The Little Vulgar Book of Mechanics (v0.10.0) - Vibration I

Last updated: March 1st 2022

Just updated this section of the book: Vibration I

Vibration I #

If vibrations didn't exist, the Universe would be literally dead.

The word "vibration" comes from the Latin vibrationem ("shaking, brandishing"). For our purposes, a vibration is when shit oscillates around its equilibrium point.

For vibrations to exist – or to even conceptualize them – first you need a state of equilibrium. Because, otherwise: What the hell is being disturbed? Disturbed from what? So we have this notion of an equilibrium point. This is when we say a body is "at rest." So that's the idea here: For a disturbance, and therefore a vibration, to exist, there must be an equilibrium point.

Vulgarly, the "equilibrium point" is the state of shit when you don't annoy it.

A vibration must happen somewhere. We call this a medium. This is why in Sound I - Medium I I talk about how sound needs a medium in which to happen. (Spoiler alert: Sound is vibration.)

But it's not just that shit has to "be in a medium."

This is what we need, more specifically: You disturb shit. Shit leaves its equilibrium point. And the medium itself starts restoring the equilibrium. You see the role of a medium now? The way equilibrium is restored depends on the medium. For shit to vibrate, it must be in a medium that has the feature of "wanting" to restore equilibrium, some time after the disturbance.

Hit a guitar string. This is the disturbance. You've displaced shit (part of the string) from its position. You've slightly deformed it, if you will. The medium is now gonna restore the equilibrium, but it's not just gonna immediately clamp right back to rest. Instead, it will start to force it back to equilibrium, as lots of secondary vibrations continue to occur. The bouncing back and forth between equilibrium and non-equilibrium is called an oscillation.

The string is the medium. But there's also the air, another medium. Disturbances propagate from one medium to another, and each medium restores equilibrium in its own peculiar ways.

Of course, if you hit the string with a baseball bat, massacring the whole guitar in the process, that obviously counts as a "disturbance," but the experiment is over. It's not creating the type of phenomena we're studying, is it? So when I say "disturbance," I mean slight disturbance, or non-party-ruining disturbance. Mostly, think of disturbances as some type of slight position displacement.

Have you ever come across this equation: F = -sx?

Like I said, the medium will restore shit to its equilibrium position. This act of "restoring towards the equilibrium position" is a force. I mean, it has to be, right? Because remember what I said, Newtonianly, in Force I: Force is the cause of motion and power.

Here, the medium is producing a motion back towards the equilibrium. So there's a force at work. F is, thus, the force restoring shit back to rest.

What are -s and x, and why are we multiplying them?

x is the position. Yes, a single number to describe the position. Which means we're describing it one-dimensionally. We're reviewing the basics of 1D kinematics.

So I'm looking at the spring you installed on the lab's ceiling. You hammered one end to the ceiling. And now we can hang shit off the bottom end.

x is the position of the thing we're hanging off the spring. It's the shit's "vertical coordinate" if you will. Neither horizontal nor depth axes are relevant right now. Remember, it's 1D kinematics. x is the displacement from the equilibrium position.

Another way to view x is to say that x is the state of deformation of the spring with respect to its fixed reference configuration, or default state, or resting state, or equilibrium state.

When x is zero, F is obviously zero. Body is at rest. If x is too large, you might deform the spring, and then the experiment is over, and our model breaks, and then I scream "God damn you! God damn you all to hell!"

Not all springs in the world are the same. Springs come in different shapes and sizes. And colors, etc. What's the relevant property of a spring for us to model how it vibrates when we have shit off it?

Essentially, we care about how stiff it is. That's s in the equation. The spring's stiffness. We're trying to express something really basic. The essence of simple vibration. We'll formally call this simple harmonic oscillation later.

s is a constant number right now, which places a limit on what x can be. Because if we stretch out the spring too much, i.e. if x gets too large, we may permanently deform the spring, and then its stiffness will change, and whatever constant for s we were using will no longer be correct.

By the way, at this point you should be able to answer this question, using what we've seen in this section (and in Force I): What is a deformation? A deformation is a change in shape due to the application of force.

How much can you deform the spring before you ruin it? How much can you deform a bone in your body before you fracture it? It depends on the material. This threshold of maximum stress before fracture, is called the Ultimate tensile strength (UTS), often shortened to tensile strength (TS).

Back to F = -sx. Take a moment to think about this relation. Imagine different combinations of s and x, and think about what they are expressing.

For example, what's going on when x is 0, i.e. F = -s0? Nothing! The restoring force is zero. Shit is in equilibrium. The gravity pulling the mass down and the spring stiffness pulling it up are cancelling each other. There is no net force on the mass.

I leave you with a question, which is the one bit of the equation I haven't explained: Why the negative sign in -sx?

We'll answer this in Vibration II, where we'll delve into the notion of a simple harmonic oscillator. But here's a hint: Remember we're talking about a force that restores the equilibrium point.

Here's some additionnal questions:

  1. How is a guitar string like our spring?
  2. How about a bow and arrow?

Something to think about, as you listen to any random music, like, say...


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