Low-frequency vibration of a timber joist floor section connected by metal screws: experimental validation of FEM model Xiaoxue Shen 1 Acoustics Research Unit School of Architecture, University of Liverpool, Liverpool, UK Carl Hopkins 2 Acoustics Research Unit School of Architecture, University of Liverpool, Liverpool, UK

ABSTRACT The structural dynamics of a timber floor junction (chipboard plate screwed to a timber joist) have been investigated experimentally in the low-frequency range up to 200Hz. Experimental Modal Anal- ysis (EMA), transfer mobilities and velocity level ratios between the chipboard walking surface and the joist were measured with two different screw spacings connecting the joist and the chipboard. These experimental data were used to validate a Finite Element (FEM) model that focused on the interaction between the chipboard and the joist due to the screws. The aim is to develop a validated FEM model for timber floors that can be used to assess heavy impacts (such as the rubber ball impact source) and act as a reference for the future development of TSEA models for lightweight structures. The screws were modelled using rigid connections in the FEM model. A comparison of mode fre- quencies and mode shapes from EMA and FEM will be made along with forced vibration of the structure. Key words: Timber floor; Screw connections; Finite Element Methods; Vibration transmission

1. INTRODUCTION

For lightweight timber floors it is common to connect sheets of timber or plasterboard to the joists using screws that are spaced at regular intervals. Based on an approach by Heckl [1] and Cremer and Heckl [2], Sharp [3] proposed a model to treat stud or joist connections between lightweight panels as either line or point connections noting that point connections were “not so common”. Later work by Craik and Smith [4] for plasterboard screwed to timber studs at a regular spacing showed that (a) it was appropriate to assume rigid point connections at frequencies where there was at least half a bend- ing wavelength between adjacent screws and (b) at lower frequencies a line connection was appro- priate. The work by Craik and Smith was carried out with mechanical excitation and indicated that the assumption of a rigid point connection was reasonable on two sheets of plasterboard connected by a single timber stud as well as on full-size double plasterboard walls. However, mechanical exci- tation on one side of a double leaf structure with timber studs (e.g. see [5-7]) indicated that there can be a significant decrease in vibration level across the wall and that SEA models using rigid point

1 xiaoxue.shen@liverpool.ac.uk

2 carl.hopkins@liverpool.ac.uk

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connections might introduce errors. For this reason, it is worth considering whether screw connec- tions into timber might behave with spring-like properties over some parts of the building acoustics frequency range. Point connections can also be modelled with different combinations of lump ele- ments such as masses and springs using mechanical four-pole theory [8]. This theory has been used by Campolina et al [9] for elastomeric mounts in the aerospace industry, and Davy [10] for rigid connections between sheets of plasterboard in cavity walls. In addition to the assumption of whether rigid or spring-like connections are appropriate, there is the issue of the effect of periodicity on vi- bration transmission across point connections for which analytical models were developed by Bosmans and Vermeir [11] and Legault and Atalla [12]. Alternatively, it is possible to use Finite Element Methods (FEM). Previous work using Experimental Modal Analysis (EMA) for comparison with FEM modelling by Bolmsvik [13] considered the screw connections as ‘partly tied’ whereas Negreira et. al [14] modelled by tying the screws to the two surfaces.

Experimental work investigating the effect of point connections on lightweight timber frame con- structions tends to have focused on vibration transmission across the surface of sheet material past one or more studs/joists or from one sheet to another sheet connected by timber studs forming a frame between them. In this paper, the aim is to reduce the number of uncertainties and focus on vibration transmission from chipboard to a timber joist via a single set of screw connections on a large timber floor section forming a T-junction. Experimental results are compared with a FEM model to assess whether it is reasonable to model the screws as rigid connections which ensures both the rotation and displacement are identical between the chipboard and the joist.

2. TEST CONSTRUCTION AND MEASUREMENT PROCEDURES

2.1. T junction construction

A T-junction formed by one timber joist and six tongue & groove chipboard sheets was built – see Figure 1. The dimensions and material properties of the components were measured and are given in Table 1. To simulate a free boundary condition at the ends of the joist, the structure was suspended on elastic bungees. The center line of the joist is 0.39m along the width so that the structure is asym- metric. In the initial configuration, 24 metal screws are used to connect the joists along their mid-line to the chipboard (four screws connect each piece of chipboard to the joist); 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 – see Figure 1 and Figure 3. To investigate the effect of screw spacing, a second configuration was tested with only 12 screws by removing the middle two screws on each chipboard sheet – see Figure 3.

Table 1. Timber floor properties

Young’s modulus

Poisson’s

Component Length

Thickness

Width

Density (kg/m 3 )

Quasi-longitudinal

ratio

(m)

(m)

(m)

wavespeed (m/s)

(GPa)

(-) Joist 3.6 0.192 0.044 468 5060 11.98 0.3 Chipboard 0.6 0.022 1.2 676 2200 3.35 0.3

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Figure 1: Experimental set-up for the timber junction.

2.2. Experimental Modal Analysis

Experimental Modal Analysis (EMA) was used to determine natural frequencies, mode shapes and damping using B&K Connect software.

Before the T-junction was built, EMA was carried out on the isolated timber joist when suspended vertically and horizontally as shown in Figure 2. This allowed bending modes to be excited in two orthogonal directions to assess whether it could be modelled in FEM as an isotropic material. There were 111 grid points used for EMA (100mm spacing) as indicated on Figure 2. A force hammer was used to hit five times at each grid position. Three accelerometers were positioned on the upper surface of the joist that measured acceleration in the direction perpendicular to the upper surface.

Figure 2: EMA on the isolated joist: vertically suspended (upper left) and horizontally suspended (upper right). Lower graphic shows the EMA positions for these two conditions respectively (‘x’ indicates the excitation position used for specific analysis of transfer functions).

On the T-junction (see Figure 3) there are 504 grid points (100 mm separation distance) which can be separated into three groups: 432 points on the upper surface of the chipboard; 36 points along the side of the joist in the middle; and another 36 points along the centre line on the lower side of the joist. A B&K Type 8200 force hammer was used for excitation for these three different groups re- spectively, producing a series of pulse force signal. Five B&K 4371 accelerometers were installed to measure acceleration; Acc1 and Acc2 were on the upper surface of the chipboard; Acc3 was on the centre line on the side of the joist measuring motion in the direction perpendicular to that joist surface; Acc4 and Acc5 were on the underside of the joist measuring in the direction perpendicular to the chipboard surface (in opposite directions to Acc1 and Acc2).

2.3. Transfer mobilities

Transfer mobilities were used to assess the validity of the FEM model using the response at Acc1, Acc2, Acc4 and Acc5 due to the force at F1, and the response at Acc3 on the joist due to the force at F2.

2.4. Velocity level difference

The spatial-average, mean-square velocity ratio from the chipboard (source) to beam (receiver) was used to assess vibration transmission between them, and was defined as:

2 𝑀 𝑚=1 ) 𝑁 𝑗=1

𝑀 ∑ | 𝑣 𝑝,𝑚,𝑗

[ 1

𝑁 ∑ ( 1

2 𝑄 𝑞=1 ) 𝑁 𝑗=1 ]

𝐹 𝑝,𝑚,𝑗 |

〈𝑣 𝑝 2 〉 〈𝑣 𝑏

(1)

10lg (

2 〉 ) = 10lg

𝑄 ∑ | 𝑣 𝑏,𝑞,𝑗

1 𝑁 ∑ ( 1

𝐹 𝑝,𝑞,𝑗 |

where 𝑁 is the number of force positions (432). 𝑀 is the accelerometer number on the chipboard ( M =2), 𝑄 is the accelerometer number on the underside of the joist ( Q =2),

𝑣 𝑝,𝑚,𝑗 𝐹 𝑝,𝑚,𝑗 is the transfer mo-

bility for the velocity at accelerometer position 𝑚 on the chipboard referred to the excitation force at position 𝑗 ,

𝑣 𝑏,𝑞,𝑗 𝐹 𝑝,𝑞,𝑗 is the transfer mobility for the velocity response at accelerometer position 𝑞 on the

joist referred to the force at position 𝑗 on the chipboard.

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Figure 3: T-junction. Left: six sheets of chipboard; right: timber joist. Black dots indicate the EMA grid points, ‘x’ indicates screw positions with the screws inside the dashed red boxes indicating the screws that were removed to change from 24 to 12 screws.

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3. FINITE ELEMENT MODELLING

Finite element modelling was carried out using Abaqus v6.14-2 [15]. Undamped natural frequencies and mode shapes were determined using the LANCZOS solver in Abaqus/Standard. The forced vi- bration response was determined using Abaqus/Standard Steady-state frequency analysis simulating the response due to a point force up to 200 Hz.

Both the chipboard and joist were modelled as isotropic solids using ‘S4R’ shell elements with an element size of 15 mm, which corresponds to 1/10 of the half of the bending wave length at 1kHz and is small enough to keep the FEM accurate. The chipboard was regarded as isotropic because it has nearly the same strain-stress relationship between the two in-plane directions. The timber joists were assumed to be isotropic although wood is typically anisotropic; hence this simplification is val- idated in section 4.1.2 and 4.1.3. Free-free boundary conditions were assumed for chipboard plate edges and the ends of the joist. In the steady-state analysis, both the chipboard [16] and the joist were assigned a loss factor of 0.01.

The screws were modelled as rigid connections. For the tongue and groove chipboard, the multi- point constraint ‘PIN_MPC’ was used to give a pinned joint between two adjacent nodes on adjacent chipboard sheets. This allows independent rotation at the tongue and groove joint but ensures the displacement is identical.

As there are no spacers between the chipboard and joist in the physical test specimen, there is some interaction between them at positions in-between the screw connections; this contact might introduce additional damping but it is not well understood for timber constructions and therefore it is not in- cluded in the model.

MAC was used to correlate the experimental mode shapes from EMA and their counterparts in FEM. This parameter is defined by [17]:

oy

2

𝑇 𝜓 𝐴 ห

𝑀𝐴𝐶(𝐴, 𝑋) = ห𝜓 𝑋

(2)

𝑇 𝜓 𝐴 ൯

(ሼ𝜓 𝑋 ሽ 𝑇 ሼ𝜓 𝑋 ሽ)൫𝜓 𝐴

where 𝜓 𝑋 𝑎𝑛𝑑 𝜓 𝐴 are mode shapes which can be either complex or real (X indicates experimental and A indicates predicted using FEM). Mode shapes from experiments are usually in complex form due to the damping in practical situations, whilst these in numerical model are usually real. The MAC calculation can consider the entire structure or the components separately, however, the magnitude of the vibration of the joist may have different scales as the chipboard, especially the vibration of the joist in the direction perpendicular to it, which may undermine the strength of the correlation. Thus, only the modal vectors of the chipboard were considered.

4. RESULTS

4.1. Isolated timber joist

The dynamic properties of the isolated timber joist, including its damping, eigenfrequencies and corresponding mode shapes were obtained from EMA.

4.1.1. Loss factor of the joist

The damping of the isolated joist from EMA for bending and torsional modes is shown in Figure 4. Note that when the joist was vertically suspended, only one torsional mode was identified. The results show that the low-frequency loss factors differ for the vertical and horizontal orientations and that the damping of the torsional modes was higher than for the bending modes. On average it was as- sumed that a frequency-independent loss factor of 0.01 was reasonable in the FEM model where the focus was on one-third octave bands up to 200Hz.

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Figure 4: Loss factors of the isolated joist for individual bending and torsional modes, V: vertically suspended; H: Horizontally suspended.

4.1.2. Comparison between FEM and EMA

For the isolated joist, the correlation between EMA and FEM in terms of MAC below 1kHz is shown in Figure 5. All MAC values are >0.9 when vertically and horizontally suspended which indicates that it is reasonable to model the joist as isotropic with the measured material properties.

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Figure 5: MAC of the isolated joist when suspended vertically (left) and horizontally (right).

4.1.3. Transfer mobility

The transfer mobility on the joist at Acc3 referred to the force (see ‘x’ in Figure 2) is shown in Figure 6. The majority of the peaks in the FEM prediction are in reasonable agreement with the measure- ments, thought there is a small frequency shift and the assignment of a single damping value in the FEM model has resulted in peak values being slightly lower than the measurements. However, the assumption of an isotropic material in FEM with these measured material properties is reasonable.

titi

Figure 6: Transfer mobility of the isolated joist when vertically suspended, calculated from the ve- locity response at Acc3 for force position ‘x’ (indicated in Figure 2).

4.2. T-junction

4.2.1. Comparison between FEM and EMA

The MAC for FEM and EMA are shown in Figure 7 for the T-junction with 24 screws and 12 screws. The mode frequencies and mode shapes are shown along the axes. Below 100Hz, MAC >0.8 for eight out of 13 modes when there were 24 screws and for eight out of 12 modes when there were 12 screws. This indicates that the FEM model with rigid point connections can reasonably pre- dict the modal dynamics although there is some scope for improvement. When 12 screws were re- moved, there was a decrease in the first three eigenfrequencies from EMA indicating that the T- junction is ‘more flexible’, possibly because the chipboard is not pulled so close to the joist in-be- tween the screw positions.

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Looking at the accompanying mode shapes, the joist affects the response on the chipboard – for example, the chipboard tends to have lower vibration levels at one-third of the distance across the width of the chipboard along the line where the joist is connected.

Figure 7: MAC between FEM and EMA: 24 screws (left figure) and 12 screws (right figure). Axes also show the mode shapes and eigenfrequencies.

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4.2.2. Transfer mobility

Figure 8 shows the transfer mobility at Acc1 and Acc2 on the chipboard, and Acc3 and Acc4 on the underside of the joist referred to the force position F1. It can be seen that the FEM transfer mo- bilities are similar to the measurements. With 12 screws, there is closer agreement between FEM and measurements below 40Hz, than with 24 screws. This indicates that the FEM model with rigid point connections between the chipboard and the joist is only appropriate when the screws are not so close together that they make a ‘continuous connection’ along the joist in-between the screws. This would need consideration for a more accurate FEM model.

Another comparison of FEM and the measurement is shown in Figure 9 for the transfer mobility where both the force and the response position are on the side of the joist with 12 screws. It can be seen that the FEM model based on rigid connections for the screws has only two modal peaks below 200Hz. Hence, an improved FEM model might require the screw connections to be more flexible

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such that springs rather than a rigid connection is needed. This will be investigated in the next stage of the work.

Figure 8: Comparison of FEM and measured transfer mobilities at Acc1, Acc2, Acc4, and Acc5 for force position F1 on the chipboard. 24 screws (Left column); 12 screws (Right column)

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Figure 9: Comparison between FEM and EMA transfer mobility at Acc3 for force position F2 on side of the joist. 12 screws.

Figure 10: Comparison of velocity level ratio between the chipboard walking surface and the under- side of the joist: 24 screws (left) and 12 screws (right).

4.2.3. Velocity level difference

In terms of the spatial-average velocities, the velocity level difference between the chipboard walking surface and the beam is shown in Figure 10. For 24 screws there is reasonable agreement between FEM and measurements up to 100Hz and for 12 screws up to 50Hz. The next stage is to extend the FEM model to higher frequencies to further assess these differences.

5. CONCLUSION

This paper shows preliminary investigations into modelling screw connections between chipboard and a timber joist as rigid point connections. EMA measurements on an isolated timber joist indicate that it is reasonable to model it as an isotropic material. EMA measurements on this joist when screw- connected to chipboard indicated that rigid point connections are a reasonable assumption to give agreement between the mode shapes from EMA and FEM for two cases with 24 screws and 12 screws respectively. However, when comparing transfer mobilities below 50Hz with excitation and response in the direction perpendicular to the chipboard, EMA gave closer agreement with the FEM model when there were fewer screws (i.e. 12 screws), It is possible that when there are more screws (i.e. 24) a line connection is formed as the chipboard is pulled tight to the joist; this might require connecting

all nodes along the joist to the chipboard. In addition, when assessing the transfer mobility for lateral excitation of the joist there is evidence that a rigid point connection may not be appropriate and there- fore future work will assess the use of springs in three coordinate directions.

6. ACKNOWLEDGEMENT

This project has received funding from the European Union's Horizon 2020 research and innovation 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.

7. REFERENCE

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