Page 48 - Prosig Catalogue 2005

SOFTWARE PRODUCTS
WHAT IS RESONANCE?

implementation. After all every analysis will use addition. That is just a What Is Resonance?
mathematical operation and so, in that sense, is the use of an FFT. First, in order to explain resonance we have to explain the terms we will
A Little Mathematics
Training & Support Fourier series could be written in the forms below. In real and imaginary • A period is the amount of time it takes to complete one cycle
use.
We will not go into all the mathematical niceties except to see that a
terms we have
The number of cycles in one second is the frequency of an oscillation.
•
•
Frequency is measured in Hertz, named after the 19th century
German physicist Heinrich Rudolf Hertz
•
One Hertz is equal to one cycle per second.
A resonance occurs when a structure or material naturally oscillates at
a high amplitude at a specific frequency. This frequency is known as

a structural resonant frequency. Typically a structure will have many
resonant frequencies.
and in modulus and phase form as
A dictionary definition of resonance gives us -
Condition Monitoring The above forms are a slightly unusual way of expressing the Fourier When the damping in a structure is small, the resonant frequencies are
“the state of a system in which an abnormally large
vibration is produced in response to an external
stimulus, occurring when the frequency of the stimulus
is the same, or nearly the same, as the natural vibration
frequency of the system.”
approximately equal to the natural frequencies of the structure, which are
the frequencies of free vibrations of the molecules of the material itself.
expansion. For instance θ is in degrees. More significantly the product f t
is shown explicitly. Usually in an FFT then f is expressed as n/NΔt and t
frequency of a structure or material and the frequency at which it is
as kΔt where Δt is the time between samples. This gives the relationship
being excited are equal or very nearly equal. This results in the structure
of the form n n k k Furthermore, an individual resonance is the condition when a natural
or material vibrating strongly and is the classical resonance state. This
resonance state can often lead to unexpected behavior of the structure
or material.
The lowest natural frequency, often called the fundamental frequency, is
related to the material of which the structure is made. The greater the
Software However, the point of using f t explicitly above is to indicate that nothing mass or density of the material the lower the fundamental frequency
of vibration. The natural frequency is also related to the speed that a
waveform can propagate through the structure. This is determined largely
by the molecular make up of the material. Gas, for example, has many
n k
in the Fourier expansion inhibits the choice of actual frequency at which
we evaluate the Fourier coefficients. The FFT gains speed by being
quickly through the material. A solid has far fewer free molecules and is
selective about where it evaluates the coefficients and also restrictive in free molecules with high kinetic energy, so the waveform can move
much denser, therefore the waveform moves more slowly.
the values of N that are permitted. There are ways around these but in
most implementations, for practical purposes N is restricted to being a In order to measure a resonance of a structure or material with a system
power of 2. such as Prosig’s P8000 data acquisition hardware and DATS signal
processing software it is necessary to attach an accelerometer to the
This means that with a DFT we can actually evaluate the Fourier structure. It is then required to excite or stimulate the structure with the
coefficients at any frequency provided we obey the anti aliasing (Nyquist) frequencies that it is normally exposed to in its working life. For example,
criterion. The DFT is slower than an FFT. Another way of getting at the an automotive car tire would need to be subject to the frequencies it
Hardware Zoom FFT based on the Chirp-z transform. Again the relative advantages use of a shaker or a large heavy hammer. The tire for example would
finer detail and still getting some speed advantage is to use the so-called
would encounter whilst in use. This would normally be accomplished by
are discussed elsewhere.
need to be tested in isolation, and not connected to anything else like
the vehicle suspension or wheel rim as these other parts have their own
As a historical note it is perhaps interesting to recall that Fourier did not
resonant frequencies and would make the capture and analysis of the tire
generate his series in order to carry out frequency analysis but rather
to determine a least squares error approximation to a function. resonant frequency difficult.
The measured response from the accelerometer will be relative to
the excitation and will only exhibit frequencies that are present in the
excitation. The excitation must be an acceptable representation of the
normal working frequencies applied to the structure or material. If the
structure has a resonance in this frequency range there will be a large
System Packages is detected then the resonant frequencies lie outside the operating range
peak in the response spectrum. The frequency of this peak will correspond
to one of the resonant frequencies of the structure or material. If no peak
of the structure or material. In order to find the resonant frequencies
of a structure or material it may be necessary to apply a wider range of
frequency excitation.
Figure 1 shows a frequency spectrum, this spectrum is a response
of a structure to its excitation. A large spike can clearly be seen at
approximately 250 Hz.
Figure 2 shows a frequency spectrum, this spectrum as in Figure 1 shows
a frequency response. However, Figure 2 shows, using cursors, the exact

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