If we summarize the behavior of transducers and sensors, they both have a common purpose—generating signals related to a particular parameter under study. The traditional way of observing signals is to view them in the time domain. That is to say, observation in the time domain is a record of what is happening to a particular parameter over the period of time.
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As an elementary example, ill. 30-1 shows a weight suspended by a spring that, if set in motion, will combine to oscillate up and down. The up-and-down motion gradually diminishes with time. A pen attached to the weight bears against a strip of paper mounted on a cylinder that is rotated at a constant rate. A trace will then be produced on the paper, representing the displacement of the weight plotted against time, or a time-domain view of displacement.
Although this form of time-domain recording is commonly used, the more common method is to measure the parameter (in this case, displacement) in such a way that it can be turned into electrical signals. These are much easier to record or display, and they usually present a more practical method of measurement. The direct recording system of ill. 30-1 would not work in practice, e.g.,. It needs to be associated with a transducer to render mechanical displacement in terms of proportional electrical signals.
ill. 30-1. A simple method of recording mechanical oscillation.
In terms of electrical signals and parameters that vary in value with time, virtually any type of record is then essentially a waveform, although it may not look like one. The picture presented by the time-domain record may look quite irregular or ragged, as in ill. 30-2A. However, any waveform that exists in the real world can be broken into a series of component parts, each of which is a simple sine wave differing in amplitude (displacement), frequency, and position relative to each other in terms of time. Thus, a ragged appearance of ill. 30-2A is simply a (complex) waveform that could be duplicated by adding a large number of individual sine waves, the exact number depending on how many individual component waves are necessary to generate the final complex form.
If this seems far fetched, it's not. This sort of thing can be worked out mathematically by a method, developed more than 100 years ago by Jean Baptiste Fourier, known as Fourier analysis or Fourier transform, although the mathematics involved is highly complex (and nowadays normally done by a computer). and the idea of analysis in terms of component simple waveforms is further sup ported by the quantum theory, which, basically, holds that everything (even solid, stationary objects) can be represented in terms of waveforms.
Let’s make things simple by accepting that Fourier analysis does logically work, and let’s take as an example the quite simple waveform of ill. 30-2B, which can be reproduced by adding just two waves together. The same principle then holds no matter how many individual sine waves are involved in the makeup of the original waveform being considered.
ill. 30-2. At A, a complex waveform. Despite its complexity it can
be reduced to a combination of sine waves. At B, a less complicated
waveform. It consists of just two sine waves.
The original waveform, as it might be measured, is shown in terms of amplitude plotted against time. If the two component sine waves are extracted, they would appear as shown in the ill. 30-3.
ill. 30-3. Composition of the waveform at ill. 30-2B. It consists
of two waves that are equal in amplitude, with one having twice the
frequency of the other (f, 2f)
Now suppose, the record is redrawn as a three-dimensional design with a frequency axis added at right angles to amplitude and time (ill. 30-4A). In the conventional two-dimensional design this axis was already there but invisible, because we were looking directly along it. But if we now change one direction of view through 90 deg. to look directly at the frequency axis, the time axis disappears, and we see a record of amplitude against frequency (ill. 30-4B). The picture in this case is quite different: just two straight lines showing the amplitude of the component sine waves at their specific frequencies. This is the frequency-domain view, where each sine wave component is separated and individually displayed.
ill. 30-4. At A, addition of frequency-domain axis to the graph
of ill. 30-3. (The frequency-domain axis is at right angles to both
the time and amplitude axes.) At B, graph of amplitude as a function
of frequency.
In changing from one domain to another we have neither gained nor lost information (except phase information), but we have only expressed this information differently. In other words, it gives an other look at the subject, which can be very useful. In particular, the frequency-domain view can often show the presence of small components not visible in the time-domain view because they are masked by larger ones. Also, if necessary, phase information can be recovered.
THE MODAL DOMAIN
There is a third domain, the modal domain, that is of particular interest for analyzing the behavior of mechanical structures (seeing “logically” how they behave). To describe this simply, we will start with the behavior of a tuning fork, which generates a simple form of vibration and , as a result, an (apparently) single tone note. In a time-domain view the sound will appear as a sine wave lightly damped (ill. 30-5A). However, a view in the frequency domain (ill. 30-5B) will show that in addition to a predominant single frequency, the tuning fork also generates a number of smaller peaks at equally spaced frequency intervals; these are called harmonics. The mean tone, in fact, is generated by the first mode of vibration of the tuning fork, the first harmonic peak by the second mode of vibration of the tuning fork, the first harmonic peak by the second mode of vibration, and so on.
ill. 30-5. At A, waveform from a tuning fork as it would appear
on an oscilloscope. At B, amplitude-vs.-frequency rendition shows that
harmonic energy is present.
To determine the total vibration of the tuning fork, you would have to simultaneously measure the vibration at several points along its structure. For simplicity, we will use just three. Figure 30-6 shows a frequency-domain picture of the individual frequencies and their appearance at points 1, 2, and 3. Sharp peaks all occur at the same frequencies, independent of the measuring points, the only difference being the relative size or amplitude of their progress.
Another interesting point to emerge from the simple tuning form model is the effect of adding weights to the two tines to pro vide damping (more rapid decay of vibration). Adding weights to the ends of the tines would provide damping of the first mode of vibration (the main tone) without affecting the amplitude of the harmonica. Adding weight to the middle of the tines would provide minimal damping of the main tone but marked damping of the harmonics. This is a principle of vital importance in providing damping in structures that vibrate.
MODAL ANALYSIS
Because any real waveform can be represented by the sum of much simpler waveforms, any real vibration in a structure or body can be represented by the sum of much simpler vibrations. The method of delivering the shape and magnitude of the structural deformities in each vibration mode is known as modal analysis. There are two basic techniques used:
-- Exciting only one mode at a time.
-- Computing the individual modes of vibration from the total vibration Let’s take a tuning fork as a simple example. To excite just the first mode, we need two shakers driven by a sine wave and attached to the ends of the tines. Varying the frequency of the generator near the first-mode resonance frequency would then give its frequency, damping, and mode shape.
In the second mode the ends of the tines don't move, so to excite the second mode, we must have the shakers to the centers of the tines. If the ends of the tines are anchored, vibrations will then be constrained to the second mode alone.
In more realistic, three-dimensional problems, it's necessary to add many more shakers to ensure that only one mode is excited. The difficulties and expense of testing with many shakers has limited the application of this traditional modal analysis technique.
To determine the modes of vibration from the total vibration of the structure, you need to determine the frequency response of the structure at several points and compute at each resonance the frequency, damping, and what is called the residue (which represents the height of the resonance). This is done by a curve- fitting routine to smooth out any noise or small experimental errors. From these measurements and the geometry of the structure, the mode shapes are computed and drawn on a CRT display or a plotter. If drawn on a CRT, these displays may be animated to help the user understand the vibration mode.
It now becomes obvious that a modal analyzer requires some type of network analyzer to measure the frequency response of the structure and a computer to convert the frequency response to mode shapes. This can be accomplished by connecting a dynamic signal analyzer through a digital interface to a computer furnished with the appropriate software. This capability is also available in single instruments called structural dynamics analyzers. In general, computer systems offer more versatile performance because they can be programmed to solve other problems. However, structural dynamics analyzers generally are much easier to use than computer systems.
EXCITATION METHODS
There are three distinct test techniques available: random ex citation, sinusoidal excitation, and transient testing. Each has alter native approaches. Of these only transient testing uses transducers for impact testing (random and sinusoidal techniques use electromagnetic and hydraulic vibrators to excite the structure).
The method involved with transient (impact) testing is to use a hammer with a load cell mounted close to the striking face to measure the force applied to the structure, with an accelerometer mounted on the structure to measure its response. This technique has some important advantages:
- The structure requires no elaborate mountings.
- it's extremely fast, up to 100 times faster than some sinusoidal tests.
- No vibrator is required.
The main drawback with this method is that the input-force power spectrum is not as easily controllable as when a vibrator is used, and there can be significant variations between successive blows, which can cause nonlinearities to be excited. To some ex tent the bandwidth of the input-force spectrum can be selected by changing the material of the hammer head; a softer head will give a longer impulse and , hence, more energy at the lower frequencies. If a hard head is used, the total energy will be spread over a wide frequency range, and the excitation energy density will be low, leading to measurements with poor signal-to-noise ratios, particularly for massive, heavily damped structures.
HAMMER TECHNOLOGY
The basic technology involved in hammer testing involves measurement of the force and response, yielding signals that can be processed in an FET analyzer to display the amplitude of the vibratory random force ration versus frequency. Force is applied to the structure by the hammer through a load cell, with response measured by a suitable response transducer (such as an accelerometer), signal information being fed to suitable signal conditioning equipment. Briefly, the conditioning equipment amplifies and digitizes the signals, which are then Fourier transformed and sampled, and the cross spectrum and two power spectrums are computed and averaged. Frequency response and coherency functions are then computed from the averaged power and cross spectrum.
Hammers
The two most important characteristics of the hammer are its weight and tip hardness. The frequency content of the force produced by a hammer blow is inversely proportional to its weight and directly proportional to the hardness of the tip. Hammer weight also determines the magnitude of the force power, so weight is the primary parameter governing choice of hammer. Tip hardness can then be chosen to achieve the required pulse-time duration.
Actual sizes of hammers used may range from a fraction of an once to many pounds, depending on the size of the object or structure to be tested. Force and response transducers are incorporated in the hammer head. These need to be of special rigid type, normally piezoelectric. Resonances in the hammer structure are a, source of spurious spikes in the force signal spectrum that may need correction in the computer circuitry, or that can be eliminated by using a modaling tuned hammer, a recent development.
Hopkinson Bars
A related device is the so-called Hopkinson bar, also used for impulse testing as well as calibrating sensors, measuring proportions of materials, and researching fast strain effects. This consists of a log, straight, circular bar of metal with a stress sensor (transducer) at one end (ill. 30-7).
ill. 30-7. A Hopkinson-bar apparatus for measuring mechanical stress
resulting from acoustic propagation.
If the sensor end is rested on a rigid surface and the other end of the bar is tapped with a (plain) hammer, a compression stress- strain wear is produced in the bar traveling down it at the speed of sound through the bar material. The original compression wear is then reflected from the fixed end, then as a (united) tension wave from the free end, putting on the stress sensor on the second pass. This process is repeated until all the energy is finally described as friction. Figure 30-8 shows typical signal data generated by the stress sensor (transducer).
ill. 30-8. Stress-vs-time plot for Hopkinson bar.
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