3.1. Validation of the Acoustic Model
Within the finite element framework, the acoustic FSI model is conceived as a room-wall-room framework considering anechoic termination at the boundaries. The model is used to predict the sound insulation of the drywall composed of plain and perforated studs sandwiched between two gypsum plasterboards of finite thickness. The sound originating room (Room 1) and the transmitted room (Room 2) are both modeled with elements featuring fluid pressure
, while the elements in contact with the wall are modeled with the displacement components
,
and
in addition to
. As such, the finite element shape function for the spatial variation of the acoustic pressure and displacement was modeled as shown in Equations (13) and (14), respectively:
where
and
are the element shape function for pressure and displacement, respectively.
is the nodal pressure and
are the nodal displacement component vectors
,
and
. From Equations (13) and (14), the second time derivative of the variables and the virtual change in the pressure was derived as shown in Equations (15)–(17):
when the matrix operator
applied to the element shape function
, the expression becomes Equation (18):
Substituting Equation (13) through (18) and into (12), the finite element expression of the wave equation can be written as:
where {
n} is normal at the fluid boundary. Terms that do not vary over the element are taken out of the integral sign.
is an arbitrarily introduced virtual change in nodal fluid pressure which can be factored out in Equation (18); since
is not equal to zero, Equation (19) can be rewritten as shown in Equation (20):
Subsequently, Equation (20) can be expressed in matrix form to get the discretized wave equation as shown in Equation (21):
where
is the fluid mass matrix given by Equation (22),
is the fluid stiffness matrix given by Equation (23) and
is the coupling mass matrix enabling fluid-structure interaction at the interface given by Equation (24), which is solved to obtain the nodal pressure values at either side of the stud wall. In the context of acoustic insulation, the quantity of interest is the sound pressure loss, which relates directly to the sound reduction index (
R), which defines the sound insulation of the wall and hence the influence of the stud and perforation configuration.
The finite element prediction model has been validated on a plasterboard wall featuring the C-channel stud as shown in
Figure 2b. The validated prediction model was subsequently used to analyze the drywalls featuring perforated studs. The experimental measurements used for validation were tested in the acoustic transmission suite at the Sound Research Laboratory (SRL), UK. The numerically predicted
R values in comparison to experimental test measurements observed for the non-perforated C-channel stud (P) is shown in
Figure 4. Overall, the finite element prediction can be seen to closely follow the physical test curve at a frequency of 100–3150 Hz. Nevertheless, a 6 dB dip can be seen at 500 Hz for the FE model which was not found in experimentally measured data, indicating the likelihood of critical frequencies generally observed in harmonic analysis.
To identify the localized variation in the sound reduction index between the experimental and numerical results, a modal analysis of the same model was carried out. This aids in identifying the inherent resonant characteristics of the system in the forms of eigenmode frequencies. As such the analyses were carried out to identify all eigenmodes at the 1/3rd octave range. The results showed that at Eigenmodes 80 and 172, the natural frequencies of the wall were at 499.8 Hz and 1002.1 Hz, respectively. This indicates that the localized dip at 500 Hz and 1000 Hz are due to the influence of neighbouring critical frequencies of the numerical model.
Other than for the two frequencies 500 Hz and 1000 Hz, the differences between the curves were around 2.5 dB, which is consistent with accepted variation in measurement reproducibility for acoustic measurements [
56]. Consequently, the sound reduction index at 1/3rd octave was used to compute the weighted index (
), which is the single number quantity that is used to represent the acoustic insulation of the wall. The higher the weighted index the superior the acoustic insulation of the structure. Generally, the weighted index is accompanied with the spectrum adaptation terms
and
referring to the insulation against A-weighted pink noise and A-weighted urban traffic noise, respectively. For characterizing the acoustic insulation of a structure, the spectrum adaptation terms are important as they contextualize the performance based on different scenarios.
A-weighted pink noise defines the insulation of the wall against sound from activities such as high-speed train noise and cars travelling at speeds above 50 mph, in addition to medium and high-frequency noise from industrial activities. The
term, on the other hand, characterizes performance at low to medium frequency sounds such as urban traffic, low-speed trains and machinery.
Table 3 lists the
between the finite element acoustic model and experimental measurement for the wall featuring the C-channel stud (P). The predicted and experimental single number ratings were found to be in excellent agreement, demonstrating that the numerical model is suitable for valid prediction of the sound reduction indices.
3.2. Influence of Perforated Studs on Sound Insulation
To identify the potential of the perforated studs, the sound reduction index was predicted using the validated finite element model. The sound insulation of each of the perforated stud configurations sandwiched between 15 mm plasterboard featuring air cavities was evaluated. Other than for the perforations all geometrical dimensions of the stud, wall and the model were kept constant. This means that the only difference between the different FE acoustic models is the perforated web configurations of the C-Channel stud.
The airborne sound reduction index for all the different slot configurations at 1/3rd octaves are shown in
Figure 5a–f. The results show that the acoustic performance of the wall is influenced by the perforated web. Despite having the perforated to the non-perforated area of the web constant for all the configurations, a difference in the sound reduction index can be seen depending upon the way the perforations are arranged on the web. It was observed that the perforation configuration Dt outperformed all other configurations other than for localized frequency effect observed at 125 Hz and 315 Hz.
For the overall sound insulation, the performance of Dt was closely followed by perforation configuration C. The performance of configuration-F was found to be the worst in comparison to the insulation demonstrated by all the other perforation configurations. Overall, the data show that both the placement and arrangement of perforations influences the acoustic response of the wall. The second-best insulation was demonstrated by the perforation configuration C, which may be favorable from an installation point of view as the performance is not dependent on the stud orientation.
Although the sound reduction index at 1/3rd octave provides a comprehensive description of the acoustic performance, when it comes to rating the sound insulation of the stud wall, the single number rating bears a higher significance. There are different single-number ratings featured in describing sound insulation of a building wall, out of which the method makes it more reliable and unambiguous for reporting the acoustic insulation using a single number that is directly proportional to the amount of sound transmission resisted by the partition.
Figure 6 shows the weighted sound reduction index observed for the wall featuring different perforation configurations in comparison to the un-perforated stud. Comparing the
, the plain stud P exhibited the lowest performance of all the designs tested. The perforated stud designs exhibit a slightly improved performance compared to the plain stud. The stud design Dt exhibited the best performance offering a +4 dB increase in insulation and F exhibited the lowest perforated stud performance with only a +1 dB increase compared to plain stud. Studs C, B and G exhibited similar performance with an improvement of +3 dB insulation. Stud E performed slightly better than F demonstrating an improvement of +2 dB in comparison to the non-perforated stud.
Comparing the , values, the perforated studs provide slightly superior acoustic insulation compared to a plain stud with similar dimensions. Although the perforation to non-perforation ratio was constant for the perforated configurations, the studs exhibited different performances depending on the location of perforation in the web. This shows that optimizing the perforation location is critical in achieving the best possible sound insulation for perforated studs. Although the overall trend is similar to what was observed when comparing the R values, the offers a clear differentiation in performance rating between the stud configurations allowing to identify the best performing design.
Comparing the acoustic performance of the perforated studs with that of literature; Hongisto et al. [
57] experimentally evaluated the acoustic performance of perforated studs without observing notable improvements. This could be due to the requirement of a high perforation ratio to sufficiently lower the stiffness to influence the sound insulation. This is consistent with the observations of Wyngaert [
58] where a high perforation ratio is called for to sufficiently lower the stud stiffness to influence the sound insulation. This study has also shown that the patterns and placement of perforations are also important parameters affecting the acoustic performance. Overall, the performance of the best performing perforated design in this study is consistent to that of inclined web perforations [
58] which showed a gain of up to 3.8 dB in comparison to non-perforated studs. However, it is acknowledged that the experimental validation of the perforated stud designs is required before actual improvements can be characterized.
3.3. Correlation between Flexural Stiffness and Acoustic Response
Although the acoustic finite element model can be used to accurately predict the sound insulation offered by stud designs, the solution time is rather long. As such, an attempt to use a suitable structural parameter to offer an indication of the acoustic performance was sought. To identify the potential of using the stiffness of the stud as a suitable parameter in developing acoustic studs, the flexural stiffness at the flanges of the studs were evaluated using the finite element method.
The highest
was exhibited by the non-perforated stud (P) that showed the lowest acoustic insulation for all parameters tested, as shown in
Figure 7. The perforated stud design that demonstrated the highest weighted sound reduction index exhibited the lowest flexural stiffness. Stud designs C and B exhibit comparatively close stiffness values resulting in
of 42 (−2; −7) and 42 (−3; −7), respectively. The
values for studs G and E were also gradually decreasing consistently with an increase in stiffness from 16.118 kN/m and 18.489 kN/m, respectively. However, when the spectrum adaptation terms were evaluated, C showed the highest insulation against A-weighted pink noise. Both perforation configurations C and B showed the highest insulation against A-weighted urban noise.
Based on the relationship between the stiffness and sound reduction index, it can be suggested that for the highest weighted sound reduction index, the perforations should be designed to reduce the stud’s flexural stiffness. For the current analysis, an overall improvement in sound insulation of +4 dB was observed when the flexural stiffness of the stud was almost halved. When it comes to the performance against A-weighted pink and urban noise, the flexural stiffness is not a reliable indicator, and the full numerical model should be considered. This is primarily due to the non-linear frequency weighting considered for the spectrum adaptation terms.