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Harmonics + Harmonic Cursor in Test.Lab

Siemens Experimenter Siemens Experimenter
Siemens Experimenter



Complex machines that feature many moving parts and systems often generate many different types of noise and vibration simultaneously.  Measuring noise and vibration signals from the machines, we can break the signals down into their different frequency components.  By investigating this frequency domain information, we can then use what we know about the design of the machine to figure out what frequencies are generated by the various different parts of the machine.

gm-4-3l-v6-ecotec3-lv3-engine-1.jpgFigure 1: A) automotive internal combustion engine



turbine_engine.jpgB) turbine engine



hydraulic_pump.gifC) hydraulic pump Each part and system of a complex machine produces characteristic frequencies.


Sometimes components will generate multiple, interrelated frequencies all from the same phenomenon (ie – a vibrating string, the acoustic modes of a room, or combustion in an engine (Figure 2)).  These interrelated frequencies are called “harmonics” or “partials” of the main frequency of excitation, usually referred to as the “fundamental” frequency.  It can be very helpful to identify harmonic frequency content when trying to address a noise or vibration problem: by modifying one component that produces many harmonics, one may be able to address or affect multiple areas of the frequency domain.


EngHarmonics.pngFigure 2: Fundamental firing order for a 4-cylinder engine, along with first three harmonics. Order numbers shown are 2.0 (fundamental), 4.0, 6.0, and 8.0.One of the most familiar mathematical examples of harmonic content is the square wave (Figure 3).  Unlike a sine wave, which contains only a single frequency, a square wave is made up (theoretically) of an infinite number of sine waves, all integer multiples of the fundamental frequency.  In the square wave case, only the odd-harmonics are included, as seen in Equation 1. 


sqwaveeqn.pngEquation 1: Infinite Fourier series expansion for an ideal square wave.


sqwave.pngFigure 3: Left) Square wave in the time domain Right) Frequency content of the same square wave contains an infinite number of odd-numbered harmonics.


Similarly, if we pluck a string (on a guitar for instance) we will excite multiple harmonics.  Each harmonic is the result of a different mode of vibration, or mode shape.  When we pluck the string, we hear the combination of all the modes excited at the same time.  Because of the way the string is constrained at the ends, all of these harmonic modes are found at integer multiples of the lowest mode of the string, which is the fundamental.  Figure 4 below shows the mode shapes associated with our plucked string from the fundamental up to the sixth partial. 



stringmodes.JPGFigure 4: First seven mode shapes for a vibrating string. Image credit: Wikipedia


Like our internal combustion engine and square wave, the frequencies associated with the harmonics are multiples of the fundamental frequency. 

Depending on how and where we pluck the string, we may excite all of these harmonics, or perhaps only a few of them.  The number and relative strength of the harmonics excited will determine how the vibrating string sounds to a human listener.  If the string is plucked near the midpoint for example, those harmonics without a nodal point in the center (f, 3f, 5f, 7f) will be excited to a greater degree than those modes with a nodal point at the midpoint (2f, 4f, 6f).

In the video below, the string is plucked near the midpoint, but slightly off-center so as to excite as many modes (and harmonics) as possible.


(view in My Videos)


If we look at an averaged frequency spectrum of the guitar string, we see the harmonics and their relative strength compared to the fundamental.  Using a Harmonic Cursor in a Test.Lab display makes identifying harmonics easy.


averagedspectrum.JPGAveraged frequency spectrum of guitar string with Harmonic Cursor added. Each cursor represents a multiple of the fundamental frequency.

To add a Harmonic Cursor to a display, right click in the display and choose "Add Harmonic Cursor -> X".


(view in My Videos)



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Siemens Legend

In the last video, around the 57 second mark, it looks like the harmonic cursor did not line up with the harmonics of the guitar string, especially at higher frequencies.


Is that to normal, or to be expected?

Siemens Experimenter Siemens Experimenter
Siemens Experimenter

Very, very good eye Peter!  


When I place the Harmonic Cursor on the true fundamental frequency at the end of the video, the first few harmonics align quite well with the peaks in the frequency spectrum.  However, after a few harmonics, the cursors no longer line up very well with the peaks in the spectrum.  In fact, the higher the harmonic, the further the peak is from the cursor (see plot below).  This is due to an effect called "inharmonicity." From Wikipedia the definition of inharmonicity is "the degree to which the frequencies of overtones (also known as partials or partial tones) depart from whole multiples of the fundamental frequency."




Inharmonicity is a result of the stiffness properties of the guitar string, particularly at the ends of the string, where it is constrained to the guitar body.  If the string were infinitely thin and infinitely flexible, then all the harmonics of the string would be exact whole multiples of the fundamental frequency.  Because the string is not infinitely thin or flexible, it exhibits something called "dead length" at the ends of the string.  Dead length is the portion of the string that has little or no displacement during vibration - it does not participate in the modes of vibration, and acts to locally stiffen the string.  This is illustrated in the graphic below.  

Deadlength pic.JPG


As the modes grow in complexity, the nodal point(s) get closer and closer to the dead length, as does the point of maximum displacement (also called “anti-nodes”). 

As the anti-nodes move closer to the dead length, a greater portion of the mode shape falls within the dead length.  The stiffening effect of the dead length therefore increases with each mode, and the higher harmonics will exhibit a greater amount of inharmonicity than lower modes.  This is sometimes referred to modal "stretching."    


To be honest, this phenomenon of inharmonicity was unfamiliar to me when I began work on the Harmonic Cursor post.  Seeing the harmonics of the guitar string not line up with the harmonic cursor is what sent me off to find out why.  Surprisingly there was very little information available on this effect, most websites describing harmonics only describe the lowest harmonics which tend to be very close to whole-number multiples of the fundamental.  Another example of why it's always good to check into things yourself, and not just accept the common answer.  Thanks for commenting!   -SM