# Vibration isolation of a plate

The topic of this blog post is one I started thinking of when a friend of mine presented the idea of whether the flexibility of a relatively thin concrete plate affects the response of a vibration-isolated foundation. In other words: should the behaviour of the plate in itself be taken into account in a situation like the one described here (without taking into account any possible other objects attached to the plate itself)?

Well, let's do some testing.

## The setup

The dimensions of the plate (whoops, t should be 8 cm)

These are the dimensions of the plate. The vibration isolators are chosen to be 10cm x 10cm & have a uniform dynamic stiffness of $E = 3 \mathrm{MPa}$ and a thickness of $t = 2.5 \mathrm{cm}$. The values I chose from concrete are $\rho = 2.5 \mathrm{kg}/\mathrm{m}^3$ and $E = 27 \mathrm{GPa}$.

## A cool gif

Steady state response / sine sweep

What you're seeing above is basically the result of a sine sweep on a damped system, where a sinusoidal force is placed at 20% from the outer right corner of the plate. So, we have a point force with a constant amplitude and a specific frequency attempting to vibrate the plate.

Note that the deformations are waaaayy exaggarated throughout this post.

One can clearly see that the deformation gets smaller as the frequency increases. Thus, a smaller portion of the exciting force is transfered to the foundations. Also note that in the beginning, mostly one corner vibrates but as we approach the modes at ~36 Hz (the modes will be shown later in this post) the plate starts to pivot/swing around the middle axis. So we're moving from a mix of modes 1 and 2/3 towards modes 2/3.

## Some theory

Starting from $\mathbf{M}\ddot{x} + \mathbf{C}\dot x + \mathbf{K} x = \mathbf{f}$, I solved the dynamic system $(-\omega^2\mathbf{M}+{i\mkern1mu}\mathbf{C}+\mathbf{K})\mathbf{x}=\mathbf{f}$ describing the steady state solution with a sinusoidal excitation. I used python, and obtained the mass/stiffness matrices using finite element analysis. The eigenfrequencies are calculated using regular analysis (from $\textbf{M}^{-1}\textbf{K}$).

The Mindlin–Reissner plate theory with bilinear elements was used for the plate. Also, I used regular spring elements lumped at the nodes for the vibration isolation (vertical shear deformations of the isolators are not central at the lowest frequencies). For the damping matrix, I used rayleigh damping with nothing but the stiffness matrix-related term with a value of 0.0075. I applied the damping directly to the global system.

Only vertical movement is taken into account here.

## The eigenmodes & resonant frequencies

Alright! What are the lowest frequencies the system will resonate on? Or, to put it another way, what are the lowest eigenmodes, and the corresponding eigenfrequencies? The lowest mode corresponds to the one obtained from the formula for a mass-spring system$f_0 = \frac{1}{2 \pi}\sqrt{\frac{k}{m}}$, which in this case gives $f_0 = 22.85 \mathrm{Hz}$.

Mode 1 @ 22 Hz

Mode 2 @ 36 Hz

Mode 3 @ 37 Hz

Mode 4 @ 188 Hz

Mode 5 @ 201 Hz

## The response of the system

How well does this system perform? What are the effects of these modes?

First, let's try placing the force in the absolute middle. This would be the ideal case, when we can be sure that everything works as well as it can. Let's also, to begin with, consider an undamped system. Note that concrete in itself definitely has some structural damping, so the response is by no means correct here:

Centered force, no damping

Alright! We can see a sharp peak at around mode 4. Mode 5 isn't visible, as the force in the absolute middle has no way of waking it up. Note that we're examining an undamped system here, so this is something which would happen if the concrete in itself wouldn't dissipate energy at vibrations around ~200 Hz (which it in reality does).

Ok then! What happens if we add some damping to the system? Let's also apply the same kind of damping to the isolating pads.

Center force, with damping

Suddenly we can see that the mode at around 188 Hz vanishes completely. This is because the type of damping we added affects higher frequencies quite a bit. I chose a coefficient which very roughly should correspond to the damping behaviour of concrete, but I did not do any thorough research here so please note that the resonance could in reality very well be somewhere between the two previous figures. Still, the results seem to very strongly indicate that the mode at around 188 Hz shouldn't be visible in a situation like the one described here.

Ok then! Let's try moving the force a bit. Let's move it so it's 20% from both sides. First, let's examine the case without damping. What does the response look like now?

Noncentric force, no damping

Apparently the force is situated in such a place that mode 4 isn't really excited. Instead, we can see that mode 5 is clearly visible.

Once again, let's see what happens when we add some damping.

Noncentric force, with damping

We can see that only the three first modes seem to affect the result. Still, I'd say that it's really important to notice that the first three modes actually affect the result quite a bit. This is also clearly visible in the animated gif at the beginning of the post.

## Conclusion

The modes of the plate in itself are very central when the structure in itself doesn't have strong damping characteristics. This would be the case if the plate was made of steel, for example. If the plate is made of homogenous concrete (no rebar is taken into account here), the results strongly indicate that only the modes where the structure can be assumed to act as a rigid body seem to affect the result.

Thus, in a case like the one above, it should be enough to consider the plate as a perfectly rigid structure. Still, the results strongly indicate that if the excitation isn't perfectly centered, other modes than the lowest one can have a strong effect on the performance of the isolating system. Also, if there is reason to suspect that the modes of the plate in itself aren't damped, it might be useful to examine them as part of the system.

# Why a tree can fall the wrong way when you're cutting it down

I was recently cutting down a tree and, even though I'm ashamed to admit it, it fell in the wrong direction. It fell in the opposite direction as to what was intended. Well, lesson learned. Next time I will be more careful and use a rope if I want to be completely sure it falls in some specific direction.

Cutting down a tree (from google, that's not me)

So, let's investigate things. Not by cutting down more trees, but instead by simulating doing it!

### So, what's happening?

The tree in static equilibrium. The center of mass is marked to the left of the trunk.

To simulate this, i drew a very simple tree-like structure in 2D (to defend myself I'll have to say that the tree I cut down looked a lot more centered than the one above, where it looks quite obvious that the tree would fall to the left). The tree has a lot more branches on one side as compared to the other. The center of mass is shown to to the left of the trunk. The resulting average force will force the tree to tip over to the left as it's located to the left of the trunk, no matter how we cut the main trunk (if there is some magic trick I'm not aware of, please let me know).

The deformation of the tree after being cut

After cutting a dent in the trunk with the purpose of allowing the tree to fall to the right, we can see that it starts to lean to the left. Whoops. This moves the centre of mass even further to the left. From the figure above, you can also see that the tiny narrow partition to the the left of the dent is carrying the whole tree; all the major forces (in red) are now centered around the dent.

### Conclusion

As long as the centre of mass is located to the right of the tiny partition left after the dent, we should be fine. This is not always easy to see, especially if the tree has a lot of branches.

### The theory

I used my solver in python with plane elements to calculate the figures above. The colors represent the von Mises stress.

# Simulating the physics behind a music box

Here's what the simulated sound of a music box sounds like:

## Some background

The comb teeth [3] in the picture below are the ones responsible for the notes you can hear.

The ratchet lever [1] rotates the cylinder [2], the pins pluck the comb teeth [3] which produces the music. The whole thing rests on the bedplate [4]. Source: wikipedia

## Version 1

The first model I made consisted of a uniform thin metallic plate (see the really cool gif below!). Problematically, there's some really strong coupling between the pins. Here's what happens when you apply a static deflection to one of the pins and let it go:

FEM model of a music box with a uniform plate thickness. Note the coupling of the pins.

As you can see, the adjacent pins start to vibrate after some time. The resulting tone is decent, but it's a bit "wobbly" due to the coupling between the pins:

## Version 2

The problem with the coupled pins must be something that would happen with real music boxes as well. Well then, how has this problem been solved before? What I noticed was that the thickness of the plate seems to vary. OK, so let's double the thickness of the plate everywhere except at the pins. Additionally, let's move the stiff supports a bit closer to the pins. Combined, these actions decrease the amount of sound that travels from one pin to another significantly. So, in practice, we get a tone that is a lot clearer. Great! After some minor additional changes, this results in the sound you can hear in the example at the very beginning of the post. There's still some coupling between some of the pins, but it's small enough not to be bothersome.

Refined music box FEM model. Note the absence of the strong coupling present in the previous model.

## Some theory

I used the Mindlin–Reissner plate theory, which should quite accurately describe the vibrating thin plate that causes the sound. I solved the problem numerically using finite element analysis using my own solver in python.

I started by calculating the static deflection of each pin when a force is applied at the very tip. I then "released" the pin, and calculated the resulting deformation in small time increments for each pin (using the GN22 algorithm). Then, I calculated the average position, velocity and acceleration of the plate at each time frame; if the whole plate is simplified and considered as a single mass with one degree of freedom the forces the plate directs at surrounding structures (such as a table, which I assume radiates most of the sound caused by the box to simplify things) should be somewhat proportional to the average acceleration of the plate.

The frequencies the pins vibrate at can be calculated according to the formula for the lowest natural frequency of a cantilever beam:

$\frac{3.52}{2\pi L^2}\sqrt{\frac{EI}{m}}$

where L is the length of the pin, E is the elastic modulus of the material, I is the moment of inertia and m is the mass per unit length. If $L_0$ is the length of the longest pin, we can now get the lengths of each subsequent pin using $\frac{L_0}{\sqrt{2^{i/12}}}$, where $i$ is how many semitones higher we wish to tune the pin to.

## Some thoughts

I'm really happy with the result, it sounds almost as good as the real thing. I added a bit of reverb, which is quite important for a natural sound. I think this is as far as I'll go with my music box simulations, but we'll see. 🙂

Mode 1

Mode 2

Mode 3

Mode 4

# Vibrating windows

Here are some nice images of vibrating window frames (ooh, wow)! The finite element model I made with Python will hopefully enable me to do two things: calculate the radiation coefficient of the window frame, and calculate the effect of coupling the wooden frame with the windowpane (although shear locking is definitely a problem for the model I'm using at the moment). These things should be very central when one considers the computational airborne sound insulation of the structure as a whole.

I'm also thinking of making a more detailed model of the music box I wrote about earlier using this model at some point. That should make a fun topic for the blog.