# Effect of bracing on a music instrument top - is FEM modal analysis enough to predict this?

I'm currently building a music instrument (string instrument), and I am at the step where one applies certain braces to the soundboard. Since I actually don't know how the bracing will affect the sound (or basically anything) I thought it would be a good starting point to model the soundboard and the applied braces with CAD, and then apply an FEM simulation that will show to me the effect of the braces.

To my Background, I'm a physicist, but besides a course in technical mechanics 1 (statics), and some time selfstudying continuum-mechanics and the cauchy-stress-tensor, I don't have a background in engineering.

The program I'm using can perform a modal analysis on the soundboard, and modal analyses are also is the predominantly used way to model e.g. violin bodys with FEM technology (for example in this publication]1, or in this youtube video).

But of what worth are those simulations actually? In the end, I roughly want to know how well the soundboard is able to transmit frequencies to the air that have been present in the strings before, that means how well an amplitude in a frequency in the strings will translate to an amplitude in air.

A modal analysis will show me in what ways (and with what frequencies) the body will vibrate on its own (without external force applied). But it won't show me the amplitude of the soundboard (because it's abitrary): following (this stack exchange answer on modal analysis, the amplitudes of the eigenmodes can have any value, similar to the length of an eigenvector that also can have any value.

Granted, a periodic force will excite a vibrational modes amplitude the more if it meets its frequency, but doesn't it also play a role where this force is applied? When I apply period forces at a point on the soundboard where a certain mode of that soundboard has a node, I would expect the mode not to be excited at all for example.

Additionally, I know from a simple harmonic oscillator that its amplitude, subject to an external driving force, will reach the maximum at its resonance frequency. Is the same true here? When I apply a driving force to the soundboard, will it be sufficient to check how it affects the eigenmodes of the soundboard, because this will be the strongest excitations anyway?

As an addendum, I am aware that I don't model the transfer from soundboard to air at all. For now, I only want to model the transfer from amplitudes in the strings to amplitudes in the soundboards vibrations.

So to make the question short and concise: What can (and what can't) modal analysis tell me about the transfer of periodic motion from the bridge (where the strings are attached) to the soundboard? And since this information alone is probably not enough - When modelling the effect of external forces on the soundboard, will it be enough to only look at the frequencies of the previously found out eigenmodes?

• Are you asking about the capability of the software or the effect of the soundboard? Jan 1, 2023 at 11:45
• @SolarMike I'm asking about the capability of the method (finding vibrational eigenmodes). The software doesn't matter at this point. What do you mean by "effect of the soundboard"? Jan 1, 2023 at 13:03
• So have a soundboard that you can excite with a range of frequencies — useful to choose a range that matches the frequencies of use, then have a movable brace that can be controlled so that you can record the results as the brace is moved. Be a nice experiment, then you can see if the theory matches practise. Jan 1, 2023 at 13:14
• @SolarMike I don't want to determine this experimentally. The whole point of the FEM is to not have to do an experiment. I can't build 20 soundboards with different brace position, and I can't determine the response to all the frequencies. Jan 1, 2023 at 14:39
• I said movable brace… And most theory is validated by experimental results - for accuracy. Jan 1, 2023 at 15:03

What can (and what can't) modal analysis tell me about the transfer of periodic motion from the bridge (where the strings are attached) to the soundboard?

Very often in engineering, we find ourselves in a situation where getting the exact complete answer is difficult and time consuming. But often there is a simplified analysis method that gets us part of the answer quickly and easily. Often time part of the answer is actually enough to solve the problem, so we don't even need to bother with the complicated analysis.

For example, in your problem, you really want to know transmissibility (and/or resonant amplification). A forced response analysis will tell you that, but it can be long and complicated (and often requires inputs that you might not even know, like the damping). A modal analysis, only tells you natural frequency and mode shape. But we know that resonant amplification is related to the frequency ratio between the excitation and the natural frequency. So we can use this to get part of the answer.

E.g. let's say you are concerned with an excitation frequency of 440 Hz. If the natural frequency is 40,000 Hz then you don't even need to run a forced response analysis to know that the resonant amplification is basically nothing. Maybe it's 1.0001 and maybe it is 1.0002 but who cares. Further, if brace A gives you a natural frequency of 40,000 Hz and brace B is 41,000 Hz, then they are both who cares conditions, there is no reason to prefer one over the other. You can save yourself the hassle of running a forced response analysis.

Now on the other hand, if the excitation frequency is 440 Hz, and the natural frequency is 430 Hz, well now you know that the resonant amplification could be significant. You don't know the exact answer until you run the full forced response analysis, but you know it is definitely more than nothing. And if brace A gives you 430 Hz and brace B gives you 40,000 Hz, well that alone might be enough to tell you to prefer one over the over, even without running forced response.

In my work (not musical instruments but very much vibration), I use modal analysis as a first screening criteria. When I come up with a design, I first look at the natural frequencies. If they are not where I want then, then I start tweaking the design to move them around. When they start to get close to where I want them, then I start looking at forced response analysis.

When I apply a periodic force at one point, is there a simple way to determine how mutch an eigenmode will be driven by that point? I would guess that the bigger the eigenmode vibrates at that point, the better the force couples to the eigenmode.

Yes, that is it exactly. In mathematical terms, the response will be the dot product of an excitation force vector and the mode shape vector.

Overall I think you might benefit from an undergraduate textbook in vibration theory. Rao could be good. The most recent edition is typical textbook expensive, but a used copy of the previous edition is reasonable: https://www.amazon.com/Mechanical-Vibrations-5th-Singiresu-Rao/dp/0132128195

• To get it right, for the forced input analysis, I would specify exactly how the vibration is transmitted to the soundboard, at what position(s) and in what strength? Jan 3, 2023 at 2:46
• Correct. You also need to know the damping, which is extremely hard to predict. Best way is just to build one and measure it (or if you have a similar instrument build by someone else, measure that) Jan 4, 2023 at 1:19

Bracing in the simplest way, say a large flat sheet extended reasonably long in both directions will increase the stiffness of the sheet and hence reduce its natural frequency. This will lead to response spectra favoring the forcing function, string vibration. $$\frac{1}{(\omega n)^2} \frac{d^2x}{dt^2}+\frac{2\zeta}{\omega n} \frac{dx}{dt}+x=KF_0 sin(\omega t)$$

• $$\omega n$$ natural frequency
• $$\zeta$$ damping ratio
• K=1/k

But the complex curved surface of a string instrument and its geometry and many other factors will affect its response spectrum and even cause random beating and unpredictable behavior.

I would imagine you need a supercomputer and sophisticated modeling to be able to achieve what you are seeking.

As seen in Chladny patterns, even on a square aluminum plate the location and frequency of perturbation (violin bow) create wildly different patterns. here is a Youtube clip on the effect of excitation on a flat plate. falt plate patterns

• Are the eigenmodes of the aluminum plate linked to the observation that different bow positions create different patterns? Or more mathematically: When I apply a periodic force at one point, is there a simple way to determine how mutch an eigenmode will be driven by that point? I would guess that the bigger the eigenmode vibrates at that point, the better the force couples to the eigenmode. On a more general level: When you guess that I need a supercomputer, do you think that a usual desktop pc won't even suffice for a qualitative observation? Not enough precision? Or not enough accuracy? Jan 2, 2023 at 11:04
• yes, of course, they are two variables and are many times complex variables! Jan 2, 2023 at 13:25