PROCEEDINGS of the23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Experimental validation of a finite element model for a heavy impact
from the standard rubber ball on a timber floor
Xiaoxue SHEN1; Carl HOPKINS2;
Acoustics Research Unit, School of Architecture, University of Liverpool, Liverpool, UKABSTRACT
In lightweight buildings there can be problems due to heavy impacts, such as from footsteps in bare feet orchildren jumping. For this reason the standard rubber ball is used in measurement standards to assess heavyimpacts. In this paper, experimental work on a mock-up timber floor is used to provide benchmark data tovalidate a model using Finite Element Methods (FEM) of the standard rubber ball exciting a timber floor.
Experimental Modal Analysis was initially used to extract damping information and the Modal Assurance
Criterion was used to confirm that the FEM model of the timber floor was reasonable. A FEM model wasthen used to predict the time-domain response of the timber floor when excited by a single impact from thestandard rubber ball which was included in the FEM model. Comparison of these FEM predictions withexperimental data was assessed using the Frequency Domain Assurance Criterion which indicatedreasonable agreement with the experimental results.
Keywords: finite element, timber floor, rubber ball

1. INTRODUCTION
Heavy impacts on floors in buildings, such as from footsteps in bare feet or children jumping cancause annoyance and disturbance. For this reason, two artificial heavy impact sources, the rubber balland the bang machine, have been developed to allow field and laboratory measurements (1). The
rubber ball is referred to in International measurement standards (2-4) to assess heavy impacts throughmeasurement of the maximum Fast time-weighted sound pressure level, LFmax . For heavyweightbuildings it is possible to predict LFmax using Transient Statistical Energy Analysis (TSEA) (5-8) andwith Finite Element Methods (FEM) (9-10). However, for lightweight buildings there are no validated
prediction models that can be used to predict the performance in both the laboratory and the field. As
the first step in the development of such a model, this paper concerns experimental validation of aFEM model that can predict the vibration response of a small timber floor that is excited by thestandard rubber ball.

2. TIMBER FLOOR CONSTRUCTION
A free-standing timber floor construction was built that can be isolated from the floor by resilient
supports to minimize the influence of a supporting structure on the dynamic response. The floor iscomprised of four timber joists (European spruce) at 390 mm spacings and six chipboard sheets withtongue & groove connections between them. In total, 96 metal screws were used to connect the joists
along their mid-line to the chipboard; the distance between screws is 180 mm on the same chipboard
sheet and 60 mm between those screws at the edge of adjacent chipboard sheets. The properties of thefloor components are given in Table 1.
In terms of the boundary conditions, 30 mm thick rubber supports (105 mm  44 mm) are locatedunderneath both ends of the four joists. The dynamic properties of the rubber were measured to have aspring stiffness of 534,236 N/m and a constant damping ratio of 0.08.

12

xiaoxue.shen@liverpool.ac.uk
carl.hopkins@liverpool.ac.uk

1271

Table 1 – Timber floor properties
FloorLengthWidth

components

(m)

(m)

Thickness/ Depth(m)Joist 1
Joist 2
Joist 3

3.6

0.044

0.192

Joist 4
Chipboard

0.6

1.2

0.022

Longitudinal

Young's

wavespeed

Modulus

(m/s)

(MPa)

468

5060

11.98

417

4198

7.35

434

4508

8.82

432

4652

9.35

676

2200

3.35

Density(kg/m3)

Poisson’s
ratio (-)

0.3

0.3

3. EXPERIMENTAL PROCEDURE
3.1 Experimental Modal Analysis
Experimental Modal Analysis (EMA) was carried out on the chipboard surface of the timber floor
to identify natural frequencies, mode shapes and modal damping. A force hammer (B&K Type 8200)
was used to excite the floor on a rectangular grid of 432 points (100mm separation) with three fixed
accelerometer (B&K Type 4371) positions.
The Modal Assurance Criterion (MAC) is used to assess correlation between the EMA and FEM
predicted mode shapes using (11)2|{𝜓𝑋 }T {𝜓𝐴 }|𝑀𝐴𝐶(𝐴, 𝑋) =({𝜓𝑋 }T {𝜓𝑋 })({𝜓𝐴 }T {𝜓𝐴 })where 𝜓𝑋 𝑎𝑛𝑑 𝜓𝐴 are mode shapes for the experimental and FEM mode shapes respectively, andsuperscript T indicates the transpose. The experimental mode shapes are complex due to the damping,whereas the FEM mode shapes are real.
3.2 Rubber Ball Excitation
A standard rubber ball (RION) was dropped from a 1m height onto the timber floor. Two excitation
positions were identified (see Ref1 and Ref2 on Figure 1); the first is close to the mid-point of the floor
between joists and the second is close to a joist. Underneath each of these excitation positions was anaccelerometer (B&K Type 4371) which was fixed using cyanoacrylate glue; this formed a reference
signal to allow measurement of a complex transfer function for each response point . To measure all
response points shown in Fig.1, six accelerometers were used to cover all the 432 sample points in 72
steps. Note that this excluded the two excitation points where the accelerometer was underneath. TheFFT frequency span was 1.6k Hz with 3200 FFT lines and a 7 Hz high-pass filter.
The rubber ball was dropped five times at each excitation position to give an average transferfunction (in terms of the frequency response function H1) for each response point referenced to thesignal underneath the excitation position.
Correlation between the experimental and FEM frequency response functions was assessed using
the Frequency Domain Assurance Criterion (FDAC) as given by (11)2|{𝐻𝑋 (𝜔𝑖 )}T𝑘 {𝐻𝐴 (𝜔𝑗 )}𝑘 |𝐹𝐷𝐴𝐶(𝐴(ωj ), 𝑋(𝜔𝑖 ))k =({𝐻𝑋 (𝜔𝑖 )}T𝑘 {𝐻𝑋 (𝜔𝑖 )}𝑘 )({𝐻𝐴 (𝜔𝑗 )}T𝑘 {𝐻𝐴 (𝜔𝑗 )}𝑘 )where 𝐻𝑋 , 𝐻𝐴 are the transfer functions of experiment and FE model respectively, and the suffix 𝑘indicates the relevant transfer functions have the 𝑘 𝑡ℎ sample point as their reference.

1272Figure 1 – Sample points and two reference points

4. FINITE ELEMENT MODELLING
All finite element modelling was carried out using Abaqus v6.14-2. The undamped natural
frequencies and mode shapes were determined using the LANCZOS solver in Abaqus/Standard. The
transient response analysis with rubber ball excitation was determined using Abaqus/Explicit. This isa general ‘Hard’ contact applied to the coupling between the rubber ball and the timber floor. The total
simulation time is 1s and the fixed time step is 2 × 10−6 s.
4.1 Rubber Ball
The rubber ball is modelled as a sphere using a general purpose conventional shell element, S3R (8)with an element size of 15 mm, and shell thickness of 30 mm; this modelling approach was validatedin (9-10).
At the time when the rubber ball begins to make contact with the timber floor, a velocity of 4.43 m/s
is assigned to the ball to simulate the 1m drop height under gravity. The material properties of therubber used in the ball were taken from experimentally-derived data from Hirakawa (10); density of1188 kg/m3 , Young’s Modulus of 3.2 × 106 N/m2 and Poisson’s ratio of 0.48.
4.2 Timber Floor
Both the chipboard and joists are considered as isotropic solids using ‘S4R’ shell elements with anelement size of 15 mm. This type of element has six degrees of freedom (dofs) and allows the shell to
deform in bending direction. The chipboard is regarded as isotropic because it has nearly the samestrain-stress relationship between the two in-plane directions. To simplify the analysis, the timber
joists are assumed to be isotropic although wood is typically anisotropic.
Results from Experimental Modal Analysis (EMA) were used to identify the damping of the floor ateach mode – see Fig 2. This damping was then applied to both the joists and the chipboard based on aRayleigh damping model defined by
𝜁=𝛼
𝜔𝛽+2𝜔
21273

where 𝛼 is the parameter of mass proportional damping, and 𝛽 is for stiffness proportional damping.
The  term is usually neglected in Abaqus/Explicit because it decreases the stable time increment.
For the 96 screws, a rigid connection “JOIN” is used to join the two nodes with the same position on
chipboard and joists, which makes all their active dofs equal. For the tongue and groove structure, themulti-point constraint ‘PIN_MPC’ is used to provide a pinned joint between two nodes on different
chipboard sheets, which makes the displacement equal but leaves the rotations independent of eachother. For the interaction between chipboard and joists, the ‘Slide-Plane’ constraint is used to ensure
they have the same translation in the normal direction of the chipboard, whilst relative displacements
and rotations in other directions are not constrained (8). This is essential to ensure that the joists canonly move in the vertical direction.
4.3 Boundary Condition
Damped, elastic, 1-D spring connectors are positioned at each of the eight nodes at the end of thejoists which sat on the rubber supports. The measured spring stiffness for the material is 534,236 N/m,so each connector is assigned a stiffness of 66,779.5 N/m and a constant damping ratio of 0.08.

5. RESULTS
5.1 Experimental Modal Analysis
The damping identified in EMA was used to calculate suitable Rayleigh damping for input into theFEM model. As shown in Figure 2, Rayleigh damping using  and  terms (which can be used inAbaqus Modal Dynamics but is not advisable in Abaqus/Explicit because it decreases the stable timeincrement) shows better agreement with EMA data above 300Hz; however, in this paper the focus is on
the low-frequency performance up to 200Hz for which better agreement is obtained using =37.2 and=0.

Figure 2 – Comparison of EMA damping and fitted Rayleigh damping curves.

Comparison of the mode frequencies and mode shapes from EMA and FEM is shown in Table 2 forthe first 12 modes along with MAC values which are plotted on Figure 3. Apart from two modes, theerrors in the mode frequencies are <10% which is reasonable. In terms of MAC, the first three modes
are highly correlated and (apart from mode 11) there is reasonable correlation for all the other modes.
This agreement was considered sufficient to move on to assess the frequency response functions withrubber ball excitation. However, model updating will be implemented in the future to improve theagreement.

1274

Table 2 – Comparison between EMA and FE results
Modes

EMA (Hz)FE (Hz)Error

MAC118.6

20.9

12.4%

0.98

2

21.6

23.7

9.5%

0.96

3

33.8

36.1

6.6%

0.89

4

45.0

45.0

0.1%

0.72

5

47.1

51.5

9.5%

0.67

6

81.8

67.5

17.5%

0.62

7

108.4

109.4

1.0%

0.66

8

132.7

141.5

6.7%

0.79

9

138.1

138.1

0.0%

0.81

10

176.0

169.7

3.6%

0.62

11

181.7

191.3

5.3%

0.42

12

201.7

184.6

8.5%

0.71

Figure 3 – MAC for EMA and FEM data.

5.2 Rubber Ball Excitation
FDAC is calculated for the two reference points and plotted in Figure 4. This shows clear diagonallines which indicate that the FEM model agrees well with the experimental results below 200 Hz.

1275

Figure 4 – FDAC for the two excitation points: Ref1 (upper) and Ref2 (lower).

6. CONCLUSIONSIn this paper, FEM has been used to model a standard rubber ball impacting a timber floor.
Experimental Modal Analysis on the timber floor was initially used to extract dampi ng informationand to validate the mode frequencies and mode shapes. A FEM model was then used to predict thetime-domain response of the timber floor when excited by a single impact from the standard rubberball which was included in the FEM model. Comparison of these FEM predictions with experimental
data was assessed using the Frequency Domain Assurance Criterion which indicated reasonableagreement with the experimental results. Future work will use model updating to improve the FEM
model.

ACKNOWLEDGEMENTS
This project has received funding from the European Union's Horizon 2020 research andinnovation programme under grant agreement No.721536.
The authors are grateful to Dr Gary Seiffert in the ARU for his help with the experimental set-up.

REFERENCES1. Tachibana H, Tanaka H, Yasuoka M. and Kimura S. (1998). Development of new heavy and soft impact
source for the assessment of floor impact sound insulation of buildings, Proceedings of Internoise 98,1276

Christchurch, New Zealand.
2. ISO 10140-3:2010+A1:2015. Acoustics – Laboratory measurement of sound insulation of building
elements – Part 3: Measurement of impact sound insulation. International Organization forStandardization.
3. ISO 10140-5:2010+A1:2014. Acoustics – Laboratory measurement of sound insulation of building
elements – Part 5: Requirements for test facilities and equipment. International Organization forStandardization.
4. ISO 16283-2:2018 Acoustics. Field measurement of sound insulation in buildings and of building
elements – Part 2: Impact sound insulation. International Organization for Standardization.
5. Robinson M and Hopkins C. (2014). Prediction of maximum time-weighted sound and vibration levels
using transient statistical energy analysis – Part 1: Theory and numerical implementation. Acta Acustica
united with Acustica, 100(1), 46-56.
6. Robinson M and Hopkins C. (2014). Prediction of maximum time-weighted sound and vibration levels
using transient statistical energy analysis – Part 2: Experimental validation. Acta Acustica united withAcustica, 100(1), 57-66.
7. Robinson M and Hopkins C. (2015). Prediction of maximum fast time-weighted sound pressure levels
due to transient excitation from the rubber ball and human footsteps. Building and Environment, 94,810-820.
8. Hirakawa S and Hopkins C. Experimental determination of transient structure-borne sound power fromheavy impact sources on heavyweight floors with floating floors using an inverse form of transient
statistical energy analysis. Applied Acoustics, 2018;140:74-82.
9. Hirakawa S and Hopkins C. (2018). Prediction of heavy impact sounds using transient statistical energy
analysis and finite element methods. Proceedings of the 25th International Congress on Sound andVibration 2018, ICSV 2018, 4377-4383.
10. Hirakawa S. Prediction of impact sound transmission with heavy impact sources in heavyweightbuildings. PhD thesis, University of Liverpool (2018).
11. Ewins DJ. Modal testing: Theory, Practice and Application. Second Edition. Research Studies
Press Ltd. 2000.

1277