# Introduction to Velocity Analysis and Measurement Considerations

**Velocity Analysis:**

*Velocity analysis is a useful tool when measuring some of the components of rolling element bearings, as it can highlight the impact of small amplitude, high frequency features that might lead to early failure of the bearing.*

In this article you will learn the measurement of vibration on a finished bearing assembly, how some of the individual components could be tested by measuring on a roundness instrument and analysing using velocity analysis. We will look at the mathematical background to the analysis and the definition of two key parameters. Finally, we'll look at some of the factors that influence the measurement results.

**Testing of rotating element bearing (REB) roller bearings and ball bearings:**

**Rotating machinery – bearings**

The one method of testing finished rotating element bearings in a production environment: the rotary motion and mechanical bearings can be divided into journal or plane bearings and rotating element bearings REB’s take many forms roller bearings or ball bearings.

**Vibration**

Vibration is an important consideration for the life of a bearing. The bearing will transmit vibration, but vibration can also be generated in the bearing typically by small imperfections in the components. These vibrations will give rise to noise wear and eventually failure of the bearing. It is the aim of the bearing manufacturer to limit these vibrations by controlling the quality of the individual components. For vibrations caused internally to the bearing, the key task is to identify which vibrations are important.

Harmonic analysis can be used to identify the frequencies present to identify the likelihood of a particular frequency causing a problem. An example of this is bearing fatigue where we need to consider the energy in the frequency and that energy is related to the velocity amplitude and hence the important of velocity analysis.

**Bearing vibration**

We are going to use the example of a ball bearing. Aside from lubricant shields and seals. The bearing has four main components, the outer race, the inner race, the ball bearings and the cage. Each of these can contribute to vibration in the bearing.

**Measuring bearing vibration:**

Bearing vibration can be tested in a vibration test rig. The bearing is loaded onto an arbor that's mounted on a precision spindle, the bearing is pre-loaded by applying an axial force to the outer race. A vibration sensor is placed in contact with the outer race to measure radial vibration in the bearing as it is rotated at constant speed, speed different types of sensors can be used but the output of the center is converted into velocity units usually micrometers per second.

Analysis of the bearing is done in two ways firstly, by listening to the noise using headphones or a speaker and secondly by the measurement of rms velocity in three bands designated as low medium and high.

ISO 15 242 describes procedures for measuring vibration. A typical rotational speed is 1800s rpm, although other speeds are allowed. The Frequency bands: L – Low 50 Hz to 300 Hz; M – Medium 300 Hz to 1800 Hz; H – High 1800 Hz to 10 000 Hz. If the rotation speed is changed then these band limits should also be changed. The RMS velocity is then calculated for each band.

Definition of RMS velocity given in the ISO standard: ISO 15242-1:2015

“rms velocity” is defined as the “square root of the average of squared values of the vibration velocity within a time interval, *T*”

**Testing REB components:**

Components for REBs are manufactured to very close tolerances: Surface texture and form need to be controlled for bearing races. A suitable Form Talysurf instrument can provide cross-track form checking and surface texture measurement. Within our product range roundness can be assessed on a suitable Talyrond or Surtronic roundness instrument depending on the application. When using a Talyrond velocity analysis is available as a licensed option. Indicates the energy in user-defined bands of harmonics in a similar manner to the way the assembled bearing is tested.

**Roundness Instrument - Data acquisition path:**

Basic components of the data acquisition path of a typical roundness instrument.

The data acquisition system is responsible for: Initial signal conditioning, digitising the signal, storing the data these readings for later processing. Signal conditioning might involve amplification and filtering of the signal prior to the converter stage the filter could be a separate component or a filter that is inherent in the way that the converter works. Sampling of the gauge signal is usually triggered by some feedback of the spindle position.

**Data acquisition:**

When a roundness measurement is made variations in gauge deflection are recorded as a function of spindle angle. These variations are a combination of component errors, setup errors, rotational errors from the instrument and other factors such as noise. Small errors in eccentricity will normally be removed by the analysis. Uncertainty residual spindle errors will also contribute to the measurement uncertainty. Typical spindle errors will mainly impact the lower harmonics say 1-15 upr. Noise is another uncertainty contributor.

**Harmonic analysis:**

When measuring roundness, a complete circular component such as a bearing race the profile is inherently periodic and so the profile can be expressed as a Fourier series as

Coefficients *a _{k}* and

*b*are found by multiplying by appropriate cosine and sine functions and then integrating.

_{k}

The Fourier series can be calculated in many ways. The methods used in a digital computer aim to reduce the computation time by breaking the overall computation into a series of smaller computations and thereby reducing the number of multiplications that are done the most well-known method of reducing the computation time is the method published by Cooley and Tukey in 1965 which is often referred to as the fast Fourier transform. Winograd – uses a combination of low order matrices whose sizes are factors of the number of points being used.

**Data analysis: ****Velocity Analysis**

*Velocity analysis is defined as:** *Rate of change of displacement*. *In terms of a roundness measurement, rate of change of displacement of the gauge signal.

**Rate of change of a sine wave**

- Given a sine wave

Velocity analysis results:

Velocity analysis results:

Velocity analysis are presented in many ways. Presentation of the results is the velocity graph is shown here. The graph shows the velocity amplitudes in micrometer per second by frequency in undulations per revolution or upr in each user band. The bands are labelled l, m and h and these can be broken into three sub-bands as shown here. For the bands H1 H2 and H3, two cursors are provided to allow the user to interrogate the graph summary text. The graph shows the velocity amplitude and wave number for each cursor and the rms velocity calculated across the band bounded by the two cursors.

**Vel(RS) – single harmonic:**For a single sinusoid of amplitude, 𝑎

**Compare to single harmonic result for Vel(RS):**

**Comparison between Vel(RMS) and Vel(RS):**

**Measurement considerations:**

Velocity measurements are used in product and process control it's therefore important to have some appreciation of the uncertainties that might impact on the results. The measurement itself is the sum of component errors and system influences at low harmonic numbers. At low harmonics, residual spindle form error is important. At higher harmonics, noise from the system or environment will be important.

The data acquisition system also plays a part, we've seen that the data acquisition system will include an anti-alias filter there is potential for this filter to attenuate higher harmonics. This can be mitigated to some extent by reducing the spindle speed.

The gauge also plays a significant part. The gauge and stylus form a mass spring system which will have a natural frequency excitation of frequencies near to the natural frequency or close to sub-harmonics could excite resonance leading to erroneous results. The stylus would be unlikely to track this faithfully. again, a reduction in spindle speed might help to mitigate this effect. The stylus tip also acts as a morphological filter and must be chosen to be appropriate to the harmonic wavelength and amplitude clearly the stylus tip size needs to be specified as one of the measurement conditions.

**Multi-wave standard**

Cylindrical artefact with multiple harmonic waves machined into its surface.

There are two types of standard one:

- Single ring with multiple frequencies superimposed
- Multiple rings with individual frequencies

A key advantage of the multiple ring type is that the individual harmonics can be calibrated. The measurement system can be tested by measuring the standard at different spindle speeds and noting any variation in the reported harmonic amplitudes where the instrument has a single spindle speed. It's necessary to use a calibrated standard and compare the reported harmonic amplitudes against the certified values.

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