Fault detection by vibration analysis

Vibration is a continuous oscillatory motion, generally made up of many different frequencies present simultaneously, the amplitude and phase of which can change with time. It can be both a cause and a symptom of impending failure. Most structures and machines vibrate. The vibration can be an intrinsic part of the design and or due to one of the many possible failure mechanisms progressively degrading the system. Generally, the vibration energy associated with early symptoms of failure is low compare to "normal" vibration energy associated with the machine. In other words, the signal to noise ration is low, often less than one percent. This requires special methods of vibration analysis.

Causes of vibration

The following is a list of vibration causes in machines and structures:

Sometimes the cause of a vibration can be difficult to identify and frequently there are multiple causes that can interact. The beating effect of two vibration sources can create the appearance of a third. The situation can become very complex if there are non-linear elements in a system, causing a wide range of frequencies to be present, few of which may be attributed to any particular source.

Modes of vibration

Vibration presents in many different forms modes. Knowing the mode can assist in identifying the source:

In all cases there is movement of material in a cyclic fashion. Resonances and standing waves will tend to emphasis some frequencies more than others. Rarely is the just one mode of vibration present.

Frequency of vibrations

The frequency of vibration can tell us a lot about the cause and significance of vibration. For example, the following table related frequency to the cause in rotation machines.

Many of these frequencies can be present simultaneously, so it is vital that a monitoring system be able to discriminate between them.

Measuring vibration

Vibration detection is most commonly achieved using accelerometers - sensors that produce an electrical signal that is directly proportional to the instantaneous acceleration of oscillatory motion. Vibration can also be measured by load (force) cells, strain gauges, displacement sensors, doppler lasers, microphones and imaging methods.

Normally in a well designed structure or machine, a design objective is to minimize vibration. Excessive vibration is bad for both man and machine. Vibration causes increased peak loading on components and hence reduces component life due to wear and fatigue. Where significant vibration is an intrinsic part of a design, components should be sized accordingly.

Spectral analysis tools

• FFT's (Fast Fourier Transforms)
• FHT's (Fast Hartley Transforms)
• constant percentage bandwidth power spectra
• harmonic analysis
• order Tracking
• cepstrum space
• wavelets transforms
• envelope analysis

Order tracking in variable speed systems

When the rotational speed of a machine changes, some vibration frequencies also change. Vibration associated with gears, shafts, fans and pumps will all be related to the rotational speeds of the machine. Sometimes these relationships will change, for example, when gear ratios in a vehicle gear box change. As the rotational speed changes, vibration frequencies will pass through resonances of the machine and its coupled structures, thus emphasizing these frequencies.

Cepstrum space

The cepstrum is the forward fourier transform of the log of a power spectrum. In other words, it is the spectrum of a spectrum, and has properties that make it useful in many types of signal analysis. One of its more useful characteristic is the fact that any repeated patterns in a spectrum will be detected in the cepstrum. The cepstrum is a measure of the periodicity of a frequency response plot. The unit of the cepstrum was named quefrency, derived by inverting the syllables of frequency and is a measure of time but not the time we normally consider.

Intuitively the Cepstrum makes sense. The FFT effectively extracts any periodic characteristic in waveform. The waveform is usually in the time domain, but it is equally valid in any other domain. By applying a log to the frequency domain waveform before applying the FFT we ensure that low magnitude components are given more emphasis than they would be otherwise.

Gearboxes and rolling element bearing vibrations lend themselves especially well to cepstrum analysis.

The cepstrum is closely related to the auto correlation function. Additions in the cepstrum domain correspond to multiplication in the frequency domain and convolution in the time domain.

The RMS value of a time waveform is calculated by taking the square root of the sum of the square of every sample.

The RMS value for a spectrum is calculated by taking the square root of the sum of the square of every FFT bin. If you were to work through the maths you would find that this amounts to exactly the same calculation as for a time waveform.

But, in practice you see a slight difference that is caused by windowing . The difference between the mathematical world and the real world is that a vibration signal is continuous in time, but we only record a short period of it, with an abrupt beginning and ending. We must use window functions to get around this issue and avoid spectral leakage. As a result, the RMS value of the FFT will always be less than the RMS value of the original time waveform. This can be shown by taking the RMS value of the windowed time waveform - its very close to the FFT value.

Terminology

The terminology used in spectral work can be confusing. The single most ambiguous issue is the scaling or units of measurement. The following definitions may assist:

Spectrum

The spectrum is the fourier transform of the signal or time series amplitude. The result is an array of coefficients with units same as the signal per Hz.

Power (of a signal)

The power of a signal is equal to the square-root of the average of the squares of the magnitude of each time point of the signal. The result is a single value representing the RMS value of the signal over a particular period of time.

Power spectrum

The power spectrum is the square of the fourier transform of the signal or time series amplitude, normalized to a 1 Hz bandwidth. The result is an array of coefficients with the units of power per Hz.

Energy spectrum

Usually applied to transient signals, the energy spectrum is the square of the fourier transform of the signal or time series amplitude normalized to a 1 second duration and a 1 Hz bandwidth. The result is an array of coefficients with the units of power per Hz per second or in other words energy per Hz.

Convolution

Convolution is an operation in which the time points of two signals (or time series) are mapped to each other, then multiplied and these multiplications are summed over the set of mapped time points. The result is a single value.

Auto-covariance

The auto-covariance function is a function of lag or the shift in time of a function or time series. The function is defined as the convolution of a time series with itself, at a given lag. The result is a single value for each lag.

Auto-spectrum

An auto-spectrum is the fourier transform of a auto-covariance function of a signal or time series. The result is an array of coefficients. Square the coefficients to convert to an auto power spectrum.

Cross-covariance

The cross-covariance function is a function of lag or the shift in time between two signals or time series. The function is defined as the convolution of a time series 1 with time series  2, at a given lag in time series 2. The result is a single value for each lag.

Cross-spectrum

A cross-spectrum is the fourier transform of a cross-covariance function between two signals or time series. The result is an array of coefficients. High values will result at frequencies (1/lag) common in amplitude and phase to both signals. Units are the same as the input signals. Square the coefficients to convert to an cross power spectrum.

Coherence spectrum

The coherence spectrum is the normalize the cross-spectrum. Normalization is achieved by dividing each coefficient of the cross-spectrum by the square root of the product of the spectrum for each individual signal . The result is and array of coefficient with values between 0 and 1. Square the coefficients to convert to an coherence power spectrum.

References

B.P. Bogert, M.J. R. Healy and J.W. Tukey, The Quefrency Analysis of Time Series for Echoes: Cepstrum, Pseudo, Cross-cepstrum and Saphe Cracking, Proceedings of the Symposium on Time Series Analysis, M. Rosenblat, Ed., Wiley, NY, 1963, pp 209-243.