Damper Airflow Theory

8/13/2019 Damper Airflow Theory

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Dampers

The essential, defining, property of a damper is that it is a device which

associates each forceto a velocity. You may be a little less familiar withdampers than with springs, but dampers are simpler, so we'll start here. By

the velocity vof the damper, we mean the rate at which the damper isgetting longer. By the force fof the damper, we mean the force applied to

the damper, by whatever is applying the force.

Convention: when we talk about the force associated with a

damper we mean the force applied to the damperby somethingelse (perhaps your hands).

The best dampers are oil-filled tubes, with a plunger that has to squish its

way through the oil as you lengthen or shorten the damper. The more force

you apply to stretch the damper, the faster the plunger moves through theoil. If you stop trying to stretch the damper, i.e. you stop applying a force to

it, it immediately stops lengthening; its velocity drops immediately to zero.

l This is in marked contrast to a mass. If you get a mass moving by

pushing on it for a while, and then you suddenly stop pushing on it(f=0), it doesn't instantly stop moving (v=0)

(A)

When we talk about the "velocity vof a damper", it's important to realizethat this is the relativevelocity of the two ends, not the velocity of either

end, or of the middle. If both ends of a damper are unconnected to anything

else, there can't possibly be any force on the damper, trying to stretch it orcompress it. Nevertheless you can toss it through the air at any speed you

want, and it remains f=0, and therefore v=0. If the velocity of one end isv

1

and the velocity of the other end is v2

, the "velocity of the damper" vis

v = v2- v

1.

Convention: when we talk about the velocity associated with adamper we mean the speed at which the damper is lengthening.

If we say "this damper has a velocity of -20 cm/sec" we mean it isgetting shorter, end-to-end. (This does not tell us the absolute

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velocity of either end of the damper.)

It is very important to realize that, as defined above, the force associated

with the damper, f, is the same regardless of which end you look at. f>0means an attempt to lengthen the damper; f 0, v > 0 . . . tension,lengthening

f < 0, v < 0 . . . compression,

shortening

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(B)

Suppose you and your friend are still both pulling on the two ends of adamper, which (as a consequence) is slowly getting longer. Any place along

the damper that you point the tension instrument it will register positive.The whole damper is in tension, and if it breaks in half the pieces will

separate (which is, I suppose, what it means to be in tension). You get thesame reading on the tension instrument all along the length of the damper.

We can say that the damper transmitted a force through itself, end-to-end,unchanged. Thus we say that force is a through variable.

To understand why the magic instrument shows this result, we can turn toNewton's Second Law. You might notice that Figure B was drawn with equal

and opposite forces on the ends of the damper, already enforcing that forceis a through variable. If you have any doubts, we can prove this fact by

redrawing the figure more generally as Figure C below,

(C)

and applying Newton's Law to find

f1

+ f2= ma

By definition we take the mass of the damper to be zero and thus

f2

= - f1

So we define f2= - f1= f, yielding the original figure (B). So from now on for

dampers (and also springs in the next chapter), we will understand thatforce is a through variable.

Now apply the other magic instrument while you and your friend are still

pulling on the two ends of the damper. If you measure absolute velocity atthe left end of the damper, it is negative. If you measure absolute velocity at

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the right end, it is positive. The damper clearly does not transmit velocity

from one end to the other, because the absolute velocity is observablydifferent at the two ends. To determine the quantity that we have defined to

be the velocity (v) of the damper, you have to subtract the two absolutevelocities of its ends. You have to measure the relative velocity across the

damper. Thus we say that velocity is an across variable.

More intuition work: I hope you and your friend are still pulling on

opposite ends of a damper. If one of you reduces the force you areapplying to the damper, the other one must do so too, instantly.

You have no choice in this matter. If your friend reduces the forceand you do not, you will pull the damper out of your friend'shands. A damper has one force, end-to-end, all the way "through".

If you apply a force to a damper, that force is transmitted right

through the damper, unchanged, to the other end. It is felt bywhoever (or whatever) is connected to the other end. If you pullwith a force of 3 newtons, your friend at the other end will bepulled with a force of 3 newtons. This is so even though the

damper, which experiences a force of +3 newtons, will begin tolengthen as a consequence.

(B)

Constitutive laws

I encourage you to think of each element of a mechanical system as having

a purpose in life. Each element has a constitutive lawrelating some of thedynamic variables, and its purpose in life is to enforce that constitutive law.

(You don't have to be quite this anthropomorphic about it.) The constitutive

law of a damper mightbe

where bis called the damping constant, very much like kin Hooke's law f =k xis called the spring constant (or Hooke's constant). The units of bare

newton seconds per meter. However, not all dampers obey equation (1).

Great Truths. Equation (1) is not a Great Truth. Not all dampers

f= bv (1)

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obey it. And different dampers, even if they do more-or-less obey

equation (1), have different damping constants b. Let's distinguishamong three levels of Truth:

1. "Momentum is conserved." This is a Great Truth. It's not just

a result of the way we think about things or the definitions ofthe variables we use, it's the way the universe works. And it's

always true, not just approximately but exactly.

2. "Velocity is the time-derivative of displacement." This isalways true, but it's not very deep. It's a matter of definition;

it's what we mean by the term "velocity". It's a Truth, but itdoesn't aspire to Greatness.

3. "f = b v". This is only a "Semi-Truth". Some dampers behavein a way that can be described pretty well by this equation,

and some don't.

It may seem at first that we can only imposea force, not a velocity. If the

object you are "imposing upon" is the wall of your room, that's true: you canimpose any force you wish and observe the resulting velocity, which

(hopefully) will be zero. However, if you are a bulldozer, you can impose avelocity upon your wall -- and you might have a bit of difficulty if you

wanted instead to impose a force of only 10 pounds. So there's nothingintrinsic that says force has to be the independent variable.

l Make sure you can imagine doing either experiment to a damper: eitherimpose a variety of force levels, and measure the resulting velocity for

each; or impose a variety of velocity levels, and measure the resultingforce for each. A hypodermic syringe(no needle, no cap) makes a

pretty good damper, with a "human scale" range of forces and

More generally than equation (1), the constitutivelaw of a device (here, a damper) may be

expressed as a graph. For a damper, this graph

will have axes fand v.

I like to think of a graph as having an inputvariable and an output variable. These are also

called the independent and the dependentvariables. Conventionally the horizontal axis is

used for the independent variable (the input.) If

the graph describes the behavior of a damper, andvis the independent variable, it expresses the

idea that we imposea particular velocity von thedamper, and we observethe resulting force f.

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velocities. You can impose either a force or a velocity, and observe the

resulting velocity or force. Try it. One plastic straw stuffed insideanother plastic straw makes a so-so substitute. A wooden spoon

moving vertically in a thick milkshake works well too - but you have toask yourself "what are the two ends of this damper?" The spoon is one

end - and you may not notice that the other end is the bottom of thecup, because it is resting on the table. It helps to imagine operating adamper with your two hands. Where do you have to place your twohands to operate the damper? In this case, to use both hands you

would hold the cup with one and the spoon with the other.

In the left graph below, I've estimated the result for the hypodermic syringe.You can see this graph is not going to be well approximated by an equationof the form of equation (1). That doesn't make it a bad damper! It just

makes it a bad lineardamper. People who build dampers for a living don'tmake them out of hypodermic syringes. They try to make dampers whose

constitutive law is more like the one in the right graph below, which can bepretty well approximated by the linear constitutive law f = 2.2 v, or perhaps

a bit more accurately by the nonlinear constitutive law f = 2.2 v + .02 v3.

Moral of the story:The constitutive law of a damper might be expressed as agraph relating force to velocity, or as an equation relating force to velocity,or (if we're lucky, or if we're willing to settle for an approximation) even as a

linear equation relating force to velocity. All of these are legitimate

descriptions of a constitutive law.

Just as we tend to think of the horizontal axis of a graph as being the input,and the vertical as being the output, we often look at the right side of an

equation as being the input and the left as being the output. Instead of thisviewpoint, let's look at the constitutive law (whether in the form of an

equation or a graph), as an instruction to do something. Specifically, to take

v, act on it according to the graph or the equation, and produce fas output.As a block diagram,

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-- or --

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Damper Airflow Theory

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