Physics: Damped Simple Harmonic Motion

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Here's classic simple harmonic motion - an object on a spring, bobbing up and down. There's very little damping. There's very little loss of energy here. It just continues to bob up and down. And this would go for quite a long time. It's really quite ideal simple harmonic motion. Minutes will go by. But there is some friction in this system and sooner or later, it will slow down. Realistic harmonic motion sometimes has some friction in the system. Let me add some friction here in a way, which makes it real obvious by making it bob up and down in water. So it's the same mass. It's the same spring. We start off at the bottom. And you can see that when it's in the water, it does bob up and down. It does seem to be - well, is it simple harmonic motion? It's almost stopped vibrating already.
So, yes, you can still call it simple harmonic motion, but you have to add the word "damped". It's damped simple harmonic motion, because there's some friction force in the story - some resistive force, which is taking away the motion - the energy. And we'd really like to be able to analyze that, because as you can imagine, an awful lot of physical situations in the real world have some kind of a damping, and so you'd better be able to describe the mathematics and understand what's going on quantitatively here. To understand what's going on quantitatively, we need to draw a force diagram.
You know, that problem had both a spring and gravity in it. Let me just think about my favorite harmonic motion, or harmonic oscillator, the one's that going back and forth on a flat plane with very little friction, but now we're going to add a little bit of drag. So here's my unstretched system. And now, let's suppose that we're here and on our way out. So we've got a big velocity this-a-way. What is Newton's law look like right at this moment? Newton's law says F = mA - m, where x, remember, is measured positive in this direction from x = 0. And what are the physical forces? Well it's still simple harmonic motion. I'm assuming that I still have an ideal spring whose force is just minus a constant times displacement.
But now I'm going to add a new force - a drag force. And what should it look like? There's various kinds of friction in the world. There's sliding friction and air drag and then there's air drag when you're moving fast and when you're moving slow. And each one of these will have a slightly different solution. Let's me solve the one that's mathematically the simplest and there's lots of physical examples of it.
When the drag force - F drag - you can think of this as some kind of a friction. And it's proportional to the velocity. So the faster you go, the more drag there is. And the proportionality constant, we'll just give a name, b. It's a coefficient, which tells you how strong are these friction forces. The bigger b is, the more drag you've got for a given velocity. And it's nice to write this formula, because it's telling us something very interesting. When v is positive, this formula says the friction force is negative. And when v is negative - when we're back on our way in - this minus sign will cancel and you'll get a positive force, which is exactly what you want. You want the friction force always to be in the opposite direction of the way it's moving.
So let me add this drag force into my equation - Newton's second law. And it's minus kx minus b, and let me write velocity as . So that is Newton's second law. And let me divide through by an m and add all the negative terms to bring it all together on one side, so that we have Newton's second law written as a mathematical differential equation. This is the differential equation for motion with damping.
If the b was zero - if there was no damping - then we would just have our good old harmonic motion equation, is equal to - if you added this to the other side - negative x. So this is the new piece. It's proportional to the velocity, . Differential equation - how do you solve it? Well, once again, I can either ask you to go to a calculus textbook and learn how to solve second-order differential equation - lots of fancy words associated with it. Or you can be like a physicist and just try to guess the answer. Remember if we can guess the answer and check that it satisfies this equation, we've got the answer.
Let's think about it for a second. What I observed was that we still did have harmonic motion. In fact, it sure looked like simple harmonic motion. It's just that it was having less and less amplitude as time went by. So I'm going to guess that we're still going to have a cosine of t + . It's just that instead of A times that, we're going to have something out front which gets smaller and smaller with time. So how about an exponential function? It's what you tend to get in physical situations like this. It's a dying exponential . Now why exactly this form? Gee, what I really have to do, of course, is to just say E to the minus something t, and plug this into the differential equation and ask myself, what do I need to put in the exponential in order to make this differential equation be satisfied for all times? And this really does work. You can try it. Taking two derivatives of it is a bit of a pain. You have to use the chain rule, because you have E to the minus something and multiply by cos(t). There's times in both of these places. But you can do it.
And you learn something when you plug this in. What's this omega? It used to be, for simple harmonic motion, just the . And when you take this and you plug it in, you're going to get a new condition on omega. In order for this differential equation to be satisfied at all times, you're going to have to have omega equals, not , but the -. As long as drag, or b, is very small, you get back to the old formula that we were used to before. And I'm really mostly going to be thinking of the situation where drag is small, although it's also interesting physically to think about what happens when there's lots and lots of friction.
Let's just think about this equation a little bit more. What I'm saying is that what's multiplying the cosine is a dying exponential. At time t = 0, E[0] is 1. So you start off at Acos(t + ). But as time goes on, you get E to a negative bigger and bigger number, and so what's out front here is getting smaller and smaller. And that makes sense. The amplitude of your displacement is decreasing with time.
Let's graph this. For small b, you'll simply get something that looks like a damped sine wave. It's the red curve. That's the answer. It's wiggling like a sine wave, but its amplitude is just getting steadily smaller and smaller as time advances. And this little dashed line is really just to guide your eye. It's . In other words, this is that dying exponential amplitude. Sometimes people call it the exponential envelope, and I put it down on the bottom, just so that you can see that it's really the amplitude of motion, which is decreasing.
So this makes sense and the formula works and that's wonderful. You can really solve and analyze physics problems that have realistic friction forces.
There is a special case I'd just like to mention. If the frictional forces get large - if gets to be equal to , that's the interesting condition - when becomes the square root of zero. So if = , my formula is telling me that we just have an exponential. You need to think a little bit carefully about the mathematics now when you solve the differential equation.
But here's the graph of what you really get. It's called critical damping. And all of a sudden, instead of oscillating, you just basically go down to the equilibrium position and stop. It's like the shock absorber on a car. That's a spring, and you would expect that if you push down the car, the car should wobble up and down. There's friction in there. There's fluids which are trying damp the motion of the car. And they've designed the frictional forces so that this is the motion of the car. It doesn't overshoot. You're not bobbing up and down on the highway like this.
If they put in too much friction - if b gets really, really large - that formula, as it stands, you really have to think about what to do, because my omega becomes the square root of a negative number. And you can still handle that. And what you get is called overdamped. And this would be a graph of x versus time. And, again, it's got this exponential die-off. It doesn't do any bobbing up and down. It's kind of like if I'd done my little demo, and instead of using water, I'd used molasses. I'd let go and that mass would just slowly start to work its way down to equilibrium and that's what this green curve is showing. So this is underdamped, critically damped, overdamped. And I think this is the most common physics situation, but all three of them occur.
Energy is not conserved when you have this minus bv form. And like I said, the minus bv - that specific form - I just picked because that way I could solve my equations easily. You could have minus bv^2. If something's moving fast through air, the friction force goes like v^2 instead of v, and now you have to worry about the directions. The problem just becomes mathematically more difficult. You have to remember that friction is always slowing you down. It's always eating up energy, putting it into heat or something.
You know, that formula for the motion - I can actually think about energy principles here. Remember when I had simple harmonic motion and I said the energy of a simple harmonic oscillator - the energy of the system is kA^2. It's proportional to the square of the amplitude. And here, I'm seeing the amplitude decaying with time. And so indeed, I'm just seeing quite clearly that the energy of this system is also decaying away with time, and I can even guess how. Just square this and you'll understand how the energy of the system behaves with time.
So damped simple harmonic motion. It's of a lot of practical interest, because many, many problems have some kind of friction force. If the frictional force - if the drag force - really does look like a constant times v, then we're all set. We've got the equation. If it doesn't, you have to solve some new differential equation. But the physical idea here is going to always remain approximately the same. You'll get harmonic motion. The amplitude dies away with time. And you can really understand what's going on, based on these same fundamental starting principles.
Oscillatory Motion
Damped and Driven Oscillations
Damped Simple Harmonic Motion Page [3 of 3]