Page 61 - Prosig Catalogue 2005

SOFTWARE PRODUCTS
HOW TO CALCULATE A RESULTANT VECTOR

If one examines the phase of the noisy signals, one can see it is now How To Calculate A
all over the place and essentially no longer any value. Automatic phase
unwrap was used, if the phase had been displayed over a 360° range it
The dynamic range of the original signal with added noise is around 90dB, Resultant Vector
would have totally filled the phase graph area.
with the differentiated and integrated signals having a similar range. That
is, the added noise has dominated the range. We can distinguish between quantities which have magnitude only and
One other aspect to notice is that the background level of the noise on the those which have magnitude and are also associated with a direction in Training & Support
integrated signal rises at the lower frequencies. This is known as 1/f noise space. The former are called scalars, for example, mass and temperature.
(one over f noise). This sets an effective lower frequency limit below The latter are called vectors, for example, acceleration, velocity and
which integration is no longer viable. displacement.
To emphasise the challenge of noise the next example has a very much In this article a vector is represented by bold face type. The magnitude
larger noise content. of any vector, r, (also called the modulus of r) is denoted by r. The
magnitude is a measurement of the size of the vector. The direction
component indicates the vector is directed from one location to another.
Scalars can be simply added together but vector addition must take into
account the directions of the vectors.
Multiple vectors may be added together to produce a resultant vector.
This resultant is a single vector whose effect is equivalent to the net
combined effect of the set of vectors that were added together.
The use of a frame of reference allows us to describe the location of a
point in space in relation to other points. The simplest frame of reference Condition Monitoring
is the rectangular Cartesian coordinate system. It consists of three
mutually perpendicular (tri-axial x, y and z) straight lines intersecting at
a point O that we call the origin. The lines Ox, Oy and Oz are called the
x-axis, y-axis and z-axis respectively.
For convenience consider a two dimensional x,y coordinate system with
an x-axis and a y-axis. In this configuration any point P with respect to
the origin can be related to these axes by the numbers x and y as shown
in Figure 1. These numbers are called the coordinates of point P and
represent the perpendicular distances of point P from the axes.
Software

Figure 8: Time series with more noise added
Here the noise on the original signal is evident. The differentiated signal
is effectively useless, but the integrated signal is relatively clean. To really
illustrate the point, the noisy sinewave was differentiated twice. The
result is shown below. All trace of the original sinewave seems to have
gone or, rather, has been lost in the noise.

Figure 1
If these two measurements represent vector quantities, for example
displacement x and y, measured in the x and y directions respectively
Figure 9: Noisy signal differentiated twice then we can use vector addition to combine them into a single resultant Hardware
vector r as shown in Figure 1. In vector terms
r = x + y
The conclusion is now clear. If there are no special circumstances, then
experience suggests it is best to measure vibration with an accelerometer. Any vector can be written as r = (r/r)*r where (r/r) is a unit vector in the
However, care is required to remove the very low frequencies if any same direction as r. A unit vector is simply a vector with unit magnitude.
integration to velocity or displacement is needed. By convention we assign three unit vectors i, j and k in the directions x, y
As a final point, why should differentiation be much noisier than and z respectively. So we can write
integration? The answer is that differentiation is a subtraction process r = x + y = xi + yj
and at its very basic level we take the difference between two successive where x is the magnitude of vector x and y is the magnitude of vector y.
values, and then divide by the time between samples. The two adjacent Sometimes we are only interested in the magnitude or size of the resultant
data points are often quite similar in size. Hence the difference is small vector. Looking at Figure 1 we can use Pythagoras’ Theorem to calculate
and will be less accurate, then we divide by what often is a small time the magnitude of vector r as
difference and this tends to amplify any errors. Integration on the other
2
2
hand is addition. As any broadband noise tends to be successively, r = x + y 2 System Packages
differently-signed then the noise cancels out. In vector terms, the scalar product a.b (also known as the dot product) of
This article, of course, does not tell the whole story, but it provides a very two vectors a and b is defined as the product of the magnitudes a and b
simple guide to good practice. and the cosine of the angle between vectors a and b. Therefore,
r.r = rr cos(0) = r = (xi + yj).(xi + yj) = x i.i + y j.j + 2xyi.j
2
2
2
By definition unit vectors have unity magnitude so i.i = 1 * 1 * cos(0) =
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