅ଶ
ͳʹ
ቇ
ቆ
ଶ
ଶ െ ͳቇ ൌ  ଶ ଶଶ݄ ݇஻ǡ௄ܶ ܿௌ ܿ஻ǡெܿௌ ൌ ඥ‫ܩ‬Ȁɏ; ܿ௅ ൌ ඥ‫ܧ‬ȀɏܶൌͲǤͺ͹ ൅ ͳǤͳʹɊ
ͳ൅Ɋ

(2)

(3); (4)(5)

A simplified model for the dispersion relation was presented by Rindel (7). It is based on the idea,that for low frequencies the bending wave velocity shows asymptotical behaviour towards the thinplate model, while for high frequencies it is asymptotic towards the shearing wave velocity. In thiscase the effective bending wave velocity is given by Eq. (6). For identical elastic parameters thedifference between the models by Rindel and Mindlin is insignificant.
ͳͳܿ஻ǡ௘௙௙ ൌ ቈ ଷ ൅ ଷ ቉ܿ஻ǡ௄ ܿௌ

ିଵȀଷ

(6)

2.2 Phase shift method
An efficient method to measure the dispersion relations is a multichannel analysis. From this, thedispersion images can be directly computed by a wavefield transformation that will be referred to as
the phase shift method in this paper.
The phase shift method is elaborately described and analysed by Park et. al. (8). It is based ontransforming the recorded wavefield ܷሺ‫ݔ‬ǡ ‫ݐ‬ሻ from the space and time domains to the space andfrequency domains to obtain ܷሺ‫ݔ‬ǡ ɘሻ. Now the wavefield can be expressed as the product of theamplitude spectrum ‫ܣ‬ሺ‫ݔ‬ǡ ɘሻ and the phase spectrum ܲሺ‫ݔ‬ǡ ɘሻ which contains the information about
the dispersion characteristics. By applying another integral transformation, the dispersion imageܸሺɘǡ Ԅሻ is obtained as shown in Eq. (7).

5037ሺɘǡ Ԅሻ ൌ න ݁ ି௝ሺ஍ିԄሻ௫‫ܣ‬ሺ‫ݔ‬ǡ ɘሻ݀‫ݔ‬ȁ‫ܣ‬ሺ‫ݔ‬ǡ ɘሻȁ

(7)For every frequency, ܸ will have a maximum if Eq. (8) holds, i.e. if the measured bending wave
velocity is employed. In case of orthotropic materials, the process must be performed in the two major
directions.
(8)Ԅ ൌ Ȱ ൌ ɘȀܿ஻The phase shift method is most commonly applied in geophysics and used for the measurement of
the dispersion characteristics of surface waves (9). In acoustics similar methods have been appliedand are best known as inhomogeneous wave correlation method or spatial Fourier transform (10).

3. MEASUREMENT SETUP
3.1

Testing configurations and procedure
Table 1 - Description of the panels tested.
Thickness

Layer

EmajorEminorfc,major

fc,minor

Orthotropic[mm]composition

[MPa]

[MPa]

[Hz]

[Hz]

factor1

18040-30-40-30-40

10914

2086

73

167

5.2

2

180

40-30-40-30-40

10914

2086

73

167

5.2

3

120

40-40-40

12519

481

102

520

26.0

4

220

60-30-40-30-60

11857

1143

57

184

10.4

5

100

30-40-30

12168

832

124

475

14.6

6

100

30-40-30

12168

832

124

475

14.6

7

160

40-20-40-20-40

11578

1422

80

227

8.1

8

140

40-20-20-20-40

12015

985

89

312

12.2

Panel

In total, eight different CLT-panels of varying specifications and overall dimensions were tested.
The characteristics of the different panels are shown in Table 1. Additional differences between thepanels were the surface quality, which influences the moisture content of the wood and the woodspecies, which was spruce or pine with densities of 470 and 500 kg/m 3 respectively.
All panels were tested directly as they came out of the production line, before they were loaded
for shipping. To ensure comparable boundary conditions, all panels were simply supported on twoedges by two beams with cross sections of 9x9 cm. Two additional layers of expanded EPDM rubber
were used for decoupling between the panel and the support. Since all CLT panels are made from an
odd number of layers of orthogonal planks, they are inherently orthotropic. The stiffer direction of thepanels will be referred to as x-direction.

Figure 1 - Measurement setup for the bending wave velocity in x-direction.

5038

The measurement setup for the multichannel analysis is shown in Figure 1. In total, only fourchannels were recorded for the measurement: one instrumented hammer and three accelerometers. All
channels were recorded through a signal conditioner (PCB 482C15) connected to an RME Fireface
802 soundcard running at 192 kHz. The data was recorded in MATLAB® with the help of the ITAToolbox (11). The accelerometers (PCB 352C33) were placed in a line with 3 cm between each otherand fixed to the panels with wax. Their position was chosen with a sufficient distance to the boundaries,
hence the influence of boundary conditions and reflections could be minimized. The instrumented
hammer (APTech AU02) was equipped with a rubber tip, so no dents were made in the wood, and 21
hits were executed on a previously drawn grid as shown in Figure 1. In the case of particularly small
panels, it was made sure, that the distance between the last hitting point and the boundary was at least
0.7 m because otherwise negative effects caused by reflections showed. By aligning the recorded dataof each hit in the time domain, an equivalent to a 64-channel measurement was created.
From the measured data, the dispersion images were calculated using the phase shift method.
3.2

Intermediate data processing
1600

lower boundupper boundupper limit
unfiltered results
first selection
second selection
Rindel 1st approx
Rindel
Mindlin

1400

Velocity [m/s]

1200
1000

800
600
400
200
0

0

500

1000

1500

2000

2500

3000

3500

4000

Frequency [Hz]Figure 2 - Dispersion relation measured for panel 8 (parallel to grain) and fitting with theoretical models.
Once the dispersion images were calculated, the experimental data was fitted with theoretic almodels to provide an estimate of the E and G moduli. Figure 2 shows the dispersion image for panel
no. 8 in the x-direction. All circles represent the maximal value of the dispersion images for eachfrequency and thus the measured velocity for that frequency. It can be seen, that there are somespurious values and hence before the fitting, the data needed to be cleaned. The process is briefly
summarized below.
1) A first selection of points was performed based on the fact, that ܿ஻ is limited by ܿௌ . Becauseof that, a value sufficiently higher than the expected ܿௌ value could be used as the upper limit
above which all points got discarded. Further boundaries were chosen such that the influenceof higher modes and aliasing effects could be minimized. The points discarded in step one areplotted in grey.
2) A preliminary fitting using Rindel’s simplified model was performed.
3) All points that deviated past a certain threshold from the first fitting were discarded. These
points are plotted in black.
4) Based on the second selection of points (red), the data was fitted with several models including
Mindlin’s model and Rindel’s simplification. The Poisson ratio was kept fixed to 0.3.
5) Another fitting was performed with the same models and selection of points, only this time the
Poisson ratio was included as a variable in the fitting.

5039

4. DISCUSSION OF THE RESULTS
4.1 Comparison of E and G
All theoretical models used for the fitting of the data fit the measured dispersion images well, as
can be seen for Rindel and Mindlin models in Figure 2. The measured values of the elastic parameters
show differences between the models. The estimated values for the elastic parameters according to
the models from Rindel and Mindlin of panel no. eight are shown in Table 2. For the two models thefitting was executed both with a Poisson ratio (Ɋ) fixed to 0.3 as well as with Ɋ as a parameter for thefitting. Rindel’s simplified model showed spurious behaviour when Ɋ was used as a variable,
returning values out of the physically sensible range, while the dispersion of the elastic parameters
for repeated measurements estimated with Mindlin’s model with variable Ɋ was bigger than with
fixed Ɋ. This might be due to not restraining Ɋ during the fitting process and allowing different ratios
among the different directions. Because of that, the values for the elastic parameters with variable Ɋ
were discarded and are not included in the further discussion.
The comparison of the remaining results shows a deviation between the repeated measurements of
up to 20 %. This fact is discussed in section 4.2. All models deliver similar values with deviations
smaller than 20 % for the elastic parameters with very few exceptions. Rindel’s model generally
delivers slightly lower values for both E- and G-modulus compared to Mindlin’s. A graphicalrepresentation of this behaviour is given in Figure 3, while the tabular data for one panel is given in
Table 2. Contrary to the variations in the elastic parameters, the estimates for the critical frequencies
are very close together, both among the repeated measurements as well as between the different
theoretical models.
Table 2 - Comparison of the measured elastic parameters for panel no. 8 from two repeated measurements.
Model

Rindel

Mindlin

Measurement

1

2

1

2

Emajor [MPa]

2914

3314

3310

3734

Eminor [MPa]

1046

923

1153

1055

Gmajor [MPa]

372

352

424

402

Gminor [MPa]

173

205

200

242

Orth. factor (E)

2,8

3,6

2,9

3,5

Orth. factor (G)

2,1

1,7

2,1

1,7

fc,major [Hz]

188

180

188

176

fc,minor [Hz]

344

352

356

356

Comparing the measured data to the values given in Table 1 shows greatly different results. In
general, the tested panels can be divided into two different groups with a different behavior: the thickpanels, consisting of five layers, and the thin panels, made from three layers of planks. For one panelof each group, a graphical comparison between the measured values and the datasheet values is given
in Figure 3 in form of a radar chart.
In the group of the thick panels the measured values for E major are generally much lower than theones given in the datasheets. The measured values range from 60 to 70 % lower than the datasheet
values, with higher deviations for higher orthotropic ratios. The measured values for Eminor on theother hand are very close to the datasheet values, averagely being around 10 % higher. However, there
are deviations of up to ±15 % between the different tested panels, without any obvious correlation tothe orthotropic factor.
Looking at the measured shear moduli in the group of the thick panels, there are huge devia tions
from the datasheet values. On average the measured Gmajor is almost one order of magnitude smaller
than the datasheet value, again with larger differences for higher orthotropic factors. This differenceis smaller for Gminor with a deviation of less than -50 % from the datasheet.

5040

Emajor
10,00
0,00

Orthotropic factor (G)-10,00Panel no. 8 (5 layers)
Eminor-20,00Datasheet-30,00Rindel-40,00

MindlinRindel (rep)GmajorOrthotropic factor (E)Mindlin (rep)Gminor
Emajor
10,00

Panel no. 3 (3 layers)

0,00

Orthotropic factor (G)-10,00Eminor-20,00Datasheet-30,00Rindel-40,00

MindlinRindel (rep)GmajorOrthotropic factor (E)Mindlin (rep)GminorFigure 3 - Radar charts comparing the measured elastic parameters to the datasheet values for a 5-layer
panel (top) and a 3-layer panel (bottom). For better comparability and legibility, the axes have been
converted to a dB-scale relative to the datasheet value.
All this leads to a reduction of the measured orthotropic ratios down to between one quarter and
one third of the datasheet values in terms of Young’s moduli and an even bigger reduction to less than
one quarter of the datasheet values in terms of shear moduli. The biggest dispersion in the data can befound in the measurements of Gminor and thus also in the orthotropic factor derived from the shear
moduli. This is because the evaluation of the shear modulus is mainly influenced by the measured
velocities for higher frequencies, which are particularly noisy in the y-direction.
In the group of the thin panels the general picture is similar, but there are some exceptions. While
the measured E major is still smaller than the datasheet values, the deviations are not as drastic with
only -35 % for very high orthotropic factors. In terms of Eminor there are pronounced differences
towards the thick panels. The measured values are up to 100 % higher than the datasheet values, again
in case of the highest orthotropic factor.
Looking at G-moduli, this behaviour repeats in the less stiff direction with measured G minor being
higher than the datasheet value. In the stiffer direction the shear modulus is one order of magnitude
smaller than the datasheet value, like with the thicker panels. This leads to an orthotropic factor of
more than one order of magnitude smaller than the given value in terms of shear moduli. In terms ofYoung’s moduli, the values are again around one third of the datasheet values. All in all, for the thinner
three-layer panels the deviations between the theoretical models as well as among the repeatedmeasurements are a little smaller, than for the thicker panels consisting of five layers.

5041

One possible reason for the deviation of the measured elastic parameters from the datasheet values
lies within the fact, that for the dynamic and non-invasive testing, the achievable strains are extremely
small in comparison to the classical methods for the evaluation of the elastic parameters. Assumingthat there is nonlinear behaviour in the stiffness of the wood, this is a possible explanation for theunderestimation of the elastic parameters in the stiffer direction, which is especially apparent f or the
thicker panels.
4.2 Evaluation of the uncertainty of the measure and presentation of the hitting accuracy
As already shown in section 4.1, the values of the elastic parameters show deviations of up to 20 %between repeated measurements on the same panel. This limits the validity of the measured data, as
well as the ability to measure small changes in the parameters.
One contribution to the uncertainty of the measurement can be found in the hitting accuracy. Since
the instrumented hammer is operated by a person and the panel needs to be hit in several different
points, there will be aberrations from the ideal hitting points. To investigate the amount of aberration,
several measurements with a steel tip on the instrumented hammer were performed on a dummy panel.
The steel tip left noticeable dents in the wood, such that the distance between the centre of the dent
and the intended hitting point marked on the grid could be measured. This was done for both thelongitudinal and the transverse deviations for 210 hitting points. The results are shown in Figure 4.
The average absolute error is 1.9 mm and close to a normal distribution. Although less than two
millimetres seem to be a small error this already represents almost seven percent of the distance
between the accelerometers, making up a potential source for the noisy data and thus, the poor
repeatability. Another potential source for noisy and dispersive data is the hitting strength. Especiallyfor the thicker panels a compromise between exciting the panel properly and not leaving dents had to
be made, with the priority in not leaving any dents in the panel.

Occurrence

150
120

All
Long
Tran

90
60
30
0
0

1

2

3

Error [mm]

4

5

6

Figure 4 - Distribution of the aberrations from the actual hitting points in longitudinal, transverse andcombined directions from the grid.
Another potential problem is the placement of the accelerometers, both in terms of noise in thedata as well as the reproducibility. The accelerometers needed to be fixed to the panel for e verymeasurement and although great care was taken to place them as accurately as possible, small
variations in the sub millimetre range can not be ruled out. The problem with this error is that it isconstant for all hitting points and not statistically evening out over the course of the measurement likethe error regarding the hitting accuracy. Hence, even smallest deviations in the placement of theaccelerometers might cause a significant variation in the results, especially in combination with the
non-infinitesimal dimensions of the accelerometers and their inherent measurement uncertainty.
As noted in the beginning of this chapter, the data is especially noisy in the very low as well as in
the upper frequency range of the measurement. In the low frequen cy range this is due to the
pronounced modal behaviour of the CLT panels in that frequency range and the limited total length
available for the measurement. Less than two meters of total distance were covered by themeasurement and the dispersion image gets noticeably noisier as soon as the wavelengths start to get
into that range. As a possible reason for the suboptimal behaviour for frequencies above 2 kHz aliasing
was already mentioned. Checking with the measured data it can be seen that this should not be anissue since the shortest measured wavelengths were larger than 15 cm, while the spac ing of theaccelerometers with 3 cm is sufficiently smaller.
Summing up these factors leads to a noticeable uncertainty in the measurement of the dispersion
images and thus the bending wave velocity. Additionally, the elastic parameters are extremely
sensitive to small changes in measured velocity, for example the Young’s modulus depends on the
fourth power of ܿ஻ .

5042

5. CONCLUDING REMARKSA quick and non-inversive measurement method for the evaluation of the dispersion relation andthus, the elastic parameters of CLT-plates has been developed. By performing a multichannel analysis,that was optimised in terms of required channel count and measurement time, and processing it withthe phase shift method, the dispersion images were obtained. From this, with the hel p of theoreticalmodels for the acoustically thick plate, the elastic parameters of the tested panels were evaluated. The
different models provide similar results, suggesting that the measurement method is valid. O n theother hand, the measured values of the elastic parameters strongly deviate from the values given in
the datasheet of the panels, although the deviations seem to follow a certain pattern. Generally, themeasured values tend to be smaller than the calculated ones, which is especially apparent in the stiffer
direction, leading to smaller orthotropic factors of the panels. In terms of the shear moduli thisbehaviour is even more pronounced, than for the Young’s modulus.
As of now the accuracy and reproducibility of the proposed method is insuffi cient. Repeating
measurements on the same panels showed deviations of up to 20 % for the elastic parameters, which
renders the results of the measurements questionable. Small changes in the panel properties can not
be measured reliably, while the big deviations between the repeated measurements make finding
correlations between the orthotropic factors, the number of the panels and the elastic parameters
difficult. Therefore, the collected data are not yet sufficient for finding improved models for theprediction of the acoustic behaviour of CLT-panels. On the upside, the measurements provide accurate
results for the critical frequencies, with small deviations between different models and repeatedmeasurements.
The main reasons for the deviations between the measurements seem to be the accuracy of thehitting with the instrumented hammer, as well as the placement of the accelerometers, which may begreatly improved on, by automating the process.

ACKNOWLEDGEMENTS
Measurements were done in the Stora Enso CLT factory in Bad St. Leonhard, Austria. The authors
gratefully acknowledge the structure that hosted the measurements and the people that helped insetting up the testing configurations.

REFERENCES
1. Asdrubali F., Ferracuti B., Lombardi L., Guattari C., Evangelisti L., Grazieschi G. A review of structural,
thermo-physical, acoustical and environmental properties of wooden materials for building applications.
Building and Environment. 2017;114:307-322.
2. Caniato M., Bettarello F., Ferluga A., Marsich L., Schmid C. and Fausti P. Acoustic of lightweight timberbuildings: a review. Renewable and sustainable Energy Reviews, 2017:80C: 585-596.
3. Caniato M., Bettarello F., Ferluga A., Marsich L., Schmid C. and Fausti P. Thermal and acoustic
performance expectations on timber buildings. Building Acoustics, 2017:24(4):219-237.
4. Guan C., Zhang H., Wang X., Miao H., Zhou L., Liu F. Experimental and Theoretical Modal Analysis of
Full-Sized Wood Composite Panels Supported on Four Nodes. Materials. 2017;10;
5. Santoni A., Schoenwald S., Van Damme B., Fausti P. Determination of the elastic and stiffness
characteristics of cross-laminated timber plates from flexural wave velocity measurements. J. Sound Vib.
2017;400:387-401.
6. Graff K.F. Wave Motion in elastic solids. Dover Publications Inc. New York, 1975.
7. Rindel J.H. Dispersion and absorption of structure-borne sound in acoustically thick plates. Appl. Acoust.
1994;41:97-111.
8. Park CB, Miller RD, Xia J. Imaging dispersion curves of surface waves on multi-channel record. Societyof Exploration Geophysicists 1998;10(1):1377-1380.
9. Park CB. Imaging Dispersion of MASW Data - Full vs. Selective Offset Scheme. Journal ofEnvironmental and Engineering Geophysics 2011;16(1):13-23.
10. Van Damme B., Zemp A. Measuring Dispersion Curves for Bending Waves in Beams: A Comparison of
Spatial Fourier Transform and Inhomogeneous Wave Correlation. Acta Acustica 2018; 103.
11. Berzborn M. et al. The ITA-Toolbox: An Open Source MATLAB Toolbox for Acoustic Measurements
and Signal Processing. 43th Annual German Congress on Acoustics, Kiel, 2017.

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