# Intuition behind forced oscillations

in every book I own and on the internet, any information on forced oscillations is a version of the following definition, followed by the solution to a differential equation.

Forced oscillations: forced oscillations occur when a mechanical system is forced to vibrate by a periodic external force which continuously transfers energy into the system. The system vibrates at the forced frequency, whereas the amplitude depends on how close the frequency of forced vibration is to one of the natural frequencies.

I am looking for a more intuitive and physical explanation of what is actually going on, without blurring it with complex mathematics.

1. what is meant by the frequency of the forced vibration? I find "periodic" force a little vague. Does this mean that the force is such that it gives the oscillator impulses at regular intervals (like child on a swing) or can the force vary continuously?

2. secondly, why does the system (eventually) oscillate at the forced frequency?

3. Lastly, why does the amplitude depend on the frequency of forced vibration?

this one bothers me the most. Some books try to explain it that the closer you are the the natural frequency, the more "efficiently" the system absorbs the energy input, which sounds like hand waving to me. provided the damping is kept constant, where does the rest of the energy go and why is it absorbed the greatest at the resonant frequency?( I am aware of real life examples and am looking for a more general explanation)

I agree that this description of forced oscillations is too short and too little - it is confusing more than explaining.

Let us go point by point:

• First of all, let us note that the machanical system implied here is an oscillator, most likely a linear oscillator which has its own natural frequency (and probably also a damping coefficient)
• Periodic force means that it has a period, $$T$$, i.e. it also has frequency $$f = \frac{1}{T}$$ That is the force repeats itself after time $$T$$: $$F(t+T) = F(t)$$ Apart from that it can have arbitrary shape - it can be a pulsed force or something else. However, most likely what is meant is a harmonic force: $$F(t) = \cos(2\pi f t).$$ Anyhow, any other force can be represented as a sum of harmonic forces, using Fourier series.
• System will obviously move along with the force (like the see moves due to the periodic force produced by the Moon rotating around the Earth). Given that the system itself is an oscillator, the overall motion could be a combination of oscillations with the natural frequency and the frequency of the external force. However, given that any real oscillator is damped, the component with the natural frequency would decay, and only the oscillations at the frequency of the external force remain.
• The last point, I believe, refers to the phenomenon of resonance. External force counteracts the other forces acting on the oscillator (those which would normally cause it to oscillate at its natural frequency). Counteracting these forces means that part of the work done by the external forces goes into counteracting these internal forces. The closer the forcing frequency to the natural frequency, the less energy is wasted on this counter-action, and it is more efficiently transferred in the amplitude of oscillations.

Considering electric circuits, the Voltage is an analogy for force in mechanical systems. The inductor, capacitor and resistances are analogs to mass, spring constant and damping respectively.

1. The frequency of forced vibration is simply the frequency of the AC voltage.
2. At steady state after all the transients are died out, the energy for the system is transferred by AC voltage, the capacitor is charged and discharged at a frequency equal to AC voltage, the current in the inductor is increased and decreased at a frequency equal to AC voltage. Hence at steady state the system eventually oscillate at forced frequency.
3. As the reactance of both inductor and capacitor depends on frequency, the amplitude is also changes as frequency changes. As the frequency changes the power loss also changes, so the Amplitude changes accordingly. At resonance the inductive and capacitive reactance cancelled out hence the amplitude is high and depends only on damping element resistance. In mechanical systems, at resonance the inertial term (motion caused by force) and the spring term (motion caused by spring) are cancelled out and depends on damping element just like in electrical electrical systems.