Forced Oscillators

(Note the difference between forced and forcing)

Remember that the forced oscillator has an amplitude (in principle) equal to that of the forcing oscillator at low frequencies and zero amplitude at high frequencies. The forcing and forced oscillations are in phase at low frequencies and out of phase at high frequencies. At resonance they are 90^o apart, with the forcing oscillation leading. We'll look at all that in more detail later, and we'll write down and solve the differential equation.

A key thing to bear in mind is the mechanism by which energy is fed into the resonating system. In your experiment is was done with a Hookean coupling device. The end attached to the hacksaw blade scarcely moved, while the other end was obliged to move by the oscillator. So what was going on in the rubber/spring was F=kx rather than x=F/k: i.e. the force arose because of the extension rather than the other way around: but although the extension is 'applied' at one end only, the force gets applied to whatever is attached to both ends. In your experiment, you had to tinker with the coupling device to make it stiffer: you wanted the rather small movement imposed by the oscillator to result in as large a force applied to the blade as possible. Of course, in real life, you want the reverse to be true: you don't want oscillations from something that is vibrating transmitted to other objects, so you use a less stiff coupling. Thus, for example, you provide a washing machine with rubber feet rather than metal feet. Mind you, if you use rubber feet that are too floppy, they don't support the washing machine at all! Your car has a spring-based suspension (and a dashpot to provide damping).

Another thing to bear in mind is that the hacksaw blade had area, so there was an Arv² force providing damping, and that prevented a really good build up of oscillations. The graph of forced amplitude against forcing frequency has its maximum at the natural frequency of the forced oscillator if there is no damping, and at a slightly greater frequency if there is. The maximum amplitude on this graph is infinite for no damping and progressively less the more damping there is.