Let me throw out some, some common numbers.

Maybe for a 5,000 psi system which would be 35,000 five gigapascals.

So I've got minus 35 MPa, so that's ten to the sixth.

Divided by lets say 1.6 gigapascals 10 to the 9th.

I'm ending up with compressibility of right around 2.2%.

So minus 2.2%.

So, just to give you a feeling, you know, our, our oils that we're dealing with are

usually 1 to 3, maybe 5% compressible, depending on

how much air is in that, in that oil.

So, often reasonable assumption that it's incompressible.

But in some cases this does become important.

Now before I move on let me just mention the sign here notice

that we have a negative sign and a definition of the bulk modulus.

This is due just to how we define a

change in volume you notice that, a dV is considered

a, a reduction in volume so, that's a little bit

arbitrary, we can, we can take care of that later.

What I was mentioning, why is this important.

So, first of all, it's important for

systems where we need to precise positioning.

So, consider that I have a hydraulic cylinder and

I'm trying to very precisely position this end effector and

perhaps there's a lot of mass attached to this

and, I don't want to have much compliance to my system.

I want it to be very stiff and, that can be an issue if

our fluid is compressible, especially if there's

a lot of air trapped in the system.

So we have to be aware of that.

Second, I'll talk about more in just a moment, is resonance of the system.

The fact that we can get dynamic resonances

of, square root of k over m, sort of term.

And having that, [SOUND] influence our system,

especially with high frequency systems, often several hydraulic

systems are running at very high frequencies,

getting, getting into the resonant frequency of the.

The hydraulics.

And then, compressible energy losses.

You might say well w, what do you mean by that?

If we look at a hydraulic pump as we were talking

about before and we were looking at the piston cylinder interface.

Now, in that piston cylinder we have.

When the piston comes up to the top dead center

position, we still have some dead volume in the chamber.

Well that fluid was compressed previously, and now

we're going to open a valve to tank pressure.

And so the energy that went into compressing that fluid, is exhaust

in the tank, and then we're going to do that cycle after cycle.

And this is happening, you know, for 3600 RPMs.

This is happening 60 times a second.

So it ends up being a significant amount of energy loss over

time if we have a large amount of dead volume in our cylinders.

So, again, different areas where the small amount of

compressibility does add up over time and does become important.

So, let me talk a little bit more about resonance, as

one of the, the key areas where this, this becomes important.

So, think about a long, slender cylinder, something like

this, relatively small diameter, long stroke, and I'm going to,

[SOUND] just do an analysis of what the residents

frequency or what the national frequency of this system is.

I'm going to simplify this ever so slightly first of all by

saying that, my piston hub length is basically minuscule and so

the length of the oil column of one side plus the

oil, other oil length is the total length of the cylinder.

Second, I'm going to say that, my piston rod is very small, or that

I have a double-ended cylinder, and that the area on each side is the same.

So they're just some simplifications to make my math a little bit easier.