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Harmonics Analysis Tutorial

Monday, January 20, 2020
Category : Metrology

The separation of imperfections on round components into their circular frequencies, or “harmonics” is crucial to the analysis of what caused them. In this tutorial Dr Mike Mills - Chief Metrologist - discusses the basis and use of harmonic analysis.

What are harmonics?

When a component is rotated about an axis and gauged at a single point, any feature on the surface will pass the gauging point once per revolution of the component. The profile can be considered as periodic, with a period being equal to one revolution of the component. The profile can be broken down into a series of sinusoids whose frequencies can be expressed as 1,2…n Undulations Per Revolution (upr) of the part. These sinusoids are the “harmonics” of the profile.

How do harmonics analysis relate to the measurement process?

Harmonic analysis provides a method of relating a roundness measurement to errors in the production process or to the resulting performance characteristics, such as bearing rumble. Certain low frequency harmonics can easily be related to setup errors. For example a first harmonic (1upr) component will arise from the component centre being eccentric from the centre of rotation. Typically in roundness this eccentricity is removed and recorded as a parameter before displaying the profile. A second harmonic content (2upr) represents ovality. This could be due to a machining fault or could be due to a cylindrical component being tilted.

Other low order terms might be associated with the way the component was held in the chuck on the machining system. Often the chuck will compress the component. The component is machined to be cylindrical, but when released from the chuck it will relax and can assume a shape related to the number of jaws on the chuck. Other low to medium frequencies can represent chatter in the process. Higher frequencies will typically relate to machining marks caused by the tool being used.

Standard roundness profile shown Full harmonic histogram

 Standard roundness profile shown                 Full harmonic analysis histogram

When are harmonics used?

Most often harmonics are used implicitly. Whenever we select a filter for the analysis, we are effectively suppressing certain harmonics. For example a 1-15upr filter might be used to suppress tool marks and to help discern the setup and possible chatter errors. A 50-500upr filter might be used to suppress the chatter marks and to examine the tooling marks.

In some cases, for example bearing measurement, advantageous to look at several ranges of harmonics, different performance such as rumble or whine. The amplitudes of these frequencies at particular rotational speeds of the bearing are often used as process control parameters. Taylor Hobson has developed special software for the analysis of bearing Harmonics. This assess the contributions of different bands of harmonics to the overall profile, and set pass/fail tolerance limits in each band.

How are harmonics calculated?

In general, the calculation of harmonic content of a profile is done by means of a Fourier transform. This mathematical technique picks out the different frequencies that go to make up a particular profile and calculates the amplitude and phase of those frequencies. In a computer the analysis is done using a special algorithm known as a Discrete Fourier Transform (discrete because the data is sampled when it is stored in the computer).

The Discrete Fourier Transform is very computationally intensive, so many techniques have been developed to speed up the calculation. The most commonly used technique is the so-called Fast Fourier Transform algorithm. This algorithm requires the data set to have 2n points. Where the data set does not contain 2n points, it has to be modified by interpolation or padding the data with zeroes. Potentially these modifications can affect the harmonics themselves.

A more recent and improved technique is the Winograd transform, which does not suffer from this constraint. The technique involves breaking down the Fourier transform of a signal with a large number of data points into a sequence of Fourier transforms with small numbers of harmonics. The advantage is that this does not introduce the potential errors that can be caused by zero-padding or interpolation. This is the technique used by Taylor Hobson.

Dr Mike Mills
Chief Metrologist.

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