6.4 Driven oscillations and resonance
Finally I need to consider the situation where the oscillator is driven, as in the case of the AFM cantilever. The driving force will depend on the application, but for my mass on a spring it might be a small motor driving what was the fixed end of the spring up and down. The simplest expression for an oscillating driving force FD will be something like:
where ω is the (angular) driving frequency. Including this as an extra term in my force Equation (5.9) and solving for x gives a steady-state solution of the form:
I say ‘a steady-state solution’ because when I switch on the driving force the initial response, often referred to as the transient, can be very complicated. After a time, however, the system will settle into a steady state. My solution has two terms which are of interest: the amplitude, A0, and the phase, φ0
Box 9 Phase
In the driven case, I have to consider the amplitude of the driving force and the relationship between the driving frequency and the motion of the mass. You can think of the phase as the time difference between the maximum amplitude of the driving force and the maximum displacement of the mass. Imagine that I am holding the spring suspending a large mass. If I move the end of the spring up and down very slowly, then it won't stretch and the motion of the mass will be in exact phase with (and have the same amplitude as) my hand movement. As I begin to move my hand more quickly the motion of the mass begins to lag behind my hand motion; now there is a phase difference between my hand motion (the driving force) and the displacement of the mass.
Figure 27 shows the two plots in a single time axis, one for the driving force and the other for the motion of the mass. You can see that the maximum amplitude of the driving force occurs well before the maximum displacement of the mass. Here we would say that the phase of the driven mass lags behind that of the driving force. We could equally well say that the phase of the driving force leads that of the driven mass.
What would be the effect on the phase of using a smaller mass on the end of the spring?
The mass would tend to follow my hand movement more readily and so would tend to lag less.
Following the same procedure as for the simple and damped cases and using quite a lot of algebra, I find that the amplitude, A0, is given by:
and the phase, φ0, by:
Remember that ω0 is the natural angular frequency of the oscillator – the frequency at which it will vibrate if it is simply displaced and then released. Looking at Equation (5.17) you can see that as the driving frequency, ω, gets close to the natural angular frequency, ω0, the difference term will tend to zero. In this case, if the damping coefficient b is also zero then A0 will tend to infinity.
Using Equation (5.17), set the frequency to the natural angular frequency (ω = ω0) and derive an expression for A0.
This is what happens if an undamped harmonic oscillator is driven at its natural frequency. The amplitude of oscillation will grow out of control; something will break, or the geometry of the system will change, so that the assumptions we made in deriving this solution will no longer be valid. Even if there is some damping present, you can see from Equation (5.17) that the amplitude will peak when ω = ω0. How sharply it peaks, and what the maximum value of the displacement is, depends on how much damping is applied.