Using Mechanical Metamaterials in Guitar Top Plates: A Numerical Study

Round 1

Reviewer 1 Report

 

The paper suggests another distribution of masses on a classical guitar top plate compared to a traditional fan-bracing, now using a grid of oval holes in a vertical or horizontal direction. Its reasoning is that this distribution is not altering the eigenfrequencies of the instrument but increases stability and therefore leads to a louder instrument. The regular hole grid is called a metamaterial structure.

The paper’s reasoning that the instrument sound is not altered using a hole grid is only taking the eigenmodes into consideration. Still, due to the large damping of wood, the internal damping is of much more importance over the whole frequency range, above about 300 Hz it might be the only important property for sound radiation, transient behaviour, or spectral shape. The paper only discusses the eigenvalues and does not estimate any changes in damping due to top plate alterations.

Furthermore, the paper has a faulty idea of metamaterials in general. The literature on metamaterials finds two main points for a material to be a metamaterial: a) it needs to show behavior not present with natural materials like negative stiffness, negative density, cloaking, etc. b) it is built as a complex geometry with graining or granulation on a sub-wavelength of the waves it acts on. The paper adds a complex geometry to a guitar soundboard, still, the only effect is one of changing mode frequencies and shapes. This always happens when changing the fan bracing of a guitar top plate or other plates and is not a metamaterial description. The paper does also not discuss any metamaterial effects, like cloaking, band-gaps, acoustic lens behavior, or the like. In this respect, the fan bracing of a guitar top plate is already a metamaterial, as it is a regular structure on the plate. The authors do build a plate with a geometry acting on a sub-wavelength level but without metamaterial effects. I would suggest leaving out the word ‘metamaterial’ altogether in the paper and only talk about altered top plate design.

The literature on metamaterials for musical acoustics is not at all sufficient. The authors cite two papers, of which one is not published and not part of the submission. Others would be

For wind instruments:

Petersen, E., Guillemain, P., ,Kergomard, J. & Colinot, T.: The effect of the cutoff frequency on the sound production of a clarinet-like instrument. JASA 145(6), 3784-3794, 2019.

 

Petersen, E., Colinot, T. Guillemain, Ph. & Kergomard, J.: The link between the tonehole lattice cutoff frequency and clarinet

sound radiation: a quantitative study. Acta Acustica 2020, 4, 18, 2020.

 

For drums:

 

Bader, R., Fischer, J. L., Münster, M. & Kontopidis, P.: Metamaterials in Musical Acoustics: A modified frame drum. JASA 145 (5), 3086-94, 2019.

 

For plates:

Bader, R., Fischer, J. L., Münster, M. & Kontopidis, P.: Metamaterials in Musical Instruments. Proceedings International Symposium of Musical Instruments (ISMA), 1-19, 2019.

Next to others.

In ‘Materials and Methods’ a ‘Torres’ model is introduced and associated with a classical guitar of seven fan-bracings. Still, such a design was introduced before Torres, while Torres only add-on was the tinning of the rims of the top plate. The authors say: “the plate here is not cut all the way through: indeed, the total thickness of the soundboard is 2.5 mm, which leaves 0.5 mm of solid material on top.“ Is this following the Torres idea? What does this sentence mean?

 

How is the FEM analysis performed by commercial software and native code? Which differential equations are used for the plates? The method section needs to be improved considerably.

 

In 3.1.1 the boundary conditions of the guitar are mentioned to be simply supported. In reality, boundary conditions are much more complicated. Simply supported means they do not move at the boundaries. Still, this is not true. As the boundary conditions crucially alter eigenmodes and internal damping, more realistic boundary conditions should be chosen.

 

In Fig. 3 the stress distribution for the three cases of normal fan-bracing, longitudinal and horizontal hole distribution are shown, and it is reasoned that the longitudinal hole distribution leads to a better stability of the instrument due to a larger distribution of stresses, away from the bridge. Still, the bridge is the most stable part of the top plate due its much larger mass. Therefore we might argue the other way round and find the longitudinal hole distribution to be less stable. A more robust method needs to be given to estimate at which tension and at which position the top plate might crack.

 

The driving-point motility in Fig. 4 would need to be transferred to a radiation sound pressure level (SPL) in dB. Only then we can reasonable see if the difference in mobility is audible. As the main reason is that the instrument is getting louder due to the alternation, this point is crucial.

Author Response

see attached file

Author Response File: Author Response.pdf

Reviewer 2 Report

The work numerically investigates a periodic array of holes to improve the acoustic performance of a guitar. The work is well written with clear images. However, it is important the authors improve some points:

1 – It’s not clear the metamaterial unusual property of the proposed plate with periodic holes. Holes have been used to improve the mechanical properties of dynamic structures even before the emergence of the metamaterial term and its unusual properties. By the mobility response (Fig. 4), the holes do not change the dynamic response in a strong way as expected to occur by using a metamaterial configuration. Please, provide some comments on the previous phrases. And please, revise the use of the metamaterial term by clarifying it or by removing it from the manuscript, maybe it can be replaced by periodic array of holes.

2 – Can you please provide the dispersion diagrams for the different plate configurations? How the elastic waves propagate in the different plate scenarios without and with holes?

3 – A high vibration level not necessary means a louder instrument. Can you please provide the sound pressure level around the guitar?

4 – Typo in Section 3.1.1: “diaplcement”.

Author Response

see attached file

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

The authors have submitted a new version of the manuscript, however it presents some issues that become the manuscript difficult to understand:

1 - The literature references are missing;

2 -  Fig. 4 is missing;

3 - The old Figures were replaced by new Figures, which are out of context in the revised version. The new Figures don't match with the description in the text and in the labels.

Due to these issues is not possible to check if all the reviewer's comments were addressed. 

 

 

 

Author Response

1 - However, some points regarding the term metamaterial or mechanical
metamaterial are still not clear:
- The cited references to support the use of the term metamaterial
[18,19] are from auxetic and phononic materials/structures that are not
considered metamaterials.

We have removed ref [18] and added two new references, in particular a review by Bertoldi et al. in Nature Reviews entitled "Flexible mechanical metamaterials" that shows that indeed those simple phononic materials are considered mechanical metamaterials. They were not considered as such when they were discovered yet is the common term used now. 

- The concept of mechanical and acoustic metamaterials are mixed through
the manuscript. Usually, the first is employed for mechanical behavior
such as strain-strain, buckling, ... while the second is used for wave
behavior such as frequency response. Can you please clarify this point?

First of all, we never use the term acoustic metamaterial. The use of the term acoustic is only for the guitar, for which we compute the frequency response. We have changed the text to make explicit that we use a mechanical metamaterial for the plate of the instrument. This mechanical metamaterials has different equivalent density and stiffness from a solid plate as we show in [17]. The difference in mechanical parameters implies a difference in the acoustic response of the whole guitar, and this is what we study in this article. 

It may be possible that the metamaterial is also an acoustic metamaterial, but preliminary results in plates have shown that the bandgap are in very high frequency range so we assume they are not as relevant as the mechanical aspect. 

- Again, it would be interesting to show the band structure for the
different configurations to better understand their differences around
600 Hz.

We agree with the referee that this is interesting but we think that an study like that goes beyond the scope of this paper for the following reasons. First, from our previous results in 2D plates with through holes the first bandgap is around 30KHz so it's not audible. Second, doing simulations of a 3D structure like we use in this article would require much more expensive simulations for which the timeframe of the this response is not enough. Third, the change in the SPL is clearly a variation in an eigenfrequency, and not a bandgap, since the power increases and changes its peak location.

Studying the bandgap is something that we are currently planning though and will be part of a future article dedicated to the acoustic properties of these metamaterials and how they change with cell geometry. 

2 - In the results of Fig. 4, the elastic and acoustic damping were
included in the simulations? If not, how does the damping affect these
results?

We have added the relevant text. 

Minor points:
- Pag. 4, section 3.1.3, instead Fig. 2(b) is Fig. 2(c)? Corrected
- Letters "a" and "b" are not specified in the label of Fig. 4 " Corrected.