Page 59 - Prosig Catalogue 2005

SOFTWARE PRODUCTS
VIBRATION ANALYSIS: SHOULD WE MEASURE ACCELERATION, VELOCITY OR DISPLACEMENT

acceleration, velocity or displacement then it is simple mathematics to
convert between them by a judicious use of integration or differentiation
as illustrated below.
Measured Signal Operation Result
Type
Displacement differentiate Velocity
Displacement double differentiate Acceleration Training & Support
Velocity differentiate Acceleration
Velocity integrate Displacement
Acceleration integrate Velocity

Figure 16: Recalculated S-N curve Acceleration double integrate Displacement
So now let us look at this with a classical sine wave signal and see the
effects of either differentiating or integrating it. To avoid other side effects
the example uses a 96Hz sinewave of unit amplitude with 32768 samples
generated at 8192 samples/second. It is useful to look at these as time
histories and as functions of frequency. That is, the original generated
sinewave was processed using a DATS worksheet as illustrated in Figure
1. Condition Monitoring

Figure 17: Recalculated life prediction
expected level of 10 hours use per day this component could be expected
to fail. Clearly this is a fragile component that is likely to fail in an
unacceptably short amount of time for an automotive application and
both a technical design change and further testing are required. Figure 1: Worksheet calculating frequency spectra
It is evident that the more testing to failure that is carried out the more
accurate the final life prediction will be. Looking at a section of the wave forms, we have a classical result as Software
[Note: This article has been reduced in complexity compared with the shown in Figure 2.
original test report and uses deliberately modified initial strain values]

Vibration Analysis:

Should We Measure

Acceleration, Velocity Or

Displacement? Hardware

When using vibration data, especially in conjunction with modelling
systems, the measured data is often needed as an acceleration, as a
velocity and as a displacement. Sometimes different analysis groups Figure 2: Original, differentiated and integrated time histories
require the measured signals in a different form. Clearly, it is impractical
to measure all three at once even if we could. Physically it is nigh on In mathematical terms if y(t) = A sin(2πft) then the differential is
impossible to put three different types of transducer in the same place. {2πAfcos(2πft)} and the integral is {-(A/2πf)cos(2πft)} + C where C is the
so called ‘constant of integration’. In both cases there is a phase shift
Accelerometers are available in all types and sizes and there is a very of 90° which turns the sine into a cosine. The differential is multiplied by
large choice. Some types will measure down to DC (0Hz), others handle 2πf. The integral is divided by 2πf, is also negated and has had an offset
high shock loading and so on. added to it, which in this case is half the resultant amplitude, resulting in
True velocimeters are quite rare, but they do exist. One interesting class the integrated signal being entirely positive. If, for example, the original
based on a coil and magnet scheme is self powered. signal had represented an acceleration then the integrated signal is a
Direct displacement measurement is not uncommon. Some use velocity, and clearly we would not expect that to be entirely positive. This System Packages
strain gauges, but many others use a capacitive effect or induced radio integration constant is an artefact of the standard integration methods.
frequency mechanism to measure displacement directly. The capacitive For the mathematically inclined, it is the result of carrying out of what
and inductive types have the advantage that they are non-contacting is usually referred to as an indefinite integration. The solution is quite
probes and do not affect the local mass. simple. After doing a standard time based integration then we should
But in any case it doesn’t matter, because if we measure either automatically reduce the result to have a zero mean value. That is,

http://prosig.com +1 248 443 2470 (USA) or contact your 59
sales@prosig.com +44 (0)1329 239925 (UK) local representative
A CMG Company