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

Investigations regarding the influence of static load and airflow
resistance on the measurement of dynamic stiffness
Heinrich BIETZ1; Martin SCHMELZER1; Volker WITTSTOCK1; Spyros BREZAS
1

Physikalisch-Technische Bundesanstalt, Braunschweig, Germany

ABSTRACT
Within the scope of a research project regarding the physical properties of insulation materials made fromrenewables, one of PTB’s tasks is to determine the dynamic stiffness of such materials used as elastic layers.
The common measurement procedure is described in DIN EN 29052-1. As a closer examination of thisstandard bears some open questions, several investigations have been conducted recently. This presentation
focuses on two aspects: The first one is the influence of the static preload. This might be relevant for
insulation materials used for thermal insulation composite systems, where, compared to floating floors, thestatic preload is significantly smaller. As the standardized measurement procedure is not suited for small
static preloads, an alternative measurement set-up was designed to investigate this topic. The second aspectis the effect of airflow resistance on the resulting dynamic stiffness. The measurement setup as devised in thestandard does not necessarily reflect the boundary conditions the material is exposed to when it is mounted
in situ. The standard describes a procedure to handle the influence of airflow resistance, but this is partially
incomplete and not applicable for some materials. Some theoretical considerations and measurement results
regarding this topic will also be presented.
Keywords: Dynamic stiffness, insulation materials, airflow resistance, sound insulation

1. INTRODUCTION
Insulation materials are often used as elastic layers in floating floors or thermal insulation
composite systems. One of the relevant parameters to predict the structure-borne or airborne soundinsulation of such constructions is the dynamic stiffness of the elastic layer. For materials used infloating floor constructions, the measurement procedure to determine the dynamic stiffness is
described in the international standard ISO 9052-1 (1) and the corresponding German standard DIN
EN 29052-1 (2). It is one of PTB’s tasks within the framework of a research project regarding theproperties of insulation materials made from renewables to measure the dynamic stiffness of a wide
range of samples. The standardised measurement set-up was realised, and some preliminary tests were
performed. During these tests, the standard procedure soon revealed to bear some open questions,which were already partially investigated in the past (3,4). Chapter 2 presents a brief description of
the standardised procedure and some of its shortcomings. A pivotal aspect is the influence of the static
preload on the dynamic stiffness. This is especially important if the insulation material is used in athermal insulation composite system and not in a floating floor, as in this case it is exposed tosignificantly smaller static loads. As the standardised set-up is not suited to handle small static loads,
an alternative measurement set-up was designed and tested. The alternative set-up and the obtained
results are discussed in chapter 3.
If an insulation material is employed in a floating floor construction to create an elastic layer, it iscontained in an approximately airtight enclosure. This is not reflected by the standardised
measurement set-up, where the edges of the sample are uncovered, allowing the surrounding air totravel freely in and out. Thus, depending on the physical characteristics of the sample, the stiffness ofthe enclosed air may not be considered correctly. The standard contains some directions on how toapproach this problem, but these are in some ways inconsistent. A detailed discussion of this problem
and a suggestion for a possible workaround are given in chapter 4. Finally, chapter 5 presents someinvestigations regarding the influence of the excitation force.

1

1402

heinrich.bietz@ptb.de2. STANDARDISED MEASUREMENT PROCEDURE
For the standard procedure, a sample with the size of 200x200 mm² is prepared from the insulationmaterial. The sample is placed on the floor or on a heavy base plate, and a steel plate with a thicknessof 25 mm is put on top of the sample as mechanical load. Together with a layer of gypsum, which acts
as a levelling course, the overall surface mass load adds up to 200 kg/m², which is a good
representation for a typical concrete screed. Figure 1 shows a sketch of the arrangement.
F

Steel plate
Elastic material
Base plate

Figure 1 – Sketch of standard arrangement
As the base plate is considered stationary, the whole arrangement forms a spring-mass system, withthe sample acting as a spring with a stiffness s, while the steel plate and the gypsum layer act as amass m. By determining the resonance frequency ݂୰ , the dynamic stiffness s of the sample, which isusually expressed per unit area, can be calculated using‫୲ݏ‬ᇱ  ൌ Ͷߨ ଶ ή ݉୲ᇱ ή ሺ݂୰ ሻ;(1).

The index “ t ” is used to indicate that the influence of the enclosed air is perhaps not considered
properly, and that a correction term must be added. This will be comprehensively discussed inchapter 4.
The resonance frequency is measured by excitation of the system with a periodical force F. This
must be applied in a way that the system oscillates only in vertical direction, which is usually best
achieved by applying the excitation force at the middle of the steel plate. Stationary excitation(sinusoidal or noise) using a shaker or impulse excitation with a hammer are possible. When using
sinusoidal excitation, the resonance frequency is determined by varying the excitation frequency while
keeping the excitation force unchanged. A special procedure must be followed if the stiffness of thematerial is dependent on the excitation force. Chapter 5 takes a look at this topic.

3. ALTERNATIVE SET-UP FOR SMALL STATIC LOADS
3.1 Measurement Set-up
As already mentioned, an insulation material is exposed to a much smaller static load when used
in a thermal insulation composite system. The measurement standard states that in this case, avariation of not more than 20 % for the dynamic stiffness, with respect to a load of 200 kg/m³, can be
expected, which is regarded as negligible for practical applications. During the ongoing research
project, it was nevertheless desired to verify this effect by measurements. As an excitation of a thinand lightweight load is likely to be difficult when using point excitation, an alternative me asurement
set-up was designed. Here, a base plate with a mass ݉ଵ of 13 kg is mounted on a robust vibration
exciter (shaker), which is capable to bear high mass loads. On this base plate, the sample and the massload are mounted as in the standard set-up, but with an aluminium composite panel with a total mass݉ଶ (including the still required gypsum layer) of 0.3 kg, instead of the steel plate. The whole system
is now excited by the shaker using a swept sine signal, and the measurement delivers the acceleration
spectra ܽଵ ሺ߱ሻ and ܽଶ ሺ߱ሻ . Eventually, the complex transfer function ܽଶ ሺ߱ሻȀܽଵ ሺ߱ሻ is calculated.
Figure 2 shows a schematic diagram and the realisation of the alternative measurement set-up.

1403

Figure 2 – Schematic diagram (left) and realisation (right) of the alternative test set-up
As wobbling and tilting movements of both the base plate and the preload cannot be completely
excluded, it is generally desirable to measure the acceleration at the centre spot of the plates. As thisis not possible for the base plate, the measurement is performed on all four corners of the plate, andsubsequently a complex average of the obtained acceleration spectra is calculated. This approach wasvalidated by measuring the acceleration response of the base plate at the centre and on the corners(without a test sample). The result is shown in Figure 3. Up to a frequency of 300 Hz, the averaged
measurements at the corners produces the same results as the measurement at the centre. The
deviations at higher frequencies are caused by the fact that the base plate does not act as a rigid body
anymore.
100

100

dB

dB
A2

95

A3

A4

A5

A1(Center)

Average A2-A5

A1(Center)

95

90

La

La

90

85

85

A5

A2
A1

80
A3

A5

A2
A1

80
A3

A4

A4

75

75
10

100

Hz

1000

10

100

Hz

1000

f

f

Figure 3 – Acceleration responses on the corners measured individually (left graph) and averaged (rightgraph) vs. response at centre of the base plate

While the procedure according to the measurement standard assumes a simple spring-mass system,
we now have a slightly more complex mass-spring-mass system, which is the reason why it is not
sufficient to measure the acceleration on the load plate (ܽଶ ሺ߱ሻ) only, but also on the base plate and to
determine the stiffness from the transfer function ܽଶ ሺ߱ሻ / ܽଵ ሺ߱ሻ. As Figure 4 illustrates, the transfer
function shows a different resonance frequency, which is the correct value for this set-up.

1404

10,0

10,0

1,0

1,0

a2/a1

a1

a2

0,1
10

a2/a1

a

m/s²

100

0,1
Hz 1000f
Figure 4 – Acceleration response of the base plate (ܽଵ ሺ߱ሻ) and the load plate (ܽଶ ሺ߱ሻ), and correspondingtransfer function ܽଶ ሺ߱ሻ / ܽଵ ሺ߱ሻ
3.2 Evaluation Procedure
The dynamic stiffness may be determined the same way as in the standard procedure by detecting
the resonance frequency from the transfer function. With this setup, however, it is also possible now
to calculate the stiffness for every frequency Z, at least in the frequency range where the measurement
set-up works properly. The simplest approach is a mass-spring-mass system with a lossless spring:‫ݏ‬Ԣ୲ ൌ  ݉Ԣଶ ߱ଶ ͳͳ െ ܽଵ Ȁܽଶ

(2)

At the resonance frequency ߱୰, the fraction on the right-hand side approaches 1, and Eq. (2) equals
Eq. (1), which is used in the standard procedure. A more realistic approach is to model the test sample
as a spring with losses. The resulting equation is‫ݏ‬Ԣ୲  ൌ ݉Ԣଶ߱ଶ ͳ െ ȁܽଵ Ȁܽଶ ȁ ߮ሺͳ െ ȁܽଵ Ȁܽଶȁ ߮ሻଶ ൅ ሺȁܽଵ Ȁܽଶ ȁ߮ሻଶ

(3)

Both approaches assume lumped elements (mere masses and springs). As the mass of the test
sample is not negligible anymore when using small static loads, a modelling of the test sample as an
expansion bar, with or without losses, can also be considered. The resulting equations can probably
only be solved with a numerical approach, which was also used but will not be described in detail.
3.3 Measurement Results
At the time of writing, only one test sample (80 mm EPS foam) was tested. This is, however, avery common material used for thermal insulation composite systems. Apart from the low preload of
0.3 kg, an added mass of 2.5 kg and the steel plate as used in the standard procedure were also tested.
For all measurements, the excitation force was varied in a range of 30 dB. A selection of the results
for a mass load of 0.3 kg and 8 kg is shown in Figure 5 and Figure 6.

1405

17
Spring, losslessSpring, with losses
using resonance freq.
Exp. Bar, losslessExp. Bar, with losses

70

60
50

40

Dyn. Stiffness in MN/m³

Dyn. Stiffness in MN/m³

80

30
20
10
0
0

100

200

300
400
500
Frequeny in Hz

600

16
15
14

Spring, with losses
Resonance frequency
using resonance freq.
Exp. Bar, with losses

13

12

700

180

190

200
210
Frequeny in Hz

220

230

Figure 5 – Left: Measurement with ݉ଶ = 0.3 kg mass load. Right: An excerptof the left-hand graph without the lossless solutions
80

Dyn. Stiffness in MN/m³

Dyn. Stiffness in MN/m³

100
80
60
40

Spring, losslessSpring, with losses
using resonance freq.
Exp. Bar, losslessExp. Bar, with losses

20
0
0

100

200

300
400
500
Frequeny in Hz

600

700

Spring, with losses
Resonance frequency
using resonance freq.
Exp. Bar, with losses

70
60

50
40
60

70

80
90
Frequeny in Hz

100

110

Figure 6 – Left: Measurement with ݉ଶ = 8 kg mass load. Right: An excerptof the left-hand graph without the lossless solutions
In most cases, the lossless solutions do not deliver reasonable results. However, if losses are
considered, the “spring” and the “expansion bar” approach both do not only show very similar resultsover a wide frequency range, but also a good match with the dynamic stiffness obtained by using theresonance frequency according to Eq. (1). Especially with the small mass load, one could expect aneven better match when the expansion bar model is used. On the other hand, the difference between
the spring and expansion bar solution is quite small, and the determination of the resonance frequency
also has a considerable uncertainty.
The major topic to be investigated with the alternative set-up is the relation between dynamic
stiffness and mass load. Table 1 shows the results for the sample under test. The displayed stiffnesses
were calculated using the resonance frequency. The measurements were performed keeping the current
fed into the shaker on a constant level.
Table 1 – Relation between dynamic stiffness and mass load, EPS 80 mm

m in kg

0,3

2,8

8,0

f in Hz

212,0

95,7

80,1

‫ ୲ݏ‬Ԣ in MN/m³

14,8

25,6

50,7

2

r

1406

For this test, the dynamic stiffness decreases by a factor of about 3 in relation to the standard
procedure when a small mass load is employed. This is initially in contradiction to the statement in
the measurement standard, where a deviation in the maximum range of 20 % is proposed. Here, ofcourse, more research is needed to verify the observed effect.

4. INFLUENCE OF THE ENCLOSED AIR VOLUME
The current version of ISO 9052-1 (1) describes how the stiffness due to the air enclosed in the
porous elastic layer should be considered if the measurement is performed according to Figure 1. Thisis based on work by Kraak (5) which is briefly sketched in the following.
Starting point are the equations of motion for the porous mediaെ‫ ݌‬ൌ Fߩ଴ 

డ࢜
డ௧

 ൅ ߝߪ࢜Ǣെ࢜ ൌଵ డ௣N௣బ  డ௧

െ

௩೛

(4)

ௗఌ

with the fluid pressure p, fluid velocity v, mean density U 0, static pressure p 0, porosity H, airflow
resistivity V, structural factor F , ratio of specific heats N , thickness of the porous layer d and thevelocity of the cover plate v p. Using harmonic excitation with angular frequency Z gives௩೛ଵെ‫ ݌‬ൌ ݆߱Fߩ଴ ࢜ ൅ ߝߪ࢜Ǣെ࢜ ൌ ݆߱ N௣  ‫ ݌‬െ ௗఌ
బ

(5)which leads to the wave equationο‫ ݌‬െ ݆߱ఌఙN௣బ ఙ‫ ݌‬ൌ െ ௗ ‫ݒ‬௣ (6).

Using polar coordinates, i.e. a circular cover plate, normalising the radial coordinate byߙ ൌ  ඥെ݆ට

ఠఌఙ

N௣బ

(7)



and averaging the pressure over the cover plate finally givesଶ ௃ ሺఈோሻ

N௣బ

బ

ఌௗ

‫ݏ‬ఙᇱ ൌ ‫ݏ‬௔ᇱ ܴ݁ ቂͳ െ ఈோ ௃భ ሺఈோሻቃǢ‫ݏ‬௔ᇱ ൌ(8)with the cover plate radius R and the dynamic stiffness of an air layer s a '. The same calculation was
also performed for a quadratic plate in (5). The ratio between the stiffness of an air layer s a ' and thestiffness due to the airflow resistivity s V' is plotted as a function of the generalised coordinate D x
where x is the radius for the circular cover plate and the half of the edge length for the quadratic coverplate (Figure 7, left graph). For D x < 1.5, the effect of air stiffness does not change the result of thestiffness measurement according to Figure 1. Therefore, the stiffness of the final assembly (complete
floor) can be calculated by adding the air stiffness to the measured stiffness in this range. With theestimates x ൎ 0.1 m; p0 ൎ 10 5 Pa; HN ൎ 1 and Z ൎ 200, the condition D x < 1.5 turns out to beequivalent to V <100 kPa s/m². This is the value mentioned in (1). For larger airflow resistivities, themeasurement automatically includes the effect of the air stiffness and thus, no further correction isrequired. ISO 9052-1 (1) states that this is the case for V > 100 kPa s / m². Nevertheless, a national
foreword in Germany (2) changes this to V > 1000 kPa s / m² without mentioning how to handle therange between 100 and 1000 kPa s / m². Kraak (5) himself mentions in his paper the criterion D x > 15,
which would correspond to an airflow resistivity of 10000 kPa s / m² with the above-mentioned
estimates.
To perform test measurements on this effect, a special test assembly was used at PTB. It consistsof an acrylic glass frame which is mounted around the standardised test set-up with an airgap of about
1 mm (Figure 7). Measurements of the dynamic stiffness were performed for two different materials
with a lateral airflow resistivity of about 40 kPa s / m² and 4 kPa s / m² once with the frame and once
without the frame. The presence of the frame generally increases the measured stiffness. The increase
was about 30 % for the first material and about 60 % for the second. This follows the general theory
that the low resistivity material is more affected. Nevertheless, further detailed investigations arenecessary.

1407

Another problem is the measurement of the airflow resistivity, which in this case must be
determined in lateral direction. Unfortunately, most existing measurement devices are only suited formeasurements in orthogonal direction, which may deliver significantly different results .
1,0
circular plate
quadratic platesV' / sa'

0,8
0,6
0,4
0,2
0,0

0,1

1

10

Dx

Figure 7 – Left: Calculated ratio of air stiffness and stiffness due to resistivity. Right: Test assembly withadjustable frame

5. INFLUENCE OF THE EXCITATION FORCE
The dynamic stiffness of most of the common insulation materials varies with the excitation force
that acts on it. For sinusoidal excitation, the measurement standard requires the measurement to be
performed in a range of 0,2 N ≤ F ≤ 0,8 N for ‫ ݏ‬ᇱ > 50 MN/m³ or 0,1 N ≤ F ≤ 0,4 N for ‫ ݏ‬ᇱ ≤ 50 MN/m³.
No hint is given if the required forces relate to peak or rms values. The measurement must be
performed at three different excitation forces in the respective range, and the final value for the
resonance frequency is then found by extrapolation to 0 N. This approach has several drawbacks. As
shown in Figure 8, the sensitivity of the resonance frequency to variations of the excitation force may
be quite pronounced when approaching 0 N. Furthermore, an excitation force of 0 N is obviously
unrealistic for built-in situations. A previous publication of the authors (3) offers a pragmaticsuggestion to perform the measurement with a peak force of 1 N. As the extrapolation to 0 N represents
some sort of “safe side” approach, a question raises about the effect on the stiffness if 1 N is used.
This was investigated for different insulation materials; the results are shown in Table 2.
27

Resonance Frequency in Hz

25

Mineral wool 40 mm
23

21
19

17
15
0

2

4

6
8
Peak Force in N

10

12

Figure 8 – Dependency of the resonance frequency to the excitation force

1408

Table 2 – Comparison of extrapolation to 0 N vs. measurement at 1 N‫ݏ‬Ԣ୲ǡ଴୒ in‫ݏ‬Ԣ୲ǡଵ୒ in

MN/m³

MN/m³

30

295

290

Minera wool 1

40

3,4

2,7

Mineral wool 2

40

34,8

29,1

Mineral wool 3

40

5,7

4,5

Coconut fibre

35

3,5

2,7

Material

d in mm

XPS

As expected, the dynamic stiffness decreases up to 30 % with an excitation force of 1 N. As thisforce is still at the very low end of a force range that can be expected in built -in situations (3), thiscan still be considered as a conservative estimation. Regarding the given problems that come with theextrapolation to 0 N, a measurement with 1 N still appears to be a viable option.
It is still unclear, however, how to determine the excitation force at the resonance frequency whenusing noise or even impulse excitation signals, even though this is permitted in the standard.

6. CONCLUSIONSAn alternative measurement set-up for low static loads was successfully designed and tested. The
first results indicate that the dependency of the dynamic stiffness with respect to static load is probably
larger than expected, but more measurements are necessary to verify this observation.
The theoretical background regarding the effect of the enclosed air volume in relation to the airflow
resistance was discussed. The current regulations in the measurement standard regarding this effectare in some points confusing and inconsistent. A possible solution would be the simulation of the insitu situation by covering the edges of the sample with an adjustable frame. Measurement results using
this approach were presented.
The handling of the interdependence between dynamic stiffness and excitation force according to
the standard is also problematic. It has been shown that the use of a fixed excitation force of 1 N could
be a realistic solution.
The results of this contribution could be considered in the next revision of the measurement
standard.

ACKNOWLEDGEMENT
The project is funded by the Federal Ministry of Food and Agriculture due to a decision of the German
Bundestag.

REFERENCES
1. ISO 9052-1:1989-02. Acoustics-determination of dynamic stiffness; part 1: materials used under floating
floors in dwellings
2. DIN EN 29052-1:1992-08. Akustik; Bestimmung der dynamischen Steifigkeit; Teil 1: Materialien, die
unter schwimmenden Estrichen in Wohngebäuden verwendet werden
3. Bietz H, Wittstock V. Investigations to determine the dynamic stiffness of elastic insulating materials,
Euronoise proc. 2018, Crete, May 2019, pp. 675-681
4. Bietz H, Wittstock, V Schmelzer, M. Messung der dynamischen Steifigkeit von Dämmstoffen: Einfluss
von statischer Last, Nichtlinearitäten und Strömungswiderstand, DAGA proc. 2019, Rostock, Mar 2019,
pp. 603-606
5. Kraak W. Der Einfluss des Strömungsstandwerts bei der Bestimmung der dynamischen Steifigkeit poröser
Dämm-Materialien. Hochfrequenztechnik und Elektroakustik, Volume 67 (1959), p. 111

1409