Combination of two vibration signals (sum of two signals)

Combination of two signals - what happens if there are two sources of vibrations in a machine?

Let’s assume each source creates a signal with a single sine wave.

We will cover four possibilities. First, let’s look at two waves of a similar frequency. One of them is a 25 Hz wave.

A time signal measurement displays a sine wave. A spectrum measurement would show wave as a single peak at 25 Hz.

The other source of vibrations in this example has a slightly different frequency. It is equal to 24 Hz.

What happens if we measure these two sources together?

A spectrum shows two peaks. But a time signal looks a little bit more complicated.

The two waves are added together. A beating waveform is created. This is a specific example of amplitude modulation. The amplitude of the combined signal changes in time. We can calculate how much.

When the amplitude reaches its maximum, it is the sum of the amplitudes A1 and A2.

When the amplitude decreases to its minimum, it is the difference between the amplitudes A1 and A2. If the amplitudes of both waves are the same value, this difference is zero. For a moment, vibrations disappear completely.

If the two waves have different amplitudes, the resulting vibrations are weaker when the beating wave is at its minimum, but not zero. In this example, the spectrum clearly shows their amplitudes are different. 

The modulation is periodic and repeats with a frequency equal to the difference between the frequencies f1 and f2. It was 1 Hz in this example.

Something similar happens if the two frequencies are close to being multiples. If there was a 25 Hz wave and a wave with half this frequency at 12.5 Hz - the resulting signal is not variable. But let’s change one of the frequencies slightly, for example the lower frequency to 13 Hz.

We see a beating signal again. But it does not sound like a beat. It sounds almost the same as a single sine wave:

The beat repeats with a frequency equal to the difference between the higher frequency f2 and a multiple of the lower frequency f1 closest to it. In our case, the closest multiple of the lower frequency is two times 13 Hz, so 26 Hz and our repeating frequency of the beat is 1 Hz.

If there is no relationship between the two waves, one of them may have a very low frequency compared to the other, like 1 Hz and 25 Hz.

This high frequency wave rides the low frequency wave and the overall pattern repeats with the frequency of the low frequency wave.

This example would sound like the higher frequency wave alone, because the lower frequency is too low.

Lastly, if the two frequencies are arbitrary, like 9 Hz and 25 Hz, we cannot even see an obvious repeating pattern in the signal. How it would sound would depend on how high each frequency is.