Simcenter Testing Solutions
Harmonics + Harmonic Cursor in Simcenter Testlab
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.
Figure 1: A) automotive internal combustion engine
B) turbine engine
C) 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.
Figure 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.
Equation 1: Infinite Fourier series expansion for an ideal square wave.
Figure 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.
Figure 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.
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 Simcenter Testlab display makes identifying harmonics easy.
Averaged 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".