Prediction models for sound, structure-borne sound and sound radiation are essential at the design stage, particularly for construction, automotive, marine, and aeronautic industries. Research in the ARU focuses on the development and validation of models and procedures such as Statistical Energy Analysis (SEA), Advanced Statistical Energy Analysis (ASEA), Experimental Statistical Energy Analysis  (ESEA), Transient Statistical Energy Analysis (TSEA), Finite Element Methods (FEM) and Finite Difference Time Domain (FDTD). 

FEM, ESEA and Monte Carlo methods have been used to determine coupling loss factors for use in SEA [1]. The aim was to use the concept of an ESEA ensemble to facilitate the use of SEA with plate subsystems that have low modal density and low modal overlap. An advantage of the ESEA ensemble approach was that when the matrix inversion failed for a single deterministic analysis, the majority of ensemble members did not encounter problems. Failure of the matrix inversion for a single deterministic analysis may incorrectly lead to the conclusion that SEA is not appropriate. However, when the majority of the ESEA ensemble members have positive coupling loss factors, this provides sufficient motivation to attempt an SEA model. Ensembles were created using the normal distribution to introduce variation into the plate dimensions. For plate systems with low modal density and low modal overlap, it was found that the resulting probability distribution function for the linear coupling loss factor could be considered as lognormal. This allowed statistical confidence limits to be determined for the coupling loss factor. The SEA permutation method was then used to calculate the expected range of the response using these confidence limits in the SEA matrix solution. For plate systems with low modal density and low modal overlap, relatively small variation/uncertainty in the physical properties caused large differences in the coupling parameters. For this reason, a single deterministic analysis is of minimal use. Therefore, the ability to determine both the ensemble average and the expected range with SEA is crucial in allowing a robust assessment of vibration transmission between plate systems with low modal density and low modal overlap.

For heavyweight buildings, a study has been made to assess the prediction of vibration transmission between masonry walls using FEM and SEA through a comparison of measured and predicted data [2]. Masonry walls typically have low modal overlap and low modal density at low frequencies; hence FEM is used to predict the large fluctuations in the vibration level difference. The issue of uncertainty in the physical description of masonry walls is addressed by using Monte Carlo methods with FEM. Two types of masonry wall junction are studied, an L-junction and a T-junction. The importance of in-plane wave transmission was demonstrated for L-junctions with apertures and T-junctions. The agreement between FEM, SEA and measured data at high frequencies indicated that the conversion between bending and in-plane waves in the FEM models was correct. Further work used numerical experiments with FEM to calculate the vibration transmission between masonry walls with window apertures at different positions in the flanking wall(s) [3,4]. Results from the numerical experiments were used to assess a simple rule-of-thumb estimate for calculating the change in the coupling parameters due to the introduction of an aperture into a flanking wall. 

To build SEA models, coupling loss factors for bending wave transmission between coupled plates can be determined using experimental statistical energy analysis. However, some types of plate junctions introduce significant wave conversion such that the assumption of an SEA system that supports only bending waves is no longer appropriate. Three methods have been assessed to identify the existence of wave conversion between bending and in-plane waves when experimental SEA is used to analyse a system that is assumed to consist of only bending wave subsystems [5]. These methods were based on errors in the internal loss factor, matrix condition numbers, and a failure to satisfy the consistency relationship. Only the former method that calculated errors in the predicted internal loss factors was found to be of use in identifying wave conversion.

An experimentally validated analytical model has been developed in order to investigate the effect on impact sound transmission at low frequencies of location of the impact, type of floor, edge conditions, floor and room dimensions, position of the receiver and room absorption [6,10]. The model uses normal mode analysis to predict the sound field generated in rectangular rooms due to point excitation of homogeneous rectangular plates with different edge conditions. Laboratory and in situ measurements confirm that the models can be used to estimate impact sound transmission at low frequencies.

Ribbed plates have been investigated theoretically and experimentally, with respect to the mobility of lightweight building elements and the effect on structure-borne sound transmission from attached vibrating sources [7]. Prediction of structure-borne sound transmission on built-up structures at audio frequencies is well suited to SEA although the inclusion of periodic ribbed plates presents challenges. Research has investigated ASEA to incorporate tunnelling mechanisms within a statistical approach to address this issue [8]. The coupled plates used for the investigation form an L-junction comprising a periodic ribbed plate with symmetric ribs and an isotropic homogeneous plate. ESEA is carried out with input data from FEM. This indicates that indirect coupling is significant at high frequencies where bays on the periodic ribbed plate can be treated as individual subsystems. SEA using coupling loss factors from wave theory leads to significant underestimates in the energy of the bays when the isotropic homogeneous plate is excited. This is due to the absence of tunnelling mechanisms in the SEA model. In contrast, ASEA shows close agreement with FEM and laboratory measurements. The errors incurred with SEA rapidly increase as the bays become more distant from the source subsystem. ASEA provides significantly more accurate predictions by accounting for the spatial filtering that leads to non-diffuse vibration fields on these more distant bays.

FDTD models have been used to predict low-frequency sound fields in small volumes containing a limp panel formed from a porous material which partially or completely subdivides the volume [9]. This porous panel is incorporated into FDTD using a Rayleigh model as proposed by Suzuki et al. However, to accurately reproduce the low-frequency sound field it is found necessary to introduce an additional Moving Frame Model (MFM) to account for motion of the porous panel. For spaces that are completely subdivided by a porous panel, the MFM accounts for a spring-mass-spring resonance that can occur below the lowest acoustic cavity mode. The MFM assumes lumped mass behaviour of the porous panel which is coupled to the FDTD update equations that incorporate the Rayleigh model. FDTD is compared against measurements using transient excitation with a pulse input to a loudspeaker in a small reverberant room under three different conditions: (1) empty room, (2) with a mineral fibre panel partially dividing the room, and (3) with a mineral fibre panel completely dividing the room. Close agreement is obtained between experimental results and FDTD incorporating the MFM; this validates the models as well as implementation of the loudspeaker as a hard velocity source. 

Transient sounds often have an adverse effect on human occupants in the built environment for which annoyance and sleep disturbance are assessed using the Fast or Slow time-weighted maximum sound pressure level. For this reason there is a need for a validated model that can predict maximum levels due to transient excitation in buildings over the audio frequency range. In this paper, TSEA has been developed to predict both maximum sound pressure levels and maximum vibration levels [11]. Three key aspects of TSEA are developed: the requirements for the time interval; the implications of using steady-state SEA coupling loss factors; and definitions of measured, hybrid and synthetic ’transient power’ inputs. A proposal is also made to modify the signal processing when measuring maximum vibration levels on source subsystems such as walls/floors where the measurement needs to be compared with predictions using TSEA. Numerical simulations and measured force and vibration data are used to demonstrate the TSEA modelling requirements and to quantify potential errors. Experimental validation of TSEA has used two experimental designs to ensure coverage of a range of building scenarios where the rooms, walls and floors have a wide range of mode counts and modal overlap factors [12]. Close agreement between measured and TSEA predicted maximum levels confirm the validity of the following aspects: (1) determining the appropriate time interval using the prescribed lower and upper limits, (2) using steady-state radiation efficiencies in TSEA models, (3) using ‘transient power’ in its three forms (measured, hybrid and synthetic) as well as the use of a Blackman window function to produce a synthetic impulse from a force hammer, and (4) modifying the signal processing when measuring maximum time-weighted vibration levels on source subsystems for comparison with  SEA. It is concluded that TSEA can be used to predict maximum Fast time-weighted sound and vibration levels in buildings with a similar accuracy to SEA for steady-state excitation.

The inclusion of rib-stiffened plates within the framework of Statistical Energy Analysis (SEA) is a challenge in the field of engineering noise control for the low- and mid-frequency ranges. Hence research was carried out to focus on periodic ribbed plates with symmetric ribs and assesses different approaches using SEA to model bending wave transmission when one or both of the rectangular plates that form an L-junction are a periodic ribbed plate [13]. SEA is compared with measurements and Finite Element Methods (FEM) with all plate boundaries pinned to give boundary conditions that are representative of engineering structures typically used for noise control. When one or both plates are ribbed, and the ribs are parallel to the junction, the closest agreement between measurements and FEM is with SEA models that use a combination of Bloch theory and wave theory to determine the coupling loss factors. However, when both plates are ribbed plates, one with ribs orientated perpendicular to the junction and the other with ribs parallel to the junction, the available SEA models which assume an effective isotropic plate, or an equivalent isotropic plate or angle-dependent bending stiffness all underestimate the energy level difference.

Prediction of bending wave transmission across systems of coupled plates which incorporate periodic ribbed plates is considered using Statistical Energy Analysis (SEA) in the low- and mid-frequency ranges and Advanced SEA (ASEA) in the high-frequency range [14]. Research has investigated the crossover from prediction with SEA to ASEA through comparison with Finite Element Methods. Results from L-junctions confirm that this crossover occurs near the frequency band containing the fundamental bending mode of the individual bays on the ribbed plate when ribs are parallel to the junction line. Below this frequency band, SEA models treating each periodic ribbed plate as a single subsystem were shown to be appropriate. Above this frequency band, large reductions occur in the vibration level when propagation takes place across successive bays on ribbed plates when the ribs are parallel to the junction. This is due to spatial filtering; hence it is necessary to use ASEA which can incorporate indirect coupling associated with this transmission mechanism. A system of three coupled plates was also modelled which introduced flanking transmission. The results show that a wide frequency range can be covered by using both SEA and ASEA for systems of coupled plates where some or all of the plates are periodic ribbed plates.

For bending wave transmission across periodic box-like arrangements of plates, the effects of spatial filtering can be significant and this needs to be considered in the choice of prediction model. Research investigates the errors that can occur with Statistical Energy Analysis (SEA) and the potential of using Advanced SEA (ASEA) to improve predictions [15]. The focus is on the low- and mid-frequency range where plates only support local modes with low mode counts and the in situ modal overlap is relatively high. To increase the computational efficiency when using ASEA on large systems, a beam tracing method is introduced which groups together all rays with the same heading into a single beam. Based on a diffuse field on the source plate, numerical experiments are used to determine the angular distribution of incident power on receiver plate edges on linear and cuboid box-like structures. These show that on receiver plates which do not share a boundary with the source plate, the angular distribution on the receiver plate boundaries differs significantly from a diffuse field. SEA and ASEA predictions are assessed through comparison with finite element models. With rain-on-the-roof excitation on the source plate, the results show that compared to SEA, ASEA provides significantly better estimates of the receiver plate energy, but only where there are at least one or two bending modes in each one-third octave band. Whilst ASEA provides better accuracy than SEA, discrepancies still exist which become more apparent when the direct propagation path crosses more than three nominally identical structural junctions.

Advanced Statistical Energy Analysis (ASEA) has been used to predict vibration transmission across coupled beams which support multiple wave types up to high frequencies where Timoshenko theory is valid [16]. Bending-longitudinal and bending-torsional models are considered for an L-junction and rectangular beam frame. Comparisons are made with measurements, Finite Element Methods (FEM) and Statistical Energy Analysis (SEA). When beams support at least two local modes for each wave type in a frequency band and the modal overlap factor is at least 0.1, measurements and FEM have relatively smooth curves. Agreement between measurements, FEM, and ASEA demonstrates that ASEA is able to predict high propagation losses which are not accounted for with SEA. These propagation losses tend to become more important at high frequencies with relatively high internal loss factors and can occur when there is more than one wave type. At such high frequencies, Timoshenko theory, rather than Euler-Bernoulli theory, is often required. Timoshenko theory is incorporated in ASEA and SEA using wave theory transmission coefficients derived assuming Euler-Bernoulli theory, but using Timoshenko group velocity when calculating coupling loss factors. The changeover between theories is appropriate above the frequency where there is a 26% difference between Euler-Bernoulli and Timoshenko group velocities.

Selected publications

[1] Hopkins C (2002) Statistical energy analysis of coupled plate systems with low modal density and low modal overlap. Journal of Sound and Vibration vol 251 issue 2 pp 193-214.

[2] Hopkins C (2003) Vibration transmission between coupled plates using finite element methods and statistical energy analysis. Part 1: Comparison of measured and predicted data for masonry walls with and without apertures. Applied Acoustics vol 64 issue 10 pp 955-973.

[3] Hopkins C (2003) Vibration transmission between coupled plates using finite element methods and statistical energy analysis. Part 2: The effect of window apertures in masonry flanking walls. Applied Acoustics vol 64 issue 10 pp 975-997.

[4] Hopkins C (2007) Sound Insulation, Butterworth-Heinemann, Imprint of Elsevier, Oxford, 2007 ISBN: 978-0-7506-6526-1.

[5] Hopkins C (2009) Experimental statistical energy analysis of coupled plates with wave conversion at the junction. Journal of Sound and Vibration vol 322 pp 155-166.

[6] Neves e Sousa A and Gibbs BM (2011) Low frequency impact sound transmission in dwellings through homogeneous concrete floors and floating floors. Applied Acoustics vol 72 pp 177-189.

[7] Mayr AR and Gibbs BM (2011) Point and transfer mobility of point-connected ribbed plates. Journal of Sound and Vibration vol 330 pp 4798-4812.

[8] Yin J and Hopkins C (2013) Prediction of high-frequency vibration transmission across coupled, periodic ribbed plates by incorporating tunnelling mechanisms. Journal of the Acoustical Society of America vol 133 issue 4 pp 2069-2081.

[9] Ferreira N and Hopkins C (2013) Using finite-difference time-domain methods with a Rayleigh approach to model low-frequency sound fields in small spaces subdivided by porous materials. Journal of Acoustical Science and Technology vol 34 issue 5 pp 332-341.

[10] Neves e Sousa A and Gibbs BM (2014) Parameters influencing low frequency impact sound transmission in dwellings. Applied Acoustics vol 78 pp 77-88.

[11] 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 vol 100 pp 46-56.

[12] 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 with Acustica vol 100 pp 57-66.

[13] Yin J and Hopkins C (2015). Treating periodic ribbed plates with symmetric ribs as individual subsystems in Statistical Energy Analysis: Models for bending wave transmission across L-junctions in the low- and mid-frequency ranges. Journal of Sound and Vibration, 344, 221-241.

[14] Yin J and Hopkins C (2015). Modelling bending wave transmission across coupled plate systems comprising periodic ribbed plates in the low-, mid-, and high-frequency ranges using forms of Statistical Energy Analysis. Shock and Vibration. doi:10.1155/2015/467875.

[15] Wilson D and Hopkins C (2015). Analysis of bending wave transmission using beam tracing with advanced statistical energy analysis for periodic box-like structures affected by spatial filtering. Journal of Sound and Vibration, 341, 138-161.

[16] Wang, X and Hopkins C (2016). Bending, longitudinal and torsional wave transmission on Euler-Bernoulli and Timoshenko beams with high propagation losses. Journal of the Acoustical Society of America, 140(4), 2312-2332.