# harmonic waves as integer multiple in spectrum

i have a motor that is rotating with a certain frequency. If i check the frequency spectrum it contains a peak on 150 hz. Also i have peaks at 300,450,600 ... i guess that those peaks are harmonics. But i dont really understand why harmonics appear.

I think the reason is: a periodic signal can be described by sine, cosine waves (except the if periodic signal itself is a sine or cosine wave) So because a rotating motor is periodic, the frequency contains multiple cosine or sine waves that together build the signal. And the harmonics are the parts of the signal?

but I still don't understand why they occur if my machine is rotating at 150 hz. How are the other frequencies build?

• According to a troubleshooting guide for motor vibrations, mechanical looseness can also cause higher order harmonics. So it may be a real physical phenomenon rather than a waveform shape issue. See table 3 here for example: bksv.com/media/doc/bo0269.pdf – ericksonla Jul 8 '20 at 15:29

Every periodic signal can be described as sum of "basis" functions. These basis functions are the sin and cosines at frequencies of integer multiples of a fundamental freqeuency. This fundamental freqeuency is a consequence of the periodicity of the signal. So if a signal is periodic with time $$T$$, then the fundamental frqeuency is $$F_o = \frac{1}{T}$$ Hz. That's the basic Fourier series theorem for periodic signals.

It is the waveform of the signal which will define which of these sines and cosines are present.

Consider the figure below:

It is not straightforward to look at the periodic signal above and tell that it will be comprised of so many frequencies. But it is, that's just a consequence that periodic signals are made of a sum of weighted sinusoids at integer multiples of a fundamental frequency.

Just as you require basis vectors in a vector space to define "any signal" in the vector space, the contributions of these basis vectors can vary depending on the vector being defined.Similarly periodic signals live in a vector space whose basis vectors are these sinusoids (that are defined by the periodicity) of the signal and hence there are contributions (big or small) from all these frequencies to define the periodic signal.

So your observations regarding the frquencies appearing in the spectrum are justified. Infact from the fundamental freuency of the motor (150Hz) we even know how fast the motor is rotating :)

• could you also recommend a paper or a link, book where i can read this? – Khan Apr 23 '20 at 21:29
• @Khan Do a search on "fourier series representation of periodic signals" and you will find plenty of material. – Cedron Dawg Apr 23 '20 at 21:34
• @Khan chapter 3 of the book signal and systems by Allen oppenheim, very famous book easily available on the internet – Dsp guy sam Apr 23 '20 at 21:46