Efficient numerical prediction and experimental validation of impact soundradiation by timber joist floors
Pengchao WANG(1) , Cédric VAN HOORICKX(1) , Geert LOMBAERT(1) ,Edwin REYNDERS(1)(1) KU Leuven, Department of Civil Engineering, Kasteelpark Arenberg 40, 3001 Leuven, Belgium, pengchao.wang@kuleuven.be

Abstract
Timber joist floors are widely applied in residential buildings. The accurate impact sound radiation predictionof a bare timber joist floor is characterized by three main challenges. First, the prediction of the impact force
on a timber joist floor is more difficult than that for a traditional heavyweight floor, since the low impedance ofthe timber joist floor induces more complex interaction between the excitation source and the floor. Second, thedynamic behavior of the floor is strongly affected by several characteristics, such as the finite dimensions, platejoist coupling, orthotropic mechanical properties, boundary conditions, etc. Third, it is difficult to efficiently
compute the precise acoustic field for the whole building acoustic frequency range. This work proposes a twostep numerical prediction model for impact sound radiation. First, the impact force of the ISO tapping machineis computed, considering the floor-hammer interaction. Second, the vibration field is accurately computed bya detailed finite element model of the floor, and the sound field in the room is subsequently computed by anefficient diffuse field model. This prediction model is experimentally validated in a case study of a timber joist
floor. Differences between the measured and computed normalized impact sound pressure level according toISO 717-2 did not exceed 2 dB.
Keywords: impact sound radiation, lightweight timber joist floor, ISO tapping machine, finite element
method, diffuse field model1

INTRODUCTION

The market share of timber joist floors in residential buildings has steadily increased in recent years [1]. However, it is challenging to achieve good impact sound insulation with a bare timber joist floor, due to its lightweight [2]. The sound insulation of bare timber joist floors can be improved by adding components such asa floating screed or a suspended ceiling, as demonstrated in experimental studies [3]. Homb et al. [4] categorized timber joist configurations into different groups, and experimentally investigated the effects of various
configurations on the impact sound insulation.
The efficient and cost-effective development and optimization of lightweight floor systems requires a computationally efficient and sufficiently accurate mathematical prediction of their impact sound radiation, because theexperimental testing of a large number of alternative configurations is usually expensive and time-consuming.
However, such a prediction generally needs to address several challenges, including the predictions of (i) the
impact force, (ii) the vibration field of the timber joist floor, and (iii) the sound power radiated by the floor
into a room.
The first challenge lies in the prediction of the impact force that is exerted onto the floor. Even for the
standardized ISO tapping machine [5], predicting the impact force with sufficient accuracy is challenging because, in contrast to heavy floors, the interaction between the actuator and the lightweight timber joist floor
cannot be neglected. Timoshenko and Goodier [6] investigated the local reaction of a rigid hammer excitation
on an infinite plate, and derived expressions of local force mobility. Brunskog and Hammer [7] analyzed thehammer-floor interaction using a lumped mass-spring-dashpot model. However, the precise prediction of the
ISO tapping machine impact force on a lightweight floor remains challenging. The impact force is for example
determined by the hammer momentum change and the floor impedance [7, 8], which are generally difficult toquantify precisely.
The second challenge is predicting the vibration field of timber joist floors. Analytical solutions for the vi-

5126

bration of lightweight periodical rib-stiffened plates have been provided, considering plate-joist coupling effects[9]. Since the amount of complexity that can be handled by analytical prediction models is limited, numerical approaches have also been presented. Kohrmann [1] constructed detailed finite element (FE) models toextensively analyze vibrations of various Cross-Laminated Timber (CLT) plates with lumber joists, hollow box
girders, floating floors, and suspended ceilings. Paolini et al. [10] developed a high-order FE model of a CLT
structure, and the unknown model parameters were determined by matching simulated and measured modal results in a model calibration process. Nevertheless, also the predictive accuracy of the numerical models heavily
depends on the accuracy of the dynamic material properties of the floor components, and on whether the assumptions regarding the connections between the different components and the boundary conditions are realistic
or not.
The third challenge, i.e., the prediction of the sound power that is radiated by the floor into the receivingroom, requires the consideration of different frequency regimes. Neves e Sousa and Gibbs [11] developed ananalytical model to compute the acoustic response in the room under a point-loaded floor at low frequencies.
They highlighted the effects of floor-room modal coupling on the acoustic response. Bard et al. [12] analyzed
the pressure field in the receiving room using a basis of analytical solutions of the Helmholtz equation. At
medium and high frequencies, the sound field in the room is usually modeled as diffuse [13]. Rayleigh integral
is typically available to numerically evaluate the radiated sound power [14].
The aim of the this work is to investigate to which degree of accuracy the impact sound insulation of timber
joist floors can be predicted with currently available, computationally efficient state-of-the-art approaches. A
practical numerical model is elaborated in this work for the prediction of impact forces, structural vibrations
and impact sound radiation. A case study of a timber joist floor is presented, in which the numerical model isexperimentally validated.
The outline of the paper is as follows. Section 2 introduces the proposed numerical prediction model for
impact sound radiation prediction of timber joist floors. Section 3 performs an experimental validation of thenumerical model in a case study of a timber joist floor. Finally, section 4 concludes the work.

2

NUMERICAL PREDICTION MODEL

This section presents a two-step numerical prediction model for impact sound radiation by a lightweight timber
joist floor. First, in section 2.1, the impact force of the ISO tapping machine is computed using the massimpedance model proposed by Brunskog and Hammer [7]. In the second step, the structural response of thefloor and sound pressure levels in the underlying room are subsequently computed by a FE-diffuse field model,
as given in section 2.2.
2.1 ISO tapping machine impact force
As commonly assumed, the tapping machine is modeled as a point excitation source with an impact period
T0 = 0.1 s. Figure 1 shows a model for the tapping machine excitation at an angular frequency ωn , with n ∈ Z.
The hammer is described by its mass mh = 0.50 kg and a blocked force with an amplitude of fˆbl,n . Using the
momentum-impulse theorem, the blocked force is given by [8]fˆbl,n = mh ∆v/T0 = mh ∆v f0(1)

where f0 = 1/T0 = 10 Hz is the impact frequency, and ∆v is the hammer velocity change during an impact. ∆vequals v0 and 2v0 for perfectly plastic and elastic collisions, respectively. An assumption is often made basedon the type of floor; alternatively, it is numerically approximated [7, 8].
As shown in Figure 1, the driving point impedance Zdp consists of a local part ZL and a global part ZG ,representing the resistance to local deformation at the hammer-floor contact surface, and the resistance to global
floor motion, respectively. For periodical excitation of the hammer, the impact force in the frequency domain is

5127

written as Fourier series coefficients:fˆimp,n =fˆbl,nmh ∆v f0=
1 + iωn mh /Zdp
1 + iωn mh (ZG + ZL )/(ZG ZL )(2)

The global impedance ZG for a finite floor is complex, with its real and imaginary part representing the energy
dissipating part and stiffness part, respectively. In this work, the global impedance ZG is computed using thedetailed FE model of the floor presented in section 3. On the other hand, the local impedance ZL is analyticallycomputed using Timoshenko and Goodier’s approach [6]:
ZL =

2Erh
iω(1 + ν)(1 − ν)(3)where E and ν are the Young’s modulus and Poisson’s ratio for isotropic floor materials, and rh = 1.5 cm is theradius of the tapping machine hammer.
2.2 Impact sound radiation
This section describes a FE-diffuse field model to compute the structural vibration of the floor and resulting
impact sound pressure level in the underlying room. The structural vibration is predicted by a fully detailed FE
model of the floor. The sound field in the room is regarded as a combination of a direct field and a diffuse
field, such that the radiated sound power flows into the direct field, and then is reflected and dissipated at the
random boundaries of the diffuse field. The diffuse field model is computationally efficient, and its predictive
accuracy is satisfactory at middle and high frequencies.

beam 7beam 6

beam 5beam 4fˆimp,nbeam 3mhbeam 2

beam 1fˆbl,n150

plate 2

10

300

y

ZG

z

150

plate 1

ZL
Zdp

7

y
z

x

1.8

x

Figure 1. Model for the ISO tapping machine hammer excitation on a floor.

9.5

38.5

fˆimp,n19

Figure 2. Top, front, and side view of the timber floor. The dimensions are in [cm].

At an angular frequency ω, the structural displacement at an arbitrary coordinate x due to an excitation at
the coordinate x0 is computed using modal decomposition:
φ j (x)φ j (x0 ) fˆ(x0 )22
j=1 −ω + ω j + iηs ωω jnmû(ω, x) = ∑(4)

where fˆ(x0 ) is the impact force at the coordinate x0 , φ j (x) and ω j are the mode shape at the coordinate x andangular natural frequency of the structural mode j, nm is the number of structural modes that are considered
to contribute to the structural response, and ηs is the structural damping loss factor. Next, the sound powerradiated by the infinitely baffled floor into the direct field is computed byP̂rad = (ω/2)Im{ûH (ω)Ddir û(ω)}

5128

(5)

in which û ∈ Cndof is the displacement vector, with ndof the number of structural DOFs along the floor-room
interface. Ddir ∈ Cndof ×ndof is the dynamic stiffness matrix of the direct field, as seen from the floor-room
interface [17]. The notation (·)H denotes the Hermitian transpose, i.e., ûH = û∗T . Next, the radiated power P̂radis dissipated in the diffuse field of the room:P̂rad = P̂diss(6)
in which the dissipated power P̂diss is computed by [18]P̂diss = ωηa Êa(7)

where Êa is the sound energy averaged over an ensemble of rooms with the same volume V and acoustic
damping loss factor ηa , but with random wave scattering in the room and at its boundaries. The ensemble
mean of the sound energy Êa is easily computed using equations (5) to (7). The normalized impact soundpressure level in the receiving room equals
(8)L̂n = 10 log Êa ρ0 c2 /(V p20 ) + 10 log(A/A0 )where c is the sound speed in air, ρ0 is the air density, p0 = 2 × 10−5 Pa, and A0 = 10 m2 are the reference
sound pressure and the reference absorption area, respectively.

3

IMPACT SOUND PREDICTION AND EXPERIMENTAL VALIDATIONIn this section, the numerical model presented in the previous section is experimentally validated in the following case study of a timber joist floor. Section 3.1 describes the considered floor. Section 3.2 illustrates the FEmodel of the floor. Section 3.3 describes the measurement setups of an impact sound radiation test, an acoustic
reverberation test, and a structural reverberation test. Section 3.4 discusses the experimental validation of thecomputed sound pressure levels for the ISO tapping machine excitation.
3.1 Description of the timber joist floor
Figure 2 shows the top, front, and side view of the considered lightweight timber joist floor, which is assembled
in a 3 m × 3 m opening in the KU Leuven Laboratory of Acoustics. The receiving room below the floor has
a volume of V = 87 m3 . The floor consists of two timber plates supported by a total of seven joist beams.
Each plate has nominal dimensions of 3 m × 1.5 m × 0.018 m, and consists of seven thin timber layers. The fiber
orientations of two adjacent thin layers are orthogonal. Each beam has nominal dimensions of 3 m × 0.07 m ×
0.19 m, and is connected with the two plates by ten screws along the central longitudinal axis of the beam.
The floor components are assumed to be homogeneous, despite the inhomogeneous nature of the wooden
texture. Table 1 lists the densities ρ of the floor components, the Young’s moduli Ex and Ey of the sevenbeams, and the Young’s moduli Ex and Ey , the major Poisson’s ratio νxy , and the shear modulus Gxy of twoplates. The stiffness parameters in table 1 were systematically calibrated using the experimentally determinedeigenmodes of separate floor components. The stiffness parameters of floor components that are not shown intable 1 were either assumed according to [19], or computed from the determined parameters [20].
3.2 FE model of the floor
Figure 3 shows the FE model of the floor, which is assembled from the seven beam models and two platemodels, respectively. The beams and plates were constructed in ANSYS with 3D eight-node linear solid elements (of the SOLID45 type) of size 2.0 cm × 2.0 cm × 2.0 cm, and with four-node linear thin shell elements (ofthe SHELL181 type) of size 2.5 cm × 2.5 cm, respectively. The neutral plane of the two plates is 9 mm higher
than the top surfaces of the beams. It is assumed that, at a screw point, the plate is strongly coupled to thebeam, such that the plate and the beam are vibrating without any relative translation or rotation. To achieve
such a plate-beam connection through the screw, all translational DOFs of the plate within a radius of 4 cm tothe screw are coupled to the closest translational DOF of the beam.

5129

ComponentNo.
ρ [kg/m3 ]Ex [GPa]Ey [GPa]Gxy [GPa]νxy [−]

plate
1
2
515 535
5.24 4.96
4.53 5.06
0.30 0.33
0.31 0.31

1
489
0.21
13.99

2
511
0.19
13.42

3
593
0.28
16.28

beam
4
561
0.25
15.33

5
484
0.21
14.02

6
524
0.18
15.06

7
534
0.17
14.84

Table 1. Densities and calibrated stiffness parameters of the floor components.
It was assumed that some beam ends were lifted from their support when screwing the plates onto thebeams. Following this assumption, the boundary conditions were investigated by evaluating the effect of eachbeam end on the modal characteristics. The boundary conditions were finally specified by keeping four beam
ends free, while constraining the vertical translational DOFs at the other beam ends. These boundary conditions
achieve better correspondences between the measured and computed modal characteristics than other ones that
had been investigated.
3.3 Measurement setup
An impact sound radiation test was performed, consisting of 16 setups. In eight setups, the ISO tapping machine
was placed above a beam, and in the other eight setups, the machine was placed at midspan between two beams.
In each setup, the sound pressure level was measured at eight microphone positions in the central zone of thereceiving room.
An acoustic reverberation test was performed, and the acoustic reverberation time Ta in the receiving room
was evaluated using the interrupted noise method as standardized in ISO-3382 [21]. The acoustic damping loss
factor ηa at angular frequency ω was computed byηa = 4.4π/(ωTa )(9)

Analogously, a structural reverberation test was performed, and the structural reverberation time Ts was evaluated using the integrated impulse response method as standardized in ISO-3382 [21]. Next, the structure
damping loss factor ηs at angular frequency ω was computed as follows:ηs = 4.4π/(ωTs )(10)

Figure 2 shows the measured acoustic reverberation time Ta of the receiving room and the damping loss
factor ηs of the floor.
f [Hz]Ta [s]
ηs [%]

63
2.18
3.70

125
1.16
2.32

250
1.03
2.21

500
1.31
1.47

1000
1.52
2.31

2000
1.49
2.18

4000
1.27
1.59

Table 2. Measured reverberation time Ta of the receiving room and the damping loss factor ηs of the floor.

3.4 Validation for tapping machine excitation
The ISO tapping machine impact force is computed using the approach in section 2.1. in which the hammer
excitation on the timber floor is assumed as a perfectly plastic collision. Figure 4 shows the computed equivalent
power spectra densities of the ISO tapping machine impact force. At frequencies below 100 Hz, the magnitudesof the impact force spectra are controlled by the floor stiffness, and are close to the pseudo force spectra. For
frequencies between 100 Hz and 1000 Hz, the results for excitation on top of a beam are different from the ones

5130

for excitation between beams. This is mainly due to the considerable difference of the point impedances at beam
and inter-beam positions. For excitation on top of a beam, a resonance peak appears at 550 Hz, corresponding
to the mass-spring resonance that is determined by the hammer mass and driving point stiffness. At frequencies
above 1000 Hz, the impact force spectra are controlled by the hammer mass.

Y

Z
X

Figure 3. FE model of the assembled floor
in ANSYS.

Figure 4. Computed power spectra densities of squared impact force spectra of excitations on eight beam positions(black solid line) and eight inter-beam positions (gray solidline) in the sound radiation test.

Figure 5 shows the measured and computed normalized sound pressure levels in the receiving room that are
integrated over 1/48-octave bands and 1/3-octave bands, and averaged over eight beam positions and eight interbeam positions of the tapping machine, respectively. In general, the measured and computed sound pressurelevels have a similar order of magnitudes. The discrepancies between the measured and computed 1/48-octave
band integrated results are attributed to the moderate accuracies of the FE model of the floor, the estimatedstructural and acoustic damping loss factors, and the measurement errors. The discrepancies at low frequencies
are also attributed to acoustic resonances, which are not considered in the diffuse room model, but influence theacoustic response in this frequency range. Furthermore, the impact force of the tapping machine was computed
using the assumptions in section 2.1, resulting in additional model errors.
The correspondence between the measurements and computations is further evaluated by comparing single
number ratings in accordance with ISO-717-2-1996 [22]. The plots in Figure 5(c-f) show the single number
rating values in the form of Ln,w (CI ), for the measured and computed normalized sound pressure levels integrated in 1/3-octave bands. For both the beam and inter-beam excitation, the computed single number rating
differs only 0 − 2 dB from the measured one. Therefore, the numerical model given in section 2 gives a goodprediction of sound radiation induced by the ISO tapping machine.

4

CONCLUSION

This work proposes a two-step numerical prediction model for impact sound radiation by timber joist floors.
First, the impact force of the ISO tapping machine is computed using both the local and global impedance of thedriving point, and the latter is determined from a detailed FE model of the floor. Second, the structural vibration
and radiated sound power are computed using an FE-diffuse field model, which is both computationally efficientand accurate at middle and high frequencies. This prediction model is experimentally validated in a case studyof a timber joist floor. Differences between the measured and computed normalized impact sound pressure levelaccording to ISO 717-2 did not exceed 2 dB.

5131

(a)

(b)

(c)

(d)

(e)

(f)

Figure 5. Measured (red dashed line) and computed (blue solid line) normalized sound pressure levels in thereceiving room integrated over (a,b) 1/48-octave bands and (c-f) 1/3-octave bands, and averaged over tapping
machine excitations at (a,c,e) eight beam positions and (b,d,f) eight inter-beam positions of the floor. The
measured results are also averaged over the microphone positions in the room. Plots (c-f) additionally display
shifted reference curves (black) and single number rating values Ln,w (CI ) in accordance with ISO-717-2-1996.
The gray zones indicate the frequency range for single number ratings.

Acknowledgments
This research was funded by the European Research Council (ERC) Executive Agency, in the form of an ERC
Starting Grant under the Horizon 2020 framework program, project 714591 VirBAcous, and by the KU Leuven

5132

Dept. of Civil Engineering. The financial support from the European Commission and from KU Leuven is
gratefully acknowledged. The authors would also like to thank Jan Van den Wyngaert for his help with theimpact sound radiation measurement, and Bernd Salaets, Jimmy Van Criekingen, and Willy Bruyninckx for
their contribution in the installation of the timber joist floor.

References[1] Kohrmann, M. Numerical methods for the vibro-acoustic assessment of timber floor constructions. PhD thesis, Technical
University Munich, 2017.
[2] Negreira, J. Vibrations in lightweight buildigns - preception and prediction. PhD thesis, Lund University, 2013.
[3] Ingelaere, B. COST Action: Net-acoustics for timber based lightweight buildings and elements - Chapter 4: acoustic designof lightweight timber frame constructions. FP0702, BBRI, Brussels, 2012.
[4] Homb, A.; Guigou-Carter, C.; Hagberg, K.; Schmid, H. Impact sound insulation of wooden joist constructions: collection oflaboratory measurements and trend analysis. Build. Acoust., Vol 23(2), 2016, pp 73-91.
[5] International Organization for Standardization. ISO 10140-5: Acoustics - Laboratory measurement of sound insulation ofbuilding elements - Part 5: Requirements for test facilities and equipment, 2010.
[6] Timoshenko, S.P.; Goodier, J.N. Theory of elasticity. McGraw-Hill, 3rd edition, 1970.
[7] Brunskog, J.; Hammer. P. The interaction between the iso tapping machine and lightweight floors. Acta Acust united Ac.,
Vol 89(2), 2003, pp 296-308.
[8] Wittstock, V. On the spectral shape of the sound generated by standard tapping machines. Acta Acust united Ac., Vol 98,
2012, pp 301-308.
[9] Brunskog, J.; Hammer. P. Prediction model for the impact sound level of lightweight floors. Acta Acust united Ac., Vol89(2), 2003, pp 309-322.
[10] Paolini, A.; Kollmannsberger, S.; Winter, C.; Buchschmid, M.; Müller, G.; Rabold, A.; Mecking, S.; Schanda, U.; Rank,
E. A high-order finite element model for vibration analysis of cross-laminated timber assemblies. Build. Acoust., Vol 24(3),2017, pp 135-158.
[11] Neves e Sousa, A.; Gibbs, B.M. Low frequency impact sound transmission in dwellings through homogeneous concrete
floors and floating floors. Appl Acoust., Vol 72(4), 2011, pp 177-189.
[12] Bard, D.; Negreira, J.; Guigou-Carter, C.; Borello, G.; Kouyoumji, J.; Speranza, A.; Coguenanff, C.; Hagberg, K. Modelling
prerequisites - fem/sea impact and airborne sound. Report 56, RISE Research Institutes of Sweden AB, 2017.
[13] Hopkins, C. Sound insulation. Elsevier Ltd., Oxford, 2007.
[14] Cremer, L.; Heckl, M.;Petersson, B.A.T. Structure-borne sound: Structural vibrations and sound radiation at audio frequencies. Springer, Berlin, 3rd edition, 2005.
[15] Vér, I.L. Impact noise isolation of composite floors. J. Acoust. Soc. Am., Vol 50(4), 1971, pp 1043-1050.
[16] Langley, R.S.; Bremner, P. A hybrid method for the vibration analysis of complex structural-acoustic systems. J. Acoust.
Soc. Am., Vol 105(3), 1999, pp 1657-1671.
[17] Langley, R.S.; Cotoni, V. Response variance prediction for uncertain vibro-acoustic systems using a hybrid deterministicstatistical method. J. Acoust. Soc. Am., Vol 122(6), 2007, pp 3445-3463.
[18] Langley, R.S.; Cordioli, J.A. Hybrid deterministic-statistic analysis of vibro-acoustic systems with domain couplings on
statistical components. J Sound Vib, Vol 321(3-5), 2009, pp 893-912.
[19] Ross, R. Wood handbook: wood as an engineering material. Technical Report 190, USDA Forest Service, Forest Products
Laboratory, 2010.
[20] Huber, M. The theory of crosswise reinforced ferroconcrete slabs and its application to various important constructional
problems involving rectangular slabs. Der Bauingenieur, Vol 4(23), 1923, pp 354-360.
[21] International Organization for Standardization. ISO-3382: Acoustics - Measurement of the reverberation time of rooms withreference to other acoustical parameters, 1997.
[22] International Organization for Standardization. ISO 717-2: Acoustics - Rating of sound insulation in buildings and of building elements - Part 2: Impact sound insulation, 1996.

5133