4.7, in which we discuss more physics (er, harmonic motion)

Section 4.7 is entitled “Forced Harmonic Motion.”

So back in section 4.4 (http://differentialequationsjourney.blogspot.com/2013/10/44-lets-talk-physics-er-harmonic-motion.html) we derived the model for harmonic motion:

We’re first going to deal with forced undamped harmonic motion, where c = 0:

We need to consider two cases when dealing with undamped harmonic motion: when the driving frequency ω is equal to the natural frequency and when it is not.

Case one: the driving frequency is not equal to the natural frequency

We use the method of undetermined coefficients to find the solution to this. I’m going to briefly summarize it.

We start by looking for a particular solution of the form

In this case, a and b are the undetermined coefficients. We substitute this particular solution into our inhomogeneous equation for undamped harmonic motion (i.e. the inhomogeneous harmonic motion model where c = 0), and we find what a and b are equal to. This gives us our particular solution, and now we can create our general solution x(t) = x_h(t) + x_p(t).

If we look at the solution where the motion starts at equilibrium, then we have the initial conditions x(0) = 0, x’(0) = 0.

Just as an example, here’s what forced and undamped harmonic motion looks like. This also shows what the book calls beats, which occur whenever two frequencies that are very similar interfere with each other. In order to understand beats, let’s introduce a couple of new things:

This then leads us to the equation

So then we have what’s called an envelope, where there’s a slow oscillating amplitude and a faster oscillation.

(Wikipedia)

The example I have is not for the cos(11t) – cos(12t) business. It’s just to illustrate what an envelope looks like. The actual envelope is the red dotted line. It's through the maxima of the faster oscillation (in blue).

Case two: driving frequency equals natural frequency.

Now we look for a particular solution of the form

Again, we insert this into the inhomogeneous equation for undamped harmonic equation, find a and b, and then our particular solution would be

We have a special case where x(0) = C₁ = 0, x’(0) = C₂ = 0. In this next picture, A = 8 and ω₀ = 4.

This type of behavior is called resonance.

Now, let’s add a damping term to the system. We’re now dealing with the equation

Suppose we’re dealing with an underdamped case, where c < ω₀. Back in section 4.4 we found that the general solution is

In order to find a solution to the inhomogeneous equation, we once again use the method of undetermined coefficients, but it’s easier to use the complex method. We’re now looking for the solution z(t) = ae^iωt to the equation

We substitute out z(t) into that equation, we’ll get

We then get

The function H is called the transfer equation.

Looking back at P for a second, we see that

We want to write this in polar form, so we’ll need a positive number R and an angle φ such that

So then we can write the transfer function as

We now can see that the solution to our complex equation is

The homogeneous part of our general solution has the factor e^-ct which is called the transient term because it quickly decays to 0 as t goes to infinity. After a factor T_c = 1/c, this factor decreases to e^-1 and the amplitude has decreased to e^-1 times its original size. T_c is called the time constant. Because x_p does not decay, it is called the steady-state. This term is driven by the external force Acos(ωt).

Finally, let’s discuss lumping parameters. We do this in order to see how the gain G depends on ω while we study the steady-state term.

This is still complicated, but this expression for gain is much easier to understand than before. Because the natural frequency is fixed, let’s look at

Something to note about this is “…the maximum gets larger as the damping constant decreases. This is another example of resonance…the resonance increases as the damping constant decreases” (185).

That’s it for section 4.7 and for chapter 4 as a whole. We have 7 sections from chapter 5 and possibly some selections from chapter 10. In any event, we’re close, my friends.

I’ll see you in chapter 5.